A generalized nonlocal vector calculus
NASA Astrophysics Data System (ADS)
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
2015-07-22
We develop the generalized Cartan Calculus for the groups G = SL(2,R) × R+, SL(5,R) 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 Poincar\\'e lemmas in this new differential geometry is also discussed. Lastly, we explore some possible extension of the generalized Cartan calculus beyond the exceptional series.
Generalized Cartan Calculus in general dimension
Wang, Yi -Nan
2015-07-22
We develop the generalized Cartan Calculus for the groups G = SL(2,R) × R^{+}, SL(5,R) 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 Poincar\\'e lemmas in this new differential geometry is also discussed. Lastly, we explore some possible extension of the generalized Cartan calculus beyond the exceptional series.
Generalized vector calculus on convex domain
NASA Astrophysics Data System (ADS)
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 Laplace Transforms and Extended Heaviside Calculus
ERIC Educational Resources Information Center
Deakin, Michael A. B.
2008-01-01
An extended Heaviside calculus proposed by Peraire in a recent paper is similar to a generalization of the Laplace transform proposed by the present author. This similarity will be illustrated by analysis of an example supplied by Peraire.
Variational calculus with constraints on general algebroids
NASA Astrophysics Data System (ADS)
Grabowska, Katarzyna; Grabowski, Janusz
2008-05-01
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.
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
Enabling quaternion derivatives: the generalized HR calculus.
Xu, Dongpo; Jahanchahi, Cyrus; Took, Clive C; Mandic, Danilo P
2015-08-01
Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.
On the origins of generalized fractional calculus
NASA Astrophysics Data System (ADS)
Kiryakova, Virginia
2015-11-01
In Fractional Calculus (FC), as in the (classical) Calculus, the notions of derivatives and integrals (of first, second, etc. or arbitrary, incl. non-integer order) are basic and co-related. One of the most frequent approach in FC is to define first the Riemann-Liouville (R-L) integral of fractional order, and then by means of suitable integer-order differentiation operation applied over it (or under its sign) a fractional derivative is defined - in the R-L sense (or in Caputo sense). The first mentioned (R-L type) is closer to the theoretical studies in analysis, but has some shortages - from the point of view of interpretation of the initial conditions for Cauchy problems for fractional differential equations (stated also by means of fractional order derivatives/ integrals), and also for the analysts' confusion that such a derivative of a constant is not zero in general. The Caputo (C-) derivative, arising first in geophysical studies, helps to overcome these problems and to describe models of applied problems with physically consistent initial conditions. The operators of the Generalized Fractional Calculus - GFC (integrals and derivatives) are based on commuting m-tuple (m = 1, 2, 3, …) compositions of operators of the classical FC with power weights (the so-called Erdélyi-Kober operators), but represented in compact and explicit form by means of integral, integro-differential (R-L type) or differential-integral (C-type) operators, where the kernels are special functions of most general hypergeometric kind. The foundations of this theory are given in Kiryakova 18. In this survey we present the genesis of the definitions of the GFC - the generalized fractional integrals and derivatives (of fractional multi-order) of R-L type and Caputo type, analyze their properties and applications. Their special cases are all the known operators of classical FC, their generalizations introduced by other authors, the hyper-Bessel differential operators of higher integer
A new proof of the generalized Hamiltonian–Real calculus
Gao, Hua; Mandic, Danilo P.
2016-01-01
The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative rules. This further reinforces the theoretical foundation of the GHR calculus and provides a convenient methodology for generic extensions of real- and complex-valued learning algorithms to the quaternion domain.
Intitialization, Conceptualization, and Application in the Generalized Fractional Calculus
NASA Technical Reports Server (NTRS)
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.
Generalized Functions for the Fractional Calculus
NASA Technical Reports Server (NTRS)
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.
ERIC Educational Resources Information Center
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…
Non-signalling Theories and Generalized Probability
NASA Astrophysics Data System (ADS)
Tylec, Tomasz I.; Kuś, Marek; Krajczok, Jacek
2016-09-01
We provide mathematically rigorous justification of using term probability in connection to the so called non-signalling theories, known also as Popescu's and Rohrlich's box worlds. No only do we prove correctness of these models (in the sense that they describe composite system of two independent subsystems) but we obtain new properties of non-signalling boxes and expose new tools for further investigation. Moreover, it allows strightforward generalization to more complicated systems.
Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus
NASA Astrophysics Data System (ADS)
Kiryakova, Virginia S.
2000-06-01
The classical Mittag-Leffler (M-L) functions have already proved their efficiency as solutions of fractional-order differential and integral equations and thus have become important elements of the fractional calculus' theory and applications. In this paper we introduce analogues of these functions, depending on two sets of multiple (m-tuple, m[greater-or-equal, slanted]2 is an integer) indices. The hint for this comes from a paper by Dzrbashjan (Izv. AN Arm. SSR 13 (3) (1960) 21-63) related to the case m=2. We study the basic properties and the relations of the multiindex M-L functions with the operators of the generalized fractional calculus. Corresponding generalized operators of integration and differention of the so-called Gelfond-Leontiev-type, as well as Borel-Laplace-type integral transforms, are also introduced and studied.
ERIC Educational Resources Information Center
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…
Generalized variational calculus in terms of multi-parameters fractional derivatives
NASA Astrophysics Data System (ADS)
Agrawal, Om P.; Muslih, Sami I.; Baleanu, Dumitru
2011-12-01
In this paper, we briefly introduce two generalizations of work presented a few years ago on fractional variational formulations. In the first generalization, we consider the Hilfer's generalized fractional derivative that in some sense interpolates between Riemann-Liouville and Caputo fractional derivatives. In the second generalization, we develop a fractional variational formulation in terms of a three parameter fractional derivative. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed.
The generalization of Plackett-Burman design based on fractional calculus of complex order
NASA Astrophysics Data System (ADS)
Ibrahim, Rabha W.; Jalab, Hamid A.
2014-12-01
Response surface attitude was working to optimize the degradation situations of Aflatoxin B1(AFB1) by Rhodococcus erythropolis in liquid culture. The interesting factors that influence the degradation, as identified by a Plackett-Burman design with six variables, were temperature, liquid volume, pH, inoculums size, agitation speed and incubation time. In this work, we generalize the Plackett-Burman design, based on fractional calculus of complex order, to describe correlation between the six variables and the degradation rate of AFB1. The experimental results show the influence of the proposed method. The results demonstrated that the degradation efficiency of R. erythropolis could extent 99% in liquid culture.
Sculpture, general view looking to the seated lions, probably from ...
Sculpture, general view looking to the seated lions, probably from the American Bungalow - National Park Seminary, Bounded by Capitol Beltway (I-495), Linden Lane, Woodstove Avenue, & Smith Drive, Silver Spring, Montgomery County, MD
ERIC Educational Resources Information Center
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;…
Probability and Relative Frequency
NASA Astrophysics Data System (ADS)
Drieschner, Michael
2016-01-01
The concept of probability seems to have been inexplicable since its invention in the seventeenth century. In its use in science, probability is closely related with relative frequency. So the task seems to be interpreting that relation. In this paper, we start with predicted relative frequency and show that its structure is the same as that of probability. I propose to call that the `prediction interpretation' of probability. The consequences of that definition are discussed. The "ladder"-structure of the probability calculus is analyzed. The expectation of the relative frequency is shown to be equal to the predicted relative frequency. Probability is shown to be the most general empirically testable prediction.
Generalizing Swendsen-Wang to sampling arbitrary posterior probabilities.
Barbu, Adrian; Zhu, Song-Chun
2005-08-01
Many vision tasks can be formulated as graph partition problems that minimize energy functions. For such problems, the Gibbs sampler provides a general solution but is very slow, while other methods, such as Ncut and graph cuts are computationally effective but only work for specific energy forms and are not generally applicable. In this paper, we present a new inference algorithm that generalizes the Swendsen-Wang method to arbitrary probabilities defined on graph partitions. We begin by computing graph edge weights, based on local image features. Then, the algorithm iterates two steps. 1) Graph clustering: It forms connected components by cutting the edges probabilistically based on their weights. 2) Graph relabeling: It selects one connected component and flips probabilistically, the coloring of all vertices in the component simultaneously. Thus, it realizes the split, merge, and regrouping of a "chunk" of the graph, in contrast to Gibbs sampler that flips a single vertex. We prove that this algorithm simulates ergodic and reversible Markov chain jumps in the space of graph partitions and is applicable to arbitrary posterior probabilities or energy functions defined on graphs. We demonstrate the algorithm on two typical problems in computer vision--image segmentation and stereo vision. Experimentally, we show that it is 100-400 times faster in CPU time than the classical Gibbs sampler and 20-40 times faster then the DDMCMC segmentation algorithm. For stereo, we compare performance with graph cuts and belief propagation. We also show that our algorithm can automatically infer generative models and obtain satisfactory results (better than the graphic cuts or belief propagation) in the same amount of time.
An infinite-dimensional calculus for generalized connections on hypercubic lattices
Mendes, R. Vilela
2011-05-15
A space for gauge theories is defined, using projective limits as subsets of Cartesian products of homomorphisms from a lattice on the structure group. In this space, non-interacting and interacting measures are defined as well as functions and operators. From projective limits of test functions and distributions on products of compact groups, a projective gauge triplet is obtained, which provides a framework for the infinite-dimensional calculus in gauge theories. The gauge measure behavior on non-generic strata is also obtained.
Polynomial Calculus: Rethinking the Role of Calculus in High Schools
ERIC Educational Resources Information Center
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…
NASA Technical Reports Server (NTRS)
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.
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
Polynomial calculus: rethinking the role of calculus in high schools
NASA Astrophysics Data System (ADS)
Grant, Melva R.; Crombie, William; Enderson, Mary; Cobb, Nell
2016-08-01
Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.
Generalized calcinosis cutis associated with probable leptospirosis in a dog.
Munday, John S; Bergen, David J; Roe, Wendi D
2005-12-01
A 6.5-year-old male German Shepherd acutely developed renal and hepatic disease. Serology revealed high concentrations of antibodies against Leptospira copenhageni, and a presumptive diagnosis of leptospirosis was made. The dog was successfully treated with antibiotics and supportive care over a 12-day period. Sixty-two days after the initial presentation, alopecia predominantly involving the dorsum and perineal areas developed. The skin lesions expanded over a 20-day period. Histology revealed generalized calcinosis cutis with follicular atrophy. An injection of 0.01 mg kg-1 dexamethasone suppressed serum cortisol concentrations. No treatment was given and lesions resolved over the following 30 days. This is the third case of generalized calcinosis cutis that has developed in an adult dog after severe systemic disease. Both previous cases developed calcinosis cutis in association with blastomycosis. To the authors' knowledge, this is the first report of generalized calcinosis cutis in an adult dog in association with a presumptive bacterial infection.
ERIC Educational Resources Information Center
McGivney-Burelle, Jean; Xue, Fei
2013-01-01
In this paper we discuss flipping pedagogy and how it can transform the teaching and learning of calculus by applying pedagogical practices that are steeped in our understanding of how students learn most effectively. In particular, we describe the results of an exploratory study we conducted to examine the benefits and challenges of flipping a…
ERIC Educational Resources Information Center
Cirillo, Michelle
2007-01-01
In this article, the author explores the history and the mathematics used by Newton and Leibniz in their invention of calculus. The exploration of this topic is intended to show students that mathematics is a human invention. Suggestions are made to help teachers incorporate the mathematics and the history into their own lessons. (Contains 3…
The Use of a PDP-11/20 Computer in a Non-Calculus General Physics Course.
ERIC Educational Resources Information Center
Yu, David U. L.
Computer-assisted instruction supplements traditional methods in a non-calculus physics course offered at Seattle Pacific College. Thirty-five science majors enrolled in the first quarter and 32 continued in the second term. The hardware for the course consists of a PDP-11/20 computer and eight teletype terminals; additional peripheral equipment…
Fractional calculus in bioengineering.
Magin, Richard L
2004-01-01
Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub
On realizations of exterior calculus with dN = 0
NASA Astrophysics Data System (ADS)
Abramov, V.
1998-11-01
We study realizations of the q-exterior calculus with exterior differential d satisfying d N = 0, N > 2 on the free associative algebra with one generator and on the generalized Clifford algebras. Analogs of the notions of connection and curvature are discussed in the case of the q-exterior calculus on the generalized Clifford algebra. We show that the q-exterior calculus on the free associative algebra with one generator is related to q-calculus on the braided line.
Height probabilities in the Abelian sandpile model on the generalized finite Bethe lattice
NASA Astrophysics Data System (ADS)
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.
Lattice Duality: The Origin of Probability and Entropy
NASA Technical Reports Server (NTRS)
Knuth, Kevin H.
2004-01-01
Bayesian probability theory is an inference calculus, which originates from a generalization of inclusion on the Boolean lattice of logical assertions to a degree of inclusion represented by a real number. Dual to this lattice is the distributive lattice of questions constructed from the ordered set of down-sets of assertions, which forms the foundation of the calculus of inquiry-a generalization of information theory. In this paper we introduce this novel perspective on these spaces in which machine learning is performed and discuss the relationship between these results and several proposed generalizations of information theory in the literature.
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
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
ERIC Educational Resources Information Center
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;…
Calculus ABCs: A Gateway for Freshman Calculus
ERIC Educational Resources Information Center
Fulton, Scott R.
2003-01-01
This paper describes a gateway testing program designed to ensure that students acquire basic skills in freshman calculus. Students must demonstrate they have mastered standards for "Absolutely Basic Competency"--the Calculus ABCs--in order to pass the course with a grade of C or better. We describe the background, standards, and testing program.…
Akanda, Md Abdus Salam; Alpizar-Jara, Russell
2014-01-01
Modeling individual heterogeneity in capture probabilities has been one of the most challenging tasks in capture–recapture studies. Heterogeneity in capture probabilities can be modeled as a function of individual covariates, but correlation structure among capture occasions should be taking into account. A proposed generalized estimating equations (GEE) and generalized linear mixed modeling (GLMM) approaches can be used to estimate capture probabilities and population size for capture–recapture closed population models. An example is used for an illustrative application and for comparison with currently used methodology. A simulation study is also conducted to show the performance of the estimation procedures. Our simulation results show that the proposed quasi-likelihood based on GEE approach provides lower SE than partial likelihood based on either generalized linear models (GLM) or GLMM approaches for estimating population size in a closed capture–recapture experiment. Estimator performance is good if a large proportion of individuals are captured. For cases where only a small proportion of individuals are captured, the estimates become unstable, but the GEE approach outperforms the other methods. PMID:24772290
Superpositions of probability distributions
NASA Astrophysics Data System (ADS)
Jizba, Petr; Kleinert, Hagen
2008-09-01
Probability distributions which can be obtained from superpositions of Gaussian distributions of different variances v=σ2 play a favored role in quantum theory and financial markets. Such superpositions need not necessarily obey the Chapman-Kolmogorov semigroup relation for Markovian processes because they may introduce memory effects. We derive the general form of the smearing distributions in v which do not destroy the semigroup property. The smearing technique has two immediate applications. It permits simplifying the system of Kramers-Moyal equations for smeared and unsmeared conditional probabilities, and can be conveniently implemented in the path integral calculus. In many cases, the superposition of path integrals can be evaluated much easier than the initial path integral. Three simple examples are presented, and it is shown how the technique is extended to quantum mechanics.
Asymptotic forms for hard and soft edge general β conditional gap probabilities
NASA Astrophysics Data System (ADS)
Forrester, Peter J.; Witte, Nicholas S.
2012-06-01
An infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix β-ensembles. The conditioning is that there are n eigenvalues in the gap, with n≪|t|, t denoting the end point of the gap. It is found that the entropy term in the formalism must be replaced by a term involving the potential drop to obtain results consist with known asymptotic expansions in the case n=0. With this modification made for general n, the derived expansions — which are for the logarithm of the gap probabilities — are conjectured to be correct up to and including terms O(log|t|). They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating β to 4/β.
Questions Revisited: A Close Examination of Calculus of Inference and Inquiry
NASA Technical Reports Server (NTRS)
Knuth, Kevin H.; Koga, Dennis (Technical Monitor)
2003-01-01
In this paper I examine more closely the way in which probability theory, the calculus of inference, is derived from the Boolean lattice structure of logical assertions ordered by implication. I demonstrate how the duality between the logical conjunction and disjunction in Boolean algebra is lost when deriving the probability calculus. In addition, I look more closely at the other lattice identities to verify that they are satisfied by the probability calculus. Last, I look towards developing the calculus of inquiry demonstrating that there is a sum and product rule for the relevance measure as well as a Bayes theorem. Current difficulties in deriving the complete inquiry calculus will also be discussed.
Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process
Yin, Chuancun; 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
Estimation of probable maximum precipitation for catchments in eastern India by a generalized method
NASA Astrophysics Data System (ADS)
Rakhecha, P. R.; Mandal, B. N.; Kulkarni, A. K.; Deshpande, N. R.
1995-03-01
A generalized method to estimate the probable maximum precipitation (PMP) has been developed for catchments in eastern India (80° E, 18° N) by pooling together all the major rainstorms that have occurred in this area. The areal raindepths of these storms are normalized for factors such as storm dew point temperature, distance of the storm from the coast, topographic effects and any intervening mountain barriers between the storm area and the moisture source. The normalized values are then applied, with appropriate adjustment factors in estimating PMP raindepths, to the Subarnarekha river catchment (upto the Chandil dam site) with an area of 5663 km2. The PMP rainfall for 1, 2 and 3 days were found to be roughly 53 cm, 78 cm and 98 cm, respectively. It is expected that the application of the generalized method proposed here will give more reliable estimates of PMP for different duration rainfall events.
Non-classical conditional probability and the quantum no-cloning theorem
NASA Astrophysics Data System (ADS)
Niestegge, Gerd
2015-09-01
The quantum mechanical no-cloning theorem for pure states is generalized and transfered to the quantum logics with a conditional probability calculus in a rather abstract, though simple and basic fashion without relying on a tensor product construction or finite dimension as required in other generalizations.
NASA Astrophysics Data System (ADS)
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.
NASA Astrophysics Data System (ADS)
Basu, Tania; Tarafdar, Sujata
2016-08-01
Solid polymer electrolytes with gelatin as host polymer are subjected to gamma irradiation with dose varying from 0 to 100 kGy. Two sets of samples are studied, one with and one without addition of lithium perchlorate as ionic salt. The effect of varying plasticizer content, salt fraction and radiation dose on the impedance is measured. The dc (direct current) ion-conductivity is determined from impedance spectroscopy results. It is shown that relative to the unirradiated sample, the room temperature dc ion-conductivity decreases in general on irradiation, by an order of magnitude. However on comparing results for the irradiated samples, a dose of 60 kGy is seen to produce the highest ion-conductivity. Considering the variation of all parameters, the highest dc-conductivity of 6.06x10-2 S/m is obtained for the un-irradiated sample at room temperature, with 12.5 wt% LiClO4 and 35.71 wt% of glycerol as plasticizer. The samples are characterized in addition by XRD, SEM and FTIR respectively. Cyclic voltametry is performed for the confirmation of the electrolytic performance for pristine and gamma irradiated samples. To understand the experimental results, a model incorporating normal, as well as anomalous diffusion has been applied. Generalized calculus is used to model the anomalous diffusion. It is shown that this model successfully reproduces the experimental frequency dependence of the complex impedance for samples subjected to varying gamma dose. The physical interpretation of the model parameters and their variation with sample composition and irradiation dose is discussed.
Calculus Demonstrations Using MATLAB
ERIC Educational Resources Information Center
Dunn, Peter K.; Harman, Chris
2002-01-01
The note discusses ways in which technology can be used in the calculus learning process. In particular, five MATLAB programs are detailed for use by instructors or students that demonstrate important concepts in introductory calculus: Newton's method, differentiation and integration. Two of the programs are animated. The programs and the…
Initialized Fractional Calculus
NASA Technical Reports Server (NTRS)
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.
ERIC Educational Resources Information Center
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…
SAR amplitude probability density function estimation based on a generalized Gaussian model.
Moser, Gabriele; Zerubia, Josiane; Serpico, Sebastiano B
2006-06-01
In the context of remotely sensed data analysis, an important problem is the development of accurate models for the statistics of the pixel intensities. Focusing on synthetic aperture radar (SAR) data, this modeling process turns out to be a crucial task, for instance, for classification or for denoising purposes. In this paper, an innovative parametric estimation methodology for SAR amplitude data is proposed that adopts a generalized Gaussian (GG) model for the complex SAR backscattered signal. A closed-form expression for the corresponding amplitude probability density function (PDF) is derived and a specific parameter estimation algorithm is developed in order to deal with the proposed model. Specifically, the recently proposed "method-of-log-cumulants" (MoLC) is applied, which stems from the adoption of the Mellin transform (instead of the usual Fourier transform) in the computation of characteristic functions and from the corresponding generalization of the concepts of moment and cumulant. For the developed GG-based amplitude model, the resulting MoLC estimates turn out to be numerically feasible and are also analytically proved to be consistent. The proposed parametric approach was validated by using several real ERS-1, XSAR, E-SAR, and NASA/JPL airborne SAR images, and the experimental results prove that the method models the amplitude PDF better than several previously proposed parametric models for backscattering phenomena.
Impact of Calculus Reform in a Liberal Arts Calculus Course.
ERIC Educational Resources Information Center
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…
Supercalculators and University Entrance Calculus Examinations.
ERIC Educational Resources Information Center
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…
Hermeneutic operative calculus
NASA Astrophysics Data System (ADS)
Ramakrishnan, Sivakumar; Isawasan, Pradeep; Mohanan, Vasuky
2014-07-01
The predicate calculus used currently by mathematical logic in computer science, philosophy and linguistic was found to be too restrictive and inadequate for describing the grammar of natural and artificial language. Therefore many higher order logics have been developed to overcome the limitation of predicate calculus. In this paper a new representation of logic using mathematical principles has been developed for the natural language called Hermeneutic Operative Calculus. This Hermeneutic Operative Calculus is a new language interpretive calculus developed to account for the syntactic, semantic and pragmatic features of natural language and allows removing the restrictions of any particular natural language in the semantic field its map out. The logic of Hermeneutic Operative Calculus capable of represent the syntactic and semantic of factual information of a natural language precisely in any language. The logic of this Hermeneutic Operative Calculus has two different forms of operations called object and meta-operations. The object operation allow for listing the various objects, picturing the various propositions and so forth. The meta-operation would specify what cannot be specified by the object operation like semantical stances of a proposition. The basic operative processes of linguistics and cognitive logic will be mathematically conceptualized and elaborated in this paper.
Overfitting, generalization, and MSE in class probability estimation with high-dimensional data.
Kim, Kyung In; Simon, Richard
2014-03-01
Accurate class probability estimation is important for medical decision making but is challenging, particularly when the number of candidate features exceeds the number of cases. Special methods have been developed for nonprobabilistic classification, but relatively little attention has been given to class probability estimation with numerous candidate variables. In this paper, we investigate overfitting in the development of regularized class probability estimators. We investigate the relation between overfitting and accurate class probability estimation in terms of mean square error. Using simulation studies based on real datasets, we found that some degree of overfitting can be desirable for reducing mean square error. We also introduce a mean square error decomposition for class probability estimation that helps clarify the relationship between overfitting and prediction accuracy.
Descartes' Calculus of Subnormals: What Might Have Been
ERIC Educational Resources Information Center
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…
Fractional vector calculus and fractional Maxwell's equations
Tarasov, Vasily E.
2008-11-15
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.
On the error probability of general trellis codes with applications to sequential decoding
NASA Technical Reports Server (NTRS)
Johannesson, R.
1977-01-01
An upper bound on the average probability of error for maximum-likelihood decoding of the ensemble of random L-branch binary trellis codes of rate R = 1/n with distinction between memory length and tail length is given. It is shown that the bound is independent of the length L of the information sequence if the memory length exceeds the tail length by a specified amount that depends on L. Sequential decoding simulations using the stack algorithm were conducted to test the dependence of the undetected error probability on tail length and memory length, and the results corroborated the theory.
Discrete Quantum Gravity in the Regge Calculus Formalism
Khatsymovsky, V.M.
2005-09-01
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{sup -33} cm, implying a discrete spacetime structure on these scales.
Generalizations and Extensions of the Probability of Superiority Effect Size Estimator
ERIC Educational Resources Information Center
Ruscio, John; Gera, Benjamin Lee
2013-01-01
Researchers are strongly encouraged to accompany the results of statistical tests with appropriate estimates of effect size. For 2-group comparisons, a probability-based effect size estimator ("A") has many appealing properties (e.g., it is easy to understand, robust to violations of parametric assumptions, insensitive to outliers). We review…
2013-03-01
Presented to the Faculty Department of Operational Sciences Graduate School of Engineering and Management Air Force Institute of Technology Air...University Air Education and Training Command In Partial Fulfillment of the Requirements for the Degree of Master of Science in Operations ...to estimate these unknown multinomial success probabilities, , for each of the systems [17]. Bechhofer and Sobel [18] made use of multinomial
Putting Differentials Back into Calculus
ERIC Educational Resources Information Center
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.
Cartan calculus on quantum Lie algebras
Schupp, P.; Watts, P.; Zumino, B.
1993-12-09
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.``
Using Discovery in the Calculus Class
ERIC Educational Resources Information Center
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)
ERIC Educational Resources Information Center
Harding, Simon; Scott, Paul
2004-01-01
Calculus is a mathematical concept that is fundamental to how we understand the world around us. Whether it is in the world of technology, finance, astronomy, sociology, medicine, calculus in one form or another can be found. This brief article describes the origins of calculus in Greece, further developments by Newton and Leibniz, and the…
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.
ERIC Educational Resources Information Center
SMITH, GARY R.
THE CAPACITY OF INTERMEDIATE PUPILS TO UNDERSTAND AND RETAIN GENERALIZATIONS RELATED TO SIMPLE MACHINES, ELECTRICAL ENERGY, AND HEAT ENERGY WAS INVESTIGATED. A STRATIFIED RANDOM SAMPLE OF APPROXIMATELY 1,200 FOURTH, FIFTH, AND SIXTH GRADE PUPILS WAS SELECTED FROM THE METROPOLITAN DETROIT AREA. GENERALIZATIONS FOR THE THREE PHYSICAL SCIENCE AREAS…
Area Regge calculus and continuum limit [rapid communication
NASA Astrophysics Data System (ADS)
Khatsymovsky, V. M.
2002-11-01
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.
ERIC Educational Resources Information Center
Steckroth, Jeffrey J.
2010-01-01
For nearly three decades, during which the author taught everything from basic algebra to advanced placement calculus, the author thought of himself as a secondary school mathematics teacher. The notion of teaching elementary school math never appealed to him because of its simplicity. The author stresses that anyone could teach children to count,…
Tangent Lines without Calculus
ERIC Educational Resources Information Center
Rabin, Jeffrey M.
2008-01-01
This article presents a problem that can help high school students develop the concept of instantaneous velocity and connect it with the slope of a tangent line to the graph of position versus time. It also gives a method for determining the tangent line to the graph of a polynomial function at any point without using calculus. (Contains 1 figure.)
The general theory of relativity - Why 'It is probably the most beautiful of all existing theories'
NASA Astrophysics Data System (ADS)
Chandrasekhar, S.
1984-03-01
An attempt is made to objectively evaluate the frequent judgment of Einstein's general theory of relativity, by such distinguished physicists as Pauli (1921), Dirac, Born, and Rutherford, as 'beautiful' and 'a work of art'. The criteria applied are that of Francis Bacon ('There is no excellent beauty that hath not some strangeness in the proportions') and that of Heisenberg ('Beauty is the proper conformity of the parts to one another and to the whole'). The strangeness in the proportions of the theory of general relativity consists in its relating, through juxtaposition, the concepts of space and time and those of matter and motion; these had previously been considered entirely independent. The criterion of 'conformity' is illustrated through the directness with which the theory allows the description of black holes.
NASA Astrophysics Data System (ADS)
Papadopoulos, Vissarion; Kalogeris, Ioannis
2016-05-01
The present paper proposes a Galerkin finite element projection scheme for the solution of the partial differential equations (pde's) involved in the probability density evolution method, for the linear and nonlinear static analysis of stochastic systems. According to the principle of preservation of probability, the probability density evolution of a stochastic system is expressed by its corresponding Fokker-Planck (FP) stochastic partial differential equation. Direct integration of the FP equation is feasible only for simple systems with a small number of degrees of freedom, due to analytical and/or numerical intractability. However, rewriting the FP equation conditioned to the random event description, a generalized density evolution equation (GDEE) can be obtained, which can be reduced to a one dimensional pde. Two Galerkin finite element method schemes are proposed for the numerical solution of the resulting pde's, namely a time-marching discontinuous Galerkin scheme and the StreamlineUpwind/Petrov Galerkin (SUPG) scheme. In addition, a reformulation of the classical GDEE is proposed, which implements the principle of probability preservation in space instead of time, making this approach suitable for the stochastic analysis of finite element systems. The advantages of the FE Galerkin methods and in particular the SUPG over finite difference schemes, like the modified Lax-Wendroff, which is the most frequently used method for the solution of the GDEE, are illustrated with numerical examples and explored further.
NASA Technical Reports Server (NTRS)
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.
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.
On certain realizations of the q-deformed exterior differential calculus
NASA Astrophysics Data System (ADS)
Kerner, Richard; Abramov, Viktor
1999-04-01
We investigate two particular realizations of a q-deformed differential calculus at q being a primitive root of unity, qN = 1. Particular attention is paid to the Z3-graded case N = 3. First we construct an analogue of the exterior differential calculus on a manifold, then we introduce a discrete realization of such a calculus on generalized Clifford algebras. Finally, combining both constructions, we discuss a ZN-graded generalization of gauge theory.
Multiplicative calculus and its applications
NASA Astrophysics Data System (ADS)
Bashirov, Agamirza E.; Kurpinar, Emine Misirli; Özyapici, Ali
2008-01-01
Two operations, differentiation and integration, are basic in calculus and analysis. In fact, they are the infinitesimal versions of the subtraction and addition operations on numbers, respectively. In the period from 1967 till 1970 Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. In the present paper our aim is to bring up this calculus to the attention of researchers and demonstrate its usefulness.
Fisher information, Borges operators, and q-calculus
NASA Astrophysics Data System (ADS)
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].
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
Early Vector Calculus: A Path through Multivariable Calculus
ERIC Educational Resources Information Center
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.)
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.
Calculus in the Middle School?
ERIC Educational Resources Information Center
Barger, Rita H.; McCoy, Ann C.
2010-01-01
This article presents an example of how middle school teachers can lay a foundation for calculus. Although many middle school activities connect directly to calculus concepts, the authors have decided to look in depth at only one: the concept of change. They will show how teachers can lead their students to see and appreciate the calculus…
The Basic Principle of Calculus?
ERIC Educational Resources Information Center
Hardy, Michael
2011-01-01
A simple partial version of the Fundamental Theorem of Calculus can be presented on the first day of the first-year calculus course, and then relied upon repeatedly in assigned problems throughout the course. With that experience behind them, students can use the partial version to understand the full-fledged Fundamental Theorem, with further…
ERIC Educational Resources Information Center
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…
Mohammadkhani, Parvaneh; Azadmehr, Hedieh; Mobramm, Ardeshir; Naseri, Esmaeil
2015-01-01
Objective: 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. Method: 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. Results: 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). Discussion: 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. PMID:26005482
Applications of fractional calculus to epidemiological models
NASA Astrophysics Data System (ADS)
Skwara, Urszula; Martins, José; Ghaffari, Peyman; Aguiar, Maíra; Boto, João; Stollenwerk, Nico
2012-09-01
Epidemiological spreading does not only happen from person to neighbouring person but often over wide distances, when infected but asymptomatic persons travel and carry infection to others over wide distances. Superdiffusion has been suggested to model such spreading in spatially restriced contact networks, i.e. there is still a notion of geographical distance, but spreading happens with high probability proportional to large distances. From fractional calculus several ways of describing superdiffusion are know. Here we investigate the representation in Fourier space and which is easily generalizable to higher dimensional space in order to compare with stochastic models of epidemiological spreading.
Leveraging Prior Calculus Study with Embedded Review
ERIC Educational Resources Information Center
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…
Open Calculus: A Free Online Learning Environment
ERIC Educational Resources Information Center
Korey, Jane; Rheinlander, Kim; Wallace, Dorothy
2007-01-01
Dartmouth College mathematicians have developed a free online calculus course called "Open Calculus." Open Calculus is an exportable distance-learning/self-study environment for learning calculus including written text, nearly 4000 online homework problems and instructional videos. The paper recounts the evaluation of course elements since 2000 in…
A Simple Acronym for Doing Calculus: CAL
ERIC Educational Resources Information Center
Hathaway, Richard J.
2008-01-01
An acronym is presented that provides students a potentially useful, unifying view of the major topics covered in an elementary calculus sequence. The acronym (CAL) is based on viewing the calculus procedure for solving a calculus problem P* in three steps: (1) recognizing that the problem cannot be solved using simple (non-calculus) techniques;…
Fractional calculus in bioengineering, part 2.
Magin, Richard L
2004-01-01
Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub
Stochastic Calculus and Differential Equations for Physics and Finance
NASA Astrophysics Data System (ADS)
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.
Integral calculus problem solving: an fMRI investigation.
Krueger, Frank; Spampinato, Maria Vittoria; Pardini, Matteo; Pajevic, Sinisa; Wood, Jacqueline N; Weiss, George H; Landgraf, Steffen; Grafman, Jordan
2008-07-16
Only a subset of adults acquires specific advanced mathematical skills, such as integral calculus. The representation of more sophisticated mathematical concepts probably evolved from basic number systems; however its neuroanatomical basis is still unknown. Using fMRI, we investigated the neural basis of integral calculus while healthy participants were engaged in an integration verification task. Solving integrals activated a left-lateralized cortical network including the horizontal intraparietal sulcus, posterior superior parietal lobe, posterior cingulate gyrus, and dorsolateral prefrontal cortex. Our results indicate that solving of more abstract and sophisticated mathematical facts, such as calculus integrals, elicits a pattern of brain activation similar to the cortical network engaged in basic numeric comparison, quantity manipulation, and arithmetic problem solving.
Regge calculus and observations. II. Further applications.
NASA Astrophysics Data System (ADS)
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.
Cartooning in Algebra and Calculus
ERIC Educational Resources Information Center
Moseley, L. Jeneva
2014-01-01
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
A White Noise Theory of Infinite Dimensional Calculus
1989-10-01
a general theory; however it is his hope that this attempt would be the very first step towards the study of Gaussian random fields using variational ... calculus . Contents: White noise; Generalized functionals; Rotation group and harmonic analysis; Applications to Physics; Gaussian random fields. Keywords: Statistic processes.
Affine connection form of Regge calculus
NASA Astrophysics Data System (ADS)
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.
Federal Register 2010, 2011, 2012, 2013, 2014
2011-10-13
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Naive Probability: A Mental Model Theory of Extensional Reasoning.
ERIC Educational Resources Information Center
Johnson-Laird, P. N.; Legrenzi, Paolo; Girotto, Vittorio; Legrenzi, Maria Sonino; Caverni, Jean-Paul
1999-01-01
Outlines a theory of naive probability in which individuals who are unfamiliar with the probability calculus can infer the probabilities of events in an "extensional" way. The theory accommodates reasoning based on numerical premises, and explains how naive reasoners can infer posterior probabilities without relying on Bayes's theorem.…
``Riemann equations'' in bidifferential calculus
NASA Astrophysics Data System (ADS)
Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.
2015-10-01
We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.
A Tutorial Review on Fractal Spacetime and Fractional Calculus
NASA Astrophysics Data System (ADS)
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.
The development and nature of problem-solving among first-semester calculus students
NASA Astrophysics Data System (ADS)
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
Making Implicit Multivariable Calculus Representations Explicit: A Clinical Study
ERIC Educational Resources Information Center
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…
Geometric Demonstration of the Fundamental Theorems of the Calculus
ERIC Educational Resources Information Center
Sauerheber, Richard D.
2010-01-01
After the monumental discovery of the fundamental theorems of the calculus nearly 350 years ago, it became possible to answer extremely complex questions regarding the natural world. Here, a straightforward yet profound demonstration, employing geometrically symmetric functions, describes the validity of the general power rules for integration and…
The Power of Investigative Calculus Projects
ERIC Educational Resources Information Center
Perrin, John Robert; Quinn, Robert J.
2008-01-01
This article describes investigative calculus projects in which students explore a question or problem of their own construction. Three exemplary pieces of student work are showcased. Investigative calculus projects are an excellent way to foster student understanding and interest in calculus. (Contains 4 figures.)
An AP Calculus Classroom Amusement Park
ERIC Educational Resources Information Center
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.…
ERIC Educational Resources Information Center
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…
Calculus Student Understanding of Continuity
ERIC Educational Resources Information Center
Wangle, Jayleen Lillian
2013-01-01
Continuity is a central concept in calculus. Yet very few students seem to understand the nature of continuity. The research described was conducted in two stages. Students were asked questions in multiple choice and true/false format regarding function, limit and continuity. These results were used to identify participants as strong, weak or…
Portfolio Analysis for Vector Calculus
ERIC Educational Resources Information Center
Kaplan, Samuel R.
2015-01-01
Classic stock portfolio analysis provides an applied context for Lagrange multipliers that undergraduate students appreciate. Although modern methods of portfolio analysis are beyond the scope of vector calculus, classic methods reinforce the utility of this material. This paper discusses how to introduce classic stock portfolio analysis in a…
Constructivized Calculus in College Mathematics
ERIC Educational Resources Information Center
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…
Calculus Students' Understanding of Volume
ERIC Educational Resources Information Center
Dorko, Allison; Speer, Natasha M.
2013-01-01
Researchers have documented difficulties that elementary school students have in understanding volume. Despite its importance in higher mathematics, we know little about college students' understanding of volume. This study investigated calculus students' understanding of volume. Clinical interview transcripts and written responses to volume…
Mathematical Features of the Calculus
ERIC Educational Resources Information Center
Sauerheber, Richard D.
2010-01-01
The fundamental theorems of the calculus describe the relationships between derivatives and integrals of functions. The value of any function at a particular location is the definite derivative of its integral and the definite integral of its derivative. Thus, any value is the magnitude of the slope of the tangent of its integral at that position,…
Reading the World with Calculus
ERIC Educational Resources Information Center
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…
The Pendulum and the Calculus.
ERIC Educational Resources Information Center
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…
Children, Additive Change, and Calculus.
ERIC Educational Resources Information Center
Nemirovsky, Ricardo; And Others
Students can learn to solve problems of qualitative integration and differentiation independently of their study of formal calculus or algebra. This exploratory study investigated the basic intuitions that elementary school children construct in their daily experience with physical and symbolic change. Elementary school children (n=18) were…
Conceptual Knowledge in Introductory Calculus.
ERIC Educational Resources Information Center
White, Paul; Mitchelmore, Michael
1996-01-01
Responses to rate-of-change problems were collected during and after 24 hours of conceptual calculus instruction given to first-year university students. Analysis revealed three categories of error in which variables were treated as symbols to be manipulated rather than quantities to be related. Contains test questions. (Author/MKR)
Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus
NASA Astrophysics Data System (ADS)
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.
Factors Associated with Success in College Calculus II
ERIC Educational Resources Information Center
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…
MEERSCHAERT, MARK M.; SABZIKAR, FARZAD; CHEN, JINGHUA
2014-01-01
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 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. PMID:26085690
Sabzikar, Farzad; Meerschaert, Mark M.; Chen, Jinghua
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.
Complexity and the Fractional Calculus
2013-01-01
Paolo Grigolini,, Mauro Bologna,, Bruce West 611102 c. THIS PAGE The public reporting burden for this collection of information is estimated to...dx.doi.org/10.1155/2013/498789 Research Article Complexity and the Fractional Calculus Pensri Pramukkul,1 Adam Svenkeson,1 Paolo Grigolini,1 Mauro ...8. PERFORMING ORGANIZATION REPORT NUMBER 19a. NAME OF RESPONSIBLE PERSON 19b. TELEPHONE NUMBER Paolo Grigolini Pensri Pramukkul, Adam Svenkeson
Formalization of the Integral Calculus in the PVS Theorem Prover
NASA Technical Reports Server (NTRS)
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.
Thermodynamics in Fractional Calculus
NASA Astrophysics Data System (ADS)
Meilanov, R. P.; Magomedov, R. A.
2014-11-01
A generalization of thermodynamics in the formalism of fractional-order derivatives is given. Results of the traditional thermodynamics of Carnot, Clausius, and Helmholtz are obtained in the particular case where the exponent of a fractional-order derivative is equal to unity. A one-parametric "fractal" equation of state is obtained with account of the second virial coefficient. The application of the resulting equation of state in the case of the gas argon is considered.
VEST: Abstract vector calculus simplification in Mathematica
NASA Astrophysics Data System (ADS)
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.
VEST: Abstract Vector Calculus Simplification in Mathematica
J. Squire, J. Burby and H. Qin
2013-03-12
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 scalar and vector expressions of a very general type using a systematic canonicalization procedure. In addition, utilizing properties of the Levi-Civita symbol, the program can derive types of multi-term vector identities that are not recognized by canonicalization, subsequently applying these to simplify large expressions. In a companion paper [1], we employ VEST in the automation of the calculation of 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. __________________________________________________
Staghorn calculus endotoxin expression in sepsis.
McAleer, Irene M; Kaplan, George W; Bradley, John S; Carroll, Stephen F
2002-04-01
Staghorn calculi are infrequent and generally are infected stones. Struvite or apatite calculi are embedded with gram-negative bacteria, which can produce endotoxin. Sepsis syndrome may occur after surgical therapy or endoscopic manipulation of infected or staghorn calculi. Sepsis, which can occur despite perioperative antibiotic use, may be due to bacteremia or endotoxemia. We present a child with an infected staghorn calculus who developed overwhelming sepsis and died after percutaneous stone manipulation. Endotoxin assay of stone fragments demonstrated an extremely high level of endotoxin despite low colony bacterial culture growth. This is the first reported case in which endotoxin was demonstrated in stone fragments from a child who died of severe sepsis syndrome after percutaneous staghorn stone manipulation.
Schroyens, Walter; Fleerackers, Lieve; Maes, Sunile
2010-12-15
Two experiments (N(1) = 117 and N(2) = 245) on reasoning with knowledge-rich conditionals showed a main effect of logical validity, which was due to the negative effect of counter-examples being smaller for valid than for invalid arguments. These findings support the thesis that some people tend to inhibit background inconsistent with the hypothetical truth of the premises, while others tend to abandon the implicit truth-assumption when they have factual evidence to the contrary. Findings show that adhering to the truth-assumption in the face of conflicting evidence to the contrary requires an investment of time and effort which people with a higher general aptitude are more likely to do.
Schroyens, Walter; Fleerackers, Lieve; Maes, Sunile
2010-01-01
Two experiments (N1 = 117 and N2 = 245) on reasoning with knowledge-rich conditionals showed a main effect of logical validity, which was due to the negative effect of counter-examples being smaller for valid than for invalid arguments. These findings support the thesis that some people tend to inhibit background inconsistent with the hypothetical truth of the premises, while others tend to abandon the implicit truth-assumption when they have factual evidence to the contrary. Findings show that adhering to the truth-assumption in the face of conflicting evidence to the contrary requires an investment of time and effort which people with a higher general aptitude are more likely to do. PMID:21228921
R-Function Relationships for Application in the Fractional Calculus
NASA Technical Reports Server (NTRS)
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.
R-function relationships for application in the fractional calculus.
Lorenzo, Carl F; Hartley, Tom T
2008-01-01
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, et, and its fractional derivatives are shown. From the relationships developed, some important approximations are observed. Further, the inverse relationships of the exponential function, et, 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.
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
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.
Calculus and Success in a Business School
ERIC Educational Resources Information Center
Kim, Dong-gook; Garcia, Fernando; Dey, Ishita
2012-01-01
Many business schools or colleges require calculus as a prerequisite for certain classes or for continuing to upper division courses. While there are many studies investigating the relationship between performance in calculus and performance in a single course, such as economics, statistics, and finance, there are very few studies investigating…
A Cross-National Study of Calculus
ERIC Educational Resources Information Center
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…
Educating about Sustainability while Enhancing Calculus
ERIC Educational Resources Information Center
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: An Active Approach with Projects.
ERIC Educational Resources Information Center
Hilbert, Stephen; And Others
1993-01-01
Discusses a pedagogical approach to calculus based on the question: What kinds of problems should students be able to solve? Includes a discussion of types of problems and curriculum threads for such a course. Describes a projects-based calculus with examples of projects and classroom activities. (Author/MDH)
Attendance and Attainment in a Calculus Course
ERIC Educational Resources Information Center
Meulenbroek, Bernard; van den Bogaard, Maartje
2013-01-01
In this paper the relationship between attendance and attainment in a standard calculus course is investigated. Calculus could in principle be studied without attending lectures due to the wealth of material available (in hardcopy and online). However, in this study we will show that the pass rate of students attending classes regularly (>75%…
Imagine Yourself in This Calculus Classroom
ERIC Educational Resources Information Center
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…
Calculus: An Active Approach with Projects.
ERIC Educational Resources Information Center
Hilbert, Steve; And Others
Ithaca College, in New York, has developed and tested a projects-based first-year calculus course over the last 3 years which uses the graphs of functions and physical phenomena to illustrate and motivate the major concepts of calculus and to introduce students to mathematical modeling. The course curriculum is designed to: (1) emphasize on the…
How Can Histograms Be Useful for Introducing Continuous Probability Distributions?
ERIC Educational Resources Information Center
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.…
Formal Modeling of Multi-Agent Systems using the Pi-Calculus and Epistemic Logic
NASA Technical Reports Server (NTRS)
Rorie, Toinette; Esterline, Albert
1998-01-01
Multi-agent systems have become important recently in computer science, especially in artificial intelligence (AI). We allow a broad sense of agent, but require at least that an agent has some measure of autonomy and interacts with other agents via some kind of agent communication language. We are concerned in this paper with formal modeling of multi-agent systems, with emphasis on communication. We propose for this purpose to use the pi-calculus, an extension of the process algebra CCS. Although the literature on the pi-calculus refers to agents, the term is used there in the sense of a process in general. It is our contention, however, that viewing agents in the AI sense as agents in the pi-calculus sense affords significant formal insight. One formalism that has been applied to agents in the AI sense is epistemic logic, the logic of knowledge. The success of epistemic logic in computer science in general has come in large part from its ability to handle concepts of knowledge that apply to groups. We maintain that the pi-calculus affords a natural yet rigorous means by which groups that are significant to epistemic logic may be identified, encapsulated, structured into hierarchies, and restructured in a principled way. This paper is organized as follows: Section 2 introduces the pi-calculus; Section 3 takes a scenario from the classical paper on agent-oriented programming [Sh93] and translates it into a very simple subset of the n-calculus; Section 4 then shows how more sophisticated features of the pi-calculus may bc brought into play; Section 5 discusses how the pi-calculus may be used to define groups for epistemic logic; and Section 6 is the conclusion.
Quantum stochastic calculus associated with quadratic quantum noises
NASA Astrophysics Data System (ADS)
Ji, Un Cig; Sinha, Kalyan B.
2016-02-01
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.
Fractional calculus in hydrologic modeling: A numerical perspective.
Benson, David A; Meerschaert, Mark M; Revielle, Jordan
2013-01-01
Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.
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.
Separation of noncommutative differential calculus on quantum Minkowski space
Bachmaier, Fabian; Blohmann, Christian
2006-02-15
Noncommutative differential calculus on quantum Minkowski space is not separated with respect to the standard generators, in the sense that partial derivatives of functions of a single generator can depend on all other generators. It is shown that this problem can be overcome by a separation of variables. We study the action of the universal L-matrix, appearing in the coproduct of partial derivatives, on generators. Powers of the resulting quantum Minkowski algebra valued matrices are calculated. This leads to a nonlinear coordinate transformation which essentially separates the calculus. A compact formula for general derivatives is obtained in form of a chain rule with partial Jackson derivatives. It is applied to the massive quantum Klein-Gordon equation by reducing it to an ordinary q-difference equation. The rest state solution can be expressed in terms of a product of q-exponential functions in the separated variables.
Quantum stochastic calculus associated with quadratic quantum noises
Ji, Un Cig; Sinha, Kalyan B.
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.
Using Dynamic Software to Address Common College Calculus Stumbling Blocks
ERIC Educational Resources Information Center
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…
Calculus: The Dynamics of Change. MAA Notes Number 39.
ERIC Educational Resources Information Center
Roberts, A. Wayne, Ed.
This book discusses the calculus reform effort. The first essay captures the basic themes that should characterize a calculus course that is modern in its vision as well as its pedagogy and content. The next section contains essays on the vision of calculus reform: "Visions of Calculus" (Sharon Cutler Ross); "Nonalgebraic Approaches…
NASA Astrophysics Data System (ADS)
Jaynes, E. T.; Bretthorst, G. Larry
2003-04-01
Foreword; Preface; Part I. Principles and Elementary Applications: 1. Plausible reasoning; 2. The quantitative rules; 3. Elementary sampling theory; 4. Elementary hypothesis testing; 5. Queer uses for probability theory; 6. Elementary parameter estimation; 7. The central, Gaussian or normal distribution; 8. Sufficiency, ancillarity, and all that; 9. Repetitive experiments, probability and frequency; 10. Physics of 'random experiments'; Part II. Advanced Applications: 11. Discrete prior probabilities, the entropy principle; 12. Ignorance priors and transformation groups; 13. Decision theory: historical background; 14. Simple applications of decision theory; 15. Paradoxes of probability theory; 16. Orthodox methods: historical background; 17. Principles and pathology of orthodox statistics; 18. The Ap distribution and rule of succession; 19. Physical measurements; 20. Model comparison; 21. Outliers and robustness; 22. Introduction to communication theory; References; Appendix A. Other approaches to probability theory; Appendix B. Mathematical formalities and style; Appendix C. Convolutions and cumulants.
The Impact of Taking a College Pre-Calculus Course on Students' College Calculus Performance
ERIC Educational Resources Information Center
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…
ERIC Educational Resources Information Center
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
NASA Astrophysics Data System (ADS)
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.
Fractional variational calculus in terms of Riesz fractional derivatives
NASA Astrophysics Data System (ADS)
Agrawal, O. P.
2007-06-01
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.
Real-Time Exponential Curve Fits Using Discrete Calculus
NASA Technical Reports Server (NTRS)
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.
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.
Applications of Monte Carlo Methods in Calculus.
ERIC Educational Resources Information Center
Gordon, Sheldon P.; Gordon, Florence S.
1990-01-01
Discusses the application of probabilistic ideas, especially Monte Carlo simulation, to calculus. Describes some applications using the Monte Carlo method: Riemann sums; maximizing and minimizing a function; mean value theorems; and testing conjectures. (YP)
Extending stochastic network calculus to loss analysis.
Luo, Chao; Yu, Li; Zheng, Jun
2013-01-01
Loss is an important parameter of Quality of Service (QoS). Though stochastic network calculus is a very useful tool for performance evaluation of computer networks, existing studies on stochastic service guarantees mainly focused on the delay and backlog. Some efforts have been made to analyse loss by deterministic network calculus, but there are few results to extend stochastic network calculus for loss analysis. In this paper, we introduce a new parameter named loss factor into stochastic network calculus and then derive the loss bound through the existing arrival curve and service curve via this parameter. We then prove that our result is suitable for the networks with multiple input flows. Simulations show the impact of buffer size, arrival traffic, and service on the loss factor.
One Answer to "What Is Calculus?"
ERIC Educational Resources Information Center
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)
Lexicographic Probability, Conditional Probability, and Nonstandard Probability
2009-11-11
the following conditions: CP1. µ(U |U) = 1 if U ∈ F ′. CP2 . µ(V1 ∪ V2 |U) = µ(V1 |U) + µ(V2 |U) if V1 ∩ V2 = ∅, U ∈ F ′, and V1, V2 ∈ F . CP3. µ(V |U...µ(V |X)× µ(X |U) if V ⊆ X ⊆ U , U,X ∈ F ′, V ∈ F . Note that it follows from CP1 and CP2 that µ(· |U) is a probability measure on (W,F) (and, in... CP2 hold. This is easily seen to determine µ. Moreover, µ vaciously satisfies CP3, since there do not exist distinct sets U and X in F ′ such that U
Confidence Probability versus Detection Probability
Axelrod, M
2005-08-18
In a discovery sampling activity the auditor seeks to vet an inventory by measuring (or inspecting) a random sample of items from the inventory. When the auditor finds every sample item in compliance, he must then make a confidence statement about the whole inventory. For example, the auditor might say: ''We believe that this inventory of 100 items contains no more than 5 defectives with 95% confidence.'' Note this is a retrospective statement in that it asserts something about the inventory after the sample was selected and measured. Contrast this to the prospective statement: ''We will detect the existence of more than 5 defective items in this inventory with 95% probability.'' The former uses confidence probability while the latter uses detection probability. For a given sample size, the two probabilities need not be equal, indeed they could differ significantly. Both these probabilities critically depend on the auditor's prior belief about the number of defectives in the inventory and how he defines non-compliance. In other words, the answer strongly depends on how the question is framed.
Functional calculus using wavelet transforms
NASA Astrophysics Data System (ADS)
Holschneider, Matthias
1994-07-01
It is shown how the wavelet transform may be used to compute for a function s the symbol s(A) for any (not necessarily) self-adjoint operator A whose spectrum is contained in the upper half plane. For self-adjoint operators it is shown that this functional calculus coincides with the usual one. In particular it is shown how the exponential eitA can be written in terms of the resolvent Rz=(A-z)-1 of A as follows: eitA=(1/c) ∫0∞da an-2∫-∞+∞ dbĝ¯ (at)eitbRb-ian(A), with c=-2iπ×∫0∞(dω/ω) (-iω)n-1ĝ¯(ω)e-ω, and n∈N, and the integral is understood as the Cesaro limit. This shows explicitly how the behavior for large t is determined by the behavior of Rz at Iz ≂1/t.
Teaching Special Relativity Without Calculus
NASA Astrophysics Data System (ADS)
Ruby, Lawrence
2009-04-01
I 2007 many AAPT members received a booklet that is the first chapter of a physics textbook available on a CD. This book espouses the new educational philosophy of teaching special relativity as the first item in the topic of mechanics. Traditionally, special relativity is part of one or more modern physics chapters at the end of the text,2 and very often this material is never utilized due to time constraints. From a logical standpoint, special relativity is important in satellite communications and in cosmology, as well as in modern physics applications such as atomic theory and high-energy physics. The purpose of this paper is to show that the new philosophy can be carried out in a noncalculus physics course, by demonstrating that all of the principal results of special relativity theory can be obtained by simple algebra. To accomplish this, we shall propose alternate derivations for two results that are usually obtained with calculus. Textbooks2 typically obtain the equations for time dilation and for length contraction from simple considerations based on Einstein's second postulate.3 We shall start from this point.
Dynamical Correspondence in a Generalized Quantum Theory
NASA Astrophysics Data System (ADS)
Niestegge, Gerd
2015-05-01
In order to figure out why quantum physics needs the complex Hilbert space, many attempts have been made to distinguish the C*-algebras and von Neumann algebras in more general classes of abstractly defined Jordan algebras (JB- and JBW-algebras). One particularly important distinguishing property was identified by Alfsen and Shultz and is the existence of a dynamical correspondence. It reproduces the dual role of the selfadjoint operators as observables and generators of dynamical groups in quantum mechanics. In the paper, this concept is extended to another class of nonassociative algebras, arising from recent studies of the quantum logics with a conditional probability calculus and particularly of those that rule out third-order interference. The conditional probability calculus is a mathematical model of the Lüders-von Neumann quantum measurement process, and third-order interference is a property of the conditional probabilities which was discovered by Sorkin (Mod Phys Lett A 9:3119-3127, 1994) and which is ruled out by quantum mechanics. It is shown then that the postulates that a dynamical correspondence exists and that the square of any algebra element is positive still characterize, in the class considered, those algebras that emerge from the selfadjoint parts of C*-algebras equipped with the Jordan product. Within this class, the two postulates thus result in ordinary quantum mechanics using the complex Hilbert space or, vice versa, a genuine generalization of quantum theory must omit at least one of them.
Loop calculus in statistical physics and information science.
Chertkov, Michael; Chernyak, Vladimir Y
2006-06-01
Considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series. The leading term in the series is the Bethe-Peierls (belief propagation) (BP) contribution; the rest are expressed as loop contributions on the factor graph and calculated directly using the BP solution. The series unveils a small parameter that often makes the BP approximation so successful. Applications of the loop calculus in statistical physics and information science are discussed.
27% Probable: Estimating Whether or Not Large Numbers Are Prime.
ERIC Educational Resources Information Center
Bosse, Michael J.
2001-01-01
This brief investigation exemplifies such considerations by relating concepts from number theory, set theory, probability, logic, and calculus. Satisfying the call for students to acquire skills in estimation, the following technique allows one to "immediately estimate" whether or not a number is prime. (MM)
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.
Anti-calculus and whitening toothpastes.
van Loveren, Cor; Duckworth, Ralph M
2013-01-01
In terms of novel formulations, there seems to have been a shift in emphasis from anti-caries/anti-gingivitis to anti-calculus/whitening toothpastes in recent years. The anti-calculus and whitening effects of toothpastes are to some extent based on the same active ingredients: compounds of high affinity for tooth mineral. Due to this affinity, crystal growth may be hindered (anti-calculus) and chromophores be displaced (whitening). Besides these common ingredients, both types of toothpaste may contain agents specifically aimed at each condition. Clinical studies have shown that these active ingredients can be successfully formulated in fluoride toothpastes to give significant reductions in supragingival calculus and stain formation and facilitate their removal. Some of the ingredients are formulated in toothpastes that additionally contain anti-plaque and anti-gingivitis ingredients, making these toothpastes (together with the fluoride) truly multi-functional. The development of these products is not straightforward because of interaction between formulation components and because the active ingredients must maintain their beneficial characteristics during the shelf life of the paste. Neither a therapeutic benefit (in terms of less gingivitis or less caries) nor a societal benefit (in terms of less treatment demand) has been demonstrated as a result of the anti-calculus and whitening effects of toothpastes.
Calculus domains modelled using an original bool algebra based on polygons
NASA Astrophysics Data System (ADS)
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.
Measuring Transducer Modelled by Means of Fractional Calculus
NASA Astrophysics Data System (ADS)
Luft, Mirosław; Szychta, Elżbieta; Cioć, Radosław; Pietruszczak, Daniel
The article is inspired by developments of the fractional calculus in different areas of science such as the control theory and electrical measurements. The current interest in mathematical analysis employing the fractional differential and integral calculus reflects the usefulness of this calculus in the development of more precise - closer to the actual observation - mathematical models of various phenomena. A model of a measuring transducer is presented, developed by means of fractional calculus. Tests are executed in the programming environment MATLAB-SIMULINK.
On Development of an Adaptive Tutoring System for Calculus Learning
NASA Astrophysics Data System (ADS)
Yokota, Hisashi
2010-06-01
One-on-one tutoring is known to be an effective model for learning calculus. Therefore, implementing one-on-one tutoring system into calculus learning software is a natural thing to do. The purpose of this article is to describe how to diagnose a students' knowledge structure about calculus without asking many questions and to show how an adaptive tutoring system is implemented into our calculus learning software JCALC.
Singular optimal control and the identically non-regular problem in the calculus of variations
NASA Technical Reports Server (NTRS)
Menon, P. K. A.; Kelley, H. J.; Cliff, E. M.
1985-01-01
A small but interesting class of optimal control problems featuring a scalar control appearing linearly is equivalent to the class of identically nonregular problems in the Calculus of Variations. It is shown that a condition due to Mancill (1950) is equivalent to the generalized Legendre-Clebsch condition for this narrow class of problems.
Improving Calculus II and III through the Redistribution of Topics
ERIC Educational Resources Information Center
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…
A Transition Course from Advanced Placement to College Calculus
ERIC Educational Resources Information Center
Lucas, Timothy A.; Spivey, Joseph
2011-01-01
In the Spring of 2007, a group of highly motivated mathematics graduate students conducted a review of Duke's Calculus curriculum. They focused on two main problems. The first problem is the result of a very positive trend: a growing number of students are earning AP credit for Calculus I in high school. However, this results in Calculus II…
Science 101: How Do We Use Calculus in Science?
ERIC Educational Resources Information Center
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 Historical Perspective on Teaching and Learning Calculus
ERIC Educational Resources Information Center
Doorman, Michiel; van Maanen, Jan
2008-01-01
Calculus is one of those topics in mathematics where the algorithmic manipulation of symbols is easier than understanding the underlying concepts. Around 1680 Leibniz invented a symbol system for calculus that codifies and simplifies the essential elements of reasoning. The calculus of Leibniz brings within the reach of an ordinary student…
Computer Managed Instruction Homework Modules for Calculus I.
ERIC Educational Resources Information Center
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.…
LOOP CALCULUS AND BELIEF PROPAGATION FOR Q-ARY ALPHABET: LOOP TOWER
CHERTKOV, MICHAEL; CHERNYAK, VLADIMIR
2007-01-10
Loop calculus introduced in [1], [2] constitutes a new theoretical tool that explicitly expresses symbol Maximum-A-Posteriori (MAP) solution of a general statistical inference problem via a solution of the Belief Propagation (BP) equations. This finding brought a new significance to the BP concept, which in the past was thought of as just a loop-free approximation. In this paper they continue a discussion of the Loop Calculus, partitioning the results into three Sections. In Section 1 they introduce a new formulation of the Loop Calculus in terms of a set of transformations (gauges) that keeping the partition function of the problem invariant. The full expression contains two terms referred to as the 'ground state' and 'excited states' contributions. The BP equations are interpreted as a special (BP) gauge fixing condition that emerges as a special orthogonality constraint between the ground state and excited states, which also selects loop contributions as the only surviving ones among the excited states. In Section 2 they demonstrate how the invariant interpretation of the Loop Calculus, introduced in Section 1, allows a natural extension to the case of a general q-ary alphabet, this is achieved via a loop tower sequential construction. The ground level in the tower is exactly equivalent to assigning one color (out of q available) to the 'ground state' and considering all 'excited' states colored in the remaining (q-1) colors, according to the loop calculus rule. Sequentially, the second level in the tower corresponds to selecting a loop from the previous step, colored in (q-1) colors, and repeating the same ground vs excited states splitting procedure into one and (q-2) colors respectively. The construction proceeds till the full (q-1)-levels deep loop tower (and the corresponding contributions to the partition function) are established. In Section 3 they discuss an ultimate relation between the loop calculus and the Bethe-Free energy variational approach of [3].
An Executable Calculus for Service Choreography
NASA Astrophysics Data System (ADS)
Besana, Paolo; Barker, Adam
The Lightweight Coordination Calculus (LCC) is a compact choreography language based on process calculus. LCC is a directly executable specification and can therefore be dynamically distributed to a group of peers for enactment at run-time; this offers flexibility and allows peers to coordinate in open systems without prior knowledge of an interaction. This paper contributes to the body of choreography research by proposing two extensions to LCC covering parallel composition and choreography abstraction. These language extensions are evaluated against a subset of the Service Interaction Patterns, a benchmark in the process modelling community.
Is Calculus Really That Different from Algebra? A More Logical Way To Understand and Teach Calculus.
ERIC Educational Resources Information Center
Elk, Seymour B.
1998-01-01
Discards the blinders that have hampered the traditional teaching of calculus and reexamines some of the intuitive ideas that underlie this subject matter. Analyzes the various indeterminate forms that arise through the blind application of algebraic operations. (Author/ASK)
On the Equivalence of Probability Measures.
1981-01-01
example Brownian motion) and for Y a process of the form V+ X, where V is of bounded variation . The privileged tool is then stochastic calculus. We...martingale by a process of bounded variation is again a martingale with the same local characteristics (in fact, in its generality, the theorem says that...the class of semimartingales, that is sums of martingales and processes of bounded variation , is invariant under absolutely continuous changes of
Using Matlab in a Multivariable Calculus Course.
ERIC Educational Resources Information Center
Schlatter, Mark D.
The benefits of high-level mathematics packages such as Matlab include both a computer algebra system and the ability to provide students with concrete visual examples. This paper discusses how both capabilities of Matlab were used in a multivariate calculus class. Graphical user interfaces which display three-dimensional surfaces, contour plots,…
On Online Assignments in a Calculus Class
ERIC Educational Resources Information Center
Jungic, Veselin; Kent, Deborah; Menz, Petra
2012-01-01
In this paper, we describe our experience with the creation and utilization of online assignments for several calculus classes at Simon Fraser University (SFU). We present our findings regarding available software by considering the needs and perspectives of the instructors, students, and administrators. We provide a list of questions that guide…
Are Homeschoolers Prepared for College Calculus?
ERIC Educational Resources Information Center
Wilkens, Christian P.; Wade, Carol H.; Sonnert, Gerhard; Sadler, Philip M.
2015-01-01
Homeschooling in the United States has grown considerably over the past several decades. This article presents findings from the Factors Influencing College Success in Mathematics (FICSMath) survey, a national study of 10,492 students enrolled in tertiary calculus, including 190 students who reported homeschooling for a majority of their high…
Boolean integral calculus for digital systems
NASA Technical Reports Server (NTRS)
Tucker, J. H.; Tapia, M. A.; Bennett, A. W.
1985-01-01
The concept of Boolean integration is introduced and developed. When the changes in a desired function are specified in terms of changes in its arguments, then ways of 'integrating' (i.e., realizing) the function, if it exists, are presented. Boolean integral calculus has applications in design of logic circuits.
Some Factors Effected Student's Calculus Learning Outcome
ERIC Educational Resources Information Center
Rajagukguk, Wamington
2016-01-01
The purpose of this study is to determine the factors effected calculus learning outcome of the student. This study was conducted with 176 respondents, which were selected randomly. The data were obtained by questionnaire, and then analyzed by using multiple regressions, and correlation, at level of a = 0.05. The findings showed there is the…
I Teach Economics, Not Algebra and Calculus
ERIC Educational Resources Information Center
Hey, John D.
2005-01-01
Most people learn to drive without knowing how the engine works. In a similar vein, the author believes that students can learn economics without knowing the algebra and calculus underlying the results. If instructors follow the philosophy of other economics courses in using graphs to illustrate the results, and draw the graphs accurately, then…
Students' Difficulties with Vector Calculus in Electrodynamics
ERIC Educational Resources Information Center
Bollen, Laurens; van Kampen, Paul; De Cock, Mieke
2015-01-01
Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven…
Beliefs about Proof in Collegiate Calculus.
ERIC Educational Resources Information Center
Raman, Manya
The broad aim of this research is to characterize the views of proof held by college calculus students and their two types of teachers mathematics graduate students and professors. The analysis is based on an examination of the ways in which people in all three groups produce and evaluate different types of solutions to a proof-based problem from…
A symbol calculus for Toeplitz operators.
Berger, C A; Coburn, L A
1986-05-01
We give a complete characterization of those functions on 2n-dimensional Euclidean space for which the Berezin-Toeplitz quantizations admit a symbol calculus modulo the compact operators. The functions in question are characterized by a condition of "small oscillation at infinity."
Sharks, Minnows, and Wheelbarrows: Calculus Modeling Projects
ERIC Educational Resources Information Center
Smith, Michael D.
2011-01-01
The purpose of this article is to present two very active applied modeling projects that were successfully implemented in a first semester calculus course at Hollins University. The first project uses a logistic equation to model the spread of a new disease such as swine flu. The second project is a human take on the popular article "Do Dogs Know…
Maple Graphing Tools for Calculus III
ERIC Educational Resources Information Center
Cook, Darwyn
2006-01-01
For those instructors lacking artistic skills, teaching 3-dimensional calculus can be a challenge. Although some instructors spend a great deal of time working on their illustrations, trying to get them just right, students nevertheless often have a difficult time understanding some of them. To address this problem, the author has written a series…
Using the Microcomputer to Enhance Calculus Teaching.
ERIC Educational Resources Information Center
Clayton, Debbie; And Others
1990-01-01
Discusses differences between computer-enhanced learning (CEL) and computer-aided learning (CAL), and describes a microcomputer-based graph-plotting program called Capgraph that was developed for use in a college calculus course. Results of a course evaluation are presented; student attitudes are described; and future considerations are discussed.…
ERIC Educational Resources Information Center
Bolt, Mike
2010-01-01
Many optimization problems can be solved without resorting to calculus. This article develops a new variational method for optimization that relies on inequalities. The method is illustrated by four examples, the last of which provides a completely algebraic solution to the problem of minimizing the time it takes a dog to retrieve a thrown ball,…
Students' Understanding of Functions in Calculus Courses.
ERIC Educational Resources Information Center
Monk, G. S.
1994-01-01
Reports on a study of students' responses to two types of questions on final examinations in calculus. Concludes that the two kinds of understanding--pointwise and across time--are clearly distinguishable. Discusses the differences between these two types of understanding. (ASK)
Developing Flexible Procedural Knowledge in Undergraduate Calculus
ERIC Educational Resources Information Center
Maciejewski, Wes; Star, Jon R.
2016-01-01
Mathematics experts often choose appropriate procedures to produce an efficient or elegant solution to a mathematical task. This "flexible procedural knowledge" distinguishes novice and expert procedural performances. This article reports on an intervention intended to aid the development of undergraduate calculus students' flexible use…
[Report of a case of tonsillar calculus].
García Rodríguez, M R; Cabanas López, A; García Calleja, J M; García Rodríguez, J F
1990-01-01
We describe the case of a patient with odynophagia without treatment response. The pharynx exploration shows a stony tumor in vertex of left palatine amygdala which was extirpated. The laboratory analysis signs calcium and oxalic calculus. The anatomopathological study of the amygdala bed biopsy informs like absent of malignancy images.
Bladder calculus presenting as excessive masturbation.
De Alwis, A C D; Senaratne, A M R D; De Silva, S M P D; Rodrigo, V S D
2006-09-01
Masturbation in childhood is a normal behaviour which most commonly begins at 2 months of age, and peaks at 4 years and in adolescence. However excessive masturbation causes anxiety in parents. We describe a boy with a bladder calculus presenting as excessive masturbation.
A symbol calculus for Toeplitz operators
Berger, C. A.; Coburn, L. A.
1986-01-01
We give a complete characterization of those functions on 2n-dimensional Euclidean space for which the Berezin-Toeplitz quantizations admit a symbol calculus modulo the compact operators. The functions in question are characterized by a condition of “small oscillation at infinity.” PMID:16593695
Exposing Calculus Students to Advanced Mathematics
ERIC Educational Resources Information Center
Griffiths, Barry J.; Haciomeroglu, Erhan Selcuk
2014-01-01
To ensure the competitiveness of the USA in the global economy, and its role as a leader in science and engineering, it is important to cultivate the next generation of home grown mathematicians. However, while universities across the USA offer calculus classes to thousands of undergraduate students each year, very few of them go on to major in…
Online Calculus: The Course and Survey Results.
ERIC Educational Resources Information Center
Allen, G. Donald
2001-01-01
Describes the development and implementation of a Web-based calculus course at Texas A & M University. Discusses the course design, layout of content and the contrast with textbook structure, results of course surveys that included student reactions, and how students learn form Web-based materials. (Author/LRW)
A Note on Discrete Mathematics and Calculus.
ERIC Educational Resources Information Center
O'Reilly, Thomas J.
1987-01-01
Much of the current literature on the topic of discrete mathematics and calculus during the first two years of an undergraduate mathematics curriculum is cited. A relationship between the recursive integration formulas and recursively defined polynomials is described. A Pascal program is included. (Author/RH)
Teaching Calculus Students How to Study.
ERIC Educational Resources Information Center
Boelkins, Matthew R.; Pfaff, Thomas J.
1998-01-01
Addresses the problem of poor study habits in calculus students and presents techniques to teach students how to study consistently and effectively. Concludes that many students greatly appreciate the added structure, work harder than in previous courses, and witness newfound success as a consequence. (Author/ASK)
Exploring Flipped Classroom Instruction in Calculus III
ERIC Educational Resources Information Center
Wasserman, Nicholas H.; Quint, Christa; Norris, Scott A.; Carr, Thomas
2017-01-01
In an undergraduate Calculus III class, we explore the effect of "flipping" the instructional delivery of content on both student performance and student perceptions. Two instructors collaborated to determine daily lecture notes, assigned the same homework problems, and gave identical exams; however, compared to a more traditional…
Flipping a Calculus Class: One Instructor's Experience
ERIC Educational Resources Information Center
Palmer, Katrina
2015-01-01
This paper describes one instructor's experiences during a year of flipping four calculus classes. The first exploration attempts to understand student expectations of a math class and their preference towards a flipped classroom. The second examines success of students from a flipped classroom, and the last investigates relationships with student…
A TENTATIVE GUIDE, DIFFERENTIAL AND INTEGRAL CALCULUS.
ERIC Educational Resources Information Center
BRANT, VINCENT; GERARDI, WILLIAM
THE COURSE IS INTENDED TO GO BEYOND THE REQUIREMENTS OF THE ADVANCED PLACEMENT PROGRAM IN MATHEMATICS AS DESIGNED BY THE COLLEGE ENTRANCE EXAMINATION BOARD. THE ADVANCED PLACEMENT PROGRAM CONSISTS OF A 1-YEAR COURSE COMBINING ANALYTIC GEOMETRY AND CALCULUS. PRESUPPOSED HERE ARE--A SEMESTER COURSE IN ANALYTIC GEOMETRY AND A THOROUGH KNOWLEDGE OF…
A Calculus Project that Really Makes Cents
ERIC Educational Resources Information Center
Green, Daniel L.
2006-01-01
This article describes a calculus project that exposes students to the concept of retirement annuities in both the saving and withdrawal phases, via revenue streams represented by integrals. Students use modeling skills to solve several related problems as the assumptions of the original problem are changed, and the project requires them to use a…
Characteristics of subgingival calculus detection by multiphoton fluorescence microscopy
NASA Astrophysics Data System (ADS)
Tung, Oi-Hong; Lee, Shyh-Yuan; Lai, Yu-Lin; Chen, How-Foo
2011-06-01
Subgingival calculus has been recognized as a major cause of periodontitis, which is one of the main chronic infectious diseases of oral cavities and a principal cause of tooth loss in humans. Bacteria deposited in subgingival calculus or plaque cause gingival inflammation, function deterioration, and then periodontitis. However, subgingival calculus within the periodontal pocket is a complicated and potentially delicate structure to be detected with current dental armamentaria, namely dental x-rays and dental probes. Consequently, complete removal of subgingival calculus remains a challenge to periodontal therapies. In this study, the detection of subgingival calculus employing a multiphoton autofluorescence imaging method was characterized in comparison with a one-photon confocal fluorescence imaging technique. Feasibility of such a system was studied based on fluorescence response of gingiva, healthy teeth, and calculus with and without gingiva covered. The multiphoton fluorescence technology perceived the tissue-covered subgingival calculus that cannot be observed by the one-photon confocal fluorescence method.
Characteristics of subgingival calculus detection by multiphoton fluorescence microscopy.
Tung, Oi-Hong; Lee, Shyh-Yuan; Lai, Yu-Lin; Chen, How-Foo
2011-06-01
Subgingival calculus has been recognized as a major cause of periodontitis, which is one of the main chronic infectious diseases of oral cavities and a principal cause of tooth loss in humans. Bacteria deposited in subgingival calculus or plaque cause gingival inflammation, function deterioration, and then periodontitis. However, subgingival calculus within the periodontal pocket is a complicated and potentially delicate structure to be detected with current dental armamentaria, namely dental x-rays and dental probes. Consequently, complete removal of subgingival calculus remains a challenge to periodontal therapies. In this study, the detection of subgingival calculus employing a multiphoton autofluorescence imaging method was characterized in comparison with a one-photon confocal fluorescence imaging technique. Feasibility of such a system was studied based on fluorescence response of gingiva, healthy teeth, and calculus with and without gingiva covered. The multiphoton fluorescence technology perceived the tissue-covered subgingival calculus that cannot be observed by the one-photon confocal fluorescence method.
A probability theory for non-equilibrium gravitational systems
NASA Astrophysics Data System (ADS)
Peñarrubia, Jorge
2015-08-01
This paper uses dynamical invariants to describe the evolution of collisionless systems subject to time-dependent gravitational forces without resorting to maximum-entropy probabilities. We show that collisionless relaxation can be viewed as a special type of diffusion process in the integral-of-motion space. In time-varying potentials with a fixed spatial symmetry the diffusion coefficients are closely related to virial quantities, such as the specific moment of inertia, the virial factor and the mean kinetic and potential energy of microcanonical particle ensembles. The non-equilibrium distribution function is found by convolving the initial distribution function with the Green function that solves Einstein's equation for freely diffusing particles. Such a convolution also yields a natural solution to the Fokker-Planck equations in the energy space. Our mathematical formalism can be generalized to potentials with a time-varying symmetry, where diffusion extends over multiple dimensions of the integral-of-motion space. The new probability theory is in many ways analogous to stochastic calculus, with two significant differences: (i) the equations of motion that govern the trajectories of particles are fully deterministic, and (ii) the diffusion coefficients can be derived self-consistently from microcanonical phase-space averages without relying on ergodicity assumptions. For illustration we follow the cold collapse of N-body models in a time-dependent logarithmic potential. Comparison between the analytical and numerical results shows excellent agreement in regions where the potential evolution does not depart too strongly from the adiabatic regime.
People's conditional probability judgments follow probability theory (plus noise).
Costello, Fintan; Watts, Paul
2016-09-01
A common view in current psychology is that people estimate probabilities using various 'heuristics' or rules of thumb that do not follow the normative rules of probability theory. We present a model where people estimate conditional probabilities such as P(A|B) (the probability of A given that B has occurred) by a process that follows standard frequentist probability theory but is subject to random noise. This model accounts for various results from previous studies of conditional probability judgment. This model predicts that people's conditional probability judgments will agree with a series of fundamental identities in probability theory whose form cancels the effect of noise, while deviating from probability theory in other expressions whose form does not allow such cancellation. Two experiments strongly confirm these predictions, with people's estimates on average agreeing with probability theory for the noise-cancelling identities, but deviating from probability theory (in just the way predicted by the model) for other identities. This new model subsumes an earlier model of unconditional or 'direct' probability judgment which explains a number of systematic biases seen in direct probability judgment (Costello & Watts, 2014). This model may thus provide a fully general account of the mechanisms by which people estimate probabilities.
Fractal Physiology and the Fractional Calculus: A Perspective
West, Bruce J.
2010-01-01
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. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. 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. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a
Students' difficulties with vector calculus in electrodynamics
NASA Astrophysics Data System (ADS)
Bollen, Laurens; van Kampen, Paul; De Cock, Mieke
2015-12-01
Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found that they are quite skilled at doing calculations, but struggle with interpreting graphical representations of vector fields and applying vector calculus to physical situations. We have found strong indications that traditional instruction is not sufficient for our students to fully understand the meaning and power of Maxwell's equations in electrodynamics.
Double dumb-bell calculus in childhood.
Joshi, Prashant; Sarda, Dinesh; Ahmad, Ashraf; Kothari, Paras
2009-01-01
An eight-year old male was admitted with complaints of right scrotal swelling, dysuria and intermittent retention of urine for 10 days. On per-rectal examination, a hard mass was palpable in the posterior urethra. An X-ray (KUB) of the abdomen revealed a double dumb-bell calculus at the base of bladder, extending into the posterior urethra. A cystolithotomy via the suprapubic approach was successfully curative.
Relativistic differential-difference momentum operators and noncommutative differential calculus
Mir-Kasimov, R. M.
2013-09-15
The relativistic kinetic momentum operators are introduced in the framework of the Quantum Mechanics (QM) in the Relativistic Configuration Space (RCS). These operators correspond to the half of the non-Euclidean distance in the Lobachevsky momentum space. In terms of kinetic momentum operators the relativistic kinetic energy is separated as the independent term of the total Hamiltonian. This relativistic kinetic energy term is not distinguishing in form from its nonrelativistic counterpart. The role of the plane wave (wave function of the motion with definite value of momentum and energy) plays the generating function for the matrix elements of the unitary irreps of Lorentz group (generalized Jacobi polynomials). The kinetic momentum operators are the interior derivatives in the framework of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions over the RCS.
Path integral in area tensor Regge calculus and complex connections
NASA Astrophysics Data System (ADS)
Khatsymovsky, V. M.
2006-06-01
Euclidean quantum measure in Regge calculus with independent area tensors is considered using example of the Regge manifold of a simple structure. We go over to integrations along certain contours in the hyperplane of complex connection variables. Discrete connection and curvature on classical solutions of the equations of motion are not, strictly speaking, genuine connection and curvature, but more general quantities and, therefore, these do not appear as arguments of a function to be averaged, but are the integration (dummy) variables. We argue that upon integrating out the latter the resulting measure can be well-defined on physical hypersurface (for the area tensors corresponding to certain edge vectors, i.e. to certain metric) as positive and having exponential cutoff at large areas on condition that we confine ourselves to configurations which do not pass through degenerate metrics.
Using an advanced graphing calculator in the teaching and learning of calculus
NASA Astrophysics Data System (ADS)
Leng, Ng Wee
2011-10-01
The purpose of this study was to investigate how the use of TI-Nspire™ could enhance the teaching and learning of calculus. A conceptual framework for the use of TI-Nspire™ for learning calculus in a mathematics classroom is proposed that describes the interactions among the students, TI-Nspire™, and the learning tasks, and how they lead to the learning of calculus. A design experiment was conducted in a class of 35 students from a secondary school in Singapore. Use of TI-Nspire™ was integrated into the teaching and learning of calculus concepts in the classroom with the aid of TI-Nspire™ Navigator, a wireless classroom network system that enables instant and active interaction between students and teachers. It was found that the appropriate use of graphical, numerical and algebraic representations of calculus concepts using TI-Nspire™ enabled students to better visualize the concepts and make generalizations about relevant mathematical properties. In addition, the students were able to link multiple representations, especially algebraic and graphical representations, to improve their conceptual understanding and problem-solving skills. Six roles of TI-Nspire™ in classroom mathematical practice were identified from the findings of the experiment; TI-Nspire™ was used as an exploratory tool, graphing tool, confirmatory tool, problem-solving tool, visualization tool and calculation tool. This suggests that TI-Nspire™ is a multi-dimensional tool that supports mathematics learning. Overall, the findings of the study indicate that TI-Nspire™ is an effective tool to develop mathematical concepts and promote learning and problem solving.
NASA Astrophysics Data System (ADS)
Casas-Castillo, M. Carmen; Rodríguez-Solà, Raúl; Navarro, Xavier; Russo, Beniamino; Lastra, Antonio; González, Paula; Redaño, Angel
2016-11-01
The fractal behavior of extreme rainfall intensities registered between 1940 and 2012 by the Retiro Observatory of Madrid (Spain) has been examined, and a simple scaling regime ranging from 25 min to 3 days of duration has been identified. Thus, an intensity-duration-frequency (IDF) master equation of the location has been constructed in terms of the simple scaling formulation. The scaling behavior of probable maximum precipitation (PMP) for durations between 5 min and 24 h has also been verified. For the statistical estimation of the PMP, an envelope curve of the frequency factor (k m ) based on a total of 10,194 station-years of annual maximum rainfall from 258 stations in Spain has been developed. This curve could be useful to estimate suitable values of PMP at any point of the Iberian Peninsula from basic statistical parameters (mean and standard deviation) of its rainfall series.
Improving Student Success in Calculus I Using a Co-Requisite Calculus I Lab
ERIC Educational Resources Information Center
Vestal, Sharon Schaffer; Brandenburger, Thomas; Furth, Alfred
2015-01-01
This paper describes how one university mathematics department was able to improve student success in Calculus I by requiring a co-requisite lab for certain groups of students. The groups of students required to take the co-requisite lab were identified by analyzing student data, including Math ACT scores, ACT Compass Trigonometry scores, and…
Success in Introductory Calculus: The Role of High School and Pre-Calculus Preparation
ERIC Educational Resources Information Center
Ayebo, Abraham; Ukkelberg, Sarah; Assuah, Charles
2017-01-01
Calculus at the college level has significant potential to serve as a pump for increasing the number of students majoring in STEM fields. It is a foundation course for all STEM majors and, if mastered well, should provide students with a positive and successful first-year experience and gateway into more advanced courses. Studies have shown that a…
An Evaluative Calculus Project: Applying Bloom's Taxonomy to the Calculus Classroom
ERIC Educational Resources Information Center
Karaali, Gizem
2011-01-01
In education theory, Bloom's taxonomy is a well-known paradigm to describe domains of learning and levels of competency. In this article I propose a calculus capstone project that is meant to utilize the sixth and arguably the highest level in the cognitive domain, according to Bloom et al.: evaluation. Although one may assume that mathematics is…
Instanton calculus of Lifshitz tails
NASA Astrophysics Data System (ADS)
Yaida, Sho
2016-02-01
Some degree of quenched disorder is present in nearly all solids, and can have a marked impact on their macroscopic properties. A manifestation of this effect is the Lifshitz tail of localized states that then gets attached to the energy spectrum, resulting in the nonzero density of states in the band gap. We present here a systematic approach for deriving the asymptotic behavior of the density of states and of the typical shape of the disorder potentials in the Lifshitz tail. The analysis is carried out first for the well-controlled case of noninteracting particles moving in a Gaussian random potential and then for a broad class of disordered scale-invariant models—pertinent to a variety of systems ranging from semiconductors to semimetals to quantum critical systems. For relevant Gaussian disorder, we obtain the general expression for the density of states deep in the tail, with the rate of exponential suppression governed by the dynamical exponent and spatial dimensions. For marginally relevant disorder, however, we would expect a power-law scaling. We discuss the implications of these results for understanding conduction in disordered materials.
Operator calculus for information field theory
NASA Astrophysics Data System (ADS)
Leike, Reimar H.; Enßlin, Torsten A.
2016-11-01
Signal inference problems with non-Gaussian posteriors can be hard to tackle. Through using the concept of Gibbs free energy these posteriors are rephrased as Gaussian posteriors for the price of computing various expectation values with respect to a Gaussian distribution. We present a way of translating these expectation values to a language of operators which is similar to that in quantum mechanics. This simplifies many calculations, for instance such as those involving log-normal priors. The operator calculus is illustrated by deriving a self-calibrating algorithm which is tested with mock data.
The Calculus of Responsibility and Commitment
NASA Astrophysics Data System (ADS)
Pollard, Carl
Ever since Montague (1974 [1970]) laid the foundations for formally precise analysis of natural language (hereafter NL) semantics in the late 1960's, the typed lambda calculus (hereafter TLC) and certain of its extensions have been the linguists' tool of choice for representing the meanings of NL expressions. But starting around the turn of the millenium, motivated by a range of linguistic phenomena collectively known as covert movement phenomena, logical grammarians of various persuasions have proposed the use of other semantic term calculi that embody, directly or indirectly, some notion or other of continuation.
Operator calculus for information field theory.
Leike, Reimar H; Enßlin, Torsten A
2016-11-01
Signal inference problems with non-Gaussian posteriors can be hard to tackle. Through using the concept of Gibbs free energy these posteriors are rephrased as Gaussian posteriors for the price of computing various expectation values with respect to a Gaussian distribution. We present a way of translating these expectation values to a language of operators which is similar to that in quantum mechanics. This simplifies many calculations, for instance such as those involving log-normal priors. The operator calculus is illustrated by deriving a self-calibrating algorithm which is tested with mock data.
Denoising Medical Images using Calculus of Variations.
Kohan, Mahdi Nakhaie; Behnam, Hamid
2011-07-01
We propose a method for medical image denoising using calculus of variations and local variance estimation by shaped windows. This method reduces any additive noise and preserves small patterns and edges of images. A pyramid structure-texture decomposition of images is used to separate noise and texture components based on local variance measures. The experimental results show that the proposed method has visual improvement as well as a better SNR, RMSE and PSNR than common medical image denoising methods. Experimental results in denoising a sample Magnetic Resonance image show that SNR, PSNR and RMSE have been improved by 19, 9 and 21 percents respectively.
Student Created Calculus Movies Using Computers and the TI-92.
ERIC Educational Resources Information Center
Sher, Lawrence; Wilkinson, Patricia
The Mathematics Department at Borough of Manhattan Community College (BMCC) (New York) has been actively involved since 1988 in a serious and successful program to improve instruction, understanding, and retention for women and minority students in calculus courses. One result of this work has been students creating calculus animations using…
Effects of Clicker Use on Calculus Students' Mathematics Anxiety
ERIC Educational Resources Information Center
Batchelor, John
2015-01-01
This paper reports the results of a survey study of clicker use and mathematics anxiety among students enrolled in an undergraduate calculus course during the Fall 2013 semester. Students in two large lecture sections of calculus completed surveys at the beginning and end of the course. One class used clickers, whereas the other class was taught…
Improving Student Success in Calculus at Seattle University
ERIC Educational Resources Information Center
Carter, J. D.; Helliwell, D.; Henrich, Allison; Principe, M.; Sloughter, J. M.
2016-01-01
Finding ways to improve student success in calculus is a critically important step on the path to supporting students who are pursuing degrees in STEM fields. Far too many students fail calculus 1 and are pushed to drop their majors in technical fields. One way of addressing this issue is by following a program that was pioneered at University of…
Modelling the Landing of a Plane in a Calculus Lab
ERIC Educational Resources Information Center
Morante, Antonio; Vallejo, Jose A.
2012-01-01
We exhibit a simple model of a plane landing that involves only basic concepts of differential calculus, so it is suitable for a first-year calculus lab. We use the computer algebra system Maxima and the interactive geometry software GeoGebra to do the computations and graphics. (Contains 5 figures and 1 note.)
Reflections on Our First Calculus Undergraduate Teaching Assistant
ERIC Educational Resources Information Center
Deshler, Jessica M.
2016-01-01
This article describes some reflections from the first Calculus I undergraduate teaching assistant in our department as she explored the various ways in which she was able to support both novice and experienced Calculus teachers and the effect of her experience on her academic and career plans.
On Flipping the Classroom in Large First Year Calculus Courses
ERIC Educational Resources Information Center
Jungic, Veselin; Kaur, Harpreet; Mulholland, Jamie; Xin, Cindy
2015-01-01
Over the course of two years, 2012-2014, we have implemented a "flipping" the classroom approach in three of our large enrolment first year calculus courses: differential and integral calculus for scientists and engineers. In this article we describe the details of our particular approach and share with the reader some experiences of…
Coordinating Multiple Representations in a Reform Calculus Textbook
ERIC Educational Resources Information Center
Chang, Briana L.; Cromley, Jennifer G.; Tran, Nhi
2015-01-01
Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions…
Visual Thinking and Gender Differences in High School Calculus
ERIC Educational Resources Information Center
Haciomeroglu, Erhan Selcuk; Chicken, Eric
2012-01-01
This study sought to examine calculus students' mathematical performances and preferences for visual or analytic thinking regarding derivative and antiderivative tasks presented graphically. It extends previous studies by investigating factors mediating calculus students' mathematical performances and their preferred modes of thinking. Data were…
Coordinating Multiple Representations in a Reform Calculus Textbook
ERIC Educational Resources Information Center
Chang, Briana L.; Cromley, Jennifer G.; Tran, Nhi
2016-01-01
Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions…
Transitioning from Introductory Calculus to Formal Limit Conceptions
ERIC Educational Resources Information Center
Nagle, Courtney
2013-01-01
The limit concept is a fundamental mathematical notion both for its practical applications and its importance as a prerequisite for later calculus topics. Past research suggests that limit conceptualizations promoted in introductory calculus are far removed from the formal epsilon-delta definition of limit. In this article, I provide an overview…
The Use of Technology and Visualization in Calculus Instruction
ERIC Educational Resources Information Center
Samuels, Jason
2010-01-01
This study was inspired by a history of student difficulties in calculus, and innovation in response to those difficulties. The goals of the study were fourfold. First, to design a mathlet for students to explore local linearity. Second, to redesign the curriculum of first semester calculus around the use of technology, an emphasis on…
Calculus Students' Early Concept Images of Tangent Lines
ERIC Educational Resources Information Center
Vincent, Brittany; LaRue, Renee; Sealey, Vicki; Engelke, Nicole
2015-01-01
This study explored first-semester calculus students' understanding of tangent lines as well as how students used tangent lines within the context of Newton's method. Task-based interviews were conducted with twelve first-semester calculus students who were asked to verbally describe a tangent line, sketch tangent lines for multiple curves, and…
Our Experiences with Using Visualization Tools in Teaching Calculus.
ERIC Educational Resources Information Center
Kowalczyk, Robert E.; Hausknecht, Adam O.
This paper describes two uses of the software package TEMATH (Tools for Exploring Mathematics) with calculus students: (1) as a demonstration tool in the classroom to visually explore with students the many mathematical models introduced in a first year calculus course; and (2) as a part of a lab where students use a set of laboratory explorations…
Restricted diversity of dental calculus methanogens over five centuries, France
Huynh, Hong T. T.; Nkamga, Vanessa D.; Signoli, Michel; Tzortzis, Stéfan; Pinguet, Romuald; Audoly, Gilles; Aboudharam, Gérard; Drancourt, Michel
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 sequencing identified Candidatus Methanobrevibacter sp. N13 in 44.6%, Methanobrevibacter oralis in 19.6%, a new Methanomassiliicoccus luminyensis-like methanogen in 12.5%, a Candidatus Nitrososphaera evergladensis-like in one and Methanoculleus bourgensis in one specimen, respectively. One Candidatus Methanobrevibacter sp. N13 dental calculus was further documented by fluorescent in situ hybridization. The prevalence of dental calculus M. oralis was significantly lower in past populations than in modern populations (P = 0.03, Chi-square test). This investigation revealed a previously unknown repertoire of archaea found in the oral cavity of past French populations as reflected in preserved dental calculus. PMID:27166431
Calculus: A Computer Oriented Presentation, Part 1 [and] Part 2.
ERIC Educational Resources Information Center
Stenberg, Warren; Walker, Robert J.
Parts one and two of a one-year computer-oriented calculus course (without analytic geometry) are presented. The ideas of calculus are introduced and motivated through computer (i.e., algorithmic) concepts. An introduction to computing via algorithms and a simple flow chart language allows the book to be self-contained, except that material on…
Interrater Agreement on Subgingival Calculus Detection Following Scaling.
ERIC Educational Resources Information Center
Pippin, David J.; Feil, Philip
1992-01-01
Two studies investigated interrater agreement among 10 clinical dental examiners who scored residual subgingival calculus after student scaling on 4,160 real and 92 manikin tooth surfaces. Interrater reliability was low. Results suggest a need in periodontics for effective examiner calibration methods and objective subgingival calculus detection…
Utilizing Microsoft Mathematics in Teaching and Learning Calculus
ERIC Educational Resources Information Center
Oktaviyanthi, Rina; Supriani, Yani
2015-01-01
The experimental design was conducted to investigate the use of Microsoft Mathematics, free software made by Microsoft Corporation, in teaching and learning Calculus. This paper reports results from experimental study details on implementation of Microsoft Mathematics in Calculus, students' achievement and the effects of the use of Microsoft…
Experimental Design: Utilizing Microsoft Mathematics in Teaching and Learning Calculus
ERIC Educational Resources Information Center
Oktaviyanthi, Rina; Supriani, Yani
2015-01-01
The experimental design was conducted to investigate the use of Microsoft Mathematics, free software made by Microsoft Corporation, in teaching and learning Calculus. This paper reports results from experimental study details on implementation of Microsoft Mathematics in Calculus, students' achievement and the effects of the use of Microsoft…
Evaluating the Use of Learning Objects for Improving Calculus Readiness
ERIC Educational Resources Information Center
Kay, Robin; Kletskin, Ilona
2010-01-01
Pre-calculus concepts such as working with functions and solving equations are essential for students to explore limits, rates of change, and integrals. Yet many students have a weak understanding of these key concepts which impedes performance in their first year university Calculus course. A series of online learning objects was developed to…
How Students Use Physics to Reason about Calculus Tasks
ERIC Educational Resources Information Center
Marrongelle, Karen A.
2004-01-01
The present research study investigates how undergraduate students in an integrated calculus and physics class use physics to help them solve calculus problems. Using Zandieh's (2000) framework for analyzing student understanding of derivative as a starting point, this study adds detail to her "paradigmatic physical" context and begins to address…
Modelling the landing of a plane in a calculus lab
NASA Astrophysics Data System (ADS)
Morante, Antonio; Vallejo, José A.
2012-10-01
We exhibit a simple model of a plane landing that involves only basic concepts of differential calculus, so it is suitable for a first-year calculus lab. We use the computer algebra system Maxima and the interactive geometry software GeoGebra to do the computations and graphics.
Calculus Instructors' and Students' Discourses on the Derivative
ERIC Educational Resources Information Center
Park, Jungeun
2011-01-01
Recently, there has been an increasing interest in collegiate mathematics education, especially teaching and learning calculus (e.g., Oehrtman, Carlson, & Thompson, 2008; Speer, Smith, & Horvath, 2010). Of many calculus concepts, the derivative is known as a difficult concept for students to understand because it involves various concepts…
Partial Fractions in Calculus, Number Theory, and Algebra
ERIC Educational Resources Information Center
Yackel, C. A.; Denny, J. K.
2007-01-01
This paper explores the development of the method of partial fraction decomposition from elementary number theory through calculus to its abstraction in modern algebra. This unusual perspective makes the topic accessible and relevant to readers from high school through seasoned calculus instructors.
DIFFERENTIAL AND INTEGRAL CALCULUS. A TENTATIVE CURRICULUM GUIDE.
ERIC Educational Resources Information Center
BRANT, VINCENT; GERARDI, WILLIAM
A GUIDE FOR A 1-YEAR COURSE IN DIFFERENTIAL AND INTEGRAL CALCULUS PREREQUISITED KNOWLEDGE IN ALGEBRA, ANALYTIC TRIGONOMETRY, AND ELEMENTARY ANALYSIS. EACH ASSIGNMENT CONTAINED BOTH NEW AND REVIEW WORK TO REINFORCE THE NEW WORK. THERE WERE ELEVEN UNITS OF STUDY USING THE FOLLOWING FOUR BOOKS--"CALCULUS AND ANALYTIC GEOMETRY, THIRD…
An Exploration of Definition and Procedural Fluency in Integral Calculus
ERIC Educational Resources Information Center
Grundmeier, Todd A.; Hansen, Jennifer; Sousa, Emily
2006-01-01
A survey was administered to calculus students who had previously been exposed to a course on integral calculus. The purpose of the survey was to explore students' understanding of the definition of a definite integral, their abilities to evaluate definite integrals, and their graphical interpretations of definite integrals. The analysis of…
Calculus detection technologies: where do we stand now?
Archana, V
2014-01-01
Epidemiological studies have implicated dental calculus as an ideal substrate for subgingival microbial colonization. Therefore, the main objective of periodontal therapy is to eliminate the microbial biofilm along with the calculus deposits from the root surface by root surface debridement. Over the past years, a large number of clinical and laboratory studies have been conducted to evaluate the efficacy of calculus removal by various methods. None of these conventional methods or devices was effective in completely eliminating all the calculus from the diseased root surfaces. In this context, a number of newer technologies have been developed to identify and selectively remove the dental calculus. Regarding this fact, the present article highlights a critical review of these devices based on published clinical and experimental data.
Calculus detection technologies: where do we stand now?
Archana, V
2014-01-01
Epidemiological studies have implicated dental calculus as an ideal substrate for subgingival microbial colonization. Therefore, the main objective of periodontal therapy is to eliminate the microbial biofilm along with the calculus deposits from the root surface by root surface debridement. Over the past years, a large number of clinical and laboratory studies have been conducted to evaluate the efficacy of calculus removal by various methods. None of these conventional methods or devices was effective in completely eliminating all the calculus from the diseased root surfaces. In this context, a number of newer technologies have been developed to identify and selectively remove the dental calculus. Regarding this fact, the present article highlights a critical review of these devices based on published clinical and experimental data. PMID:25870667
ERIC Educational Resources Information Center
Ferguson, Leann J.
2012-01-01
Calculus is an important tool for building mathematical models of the world around us and is thus used in a variety of disciplines, such as physics and engineering. These disciplines rely on calculus courses to provide the mathematical foundation needed for success in their courses. Unfortunately, due to the basal conceptions of what it means to…
Toeplitz, Otto
2016-12-01
In the above mentioned article [1] unfortunately the names of the translators of Toeplitz's lecture were omitted. The correct title is: The Problem of University Courses on Infinitesimal Calculus and Their Demarcation from Infinitesimal Calculus in High Schools Otto Toeplitz Translated into English by Michael N. Fried and Hans Niels Jahnke.
Graphical calculus for Gaussian pure states
Menicucci, Nicolas C.; Flammia, Steven T.; Loock, Peter van
2011-04-15
We provide a unified graphical calculus for all Gaussian pure states, including graph transformation rules for all local and semilocal Gaussian unitary operations, as well as local quadrature measurements. We then use this graphical calculus to analyze continuous-variable (CV) cluster states, the essential resource for one-way quantum computing with CV systems. Current graphical approaches to CV cluster states are only valid in the unphysical limit of infinite squeezing, and the associated graph transformation rules only apply when the initial and final states are of this form. Our formalism applies to all Gaussian pure states and subsumes these rules in a natural way. In addition, the term 'CV graph state' currently has several inequivalent definitions in use. Using this formalism we provide a single unifying definition that encompasses all of them. We provide many examples of how the formalism may be used in the context of CV cluster states: defining the 'closest' CV cluster state to a given Gaussian pure state and quantifying the error in the approximation due to finite squeezing; analyzing the optimality of certain methods of generating CV cluster states; drawing connections between this graphical formalism and bosonic Hamiltonians with Gaussian ground states, including those useful for CV one-way quantum computing; and deriving a graphical measure of bipartite entanglement for certain classes of CV cluster states. We mention other possible applications of this formalism and conclude with a brief note on fault tolerance in CV one-way quantum computing.
Approximate inference on planar graphs using loop calculus and belief progagation
Chertkov, Michael; Gomez, Vicenc; Kappen, Hilbert
2009-01-01
We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006b) allows to express the exact partition function Z of a graphical model as a finite sum of terms that can be evaluated once the belief propagation (BP) solution is known. In general, full summation over all correction terms is intractable. We develop an algorithm for the approach presented in Chertkov et al. (2008) which represents an efficient truncation scheme on planar graphs and a new representation of the series in terms of Pfaffians of matrices. We analyze in detail both the loop series and the Pfaffian series for models with binary variables and pairwise interactions, and show that the first term of the Pfaffian series can provide very accurate approximations. The algorithm outperforms previous truncation schemes of the loop series and is competitive with other state-of-the-art methods for approximate inference.
A Calculus for Boxes and Traits in a Java-Like Setting
NASA Astrophysics Data System (ADS)
Bettini, Lorenzo; Damiani, Ferruccio; de Luca, Marco; Geilmann, Kathrin; Schäfer, Jan
The box model is a component model for the object-oriented paradigm, that defines components (the boxes) with clear encapsulation boundaries. Having well-defined boundaries is crucial in component-based software development, because it enables to argue about the interference and interaction between a component and its context. In general, boxes contain several objects and inner boxes, of which some are local to the box and cannot be accessed from other boxes and some can be accessible by other boxes. A trait is a set of methods divorced from any class hierarchy. Traits can be composed together to form classes or other traits. We present a calculus for boxes and traits. Traits are units of fine-grained reuse, whereas boxes can be seen as units of coarse-grained reuse. The calculus is equipped with an ownership type system and allows us to combine coarse- and fine-grained reuse of code by maintaining encapsulation of components.
Vortex Images, q-Calculus and Entangled Coherent States
NASA Astrophysics Data System (ADS)
Pashaev, Oktay K.
2012-02-01
The two circles theorem for hydrodynamic flow in annular domain bounded by two concentric circles is derived. Complex potential and velocity of the flow are represented as q-periodic functions and rewritten in terms of the Jackson q-integral. This theorem generalizes the Milne-Thomson one circle theorem and reduces to the last on in the limit q → ∞. By this theorem problem of vortex images in annular domain between coaxial cylinders is solved in terms of q-elementary functions. An infinite set of images, as symmetric points under two circles, is determined completely by poles of the q-logarithmic function, where dimensionless parameter q = r22/r21 is given by square ratio of the cylinder radii. Motivated by Möbius transformation for symmetrical points under generalized circle in complex plain, the system of symmetric spin coherent states corresponding to antipodal qubit states is introduced. By these states we construct the maximally entangled orthonormal two qubit spin coherent state basis, in the limiting case reducible to the Bell basis. Average energy of XYZ model in these states, describing finite localized structure with characteristic extremum points, appears as an energy surface in maximally entangled two qubit space. Generalizations to three and higher multiple qubits are found. We show that our entangled N qubit states are determined by set of complex Fibonacci and Lucas polynomials and corresponding Binet-Fibonacci q-calculus.
Miniature endoscopic optical coherence tomography for calculus detection.
Kao, Meng-Chun; Lin, Chun-Li; Kung, Che-Yen; Huang, Yi-Fung; Kuo, Wen-Chuan
2015-08-20
The effective treatment of periodontitis involves the detection and removal of subgingival dental calculus. However, subgingival calculus is more difficult to detect than supragingival calculus because it is firmly attached to root surfaces within periodontal pockets. To achieve a smooth root surface, clinicians often remove excessive amounts of root structure because of decreased visibility. In addition, enamel pearl, a rare type of ectopic enamel formation on the root surface, can easily be confused with dental calculus in the subgingival environment. In this study, we developed a fiber-probe swept-source optical coherence tomography (SSOCT) technique and combined it with the quantitative measurement of an optical parameter [standard deviation (SD) of the optical coherence tomography (OCT) intensity] to differentiate subgingival calculus from sound enamel, including enamel pearl. Two-dimensional circumferential images were constructed by rotating the miniprobe (0.9 mm diameter) while acquiring image lines, and the adjacent lines in each rotation were stacked to generate a three-dimensional volume. In OCT images, compared to sound enamel and enamel pearls, dental calculus showed significant differences (P<0.001) in SD values. Finally, the receiver operating characteristic curve had a high capacity (area under the curve=0.934) for discriminating between healthy regions (including enamel pearl) and dental calculus.
Pulsed laser ablation of dental calculus in the near ultraviolet.
Schoenly, Joshua E; Seka, Wolf; Rechmann, Peter
2014-02-01
Pulsed lasers emitting wavelengths near 400 nm can selectively ablate dental calculus without damaging underlying and surrounding sound dental hard tissue. Our results indicate that calculus ablation at this wavelength relies on the absorption of porphyrins endogenous to oral bacteria commonly found in calculus. Sub- and supragingival calculus on extracted human teeth, irradiated with 400-nm, 60-ns laser pulses at ≤8 J/cm2, exhibits a photobleached surface layer. Blue-light microscopy indicates this layer highly scatters 400-nm photons, whereas fluorescence spectroscopy indicates that bacterial porphyrins are permanently photobleached. A modified blow-off model for ablation is proposed that is based upon these observations and also reproduces our calculus ablation rates measured from laser profilometry. Tissue scattering and a stratified layering of absorbers within the calculus medium explain the gradual decrease in ablation rate from successive pulses. Depending on the calculus thickness, ablation stalling may occur at <5 J/cm2 but has not been observed above this fluence.
Trygve Haavelmo and the Emergence of Causal Calculus
2014-06-01
Econometric Theory, 2014, Page 1 of 28. doi:10.1017/S0266466614000231 TRYGVE HAAVELMO AND THE EMERGENCE OF CAUSAL CALCULUS JUDEA PEARL University of...Emergence of Causal Calculus 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK...definition of a (Pearl, 1994): a = ∂ ∂x E(Y |do(x)) (3) HAAVELMO AND CAUSAL CALCULUS 3 which refers to a controlled experiment in which an agent (e.g
Emphysematous pyelonephritis with calculus: Management strategies
Goel, Tanmaya; Reddy, Sreedhar; Thomas, Joseph
2007-01-01
Objective: Emphysematous pyelonephritis (EPN) with calculus is well recognized but with very few reports on its treatment. Our aim is to elucidate our experience in its successful management. Materials and Methods: Over four years, we diagnosed seven cases (eight renal units) of EPN, out of which two patients (three renal units) had EPN with urinary calculi. After the initial conservative management of EPN, the stones were tackled appropriately. Results: EPN was initially managed effectively with antibiotics and supportive care. Once the patient was stable, the stones were cleared in a step-wise fashion. The associated postoperative complications were also tackled efficiently with preservation of renal function. Conclusion: In EPN with stones, nephrectomy is not the sole option available and they can be effectively managed with open / endoscopic measures. PMID:19718324
Regge calculus models of closed lattice universes
NASA Astrophysics Data System (ADS)
Liu, Rex G.; Williams, Ruth M.
2016-01-01
This paper examines the behavior of closed "lattice universes" wherein masses are distributed in a regular lattice on the Cauchy surfaces of closed vacuum universes. Such universes are approximated using a form of Regge calculus originally developed by Collins and Williams to model closed Friedmann-Lemaître-Robertson-Walker universes. We consider two types of lattice universes, one where all masses are identical to each other and another where one mass gets perturbed in magnitude. In the unperturbed universe, we consider the possible arrangements of the masses in the Regge Cauchy surfaces and demonstrate that the model will only be stable if each mass lies within some spherical region of convergence. We also briefly discuss the existence of Regge models that are dual to the ones we have considered. We then model a perturbed lattice universe and demonstrate that the model's evolution is well behaved, with the expansion increasing in magnitude as the perturbation is increased.
ERIC Educational Resources Information Center
Koo, Reginald; Jones, Martin L.
2011-01-01
Quite a number of interesting problems in probability feature an event with probability equal to 1/e. This article discusses three such problems and attempts to explain why this probability occurs with such frequency.
Improving Student Success in Calculus: A Comparison of Four College Calculus Classes
NASA Astrophysics Data System (ADS)
Bagley, Spencer Franklin
The quality of education in science, technology, engineering, and mathematics (STEM) fields is an issue of particular educational and economic importance, and Calculus I is a linchpin course in STEM major tracks. A national study is currently being conducted examining the characteristics of successful programs in college calculus (CSPCC, 2012). In work related to the CSPCC program, this study examines the effects on student outcomes of four different teaching strategies used at a single institution. The four classes were a traditional lecture, a lecture with discussion, a lecture incorporating both discussion and technology, and an inverted model. This dissertation was guided by three questions: (1) What impact do these four instructional approaches have on students' persistence, beliefs about mathematics, and conceptual and procedural achievement in calculus? (2) How do students at the local institution compare to students in the national database? And (3) How do the similarities and differences in opportunities for learning presented in the four classes contribute to the similarities and differences in student outcomes? Quantitative analysis of surveys and exams revealed few statistically significant differences in outcomes, and students in the inverted classroom often had poorer outcomes than those in other classes. Students in the technology-enhanced class scored higher on conceptual items on the final exam than those in other classes. Comparing to the national database, local students had similar switching rates but less expert-like attitudes and beliefs about mathematics than the national average. Qualitative analysis of focus group interviews, classroom observations, and student course evaluations showed that several implementation issues, some the result of pragmatic constraints, others the result of design choice, weakened affordances provided by innovative features and shrunk the differences between classes. There were substantial differences between the
A transition calculus for Boolean functions. [logic circuit analysis
NASA Technical Reports Server (NTRS)
Tucker, J. H.; Bennett, A. W.
1974-01-01
A transition calculus is presented for analyzing the effect of input changes on the output of logic circuits. The method is closely related to the Boolean difference, but it is more powerful. Both differentiation and integration are considered.
Teacher-Controlled Programs for Demonstrating Concepts in Calculus.
ERIC Educational Resources Information Center
Hativa, Nira; Barall, Michael
1984-01-01
Describes the principles underlying computer programs designed to improve undergraduate calculus instruction. The programs were produced for teacher use on a single microcomputer for when there are not enough microcomputers available to allow students to have access to them. (JN)
Forest Carbon Uptake and the Fundamental Theorem of Calculus
ERIC Educational Resources Information Center
Zobitz, John
2013-01-01
Using the fundamental theorem of calculus and numerical integration, we investigate carbon absorption of ecosystems with measurements from a global database. The results illustrate the dynamic nature of ecosystems and their ability to absorb atmospheric carbon.
Spontaneous bladder rupture caused by a giant vesical calculus.
Kaur, Navneet; Attam, Amit; Gupta, Ashish; Amratash
2006-01-01
Spontaneous rupture of the urinary bladder is an uncommon occurrence. A 36-year-old man had complaints of pain and progressive distension of abdomen and anuria for 2 days. His abdomen was tense, tender and distended with free fluid. Blood urea was 340 mg% and ascitic fluid urea was 337 mg%. An USG showed massive ascitis, a large vesical calculus and a left renal calculus. The urinary bladder could not be catheterized. Patient underwent hemodialysis and placement of abdominal drains. About 2 l of yellow turbid fluid was drained. Cystolithotomy showed a 6 cm size impacted calculus with a rent in the dome of the bladder, which was repaired. Subsequently patient underwent percutaneous nephrolithotrypsy for left staghorn renal calculus and nephrectomy for right non-functioning kidney.
Ants, Tunnels, and Calculus: An Exercise in Mathematical Modeling.
ERIC Educational Resources Information Center
Winkel, Brian J.
1994-01-01
Discusses an activity which models the building of a tunnel by ants using the definitions of derivative and indefinite integral from calculus. Includes a discussion of reasonableness and interpretation of the problem. (MKR)
Geometric constrained variational calculus. II: The second variation (Part I)
NASA Astrophysics Data System (ADS)
Massa, Enrico; Bruno, Danilo; Luria, Gianvittorio; Pagani, Enrico
2016-10-01
Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.
Using `min' and `max' functions in calculus teaching
NASA Astrophysics Data System (ADS)
Satianov, Pavel; Dagan, Miriam; Amram, Meirav
2015-08-01
In this paper, we discuss the use of the min and max functions in teaching calculus to engineering students. Our experience illustrates that such functions have great possibilities in the development of a student's analytical thinking. The types of problems we present here are not common in most instructional texts, which lead us to suggest that the paper will be interesting and useful to calculus lecturers.
Dental Calculus and the Evolution of the Human Oral Microbiome.
Warinner, Christina
2016-07-01
Characterizing the evolution of the oral microbiome is a challenging, but increasingly feasible, task. Recently, dental calculus has been shown to preserve ancient biomolecules from the oral microbiota, host tissues and diet for tens of thousands of years. As such, it provides a unique window into the ancestral oral microbiome. This article reviews recent advancements in ancient dental calculus research and emerging insights into the evolution and ecology of the human oral microbiome.
ERIC Educational Resources Information Center
Hall, Angela Renee
2011-01-01
This investigative research focuses on the level of readiness of Science, Technology, Engineering, and Mathematics (STEM) students entering Historically Black Colleges and Universities (HBCU) in the college Calculus sequence. Calculus is a fundamental course for STEM courses. The level of readiness of the students for Calculus can very well play a…
The relationship between species detection probability and local extinction probability
Alpizar-Jara, R.; Nichols, J.D.; Hines, J.E.; Sauer, J.R.; Pollock, K.H.; Rosenberry, C.S.
2004-01-01
In community-level ecological studies, generally not all species present in sampled areas are detected. Many authors have proposed the use of estimation methods that allow detection probabilities that are < 1 and that are heterogeneous among species. These methods can also be used to estimate community-dynamic parameters such as species local extinction probability and turnover rates (Nichols et al. Ecol Appl 8:1213-1225; Conserv Biol 12:1390-1398). Here, we present an ad hoc approach to estimating community-level vital rates in the presence of joint heterogeneity of detection probabilities and vital rates. The method consists of partitioning the number of species into two groups using the detection frequencies and then estimating vital rates (e.g., local extinction probabilities) for each group. Estimators from each group are combined in a weighted estimator of vital rates that accounts for the effect of heterogeneity. Using data from the North American Breeding Bird Survey, we computed such estimates and tested the hypothesis that detection probabilities and local extinction probabilities were negatively related. Our analyses support the hypothesis that species detection probability covaries negatively with local probability of extinction and turnover rates. A simulation study was conducted to assess the performance of vital parameter estimators as well as other estimators relevant to questions about heterogeneity, such as coefficient of variation of detection probabilities and proportion of species in each group. Both the weighted estimator suggested in this paper and the original unweighted estimator for local extinction probability performed fairly well and provided no basis for preferring one to the other.
The relationship between species detection probability and local extinction probability
Alpizar-Jara, R.; Nichols, J.D.; Hines, J.E.; Sauer, J.R.; Pollock, K.H.; Rosenberry, C.S.
2004-01-01
In community-level ecological studies, generally not all species present in sampled areas are detected. Many authors have proposed the use of estimation methods that allow detection probabilities that are <1 and that are heterogeneous among species. These methods can also be used to estimate community-dynamic parameters such as species local extinction probability and turnover rates (Nichols et al. Ecol Appl 8:1213-1225; Conserv Biol 12:1390-1398). Here, we present an ad hoc approach to estimating community-level vital rates in the presence of joint heterogeneity of detection probabilities and vital rates. The method consists of partitioning the number of species into two groups using the detection frequencies and then estimating vital rates (e.g., local extinction probabilities) for each group. Estimators from each group are combined in a weighted estimator of vital rates that accounts for the effect of heterogeneity. Using data from the North American Breeding Bird Survey, we computed such estimates and tested the hypothesis that detection probabilities and local extinction probabilities were negatively related. Our analyses support the hypothesis that species detection probability covaries negatively with local probability of extinction and turnover rates. A simulation study was conducted to assess the performance of vital parameter estimators as well as other estimators relevant to questions about heterogeneity, such as coefficient of variation of detection probabilities and proportion of species in each group. Both the weighted estimator suggested in this paper and the original unweighted estimator for local extinction probability performed fairly well and provided no basis for preferring one to the other.
Definition of the Neutrosophic Probability
NASA Astrophysics Data System (ADS)
Smarandache, Florentin
2014-03-01
Neutrosophic probability (or likelihood) [1995] is a particular case of the neutrosophic measure. It is an estimation of an event (different from indeterminacy) to occur, together with an estimation that some indeterminacy may occur, and the estimation that the event does not occur. The classical probability deals with fair dice, coins, roulettes, spinners, decks of cards, random works, while neutrosophic probability deals with unfair, imperfect such objects and processes. For example, if we toss a regular die on an irregular surface which has cracks, then it is possible to get the die stuck on one of its edges or vertices in a crack (indeterminate outcome). The sample space is in this case: {1, 2, 3, 4, 5, 6, indeterminacy}. So, the probability of getting, for example 1, is less than 1/6. Since there are seven outcomes. The neutrosophic probability is a generalization of the classical probability because, when the chance of determinacy of a stochastic process is zero, these two probabilities coincide. The Neutrosophic Probability that of an event A occurs is NP (A) = (ch (A) , ch (indetA) , ch (A ̲)) = (T , I , F) , where T , I , F are subsets of [0,1], and T is the chance that A occurs, denoted ch(A); I is the indeterminate chance related to A, ch(indetermA) ; and F is the chance that A does not occur, ch (A ̲) . So, NP is a generalization of the Imprecise Probability as well. If T, I, and F are crisp numbers then: - 0 <= T + I + F <=3+ . We used the same notations (T,I,F) as in neutrosophic logic and set.
Water content contribution in calculus phantom ablation during Q-switched Tm:YAG laser lithotripsy.
Zhang, Jian J; Rajabhandharaks, Danop; Xuan, Jason Rongwei; Wang, Hui; Chia, Ray W J; Hasenberg, Tom; Kang, Hyun Wook
2015-01-01
Q-switched (QS) Tm:YAG laser ablation mechanisms on urinary calculi are still unclear to researchers. Here, dependence of water content in calculus phantom on calculus ablation performance was investigated. White gypsum cement was used as a calculus phantom model. The calculus phantoms were ablated by a total 3-J laser pulse exposure (20 mJ, 100 Hz, 1.5 s) and contact mode with N=15 sample size. Ablation volume was obtained on average 0.079, 0.122, and 0.391 mm3 in dry calculus in air, wet calculus in air, and wet calculus in-water groups, respectively. There were three proposed ablation mechanisms that could explain the effect of water content in calculus phantom on calculus ablation performance, including shock wave due to laser pulse injection and bubble collapse, spallation, and microexplosion. Increased absorption coefficient of wet calculus can cause stronger spallation process compared with that caused by dry calculus; as a result, higher calculus ablation was observed in both wet calculus in air and wet calculus in water. The test result also indicates that the shock waves generated by short laser pulse under the in-water condition have great impact on the ablation volume by Tm:YAG QS laser.
Canonical linearized Regge calculus: Counting lattice gravitons with Pachner moves
NASA Astrophysics Data System (ADS)
Höhn, Philipp A.
2015-06-01
We afford a systematic and comprehensive account of the canonical dynamics of 4D Regge calculus perturbatively expanded to linear order around a flat background. To this end, we consider the Pachner moves which generate the most basic and general simplicial evolution scheme. The linearized regime features a vertex displacement (diffeomorphism) symmetry for which we derive an Abelian constraint algebra. This permits us to identify gauge invariant lattice "gravitons" as propagating curvature degrees of freedom. The Pachner moves admit a simple method to explicitly count the gauge and graviton degrees of freedom on an evolving triangulated hypersurface, and we clarify the distinct role of each move in the dynamics. It is shown that the 1-4 move generates four "lapse and shift" variables and four conjugate vertex displacement generators; the 2-3 move generates a graviton; the 3-2 move removes one graviton and produces the only non-trivial equation of motion; and the 4-1 move removes four lapse and shift variables and trivializes the four conjugate symmetry generators. It is further shown that the Pachner moves preserve the vertex displacement generators. These results may provide new impetus for exploring `graviton dynamics' in discrete quantum gravity models.
Geometric constrained variational calculus I: Piecewise smooth extremals
NASA Astrophysics Data System (ADS)
Massa, Enrico; Bruno, Danilo; Luria, Gianvittorio; Pagani, Enrico
2015-05-01
A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the co-rank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagin's equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagin's algorithm in Hamiltonian terms. The analysis is completed by a general result, concerning the existence of finite deformations with fixed endpoints.
Pattern formation, logistics, and maximum path probability
NASA Astrophysics Data System (ADS)
Kirkaldy, J. S.
1985-05-01
The concept of pattern formation, which to current researchers is a synonym for self-organization, carries the connotation of deductive logic together with the process of spontaneous inference. Defining a pattern as an equivalence relation on a set of thermodynamic objects, we establish that a large class of irreversible pattern-forming systems, evolving along idealized quasisteady paths, approaches the stable steady state as a mapping upon the formal deductive imperatives of a propositional function calculus. In the preamble the classical reversible thermodynamics of composite systems is analyzed as an externally manipulated system of space partitioning and classification based on ideal enclosures and diaphragms. The diaphragms have discrete classification capabilities which are designated in relation to conserved quantities by descriptors such as impervious, diathermal, and adiabatic. Differentiability in the continuum thermodynamic calculus is invoked as equivalent to analyticity and consistency in the underlying class or sentential calculus. The seat of inference, however, rests with the thermodynamicist. In the transition to an irreversible pattern-forming system the defined nature of the composite reservoirs remains, but a given diaphragm is replaced by a pattern-forming system which by its nature is a spontaneously evolving volume partitioner and classifier of invariants. The seat of volition or inference for the classification system is thus transferred from the experimenter or theoretician to the diaphragm, and with it the full deductive facility. The equivalence relations or partitions associated with the emerging patterns may thus be associated with theorems of the natural pattern-forming calculus. The entropy function, together with its derivatives, is the vehicle which relates the logistics of reservoirs and diaphragms to the analog logistics of the continuum. Maximum path probability or second-order differentiability of the entropy in isolation are
Noninvasive control of dental calculus removal: qualification of two fluorescence methods
NASA Astrophysics Data System (ADS)
Gonchukov, S.; Sukhinina, A.; Bakhmutov, D.; Biryukova, T.
2013-02-01
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 this work it was shown that fluorescence diagnostics during calculus removal can be successfully used for precise noninvasive detection of calculus-tooth 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.
The Calculus of Relativistic Temporal Geometry
NASA Astrophysics Data System (ADS)
Mayer, Alexander
2009-05-01
Richard Feynman's unpublished 1965 gedanken experiment, discussed on pages 60-62 of A. F. Mayer, On the Geometry of Time in Physics and Cosmology (April 2009), demonstrates that the principles of relativity destroy both Newton's concept of absolute time and the concept of a Newtonian gravitational equipotential surface. According to logic arising from experience, it has long been falsely assumed that no energy cost is incurred for translation over an ideally frictionless level surface in the presence of a vertical acceleration. However, that the speed of light is a limiting velocity implies that while two distinct points on such a surface can be considered to be at the same potential relative to a third point that is not on that surface, a particle translated between two such points must incur energy transfer to the accelerating field. Typically, this manifests as a redshift of electromagnetic radiation as demonstrated by ``Feynman's rocket.'' Accurate calculation of this relativistic transverse gravitational redshift (TGR) for observable phenomena in a real-world astrophysical gravitational field requires the calculus of relativistic temporal geometry. Calculations using this technique accurately predict the following empirically observed but heretofore unexplained natural phenomena: the center-to-limb variation of solar wavelength (˜1 km/ s), the K-effect for massive main sequence stars (˜2-3 km/s), and the excess redshift of white dwarf stars (˜10-15 km/s).
The National Aquatic Resource Surveys (NARS) use probability-survey designs to assess the condition of the nation’s waters. In probability surveys (also known as sample-surveys or statistical surveys), sampling sites are selected randomly.
ERIC Educational Resources Information Center
Edwards, William F.; Shiflett, Ray C.; Shultz, Harris
2008-01-01
The mathematical model used to describe independence between two events in probability has a non-intuitive consequence called dependent spaces. The paper begins with a very brief history of the development of probability, then defines dependent spaces, and reviews what is known about finite spaces with uniform probability. The study of finite…
47 CFR 1.1623 - Probability calculation.
Code of Federal Regulations, 2012 CFR
2012-10-01
... 47 Telecommunication 1 2012-10-01 2012-10-01 false Probability calculation. 1.1623 Section 1.1623 Telecommunication FEDERAL COMMUNICATIONS COMMISSION GENERAL PRACTICE AND PROCEDURE Grants by Random Selection Random Selection Procedures for Mass Media Services General Procedures § 1.1623 Probability calculation. (a)...
47 CFR 1.1623 - Probability calculation.
Code of Federal Regulations, 2010 CFR
2010-10-01
... 47 Telecommunication 1 2010-10-01 2010-10-01 false Probability calculation. 1.1623 Section 1.1623 Telecommunication FEDERAL COMMUNICATIONS COMMISSION GENERAL PRACTICE AND PROCEDURE Random Selection Procedures for Mass Media Services General Procedures § 1.1623 Probability calculation. (a) All calculations shall be computed to no less than...
Laboratory-Tutorial Activities for Teaching Probability
ERIC Educational Resources Information Center
Wittmann, Michael C.; Morgan, Jeffrey T.; Feeley, Roger E.
2006-01-01
We report on the development of students' ideas of probability and probability density in a University of Maine laboratory-based general education physics course called "Intuitive Quantum Physics". Students in the course are generally math phobic with unfavorable expectations about the nature of physics and their ability to do it. We…
Colloquium: Fractional calculus view of complexity: A tutorial
NASA Astrophysics Data System (ADS)
West, Bruce J.
2014-10-01
The fractional calculus has been part of the mathematics and science literature for 310 years. However, it is only in the past decade or so that it has drawn the attention of mainstream science as a way to describe the dynamics of complex phenomena with long-term memory, spatial heterogeneity, along with nonstationary and nonergodic statistics. The most recent application encompasses complex networks, which require new ways of thinking about the world. Part of the new cognition is provided by the fractional calculus description of temporal and topological complexity. Consequently, this Colloquium is not so much a tutorial on the mathematics of the fractional calculus as it is an exploration of how complex phenomena in the physical, social, and life sciences that have eluded traditional mathematical modeling become less mysterious when certain historical assumptions such as differentiability are discarded and the ordinary calculus is replaced with the fractional calculus. Exemplars considered include the fractional differential equations describing the dynamics of viscoelastic materials, turbulence, foraging, and phase transitions in complex social networks.
On flipping first-semester calculus: a case study
NASA Astrophysics Data System (ADS)
Petrillo, Joseph
2016-05-01
High failure rates in calculus have plagued students, teachers, and administrators for decades, while science, technology, engineering, and mathematics programmes continue to suffer from low enrollments and high attrition. In an effort to affect this reality, some educators are 'flipping' (or inverting) their classrooms. By flipping, we mean administering course content outside of the classroom and replacing the traditional in-class lectures with discussion, practice, group work, and other elements of active learning. This paper presents the major results from a three-year study of a flipped, first-semester calculus course at a small, comprehensive, American university with a well-known engineering programme. The data we have collected help quantify the positive and substantial effects of our flipped calculus course on failure rates, scores on the common final exam, student opinion of calculus, teacher impact on measurable outcomes, and success in second-semester calculus. While flipping may not be suitable for every teacher, every student, and in every situation, this report provides some evidence that it may be a viable option for those seeking an alternative to the traditional lecture model.
Two-parameter asymptotics in magnetic Weyl calculus
Lein, Max
2010-12-15
This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter {epsilon}, the case of small coupling {lambda} to the magnetic vector potential naturally occurs in this context. Magnetic Weyl calculus is adapted to incorporate both parameters, at least one of which needs to be small. Of particular interest is the expansion of the Weyl product which can be used to expand the product of operators in a small parameter, a technique which is prominent to obtain perturbation expansions. Three asymptotic expansions for the magnetic Weyl product of two Hoermander class symbols are proven as (i) {epsilon}<< 1 and {lambda}<< 1, (ii) {epsilon}<< 1 and {lambda}= 1, as well as (iii) {epsilon}= 1 and {lambda}<< 1. Expansions (i) and (iii) are impossible to obtain with ordinary Weyl calculus. Furthermore, I relate the results derived by ordinary Weyl calculus with those obtained with magnetic Weyl calculus by one- and two-parameter expansions. To show the power and versatility of magnetic Weyl calculus, I derive the semirelativistic Pauli equation as a scaling limit from the Dirac equation up to errors of fourth order in 1/c.
Dynamical Simulation of Probabilities
NASA Technical Reports Server (NTRS)
Zak, Michail
1996-01-01
It has been demonstrated that classical probabilities, and in particular, probabilistic Turing machine, can be simulated by combining chaos and non-Lipschitz dynamics, without utilization of any man-made devices(such as random number generators). Self-orgainizing properties of systems coupling simulated and calculated probabilities and their link to quantum computations are discussed. Special attention was focused upon coupled stochastic processes, defined in terms of conditional probabilities, for which joint probability does not exist. Simulations of quantum probabilities are also discussed.
Minimal entropy probability paths between genome families.
Ahlbrandt, Calvin; Benson, Gary; Casey, William
2004-05-01
We develop a metric for probability distributions with applications to biological sequence analysis. Our distance metric is obtained by minimizing a functional defined on the class of paths over probability measures on N categories. The underlying mathematical theory is connected to a constrained problem in the calculus of variations. The solution presented is a numerical solution, which approximates the true solution in a set of cases called rich paths where none of the components of the path is zero. The functional to be minimized is motivated by entropy considerations, reflecting the idea that nature might efficiently carry out mutations of genome sequences in such a way that the increase in entropy involved in transformation is as small as possible. We characterize sequences by frequency profiles or probability vectors, in the case of DNA where N is 4 and the components of the probability vector are the frequency of occurrence of each of the bases A, C, G and T. Given two probability vectors a and b, we define a distance function based as the infimum of path integrals of the entropy function H( p) over all admissible paths p(t), 0 < or = t< or =1, with p(t) a probability vector such that p(0)=a and p(1)=b. If the probability paths p(t) are parameterized as y(s) in terms of arc length s and the optimal path is smooth with arc length L, then smooth and "rich" optimal probability paths may be numerically estimated by a hybrid method of iterating Newton's method on solutions of a two point boundary value problem, with unknown distance L between the abscissas, for the Euler-Lagrange equations resulting from a multiplier rule for the constrained optimization problem together with linear regression to improve the arc length estimate L. Matlab code for these numerical methods is provided which works only for "rich" optimal probability vectors. These methods motivate a definition of an elementary distance function which is easier and faster to calculate, works on non
Today's Calculus Courses Are Too Watered Down and Outdated to Capture the Interest of Students.
ERIC Educational Resources Information Center
Douglas, Ronald G.
1988-01-01
Calculus provides the language for expressing the differential equations that govern change and also the methods for solving them. In order to insure that more Americans qualify for science-related careers, the way calculus is taught must change. (MLW)
Dental wax decreases calculus accumulation in small dogs.
Smith, Mark M; Smithson, Christopher W
2014-01-01
A dental wax was evaluated after unilateral application in 20 client-owned, mixed and purebred small dogs using a clean, split-mouth study model. All dogs had clinical signs of periodontal disease including plaque, calculus, and/or gingivitis. The wax was randomly applied to the teeth of one side of the mouth daily for 30-days while the contralateral side received no treatment. Owner parameters evaluated included compliance and a subjective assessment of ease of wax application. Gingivitis, plaque and calculus accumulation were scored at the end of the study period. Owners considered the wax easy to apply in all dogs. Compliance with no missed application days was achieved in 8 dogs. The number of missed application days had no effect on wax efficacy. There was no significant difference in gingivitis or plaque accumulation scores when comparing treated and untreated sides. Calculus accumulation scores were significantly less (22.1 %) for teeth receiving the dental wax.
Quantum Stratonovich calculus and the quantum Wong-Zakai theorem
Gough, John
2006-11-15
We extend the Ito(bar sign)-to-Stratonovich analysis or quantum stochastic differential equations, introduced by Gardiner and Collett for emission (creation), absorption (annihilation) processes, to include scattering (conservation) processes. Working within the framework of quantum stochastic calculus, we define Stratonovich calculus as an algebraic modification of the Ito(bar sign) one and give conditions for the existence of Stratonovich time-ordered exponentials. We show that conversion formula for the coefficients has a striking resemblance to Green's function formulas from standard perturbation theory. We show that the calculus conveniently describes the Markov limit of regular open quantum dynamical systems in much the same way as in the Wong-Zakai approximation theorems of classical stochastic analysis. We extend previous limit results to multiple-dimensions with a proof that makes use of diagrammatic conventions.
Analysis-based arguments for abstract data type calculus.
Rouson, Damian W. I.
2008-10-01
Increasing demands on the complexity of scientific models coupled with increasing demands for their scalability are placing programming models on equal footing with the numerical methods they implement in terms of significance. A recurring theme across several major scientific software development projects involves defining abstract data types (ADTs) that closely mimic mathematical abstractions such as scalar, vector, and tensor fields. In languages that support user-defined operators and/or overloading of intrinsic operators, coupling ADTs with a set of algebraic and/or integro-differential operators results in an ADT calculus. This talk will analyze ADT calculus using three tool sets: object-oriented design metrics, computational complexity theory, and information theory. It will be demonstrated that ADT calculus leads to highly cohesive, loosely coupled abstractions with code-size-invariant data dependencies and minimal information entropy. The talk will also discuss how these results relate to software flexibility and robustness.
Golden quantum oscillator and Binet-Fibonacci calculus
NASA Astrophysics Data System (ADS)
Pashaev, Oktay K.; Nalci, Sengul
2012-01-01
The Binet formula for Fibonacci numbers is treated as a q-number and a q-operator with Golden ratio bases q = φ and Q = -1/φ, and the corresponding Fibonacci or Golden calculus is developed. A quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given only by Fibonacci numbers. The ratio of successive energy levels is found to be the Golden sequence, and for asymptotic states in the limit n → ∞ it appears as the Golden ratio. We call this oscillator the Golden oscillator. Using double Golden bosons, the Golden angular momentum and its representation in terms of Fibonacci numbers and the Golden ratio are derived. Relations of Fibonacci calculus with a q-deformed fermion oscillator and entangled N-qubit states are indicated.
Probability state modeling theory.
Bagwell, C Bruce; Hunsberger, Benjamin C; Herbert, Donald J; Munson, Mark E; Hill, Beth L; Bray, Chris M; Preffer, Frederic I
2015-07-01
As the technology of cytometry matures, there is mounting pressure to address two major issues with data analyses. The first issue is to develop new analysis methods for high-dimensional data that can directly reveal and quantify important characteristics associated with complex cellular biology. The other issue is to replace subjective and inaccurate gating with automated methods that objectively define subpopulations and account for population overlap due to measurement uncertainty. Probability state modeling (PSM) is a technique that addresses both of these issues. The theory and important algorithms associated with PSM are presented along with simple examples and general strategies for autonomous analyses. PSM is leveraged to better understand B-cell ontogeny in bone marrow in a companion Cytometry Part B manuscript. Three short relevant videos are available in the online supporting information for both of these papers. PSM avoids the dimensionality barrier normally associated with high-dimensionality modeling by using broadened quantile functions instead of frequency functions to represent the modulation of cellular epitopes as cells differentiate. Since modeling programs ultimately minimize or maximize one or more objective functions, they are particularly amenable to automation and, therefore, represent a viable alternative to subjective and inaccurate gating approaches.
Recalling Prerequisite Material in a Calculus II Course to Improve Student Success
ERIC Educational Resources Information Center
Mokry, Jeanette
2016-01-01
This article discusses preparation assignments used in a Calculus II course that cover material from prerequisite courses. Prior to learning new material, students work on problems outside of class involving concepts from algebra, trigonometry, and Calculus I. These problems are directly built upon in order to answer Calculus II questions,…
The Development and Nature of Problem-Solving among First-Semester Calculus Students
ERIC Educational Resources Information Center
Dawkins, Paul Christian; Epperson, James A. Mendoza
2014-01-01
This study investigates interactions between calculus learning and problem-solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students' problem-solving abilities in a common US calculus course design that included traditional lecture and assessment with problem-solving-oriented labs. We investigate…
ERIC Educational Resources Information Center
Laurent, Theresa A.
2009-01-01
The purpose of this study was to investigate higher education mathematics departments' credit granting policies for students with high school calculus experience. The number of students taking calculus in high school has more than doubled since 1982 (NCES, 2007) and it is estimated that approximately 530,000 students took a calculus course in high…
An operator calculus for surface and volume modeling
NASA Technical Reports Server (NTRS)
Gordon, W. J.
1984-01-01
The mathematical techniques which form the foundation for most of the surface and volume modeling techniques used in practice are briefly described. An outline of what may be termed an operator calculus for the approximation and interpolation of functions of more than one independent variable is presented. By considering the linear operators associated with bivariate and multivariate interpolation/approximation schemes, it is shown how they can be compounded by operator multiplication and Boolean addition to obtain a distributive lattice of approximation operators. It is then demonstrated via specific examples how this operator calculus leads to practical techniques for sculptured surface and volume modeling.
The relationships between malocclusion, gingival inflammation, plaque and calculus.
Buckley, L A
1981-01-01
Certain features of malocclusion considered important in relation to periodontal health were analyzed in a study of 300 subjects. It was found that plaque and gingival inflammation were not related to vertical incisor overbite, horizontal incisor overjet or posterior cuspal interdigitation. Individual tooth irregularity measured as tilting, rotation, displacement and crowding had a low but statistically significant correlation with plaque, calculus and gingival inflammation. However, the study showed that these features of malocclusion are far less important than the extent of plaque and calculus deposits in the development of gingival inflammation.
Probability and radical behaviorism
Espinosa, James M.
1992-01-01
The concept of probability appears to be very important in the radical behaviorism of Skinner. Yet, it seems that this probability has not been accurately defined and is still ambiguous. I give a strict, relative frequency interpretation of probability and its applicability to the data from the science of behavior as supplied by cumulative records. Two examples of stochastic processes are given that may model the data from cumulative records that result under conditions of continuous reinforcement and extinction, respectively. PMID:22478114
NASA Astrophysics Data System (ADS)
Laktineh, Imad
2010-04-01
This ourse constitutes a brief introduction to probability applications in high energy physis. First the mathematical tools related to the diferent probability conepts are introduced. The probability distributions which are commonly used in high energy physics and their characteristics are then shown and commented. The central limit theorem and its consequences are analysed. Finally some numerical methods used to produce diferent kinds of probability distribution are presented. The full article (17 p.) corresponding to this lecture is written in french and is provided in the proceedings of the book SOS 2008.
Probability of satellite collision
NASA Technical Reports Server (NTRS)
Mccarter, J. W.
1972-01-01
A method is presented for computing the probability of a collision between a particular artificial earth satellite and any one of the total population of earth satellites. The collision hazard incurred by the proposed modular Space Station is assessed using the technique presented. The results of a parametric study to determine what type of satellite orbits produce the greatest contribution to the total collision probability are presented. Collision probability for the Space Station is given as a function of Space Station altitude and inclination. Collision probability was also parameterized over miss distance and mission duration.
STATISTICAL ANALYSIS, REPORTS), (*PROBABILITY, REPORTS), INFORMATION THEORY, DIFFERENTIAL EQUATIONS, STATISTICAL PROCESSES, STOCHASTIC PROCESSES, MULTIVARIATE ANALYSIS, DISTRIBUTION THEORY , DECISION THEORY, MEASURE THEORY, OPTIMIZATION
NASA Astrophysics Data System (ADS)
Steele, Diana F.; Levin, Amy K.; Blecksmith, Richard; Shahverdian, Jill
2005-10-01
The purpose of this study was to investigate the ways in which a multi-layered women's calculus course influenced the participants' learning of mathematics. This study, conducted in a state university in the Midwestern region of the United States, revealed not only that women in this particular section of calculus were likely to select careers that involved mathematics, but that the focus on peer support, psychosocial issues such as self-confidence, and pedagogy helped the young women overcome gender barriers, as well as barriers of class, poverty, and race. In this article we provide some of the relevant quantitative statistics and relate the stories of two particular women through excerpts from interviews, student artefacts, and participant observation data. We selected these young women because they faced multiple barriers to success in Calculus I and might not have completed the course or taken additional mathematics courses without the support structures that were fundamental to the course.
Anisotropic fractal media by vector calculus in non-integer dimensional space
Tarasov, Vasily E.
2014-08-15
A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.
The impact of instructor pedagogy on college calculus students' attitude toward mathematics
NASA Astrophysics Data System (ADS)
Sonnert, Gerhard; Sadler, Philip M.; Sadler, Samuel M.; Bressoud, David M.
2015-04-01
College calculus teaches students important mathematical concepts and skills. The course also has a substantial impact on students' attitude toward mathematics, affecting their career aspirations and desires to take more mathematics. This national US study of 3103 students at 123 colleges and universities tracks changes in students' attitudes toward mathematics during a 'mainstream' calculus course while controlling for student backgrounds. The attitude measure combines students' self-ratings of their mathematics confidence, interest in, and enjoyment of mathematics. Three major kinds of instructor pedagogy, identified through the factor analysis of 61 student-reported variables, are investigated for impact on student attitude as follows: (1) instructors who employ generally accepted 'good teaching' practices (e.g. clarity in presentation and answering questions, useful homework, fair exams, help outside of class) are found to have the most positive impact, particularly with students who began with a weaker initial attitude. (2) Use of educational 'technology' (e.g. graphing calculators, for demonstrations, in homework), on average, is found to have no impact on attitudes, except when used by graduate student instructors, which negatively affects students' attitudes towards mathematics. (3) 'Ambitious teaching' (e.g. group work, word problems, 'flipped' reading, student explanations of thinking) has a small negative impact on student attitudes, while being a relatively more constructive influence only on students who already enjoyed a positive attitude toward mathematics and in classrooms with a large number of students. This study provides support for efforts to improve calculus teaching through the training of faculty and graduate students to use traditional 'good teaching' practices through professional development workshops and courses. As currently implemented, technology and ambitious pedagogical practices, while no doubt effective in certain classrooms, do
ERIC Educational Resources Information Center
Barnes, Bernis, Ed.; And Others
This teacher's guide to probability and statistics contains three major sections. The first section on elementary combinatorial principles includes activities, student problems, and suggested teaching procedures for the multiplication principle, permutations, and combinations. Section two develops an intuitive approach to probability through…
Teachers' Understandings of Probability
ERIC Educational Resources Information Center
Liu, Yan; Thompson, Patrick
2007-01-01
Probability is an important idea with a remarkably wide range of applications. However, psychological and instructional studies conducted in the last two decades have consistently documented poor understanding of probability among different populations across different settings. The purpose of this study is to develop a theoretical framework for…
ERIC Educational Resources Information Center
Judson, Thomas W.; Nishimori, Toshiyuki
2005-01-01
In this study we investigated above-average high school calculus students from Japan and the United States in order to determine any differences in their conceptual understanding of calculus and their ability to use algebra to solve traditional calculus problems. We examined and interviewed 18 Calculus BC students in the United States and 26…
Calculus of One and More Variables with Maple
ERIC Educational Resources Information Center
Samkova, Libuse
2012-01-01
This is a guide to using Maple in teaching fundamental calculus of one, two and three variables (limits, derivatives, integrals, etc.), also suitable for Maple beginners. It outlines one of the ways to effective use of computers in the teaching process. It scans advantages and disadvantages of using Maple in relation to students and teacher. The…
Flipping the Calculus Classroom: A Cost-Effective Approach
ERIC Educational Resources Information Center
Young, Andrea
2015-01-01
This article discusses a cost-effective approach to flipping the calculus classroom. In particular, the emphasis is on low-cost choices, both monetarily and with regards to faculty time, that make the daunting task of flipping a course manageable for a single instructor. Student feedback and overall impressions are also presented.
On λ-Bell polynomials associated with umbral calculus
NASA Astrophysics Data System (ADS)
Kim, T.; Kim, D. S.
2017-01-01
In this paper, we introduce some new λ-Bell polynomials and Bell polynomials of the second kind and investigate properties of these polynomials. Using our investigation, we derive some new identities for the two kinds of λ-Bell polynomials arising from umbral calculus.
Non-Mathematics Students' Reasoning in Calculus Tasks
ERIC Educational Resources Information Center
Jukic Matic, Ljerka
2015-01-01
This paper investigates the reasoning of first year non-mathematics students in non-routine calculus tasks. The students in this study were accustomed to imitative reasoning from their primary and secondary education. In order to move from imitative reasoning toward more creative reasoning, non-routine tasks were implemented as an explicit part of…
On Flipping First-Semester Calculus: A Case Study
ERIC Educational Resources Information Center
Petrillo, Joseph
2016-01-01
High failure rates in calculus have plagued students, teachers, and administrators for decades, while science, technology, engineering, and mathematics programmes continue to suffer from low enrollments and high attrition. In an effort to affect this reality, some educators are "flipping" (or inverting) their classrooms. By flipping, we…
Renal artery aneurysm mimicking renal calculus with hydronephrosis.
Chen, Shanwen; Meng, Hongzhou; Cao, Min; Shen, Baihua
2013-06-01
A 51-year-old woman was found to have a left renal calculus with hydronephrosis. She underwent unsuccessful extracorporeal shock wave lithotripsy, leading to the recommendation that percutaneous lithotomy was necessary to remove the renal calculus. In view of the unusual shape of the calculus and absence of abnormalities in urine sediment, preoperative computed tomography and renal angiography were performed, which instead showed a calcified left renal artery aneurysm. Subsequent efforts to perform an aneurysmectomy also failed, eventually necessitating left nephrectomy. This case illustrates the pitfalls in the diagnosis of a renal artery aneurysm, which is a relatively common condition that may have unusual presentations. Hence, it is suggested that the possibility of a renal artery aneurysm be considered in the differential diagnosis when one detects a renal calculus with an unusual appearance. In addition, we propose that 3-dimensional reconstruction computed tomography be performed before considering surgical options for such renal calculi to rule out the possibility of a renal artery aneurysm.
Assessing Online Homework in First-Semester Calculus
ERIC Educational Resources Information Center
Callahan, Jason T.
2016-01-01
This paper describes and assesses the implementation of online homework in a first-semester calculus course. Comparing sections of the course before implementation to those after, we find statistically significant improvements in retention rates, measures of student engagement, and participation on homework. We do not, however, find statistically…
An Experiment in "Flipped" Teaching in Freshman Calculus
ERIC Educational Resources Information Center
Anderson, Laura; Brennan, Joseph Phillip
2015-01-01
At Binghamton, Calculus 1 is taught to over 1000 students each fall in sections of about 30-40 students, with graduate student instructors teaching most sections. Despite having small classrooms instead of lecture halls, the satisfaction and performance of students has historically been poor. We had hoped to improve student success by changing how…
Using Origami Boxes to Explore Concepts of Geometry and Calculus
ERIC Educational Resources Information Center
Wares, Arsalan
2011-01-01
The purpose of this classroom note is to provide an example of how a simple origami box can be used to explore important concepts of geometry and calculus. This article describes how an origami box can be folded, then it goes on to describe how its volume and surface area can be calculated. Finally, it describes how the box could be folded to…
Readiness and Attitudes as Indicators for Success in College Calculus
ERIC Educational Resources Information Center
Pyzdrowski, Laura J.; Sun, Ye; Curtis, Reagan; Miller, David; Winn, Gary; Hensel, Robin A. M.
2013-01-01
This study examined student indicators for success in entry-level college calculus. An attitude toward mathematics inventory, course performance, readiness assessment, and student interviews were used to determine characteristics and behaviors of students who succeeded in the course. In addition to student indicators, difficult topics and…
Relation of Spatial Skills to Calculus Proficiency: A Brief Report
ERIC Educational Resources Information Center
Cromley, Jennifer G.; Booth, Julie L.; Wills, Theodore W.; Chang, Briana L.; Tran, Nhi; Madeja, Michael; Shipley, Thomas F.; Zahner, William
2017-01-01
Spatial skills have been shown in various longitudinal studies to be related to multiple science, technology, engineering, and math (STEM) achievement and retention. The specific nature of this relation has been probed in only a few domains, and has rarely been investigated for calculus, a critical topic in preparing students for and in STEM…
Integrating Precalculus Review with the First Course in Calculus.
ERIC Educational Resources Information Center
Sevilla, Alicia; Somers, Kay
1993-01-01
Describes a course designed by Moravian College, Pennsylvania, to integrate precalculus topics as needed into a first calculus course. The textbook developed for the course covers the concepts of functions, Cartesian coordinates, limits, continuity, infinity, and the derivative. Examples are discussed. (MDH)
Will Discrete Mathematics Surpass Calculus in Importance? and Responses .
ERIC Educational Resources Information Center
Ralston, Anthony; And Others
1984-01-01
Ralston proposes that the decrease in the importance of calculus in the world of mathematics is accelerating and the world of applied mathematics is changing rapidly. He briefly presents arguments for discrete mathematics. Then follow reactions from McLane, Wagner, Hilton, Woodriff, Kleitman, and Lax, and a response by Ralston. (MNS)
Evaluating the Performance of Calculus Classes Using Operational Research Tools.
ERIC Educational Resources Information Center
Soares de Mello, Joao Carlos C. B.; Lins, Marcos P. E.; Soares de Mello, Maria Helena C.; Gomes, Eliane G.
2002-01-01
Compares the efficiency of calculus classes and evaluates two kinds of classes: traditional and others that use computational methods in teaching. Applies quantitative evaluation methods using two operational research tools, multicriteria decision aid methods (mainly using the MACBETH approach) and data development analysis. (Author/YDS)
Students' Exploratory Thinking about a Nonroutine Calculus Task
ERIC Educational Resources Information Center
Nabb, Keith
2013-01-01
In this article on introductory calculus, intriguing questions are generated that can ignite an appreciation for the subject of mathematics. These questions open doors to advanced mathematical thinking and harness many elements of research-oriented mathematics. Such questions also offer greater incentives for students to think and reflect.…
Designing a Telescope Mirror for Second-Semester Calculus Students
ERIC Educational Resources Information Center
Marchand, Richard J.; Rogers, Robert R.; Parker, Andrew T.
2006-01-01
The purpose of this article is to present an interdisciplinary project, developed as a collaborative effort by the authors, involving the design of a telescope mirror as it was given to second semester calculus students. The goals of the project are to provide an applied setting for the topics typically covered in this type of course including the…
Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings
NASA Astrophysics Data System (ADS)
de León, Manuel; Jiménez, Fernando; Martín de Diego, David
2012-05-01
The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called Tulczyjew triples. The results are also extended to the case of discrete dynamics and nonholonomic mechanics. Interesting applications to the geometrical integration of Hamiltonian systems are obtained.
A Measurement Activity to Encourage Exploration of Calculus Concepts
ERIC Educational Resources Information Center
McGuffey, William
2015-01-01
This article describes an activity that incorporates measurement in order to lead students to discover and explore fundamental concepts of calculus. Students are provided with an experientially real starting point involving measurement and are encouraged to engage in mathematical discussions in a low-stakes environment. I describe the activity as…
The Vector Calculus Gap: Mathematics (Does Not Equal) Physics.
ERIC Educational Resources Information Center
Dray, Tevian; Manogue, Corinne A.
1999-01-01
Discusses some of the differences between the ways mathematicians and physicists view vector calculus and the gap between the way this material is traditionally taught by mathematicians and the way physicists use it. Suggests some ways to narrow the gap. (Author/ASK)
Development of Boolean calculus and its applications. [digital systems design
NASA Technical Reports Server (NTRS)
Tapia, M. A.
1980-01-01
The development of Boolean calculus for its application to developing digital system design methodologies that would reduce system complexity, size, cost, speed, power requirements, etc., is discussed. Synthesis procedures for logic circuits are examined particularly asynchronous circuits using clock triggered flip flops.
An Application of Calculus: Optimum Parabolic Path Problem
ERIC Educational Resources Information Center
Atasever, Merve; Pakdemirli, Mehmet; Yurtsever, Hasan Ali
2009-01-01
A practical and technological application of calculus problem is posed to motivate freshman students or junior high school students. A variable coefficient of friction is used in modelling air friction. The case in which the coefficient of friction is a decreasing function of altitude is considered. The optimum parabolic path for a flying object…
Contrasting Cases of Calculus Students' Understanding of Derivative Graphs
ERIC Educational Resources Information Center
Haciomeroglu, Erhan Selcuk; Aspinwall, Leslie; Presmeg, Norma C.
2010-01-01
This study adds momentum to the ongoing discussion clarifying the merits of visualization and analysis in mathematical thinking. Our goal was to gain understanding of three calculus students' mental processes and images used to create meaning for derivative graphs. We contrast the thinking processes of these three students as they attempted to…
Airfoil Design in Multivariable Calculus: Tying It All Together
ERIC Educational Resources Information Center
Laverty, Rich; Povich, Timothy; Williams, Tasha
2005-01-01
Near the conclusion of their final term in the calculus sequence at The United States Military Academy, cadets are given a week long group project. At the end of the week, the project is briefed to their instructors, classmates, and superior officers. From a teaching perspective, the goal is to encapsulate as much of the course as possible in one…
Factors Affecting Achievement in the First Course in Calculus.
ERIC Educational Resources Information Center
Edge, Orlyn P.; Friedberg, Stephen H.
1984-01-01
American College Testing Program (ACT) scores, high school rank, high school GPA, high school algebra grades, algebra pretest score, sex, birth order, family size, and high school size were used to predict course grades in the first semester of college calculus. The best predictors were the algebra pretest and high school rank. (Author/BW)
Inertial Navigation: A Bridge between Kinematics and Calculus
ERIC Educational Resources Information Center
Sadler, Philip M.; Garfield, Eliza N.; Tremblay, Alex; Sadler, Daniel J.
2012-01-01
Those who come to Cambridge soon learn that the fastest route between Harvard and MIT is by the subway. For many students, this short ride is a quick and easy way to link physics and calculus. A simple, homemade accelerometer provides all the instrumentation necessary to produce accurate graphs of acceleration, velocity, and displacement position…
Calculus Students' Understanding of Area and Volume Units
ERIC Educational Resources Information Center
Dorko, Allison; Speer, Natasha
2015-01-01
Units of measure are critical in many scientific fields. While instructors often note that students struggle with units, little research has been conducted about the nature and extent of these difficulties or why they exist. We investigated calculus students' unit use in area and volume computations. Seventy-three percent of students gave…
Developing Reasoning through Proof in High School Calculus
ERIC Educational Resources Information Center
Perrin, John Robert
2008-01-01
Developing students' ability to reason has long been a fundamental goal of mathematics education. A primary way in which mathematics students develop reasoning skills is by constructing mathematical proofs. This article presents a number of nontypical results, along with their proofs, that can be explored with students in any calculus classroom.…
An Investigation of Calculus Learning Using Factorial Modeling.
ERIC Educational Resources Information Center
Dick, Thomas P.; Balomenos, Richard H.
Structural covariance models that would explain the correlations observed among mathematics achievement and participation measures and related cognitive and affective variables were developed. A sample of college calculus students (N=268; 124 females and 144 males) was administered a battery of cognitive tests (including measures of spatial-visual…
Multivariate Limits and Continuity: A Survey of Calculus Textbooks.
ERIC Educational Resources Information Center
Thompson, Thomas M.; Wiggins, Kenneth L.
There has been much recent discussion concerning the content of the standard calculus course for students majoring in mathematics and the sciences. Some of this discussion has focused on the available textbooks. One weakness noted in some of these books involves the definitions of limit and continuity for functions of several variables. A…
Using the Pottery Wheel to Explore Topics in Calculus
ERIC Educational Resources Information Center
Farnell, Elin; Snipes, Marie A.
2015-01-01
Students sometimes struggle with visualizing the three-dimensional solids encountered in certain integral problems in a calculus class. We present a project in which students create solids of revolution with clay on a pottery wheel and estimate the volumes of these objects using Riemann sums. In addition to giving students an opportunity for…
Assessing college students' retention and transfer from calculus to physics
NASA Astrophysics Data System (ADS)
Cui, Lili
Many introductory calculus-based physics students have difficulties when solving physics problems involving calculus. This study investigates students' retention and transfer from calculus to physics. While retention is the ability to recall your knowledge at a later point in time, transfer of learning is defined as the ability to apply what one has learned in one situation to a different situation. In this dissertation we propose a theoretical framework to assess students' transfer of learning in the context of problem solving. We define two kinds of transfer---horizontal transfer and vertical transfer. Horizontal transfer involves applying previously learned ideas in a problem. Vertical transfer involves constructing new ideas to solve the problem. Students need to employ both horizontal and vertical transfer when they solve any problem. This framework evolves through this research and provides a lens that enables us to examine horizontal and vertical transfer. Additionally, this proposed framework offers researchers a vocabulary to describe and assess transfer of learning in any problem solving context. We use a combination of qualitative and quantitative methods to examine transfer in the context of problem solving. The participants in this study were students enrolled in a second-semester physics course taken by future engineers and physicists, calculus instructors and physics instructors. A total of 416 students' exam sheets were collected and reviewed. Statistical methods were used to analyze the quantitative data. A total of 28 students and nine instructors were interviewed. The video and audio recordings were transcribed and analyzed in light of the aforementioned theoretical framework. A major finding from this study is that a majority of students possess the requisite calculus skills, yet have several difficulties in applying them in the context of physics. These difficulties included: deciding the appropriate variable and limits of integration; not
Alternative probability theories for cognitive psychology.
Narens, Louis
2014-01-01
Various proposals for generalizing event spaces for probability functions have been put forth in the mathematical, scientific, and philosophic literatures. In cognitive psychology such generalizations are used for explaining puzzling results in decision theory and for modeling the influence of context effects. This commentary discusses proposals for generalizing probability theory to event spaces that are not necessarily boolean algebras. Two prominent examples are quantum probability theory, which is based on the set of closed subspaces of a Hilbert space, and topological probability theory, which is based on the set of open sets of a topology. Both have been applied to a variety of cognitive situations. This commentary focuses on how event space properties can influence probability concepts and impact cognitive modeling.
ERIC Educational Resources Information Center
Carlson, Marilyn P.; Smith, Nanci; Persson, Joni
2003-01-01
An overview of the conceptual underpinnings, reasoning abilities and notational issues related to learning the Fundamental Theorem of Calculus (FTC) is provided. Using this theoretical framework, curricular materials were developed to promote these understandings and reasoning abilities in students. Results from a study that investigated the…
ERIC Educational Resources Information Center
Dorko, Allison; Weber, Eric
2014-01-01
We analysed multivariable calculus students' meanings for domain and range and their generalisation of that meaning as they reasoned about the domain and range of multivariable functions. We found that students' thinking about domain and range fell into three broad categories: input/output, independence/dependence, and/or as attached to specific…
NASA Technical Reports Server (NTRS)
Soneira, R. M.; Bahcall, J. N.
1981-01-01
Probabilities are calculated for acquiring suitable guide stars (GS) with the fine guidance system (FGS) of the space telescope. A number of the considerations and techniques described are also relevant for other space astronomy missions. The constraints of the FGS are reviewed. The available data on bright star densities are summarized and a previous error in the literature is corrected. Separate analytic and Monte Carlo calculations of the probabilities are described. A simulation of space telescope pointing is carried out using the Weistrop north galactic pole catalog of bright stars. Sufficient information is presented so that the probabilities of acquisition can be estimated as a function of position in the sky. The probability of acquiring suitable guide stars is greatly increased if the FGS can allow an appreciable difference between the (bright) primary GS limiting magnitude and the (fainter) secondary GS limiting magnitude.
Quantum computing and probability.
Ferry, David K
2009-11-25
Over the past two decades, quantum computing has become a popular and promising approach to trying to solve computationally difficult problems. Missing in many descriptions of quantum computing is just how probability enters into the process. Here, we discuss some simple examples of how uncertainty and probability enter, and how this and the ideas of quantum computing challenge our interpretations of quantum mechanics. It is found that this uncertainty can lead to intrinsic decoherence, and this raises challenges for error correction.
Rationalizing Hybrid Earthquake Probabilities
NASA Astrophysics Data System (ADS)
Gomberg, J.; Reasenberg, P.; Beeler, N.; Cocco, M.; Belardinelli, M.
2003-12-01
An approach to including stress transfer and frictional effects in estimates of the probability of failure of a single fault affected by a nearby earthquake has been suggested in Stein et al. (1997). This `hybrid' approach combines conditional probabilities, which depend on the time elapsed since the last earthquake on the affected fault, with Poissonian probabilities that account for friction and depend only on the time since the perturbing earthquake. The latter are based on the seismicity rate change model developed by Dieterich (1994) to explain the temporal behavior of aftershock sequences in terms of rate-state frictional processes. The model assumes an infinite population of nucleation sites that are near failure at the time of the perturbing earthquake. In the hybrid approach, assuming the Dieterich model can lead to significant transient increases in failure probability. We explore some of the implications of applying the Dieterich model to a single fault and its impact on the hybrid probabilities. We present two interpretations that we believe can rationalize the use of the hybrid approach. In the first, a statistical distribution representing uncertainties in elapsed and/or mean recurrence time on the fault serves as a proxy for Dieterich's population of nucleation sites. In the second, we imagine a population of nucleation patches distributed over the fault with a distribution of maturities. In both cases we find that the probability depends on the time since the last earthquake. In particular, the size of the transient probability increase may only be significant for faults already close to failure. Neglecting the maturity of a fault may lead to overestimated rate and probability increases.
Asteroidal collision probabilities
NASA Astrophysics Data System (ADS)
Bottke, W. F.; Greenberg, R.
1993-05-01
Several past calculations of collision probabilities between pairs of bodies on independent orbits have yielded inconsistent results. We review the methodologies and identify their various problems. Greenberg's (1982) collision probability formalism (now with a corrected symmetry assumption) is equivalent to Wetherill's (1967) approach, except that it includes a way to avoid singularities near apsides. That method shows that the procedure by Namiki and Binzel (1991) was accurate for those cases where singularities did not arise.
Widesott, Lamberto; Pierelli, Alessio; Fiorino, Claudio; Lomax, Antony J.; Amichetti, Maurizio; Cozzarini, Cesare; Soukup, Martin; Schneider, Ralf; Hug, Eugen; Di Muzio, Nadia; Calandrino, Riccardo; Schwarz, Marco
2011-08-01
Purpose: To compare intensity-modulated proton therapy (IMPT) and helical tomotherapy (HT) treatment plans for high-risk prostate cancer (HRPCa) patients. Methods and Materials: The plans of 8 patients with HRPCa treated with HT were compared with IMPT plans with two quasilateral fields set up (-100{sup o}; 100{sup o}) and optimized with the Hyperion treatment planning system. Both techniques were optimized to simultaneously deliver 74.2 Gy/Gy relative biologic effectiveness (RBE) in 28 fractions on planning target volumes (PTVs)3-4 (P + proximal seminal vesicles), 65.5 Gy/Gy(RBE) on PTV2 (distal seminal vesicles and rectum/prostate overlapping), and 51.8 Gy/Gy(RBE) to PTV1 (pelvic lymph nodes). Normal tissue calculation probability (NTCP) calculations were performed for the rectum, and generalized equivalent uniform dose (gEUD) was estimated for the bowel cavity, penile bulb and bladder. Results: A slightly better PTV coverage and homogeneity of target dose distribution with IMPT was found: the percentage of PTV volume receiving {>=}95% of the prescribed dose (V{sub 95%}) was on average >97% in HT and >99% in IMPT. The conformity indexes were significantly lower for protons than for photons, and there was a statistically significant reduction of the IMPT dosimetric parameters, up to 50 Gy/Gy(RBE) for the rectum and bowel and 60 Gy/Gy(RBE) for the bladder. The NTCP values for the rectum were higher in HT for all the sets of parameters, but the gain was small and in only a few cases statistically significant. Conclusions: Comparable PTV coverage was observed. Based on NTCP calculation, IMPT is expected to allow a small reduction in rectal toxicity, and a significant dosimetric gain with IMPT, both in medium-dose and in low-dose range in all OARs, was observed.
Probabilities in implicit learning.
Tseng, Philip; Hsu, Tzu-Yu; Tzeng, Ovid J L; Hung, Daisy L; Juan, Chi-Hung
2011-01-01
The visual system possesses a remarkable ability in learning regularities from the environment. In the case of contextual cuing, predictive visual contexts such as spatial configurations are implicitly learned, retained, and used to facilitate visual search-all without one's subjective awareness and conscious effort. Here we investigated whether implicit learning and its facilitatory effects are sensitive to the statistical property of such implicit knowledge. In other words, are highly probable events learned better than less probable ones even when such learning is implicit? We systematically varied the frequencies of context repetition to alter the degrees of learning. Our results showed that search efficiency increased consistently as contextual probabilities increased. Thus, the visual contexts, along with their probability of occurrences, were both picked up by the visual system. Furthermore, even when the total number of exposures was held constant between each probability, the highest probability still enjoyed a greater cuing effect, suggesting that the temporal aspect of implicit learning is also an important factor to consider in addition to the effect of mere frequency. Together, these findings suggest that implicit learning, although bypassing observers' conscious encoding and retrieval effort, behaves much like explicit learning in the sense that its facilitatory effect also varies as a function of its associative strengths.
NASA Technical Reports Server (NTRS)
Bollenbacher, Gary; Guptill, James D.
1999-01-01
This report analyzes the probability of a launch vehicle colliding with one of the nearly 10,000 tracked objects orbiting the Earth, given that an object on a near-collision course with the launch vehicle has been identified. Knowledge of the probability of collision throughout the launch window can be used to avoid launching at times when the probability of collision is unacceptably high. The analysis in this report assumes that the positions of the orbiting objects and the launch vehicle can be predicted as a function of time and therefore that any tracked object which comes close to the launch vehicle can be identified. The analysis further assumes that the position uncertainty of the launch vehicle and the approaching space object can be described with position covariance matrices. With these and some additional simplifying assumptions, a closed-form solution is developed using two approaches. The solution shows that the probability of collision is a function of position uncertainties, the size of the two potentially colliding objects, and the nominal separation distance at the point of closest approach. ne impact of the simplifying assumptions on the accuracy of the final result is assessed and the application of the results to the Cassini mission, launched in October 1997, is described. Other factors that affect the probability of collision are also discussed. Finally, the report offers alternative approaches that can be used to evaluate the probability of collision.
Matrix approach to discrete fractional calculus II: Partial fractional differential equations
NASA Astrophysics Data System (ADS)
Podlubny, Igor; Chechkin, Aleksei; Skovranek, Tomas; Chen, YangQuan; Vinagre Jara, Blas M.
2009-05-01
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359-386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.
Fractional calculus transmutation for the Airy WKB solutions and Stokes phenomenon
NASA Astrophysics Data System (ADS)
Kiryakova, Virginia
2016-12-01
We apply the transmutation method to give a new explanation of the Stokes phenomenon for the Airy differential equation and of the change of the coeffcients in its asymptotic solutions for large values of argument in different parts of the complex plane. As a transmutation operator, a Weyl type fractional order integral is used. But this scheme is a special case of the so-called Poisson- Sonine-Dimovski transmutation operators related to the hyper-Bessel differential equations of arbitrary integer order, and of the generalized fractional calculus operators related to differential equations of fractional multi-order and their solutions, including a number of special functions. We analyze also the previous results of other authors and suggest some perspectives to use the same method in more general cases.
The perception of probability.
Gallistel, C R; Krishan, Monika; Liu, Ye; Miller, Reilly; Latham, Peter E
2014-01-01
We present a computational model to explain the results from experiments in which subjects estimate the hidden probability parameter of a stepwise nonstationary Bernoulli process outcome by outcome. The model captures the following results qualitatively and quantitatively, with only 2 free parameters: (a) Subjects do not update their estimate after each outcome; they step from one estimate to another at irregular intervals. (b) The joint distribution of step widths and heights cannot be explained on the assumption that a threshold amount of change must be exceeded in order for them to indicate a change in their perception. (c) The mapping of observed probability to the median perceived probability is the identity function over the full range of probabilities. (d) Precision (how close estimates are to the best possible estimate) is good and constant over the full range. (e) Subjects quickly detect substantial changes in the hidden probability parameter. (f) The perceived probability sometimes changes dramatically from one observation to the next. (g) Subjects sometimes have second thoughts about a previous change perception, after observing further outcomes. (h) The frequency with which they perceive changes moves in the direction of the true frequency over sessions. (Explaining this finding requires 2 additional parametric assumptions.) The model treats the perception of the current probability as a by-product of the construction of a compact encoding of the experienced sequence in terms of its change points. It illustrates the why and the how of intermittent Bayesian belief updating and retrospective revision in simple perception. It suggests a reinterpretation of findings in the recent literature on the neurobiology of decision making.
Levin-Plotnik, D; Hamilton, R J
2004-02-07
We find the dose distribution that maximizes the tumour control probability (TCP) for a fixed mean tumour dose per fraction. We consider a heterogeneous tumour volume having a radiation response characterized by the linear quadratic model with heterogeneous radiosensitivity and repopulation rate that may vary in time. Using variational calculus methods a general solution is obtained. We demonstrate the spatial dependence of the optimal dose distribution by explicitly evaluating the solution for different functional forms of the tumour properties. For homogeneous radiosensitivity and growth rate, we find that the dose distribution that maximizes TCP is homogeneous when the clonogen cell density is homogeneous, while for a heterogeneous initial tumour density we find that the first dose fraction is inhomogeneous, which homogenizes the clonogen cell density, and subsequent dose fractions are homogeneous. When the tumour properties have explicit spatial dependence, we show that the spatial variation of the optimized dose distribution is insensitive to the functional form. However, the dose distribution and tumour clonogen density are sensitive to the value of the repopulation rate. The optimized dose distribution yields a higher TCP than a typical clinical dose distribution or a homogeneous dose distribution.
Experimental Probability in Elementary School
ERIC Educational Resources Information Center
Andrew, Lane
2009-01-01
Concepts in probability can be more readily understood if students are first exposed to probability via experiment. Performing probability experiments encourages students to develop understandings of probability grounded in real events, as opposed to merely computing answers based on formulae.
Logic, Probability, and Human Reasoning
2015-01-01
logics developed in artificial intelligence, which allow conclusions to be withdrawn [38–42]. Second, conditional assertions (e.g., ‘If she insulted him...N. (2014) Probabilistic single function dual process theory and logic programming as approaches to non- monotonicity in human vs artificial reasoning...How can we solve this crisis? Leibniz dreamed of a calculus that settles any argument. Can cognitive scientists devise such a system? Feature
Time-dependent earthquake probabilities
Gomberg, J.; Belardinelli, M.E.; Cocco, M.; Reasenberg, P.
2005-01-01
We have attempted to provide a careful examination of a class of approaches for estimating the conditional probability of failure of a single large earthquake, particularly approaches that account for static stress perturbations to tectonic loading as in the approaches of Stein et al. (1997) and Hardebeck (2004). We have loading as in the framework based on a simple, generalized rate change formulation and applied it to these two approaches to show how they relate to one another. We also have attempted to show the connection between models of seismicity rate changes applied to (1) populations of independent faults as in background and aftershock seismicity and (2) changes in estimates of the conditional probability of failures of different members of a the notion of failure rate corresponds to successive failures of different members of a population of faults. The latter application requires specification of some probability distribution (density function of PDF) that describes some population of potential recurrence times. This PDF may reflect our imperfect knowledge of when past earthquakes have occurred on a fault (epistemic uncertainty), the true natural variability in failure times, or some combination of both. We suggest two end-member conceptual single-fault models that may explain natural variability in recurrence times and suggest how they might be distinguished observationally. When viewed deterministically, these single-fault patch models differ significantly in their physical attributes, and when faults are immature, they differ in their responses to stress perturbations. Estimates of conditional failure probabilities effectively integrate over a range of possible deterministic fault models, usually with ranges that correspond to mature faults. Thus conditional failure probability estimates usually should not differ significantly for these models. Copyright 2005 by the American Geophysical Union.
Carr, D.B.; Tolley, H.D.
1982-12-01
This paper investigates procedures for univariate nonparametric estimation of tail probabilities. Extrapolated values for tail probabilities beyond the data are also obtained based on the shape of the density in the tail. Several estimators which use exponential weighting are described. These are compared in a Monte Carlo study to nonweighted estimators, to the empirical cdf, to an integrated kernel, to a Fourier series estimate, to a penalized likelihood estimate and a maximum likelihood estimate. Selected weighted estimators are shown to compare favorably to many of these standard estimators for the sampling distributions investigated.
Numerical approaches to fractional calculus and fractional ordinary differential equation
NASA Astrophysics Data System (ADS)
Li, Changpin; Chen, An; Ye, Junjie
2011-05-01
Nowadays, fractional calculus are used to model various different phenomena in nature, but due to the non-local property of the fractional derivative, it still remains a lot of improvements in the present numerical approaches. In this paper, some new numerical approaches based on piecewise interpolation for fractional calculus, and some new improved approaches based on the Simpson method for the fractional differential equations are proposed. We use higher order piecewise interpolation polynomial to approximate the fractional integral and fractional derivatives, and use the Simpson method to design a higher order algorithm for the fractional differential equations. Error analyses and stability analyses are also given, and the numerical results show that these constructed numerical approaches are efficient.
A huge pelvic calculus causing acute renal failure.
Lai, Allen Yu-Hung; Kuo, Yuh-Chen
2008-02-01
We present a 69-year-old man with repeated urinary tract infection and lower abdominal pain. Kidney-ureter-bladder (KUB) scout film showed a huge, 320-g triangular pelvic calculus that was surgically removed with excellent results. Bladder stone is a common disease, but it is rare for such a calculus to be so large as to cause bilateral hydronephrosis. Surgical intervention by cystolithotomy or endoscopic cystolithotripsy can achieve satisfactory results. Bladder outlet obstruction should be treated simultaneously. Close follow-up, however, is mandatory because the recurrence of urolithiasis is high in those patients with voiding problems and recurrent urinary infection. To the best of our knowledge, this is the largest bladder stone in a human male. This case report also illustrates the importance of radiologic evaluation of patients with repeated urinary infections.
Calculus students' early concept images of tangent lines
NASA Astrophysics Data System (ADS)
Vincent, Brittany; LaRue, Renee; Sealey, Vicki; Engelke, Nicole
2015-07-01
This study explored first-semester calculus students' understanding of tangent lines as well as how students used tangent lines within the context of Newton's method. Task-based interviews were conducted with twelve first-semester calculus students who were asked to verbally describe a tangent line, sketch tangent lines for multiple curves, and use tangent lines within the context of Newton's method. We examined students' graphical illustrations and the language they used to describe tangent lines and identified six prominent categories that described students' concept images of tangent lines. Our data show that individual students often possessed multiple concept images of tangent lines, and often these multiple concept images were conflicting. Furthermore, students were usually willing to modify their concept images of tangent lines depending on the task presented to the students.
A Unifying Probability Example.
ERIC Educational Resources Information Center
Maruszewski, Richard F., Jr.
2002-01-01
Presents an example from probability and statistics that ties together several topics including the mean and variance of a discrete random variable, the binomial distribution and its particular mean and variance, the sum of independent random variables, the mean and variance of the sum, and the central limit theorem. Uses Excel to illustrate these…
ERIC Educational Resources Information Center
Varga, Tamas
This booklet resulted from a 1980 visit by the author, a Hungarian mathematics educator, to the Teachers' Center Project at Southern Illinois University at Edwardsville. Included are activities and problems that make probablility concepts accessible to young children. The topics considered are: two probability games; choosing two beads; matching…
Univariate Probability Distributions
ERIC Educational Resources Information Center
Leemis, Lawrence M.; Luckett, Daniel J.; Powell, Austin G.; Vermeer, Peter E.
2012-01-01
We describe a web-based interactive graphic that can be used as a resource in introductory classes in mathematical statistics. This interactive graphic presents 76 common univariate distributions and gives details on (a) various features of the distribution such as the functional form of the probability density function and cumulative distribution…
Approximating Integrals Using Probability
ERIC Educational Resources Information Center
Maruszewski, Richard F., Jr.; Caudle, Kyle A.
2005-01-01
As part of a discussion on Monte Carlo methods, which outlines how to use probability expectations to approximate the value of a definite integral. The purpose of this paper is to elaborate on this technique and then to show several examples using visual basic as a programming tool. It is an interesting method because it combines two branches of…
Example Solar Electric Propulsion System asteroid tours using variational calculus
NASA Technical Reports Server (NTRS)
Burrows, R. R.
1985-01-01
Exploration of the asteroid belt with a vehicle utilizing a Solar Electric Propulsion System has been proposed in past studies. Some of those studies illustrated multiple asteroid rendezvous with trajectories obtained using approximate methods. Most of the inadequacies of those approximations are overcome in this paper, which uses the calculus of variations to calculate the trajectories and associated payloads of four asteroid tours. The modeling, equations, and solution techniques are discussed, followed by a presentation of the results.
q-CALCULUS and the Discrete Inverse Scattering
NASA Astrophysics Data System (ADS)
Karlo, T.; Jacob, H.; Tripathy, K. C.
The discrete inverse scattering in one dimension has been re-identified with lattice calculus. By transforming the deformation parameter, the coordinate and the partial derivatives from lattice space to q-space, the Schrödinger equation with a potential is systematically analyzed. The potential having explicit q-dependence is derived from the knowledge of the spectral measure. The role of q to generate simulation is also highlighted.
A formalism for the calculus of variations with spinors
Bäckdahl, Thomas; Valiente Kroon, Juan A.
2016-02-15
We develop a frame and dyad gauge-independent formalism for the calculus of variations of functionals involving spinorial objects. As a part of this formalism, we define a modified variation operator which absorbs frame and spin dyad gauge terms. This formalism is applicable to both the standard spacetime (i.e., SL(2, ℂ)) 2-spinors as well as to space (i.e., SU(2, ℂ)) 2-spinors. We compute expressions for the variations of the connection and the curvature spinors.
Bunny hops: using multiplicities of zeroes in calculus for graphing
NASA Astrophysics Data System (ADS)
Miller, David; Deshler, Jessica M.; Hansen, Ryan
2016-07-01
Students learn a lot of material in each mathematics course they take. However, they are not always able to make meaningful connections between content in successive mathematics courses. This paper reports on a technique to address a common topic in calculus I courses (intervals of increase/decrease and concave up/down) while also making use of students' pre-existing knowledge about the behaviour of functions around zeroes based on multiplicities.
Multiple multiresolution representation of functions and calculus for fast computation
Fann, George I; Harrison, Robert J; Hill, Judith C; Jia, Jun; Galindo, Diego A
2010-01-01
We describe the mathematical representations, data structure and the implementation of the numerical calculus of functions in the software environment multiresolution analysis environment for scientific simulations, MADNESS. In MADNESS, each smooth function is represented using an adaptive pseudo-spectral expansion using the multiwavelet basis to a arbitrary but finite precision. This is an extension of the capabilities of most of the existing net, mesh and spectral based methods where the discretization is based on a single adaptive mesh, or expansions.
Generalized Volterra lattices: Binary Darboux transformations and self-consistent sources
NASA Astrophysics Data System (ADS)
Müller-Hoissen, F.; Chvartatskyi, O.; Toda, K.
2017-03-01
We study two families of matrix versions of generalized Volterra (or Bogoyavlensky) lattice equations. For each family, the equations arise as reductions of a partial differential-difference equation in one continuous and two discrete variables, which is a realization of a general integrable equation in bidifferential calculus. This allows to derive a binary Darboux transformation and also self-consistent source extensions via general results of bidifferential calculus. Exact solutions are constructed from the simplest seed solutions.
Some basic results on the sets of sequences with geometric calculus
NASA Astrophysics Data System (ADS)
Türkmen, Cengiz; Başar, Feyzi
2012-08-01
As an alternative to the classical calculus, Grossman and Katz [Non-Newtonian Calculus, Lee Press, Pigeon Cove, Massachusetts, 1972] introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and bigeometric calculus. Following Grossman and Katz, we construct the field C(G) of geometric complex numbers and the concept of geometric metric. Also we give the triangle and Minkowski's inequalities in the sense of geometric calculus. Later we respectively define the sets w(G), ℓ∞(G), c(G), c0(G) and ℓp(G) of all, bounded, convergent, null and p-absolutely summable sequences, in the sense of geometric calculus and show that each of the set forms a complete vector space on the field C(G).
A huge bladder calculus causing acute renal failure.
Komeya, Mitsuru; Sahoda, Tamami; Sugiura, Shinpei; Sawada, Takuto; Kitami, Kazuo
2013-02-01
A 81-year-old male was referred to our emergency outpatient unit due to acute renal failure. The level of serum creatinine was 276 μmol/l. A CT scan showed bilateral hydronephroureter, large bladder stone (7 cm × 6 cm × 6 cm) and bladder wall thickness. He was diagnosed as post renal failure due to bilateral hydronephroureter. Large bladder stone is thought to be the cause of bilateral hydronephroureter and renal failure. To improve renal failure, we performed open cystolithotomy and urethral catheterization. Three days after the surgery, the level of serum creatinine decreased to 224 μmol/l. He was discharged from our hospital with uneventful course. Bladder calculus is thought to be a rare cause of renal failure. We summarize the characteristics of bladder calculus causing renal failure. We should keep that long-term pyuria and urinary symptom, and repeated urinary tract infection can cause huge bladder calculus and renal failure in mind.
Probability summation--a critique.
Laming, Donald
2013-03-01
This Discussion Paper seeks to kill off probability summation, specifically the high-threshold assumption, as an explanatory idea in visual science. In combination with a Weibull function of a parameter of about 4, probability summation can accommodate, to within the limits of experimental error, the shape of the detectability function for contrast, the reduction in threshold that results from the combination of widely separated grating components, summation with respect to duration at threshold, and some instances, but not all, of spatial summation. But it has repeated difficulty with stimuli below threshold, because it denies the availability of input from such stimuli. All the phenomena listed above, and many more, can be accommodated equally accurately by signal-detection theory combined with an accelerated nonlinear transform of small, near-threshold, contrasts. This is illustrated with a transform that is the fourth power for the smallest contrasts, but tends to linear above threshold. Moreover, this particular transform can be derived from elementary properties of sensory neurons. Probability summation cannot be regarded as a special case of a more general theory, because it depends essentially on the 19th-century notion of a high fixed threshold. It is simply an obstruction to further progress.
White, D J
1997-10-01
Dental calculus, both supra- and subgingival occurs in the majority of adults worldwide. Dental calculus is calcified dental plaque, composed primarily of calcium phosphate mineral salts deposited between and within remnants of formerly viable microorganisms. A viable dental plaque covers mineralized calculus deposits. Levels of calculus and location of formation are population specific and are affected by oral hygiene habits, access to professional care, diet, age, ethnic origin, time since last dental cleaning, systemic disease and the use of prescription medications. In populations that practice regular oral hygiene and with access to regular professional care, supragingival dental calculus formation is restricted to tooth surfaces adjacent to the salivary ducts. Levels of supragingival calculus in these populations is minor and the calculus has little if any impact on oral-health. Subgingival calculus formation in these populations occurs coincident with periodontal disease (although the calculus itself appears to have little impact on attachment loss), the latter being correlated with dental plaque. In populations that do not practice regular hygiene and that do not have access to professional care, supragingival calculus occurs throughout the dentition and the extent of calculus formation can be extreme. In these populations, supragingival calculus is associated with the promotion of gingival recession. Subgingival calculus, in "low hygiene" populations, is extensive and is directly correlated with enhanced periodontal attachment loss. Despite extensive research, a complete understanding of the etiologic significance of subgingival calculus to periodontal disease remains elusive, due to inability to clearly differentiate effects of calculus versus "plaque on calculus". As a result, we are not entirely sure whether subgingival calculus is the cause or result of periodontal inflammation. Research suggests that subgingival calculus, at a minimum, may expand the
Lipschitz and Besov spaces in quantum calculus
NASA Astrophysics Data System (ADS)
Nemri, Akram; Selmi, Belgacem
2016-08-01
The purpose of this paper is to investigate the harmonic analysis on the time scale 𝕋q, q ∈ (0, 1) to introduce q-weighted Besov spaces subspaces of Lp(𝕋 q) generalizing the classical one. Further, using an example of q-weighted wα,β(.; q) which is introduced and studied. We give a new characterization of the q-Besov space using q-Poisson kernel and the g1 Littlewood-Paley operator.
NASA Astrophysics Data System (ADS)
von der Linden, Wolfgang; Dose, Volker; von Toussaint, Udo
2014-06-01
Preface; Part I. Introduction: 1. The meaning of probability; 2. Basic definitions; 3. Bayesian inference; 4. Combinatrics; 5. Random walks; 6. Limit theorems; 7. Continuous distributions; 8. The central limit theorem; 9. Poisson processes and waiting times; Part II. Assigning Probabilities: 10. Transformation invariance; 11. Maximum entropy; 12. Qualified maximum entropy; 13. Global smoothness; Part III. Parameter Estimation: 14. Bayesian parameter estimation; 15. Frequentist parameter estimation; 16. The Cramer-Rao inequality; Part IV. Testing Hypotheses: 17. The Bayesian way; 18. The frequentist way; 19. Sampling distributions; 20. Bayesian vs frequentist hypothesis tests; Part V. Real World Applications: 21. Regression; 22. Inconsistent data; 23. Unrecognized signal contributions; 24. Change point problems; 25. Function estimation; 26. Integral equations; 27. Model selection; 28. Bayesian experimental design; Part VI. Probabilistic Numerical Techniques: 29. Numerical integration; 30. Monte Carlo methods; 31. Nested sampling; Appendixes; References; Index.
Presymplectic current and the inverse problem of the calculus of variations
Khavkine, Igor
2013-11-15
The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when horizontally closed on solutions, allows us to construct a variational formulation for a subsystem of the given PDE. No constraints on the differential order or number of dependent or independent variables are assumed. The proof follows a recent observation of Bridges, Hydon, and Lawson [Math. Proc. Cambridge Philos. Soc. 148(01), 159–178 (2010)] and generalizes an older result of Henneaux [Ann. Phys. 140(1), 45–64 (1982)] from ordinary differential equations (ODEs) to PDEs. Uniqueness of the variational formulation is also discussed.
Efficient Probability Sequences
2014-08-18
Ungar (2014), to produce a distinct forecasting system. The system consists of the method for eliciting individual subjective forecasts together with...E. Stone, and L. H. Ungar (2014). Two reasons to make aggregated probability forecasts more extreme. Decision Analysis 11 (2), 133–145. Bickel, J. E...Letters 91 (3), 425–429. Mellers, B., L. Ungar , J. Baron, J. Ramos, B. Gurcay, K. Fincher, S. E. Scott, D. Moore, P. Atanasov, S. A. Swift, et al. (2014
1983-07-26
DeGroot , Morris H. Probability and Statistic. Addison-Wesley Publishing Company, Reading, Massachusetts, 1975. [Gillogly 78] Gillogly, J.J. Performance...distribution [ DeGroot 751 has just begun. The beta distribution has several features that might make it a more reasonable choice. As with the normal-based...1982. [Cooley 65] Cooley, J.M. and Tukey, J.W. An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19, 1965. [ DeGroot 75
Troutman, B.M.; Karlinger, M.R.
2003-01-01
The T-year annual maximum flood at a site is defined to be that streamflow, that has probability 1/T of being exceeded in any given year, and for a group of sites the corresponding regional flood probability (RFP) is the probability that at least one site will experience a T-year flood in any given year. The RFP depends on the number of sites of interest and on the spatial correlation of flows among the sites. We present a Monte Carlo method for obtaining the RFP and demonstrate that spatial correlation estimates used in this method may be obtained with rank transformed data and therefore that knowledge of the at-site peak flow distribution is not necessary. We examine the extent to which the estimates depend on specification of a parametric form for the spatial correlation function, which is known to be nonstationary for peak flows. It is shown in a simulation study that use of a stationary correlation function to compute RFPs yields satisfactory estimates for certain nonstationary processes. Application of asymptotic extreme value theory is examined, and a methodology for separating channel network and rainfall effects on RFPs is suggested. A case study is presented using peak flow data from the state of Washington. For 193 sites in the Puget Sound region it is estimated that a 100-year flood will occur on the average every 4,5 years.
NASA Astrophysics Data System (ADS)
Bolotin, S. V.; Kozlov, V. V.
2015-10-01
For non-autonomous Lagrangian systems we introduce the notion of a dynamically convex domain with respect to the Lagrangian. We establish the solubility of boundary-value problems in compact dynamically convex domains. If the Lagrangian is time-periodic, then such a domain contains a periodic trajectory. The proofs use the Hamilton principle and known tools of the calculus of variations in the large. Our general results are applied to Whitney's problem on the existence of motions of an inverted pendulum without falls.
Radvar, M; Creanor, S L; Gilmour, W H; Payne, A P; McGadey, J; Foye, R H; Whitters, C J; Kinane, D F
1995-01-01
The aim of this study was to evaluate the effects of Nd:YAG laser treatment on subgingival calculus, cementum and dentine, in vitro at different power settings and durations. The study included 2 experiments. In the 1st experiment, 32 extracted teeth with calculus were divided into 8 laser treatment groups. Each tooth was treated on 2, 3 or 4 sites. In the 2nd experiment, 3 extracted cementum covered teeth and 3 extracted root planed teeth with exposed dentine were selected. 1 surface of each tooth was subjected to 8 different laser treatments. In both experiments, all specimens were assessed using scanning electron microscopy. Micrographs were taken from each treated site at x 100 and x 750 magnifications. An arbitrary scale (from 0 to 3) was used to score the degree of damage caused by the laser. Generally, the laser caused greater damage on calculus than either cementum or dentine. Linear regression analysis showed that higher total energy input caused a greater mean damage score on calculus (R2 = 66%, p < 0.001). 3-way analysis of variance showed that for calculus, the power setting, number of pulses per second and the duration of exposure contributed independently to the mean damage score in an additive way. Cementum specimens were not affected by treatment 1 (50 mJ, 10 pps, 1 s), treatment 2 (50 mJ, 10 pps, 5 s), and treatment 5 (50 mJ, 20 pps, 1 s). Dentine specimens were not affected by treatment 1 (50 mJ, 10 pps, 1 s).(ABSTRACT TRUNCATED AT 250 WORDS)
Retrieve Tether Survival Probability
2007-11-02
cuts of the tether by meteorites and orbital debris , is calculated to be 99.934% for the planned experiment duration of six months or less. This is...due to the unlikely event of a strike by a large piece of orbital debris greater than 1 meter in size cutting all the lines of the tether at once. The...probability of the tether surviving multiple cuts by meteoroid and orbital debris impactors smaller than 5 cm in diameter is 99.9993% at six months
Stochastic calculus for uncoupled continuous-time random walks.
Germano, Guido; Politi, Mauro; Scalas, Enrico; Schilling, René L
2009-06-01
The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy alpha -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.
Stochastic calculus for uncoupled continuous-time random walks
NASA Astrophysics Data System (ADS)
Germano, Guido; Politi, Mauro; Scalas, Enrico; Schilling, René L.
2009-06-01
The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy α -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.
SUPERSYMMETRIC INSTANTON CALCULUS: Gauge theories with matter
NASA Astrophysics Data System (ADS)
Novikov, V. A.; Shifman, M. A.; Vainshtein, A. I.; Zakharov, V. I.
Within the framework of gauge SUSY theories we discuss correlation functions of the type (W2(x),S2(0)) where S is the chiral matter superfield (in the one-flavor model). SUSY implies that these correlation functions do not depend on coordinates and vanish identically in perturbation theory. We develop a technique for the systematic calculation of instanton effects. It is shown that even in the limit x → 0 the correlation functions at hand are not saturated by small-size instantons with radius ρ ˜ x; a contribution of the same order of magnitude comes from the instantons of characteristic size ρ ˜ l/v (v is the vacuum expectation value of the scalar field, and we concentrate on the models with v > Λ where Λ is the scale parameter fixing the running gauge coupling constant). If v > Λ both types of instantons can be consistently taken into account. The computational formalism proposed is explicitly supersymmetric and uses the language of instanton-associated superfields. We demonstrate, in particular, that one can proceed to a new variable, ρinv, which can be naturally considered as a supersymmetric generalization of the instanton radius. Unlike the ordinary radius ρ, this variable is invariant under the SUSY transformations. If one uses ρinv instead of ρ the expressions for the instanton contribution can be rewritten in the form saturated by the domain ρ2inv=0. The cluster decomposition as well as x-independence of the correlation functions considered turn out to be obvious in this formalism.
Alien calculus and non perturbative effects in Quantum Field Theory
NASA Astrophysics Data System (ADS)
Bellon, Marc P.
2016-12-01
In many domains of physics, methods for dealing with non-perturbative aspects are required. Here, I want to argue that a good approach for this is to work on the Borel transforms of the quantities of interest, the singularities of which give non-perturbative contributions. These singularities in many cases can be largely determined by using the alien calculus developed by Jean Écalle. My main example will be the two point function of a massless theory given as a solution of a renormalization group equation.
Using Case Studies in Calculus-based Physics
NASA Astrophysics Data System (ADS)
Katz, Debora M.
2006-12-01
Do your students believe that the physics only works in your classroom or laboratory? Or do they see that physics underlies their everyday experience? Case studies in physics help students connect physics principles to their everyday experience. For decades, case studies have been used to teach law, medicine and biology, but they are rarely used in physics. I am working on a calculus-based physics textbook for scientists and engineers. Case studies are woven into each chapter. Stop by and get a case study to test out in your classroom. I would love to get your feedback.
Spontaneous uretero-sigmoid fistula secondary to calculus
Marzouk, Ines; Moussa, Makram; Saadallah, Lotfi; Bouchoucha, Sami; Hendaoui, Lotfi
2016-01-01
A 25-year-old man was referred to the urology department after a subacute history of left back pain, burning micturition associated with pneumaturia and fecaluria. Ultrasonography was performed showing hydronephrosis, and plain film radiography demonstrated a long vertical left pelvic calculi. Uro-computed tomography (CT) combined with a water enema CT showed a 10 cm long calculus with the cranial extremity fistulating the sigmoidal wall. Surgical treatment included left nephroureterectomy and sigmoidectomy with a colorectal anastomosis. Postoperative course was uneventful. PMID:28096928
Simplifying the modal mu-calculus alternation hierarchy
NASA Astrophysics Data System (ADS)
Bradfield, J. C.
In [Bra96], the strictness of the modal mu-calculus alternation hierarchy was shown by transferring a hierarchy from arithmetic; the latter was a corollary of a deep and highly technical analysis of [Lub93]. In this paper, we show that the alternation hierarchy in arithmetic can be established by entirely elementary means; further, simple examples of strict alternation depth n formulae can be constructed, which in turn give very simple examples to separate the modal hierarchy. In addition, the winning strategy formulae of parity games are shown to be such examples.
Advanced Jones calculus for the classification of periodic metamaterials
Menzel, Christoph; Rockstuhl, Carsten; Lederer, Falk
2010-11-15
By relying on an advanced Jones calculus, we analyze the polarization properties of light upon propagation through metamaterial slabs in a comprehensive manner. Based on symmetry considerations, we show that all periodic metamaterials may be divided into five different classes only. It is shown that each class differently affects the polarization of the transmitted light and sustains different eigenmodes. We show how to deduce these five classes from symmetry considerations and provide a simple algorithm that can be applied to decide to which class a given metamaterial belongs by measuring only the transmitted intensities.
Symbol calculus and zeta-function regularized determinants
Kaynak, Burak Tevfik; Turgut, O. Teoman
2007-11-15
In this work, we use semigroup integral to evaluate zeta-function regularized determinants. This is especially powerful for nonpositive operators such as the Dirac operator. In order to understand fully the quantum effective action, one should know not only the potential term but also the leading kinetic term. In this purpose, we use the Weyl type of symbol calculus to evaluate the determinant as a derivative expansion. The technique is applied both to a spin-0 bosonic operator and to the Dirac operator coupled to a scalar field.
Calculus students' ability to solve geometric related-rates problems
NASA Astrophysics Data System (ADS)
Martin, Tami
2000-09-01
This study assessed the ability of university students enrolled in an introductory calculus course to solve related-rates problems set in geometric contexts. Students completed a problem-solving test and a test of performance on the individual steps involved in solving such problems. Each step was characterised as primarily relying on procedural knowledge or conceptual understanding. Results indicated that overall performance on the geometric related-rates problems was poor. The poorest performance was on steps linked to conceptual understanding, specifically steps involving the translation of prose to geometric and symbolic representations. Overall performance was most strongly related to performance on the procedural steps.
Teaching calculus using module based on cooperative learning strategy
NASA Astrophysics Data System (ADS)
Arbin, Norazman; Ghani, Sazelli Abdul; Hamzah, Firdaus Mohamad
2014-06-01
The purpose of the research is to evaluate the effectiveness of a module which utilizes the cooperative learning for teaching Calculus for limit, derivative and integral. The sample consists of 50 semester 1 students from the Science Programme (AT 16) Sultan Idris Education University. A set of questions of related topics (pre and post) has been used as an instrument to collect data. The data is analyzed using inferential statistics involving the paired sample t-test and the independent t-test. The result shows that students have positive inclination towards the modulein terms of understanding.
Computational approach to Thornley's problem by bivariate operational calculus
NASA Astrophysics Data System (ADS)
Bazhlekova, E.; Dimovski, I.
2012-10-01
Thornley's problem is an initial-boundary value problem with a nonlocal boundary condition for linear onedimensional reaction-diffusion equation, used as a mathematical model of spiral phyllotaxis in botany. Applying a bivariate operational calculus we find explicit representation of the solution, containing two convolution products of special solutions and the arbitrary initial and boundary functions. We use a non-classical convolution with respect to the space variable, extending in this way the classical Duhamel principle. The special solutions involved are represented in the form of fast convergent series. Numerical examples are considered to show the application of the present technique and to analyze the character of the solution.
A miniture spectrometer using color CCD and frame calculus technique
NASA Astrophysics Data System (ADS)
Wan, Wei; Zhang, Guoping; Chen, Minghong; Liu, Minmin
2005-01-01
A design of spectrometer is presented, which uses a holographic grating and a two-dimensional color CCD camera connected with PC via video format port. And in the image post-procession, a real-time frame calculus technique and a non-linear filter were applied to provider higher image quality and better resistant to background noise. With improved designed zoom mechanics, the device has a wide resolution dynamic range and high frequency, since it can gather more spectrum information than linear black-white CCD. The spectrum analysis experiments for water quality detection indicate that the device can meet variant requirements of analysis at low cost.
Incorporating Inquiry-Based Learning in the Calculus Sequence: A Most Challenging Endeavour
ERIC Educational Resources Information Center
McLoughlin, M. Padraig M. M.
2009-01-01
A course in the Calculus sequence is arguably the most difficult course in which inquiry-based learning (IBL) can be achieved with any degree of success within the curriculum in part due to: (1) the plethora of majors taking Calculus to which the sequence relates to their majors in what is considered an "applied" manner; and (2) the…
Assessment of Peer-Led Team Learning in Calculus I: A Five-Year Study
ERIC Educational Resources Information Center
Merkel, John Conrad; Brania, Abdelkrim
2015-01-01
This five-year study of the peer-led team learning (PLTL) paradigm examined its implementation in a Calculus I course at an all-male HBCU institution. For this study we set up a strong control group and measured the effect of PLTL in the teaching and learning of Calculus I through two points of measure: retention and success rates and learning…
Analysis of Errors and Misconceptions in the Learning of Calculus by Undergraduate Students
ERIC Educational Resources Information Center
Muzangwa, Jonatan; Chifamba, Peter
2012-01-01
This paper is going to analyse errors and misconceptions in an undergraduate course in Calculus. The study will be based on a group of 10 BEd. Mathematics students at Great Zimbabwe University. Data is gathered through use of two exercises on Calculus 1&2.The analysis of the results from the tests showed that a majority of the errors were due…
The Role of Cognitive Ability and Preferred Mode of Processing in Students' Calculus Performance
ERIC Educational Resources Information Center
Haciomeroglu, Erhan Selcuk
2015-01-01
The present study sought to design calculus tasks to determine students' preference for visual or analytic processing as well as examine the role of preferred mode of processing in calculus performance and its relationship to spatial ability and verbal-logical reasoning ability. Data were collected from 150 high school students who were enrolled…
Using Dynamic Tools to Develop an Understanding of the Fundamental Ideas of Calculus
ERIC Educational Resources Information Center
Verzosa, Debbie; Guzon, Angela Fatima; De Las Peñas, Ma. Louise Antonette N.
2014-01-01
Although dynamic geometry software has been extensively used for teaching calculus concepts, few studies have documented how these dynamic tools may be used for teaching the rigorous foundations of the calculus. In this paper, we describe lesson sequences utilizing dynamic tools for teaching the epsilon-delta definition of the limit and the…
Improving Student Learning of Calculus Topics via Modified Just-in-Time Teaching Methods
ERIC Educational Resources Information Center
Natarajan, Rekha; Bennett, Andrew
2014-01-01
Although the use of traditional just-in-time teaching techniques has long been viewed positively by students and instructors in undergraduate calculus courses, past studies in this area have not addressed gains in student achievement with respect to specific calculus topics. This paper investigates the latter by administering modified just-in-time…
Factors Associated with Success in a Calculus Course: An Examination of Personal Variables
ERIC Educational Resources Information Center
Ubuz, Behiye
2011-01-01
This study examined relationships between students' personal variables (gender, prior achievements, age and academic major) and their success in the first year undergraduate calculus course. The study sample consisted of 59 first year undergraduate students taking Math 154 Calculus II course. A written test about integral, sequence and series…
The Association of Precollege Use of Calculators with Student Performance in College Calculus
ERIC Educational Resources Information Center
Mao, Yi; White, Tyreke; Sadler, Philip M.; Sonnert, Gerhard
2017-01-01
This study investigates how the use of calculators during high school mathematics courses is associated with student performance in introductory college calculus courses in the USA. Data were drawn from a nationally representative sample of 7087 students enrolled in college calculus at 134 colleges and universities. They included information about…
A Decade of Teaching "Reform Calculus" Has Been a Disaster, Critics Charge.
ERIC Educational Resources Information Center
Wilson, Robin
1997-01-01
A decade after mathematicians began a crusade to make calculus more relevant to undergraduate students ("Reform Calculus"), a backlash threatens to derail the effort and divide the profession. Critics charge the movement has watered down mathematics courses, teaching only superficial use of skills. The debate is fierce and the stakes…
A Preliminary Evaluation of Student Preparation for the Study of Calculus.
ERIC Educational Resources Information Center
Henderson, Ronald W.; Landesman, Edward M.
This report explores the student background characteristics that might be associated with success or failure in calculus and evaluates the effectiveness of remedial mathematics education. The sample consisted of two groups at the University of California, Santa Cruz: (1) all students (105) who took first quarter calculus in spring, 1985; and (2)…
Calculus structure on the Lie conformal algebra complex and the variational complex
De Sole, Alberto; Hekmati, Pedram; Kac, Victor G.
2011-05-15
We construct a calculus structure on the Lie conformal algebra cochain complex. By restricting to degree one chains, we recover the structure of a g-complex introduced in [A. De Sole and V. G. Kac, Commun. Math. Phys. 292, 667 (2009)]. A special case of this construction is the variational calculus, for which we provide explicit formulas.
Students' Conceptual Knowledge of Limits in Calculus: A Two-Part Constructivist Case Study
ERIC Educational Resources Information Center
Adams, Margaret Smolinka
2013-01-01
This case study investigated students' conceptual knowledge of limits in calculus by implementing semi-structured interviews. The constructivist learning principles of Piaget and Inhelder as well as theories of understanding by Skemp guided the study. In Phase I, a pilot study was conducted with 15 students from a Calculus III class. By using…
ERIC Educational Resources Information Center
Hare, Angela; Phillippy, Doug
2004-01-01
A program on calculus is conducted, which helps students learn about inherent differentiation through a study of mathematical functions, while simultaneously reinforcing their understanding of functional concepts. This process develops their mathematical experience in the field of calculus and in other advanced quantitative programs.
Student Achievement in College Calculus, Louisiana State University 1967-1968.
ERIC Educational Resources Information Center
Scannicchio, Thomas Henry
An investigation of freshmen achievement in an introductory calculus course was performed on the basis of high school mathematics background to find predictors of college calculus grades. Overall high school academic achievement, overall high school mathematics achievement, number of high school mathematics units, pattern of college preparatory…
Tablet PC: A Preliminary Report on a Tool for Teaching Calculus
ERIC Educational Resources Information Center
Gorgievski, Nicholas; Stroud, Robert; Truxaw, Mary; DeFranco, Thomas
2005-01-01
This study examined students' perceptions of the Tablet PC as an instructional tool for teaching Calculus. A thirteen item survey was developed by the researchers and administered to 103 students in an introductory Calculus course at a large university in the Northeast of the United States. The purpose of this survey was to collect data regarding…
Analyzing Conceptual Gains in Introductory Calculus with Interactively-Engaged Teaching Styles
ERIC Educational Resources Information Center
Thomas, Matthew
2013-01-01
This dissertation examines the relationship between an instructional style called Interactive-Engagement (IE) and gains on a measure of conceptual knowledge called the Calculus Concept Inventory (CCI). The data comes from two semesters of introductory calculus courses (Fall 2010 and Spring 2011), consisting of a total of 482 students from the…
ERIC Educational Resources Information Center
Yaacob, Yuzita; Wester, Michael; Steinberg, Stanly
2010-01-01
This paper presents a prototype of a computer learning assistant ILMEV (Interactive Learning-Mathematica Enhanced Vector calculus) package with the purpose of helping students to understand the theory and applications of integration in vector calculus. The main problem for students using Mathematica is to convert a textbook description of a…
The Introduction of Calculus in 12th Grade: The Role of Artefacts
ERIC Educational Resources Information Center
Maschietto, Michela
2004-01-01
The paper concerns the analysis of the role of artefacts and instruments in approaching calculus by graphic-symbolic calculator at high school level. We focus on an element of the introduction of calculus: the global/local game. We discus the hypothesis that the zoom-controls of calculator support the production of gestures and metaphors that…
Preparatory Year Program Courses as Predictors of First Calculus Course Grade
ERIC Educational Resources Information Center
Yushau, B; Omar, M. H
2007-01-01
This study investigates the effect of the preparatory year program courses on the first calculus course (Calculus I) at King Fahd University of Petroleum and Minerals (KFUPM). The data consists of more than 2,000 bilingual Arab university students studying in the English language, tracked over seven semesters. These students represent over 70% of…
An Excel-Aided Method for Teaching Calculus-Based Business Mathematics
ERIC Educational Resources Information Center
Liang, Jiajuan; Martin, Linda
2008-01-01
Calculus-based business mathematics is a required quantitative course for undergraduate business students in most AACSB accredited schools or colleges of business. Many business students, however, have relatively weak mathematical background or even display math-phobia when presented with calculus problems. Because of the popularity of Excel, its…
What Does It Mean for a Student to Understand the First-Year Calculus? Perspectives of 24 Experts
ERIC Educational Resources Information Center
Sofronas, Kimberly S.; DeFranco, Thomas C.; Vinsonhaler, Charles; Gorgievski, Nicholas; Schroeder, Larissa; Hamelin, Chris
2011-01-01
This article presents the views of 24 nationally recognized authorities in the field of mathematics, and in particular the calculus, on student understanding of the first-year calculus. A framework emerged that includes four overarching end goals for understanding of the first-year calculus: (a) mastery of the fundamental concepts and-or skills of…
A giant dumbbell shaped vesico-prostatic urethral calculus: a case report and review of literature.
Prabhuswamy, Vinod Kumar; Tiwari, Rahul; Krishnamoorthy, Ramakrishnan
2013-01-01
Calculi in the urethra are an uncommon entity. Giant calculi in prostatic urethra are extremely rare. The decision about treatment strategy of calculi depends upon the size, shape, and position of the calculus and 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 of the previous reported cases, giant calculi were extracted via the transvesical approach and external urethrotomy. A 38-year-old male patient presented with complaints of lower urinary tract symptoms. Further investigations showed a giant urethral calculus secondary to stricture of bulbo-membranous part of the urethra. Surgical removal of calculus was done via transvesical approach. Two calculi were found and extracted. One was a huge dumbbell calculus and the other was a smaller round calculus. This case was reported because of the rare size and the dumbbell nature of the stone. Giant urethral calculi are better managed by open surgery.
An introductory analysis of satellite collision probabilities
NASA Astrophysics Data System (ADS)
Carlton-Wippern, Kitt C.
This paper addresses a probailistic approach in assessing the probabilities of a satellite collision occurring due to relative trajectory analyses and probability density functions representing the satellites' position/momentum vectors. The paper is divided into 2 parts: Static and Dynamic Collision Probabilities. In the Static Collision Probability section, the basic phenomenon under study is: given the mean positions and associated position probability density functions for the two objects, calculate the probability that the two objects collide (defined as being within some distance of each other). The paper presents the classic Laplace problem of the probability of arrival, using standard uniform distribution functions. This problem is then extrapolated to show how 'arrival' can be classified as 'collision', how the arrival space geometries map to collision space geometries and how arbitrary position density functions can then be included and integrated into the analysis. In the Dynamic Collision Probability section, the nature of collisions based upon both trajectory and energy considerations is discussed, and that energy states alone cannot be used to completely describe whether or not a collision occurs. This fact invalidates some earlier work on the subject and demonstrates why Liouville's theorem cannot be used in general to describe the constant density of the position/momentum space in which a collision may occur. Future position probability density functions are then shown to be the convolution of the current position and momentum density functions (linear analysis), and the paper further demonstrates the dependency of the future position density functions on time. Strategies for assessing the collision probabilities for two point masses with uncertainties in position and momentum at some given time, and thes integrated with some arbitrary impact volume schema, are then discussed. This presentation concludes with the formulation of a high level design
Direct evidence of milk consumption from ancient human dental calculus
Warinner, C.; Hendy, J.; Speller, C.; Cappellini, E.; Fischer, R.; Trachsel, C.; Arneborg, J.; Lynnerup, N.; Craig, O. E.; Swallow, D. M.; Fotakis, A.; Christensen, R. J.; Olsen, J. V.; Liebert, A.; Montalva, N.; Fiddyment, S.; Charlton, S.; Mackie, M.; Canci, A.; Bouwman, A.; Rühli, F.; Gilbert, M. T. P.; Collins, M. J.
2014-01-01
Milk is a major food of global economic importance, and its consumption is regarded as a classic example of gene-culture evolution. Humans have exploited animal milk as a food resource for at least 8500 years, but the origins, spread, and scale of dairying remain poorly understood. Indirect lines of evidence, such as lipid isotopic ratios of pottery residues, faunal mortality profiles, and lactase persistence allele frequencies, provide a partial picture of this process; however, in order to understand how, where, and when humans consumed milk products, it is necessary to link evidence of consumption directly to individuals and their dairy livestock. Here we report the first direct evidence of milk consumption, the whey protein β-lactoglobulin (BLG), preserved in human dental calculus from the Bronze Age (ca. 3000 BCE) to the present day. Using protein tandem mass spectrometry, we demonstrate that BLG is a species-specific biomarker of dairy consumption, and we identify individuals consuming cattle, sheep, and goat milk products in the archaeological record. We then apply this method to human dental calculus from Greenland's medieval Norse colonies, and report a decline of this biomarker leading up to the abandonment of the Norse Greenland colonies in the 15th century CE. PMID:25429530
Direct evidence of milk consumption from ancient human dental calculus.
Warinner, C; Hendy, J; Speller, C; Cappellini, E; Fischer, R; Trachsel, C; Arneborg, J; Lynnerup, N; Craig, O E; Swallow, D M; Fotakis, A; Christensen, R J; Olsen, J V; Liebert, A; Montalva, N; Fiddyment, S; Charlton, S; Mackie, M; Canci, A; Bouwman, A; Rühli, F; Gilbert, M T P; Collins, M J
2014-11-27
Milk is a major food of global economic importance, and its consumption is regarded as a classic example of gene-culture evolution. Humans have exploited animal milk as a food resource for at least 8500 years, but the origins, spread, and scale of dairying remain poorly understood. Indirect lines of evidence, such as lipid isotopic ratios of pottery residues, faunal mortality profiles, and lactase persistence allele frequencies, provide a partial picture of this process; however, in order to understand how, where, and when humans consumed milk products, it is necessary to link evidence of consumption directly to individuals and their dairy livestock. Here we report the first direct evidence of milk consumption, the whey protein β-lactoglobulin (BLG), preserved in human dental calculus from the Bronze Age (ca. 3000 BCE) to the present day. Using protein tandem mass spectrometry, we demonstrate that BLG is a species-specific biomarker of dairy consumption, and we identify individuals consuming cattle, sheep, and goat milk products in the archaeological record. We then apply this method to human dental calculus from Greenland's medieval Norse colonies, and report a decline of this biomarker leading up to the abandonment of the Norse Greenland colonies in the 15(th) century CE.
On the construction of unitary quantum group differential calculus
NASA Astrophysics Data System (ADS)
Pyatov, Pavel
2016-10-01
We develop a construction of the unitary type anti-involution for the quantized differential calculus over {{GL}}q(n) in the case | q| =1. To this end, we consider a joint associative algebra of quantized functions, differential forms and Lie derivatives over {{GL}}q(n)/{{SL}}q(n), which is bicovariant with respect to {{GL}}q(n)/{{SL}}q(n) coactions. We define a specific non-central spectral extension of this algebra by the spectral variables of three matrices of the algebra generators. In the spectrally expended algebra, we construct a three-parametric family of its inner automorphisms. These automorphisms are used for the construction of the unitary anti-involution for the (spectrally extended) calculus over {{GL}}q(n). This work has been funded by the Russian Academic Excellence Project ‘5-100’. The results of section 5 (propositions 5.2, 5.3 and theorem 5.5) have been obtained under support of the RSF grant No.16-11-10160.
Inclusion probability with dropout: an operational formula.
Milot, E; Courteau, J; Crispino, F; Mailly, F
2015-05-01
In forensic genetics, a mixture of two or more contributors to a DNA profile is often interpreted using the inclusion probabilities theory. In this paper, we present a general formula for estimating the probability of inclusion (PI, also known as the RMNE probability) from a subset of visible alleles when dropouts are possible. This one-locus formula can easily be extended to multiple loci using the cumulative probability of inclusion. We show that an exact formulation requires fixing the number of contributors, hence to slightly modify the classic interpretation of the PI. We discuss the implications of our results for the enduring debate over the use of PI vs likelihood ratio approaches within the context of low template amplifications.
ERIC Educational Resources Information Center
Benakli, Nadia; Kostadinov, Boyan; Satyanarayana, Ashwin; Singh, Satyanand
2017-01-01
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…
ERIC Educational Resources Information Center
St. John, Michael M.; Babo, Gerard
2015-01-01
This study examined the influence of placement in a co-taught inclusive classroom on the academic achievement of general education students in grades 6-8 in a suburban New York school district on the 2014 New York State ELA and Mathematics Assessments. Propensity Score Matching (PSM) was utilized for sample selection in order to simulate a more…
Posterior Probabilities for a Consensus Ordering.
ERIC Educational Resources Information Center
Fligner, Michael A.; Verducci, Joseph S.
1990-01-01
The concept of consensus ordering is defined, and formulas for exact and approximate posterior probabilities for consensus ordering are developed under the assumption of a generalized Mallows' model with a diffuse conjugate prior. These methods are applied to a data set concerning 98 college students. (SLD)
ERIC Educational Resources Information Center
Falk, Ruma; Kendig, Keith
2013-01-01
Two contestants debate the notorious probability problem of the sex of the second child. The conclusions boil down to explication of the underlying scenarios and assumptions. Basic principles of probability theory are highlighted.
Coherent Assessment of Subjective Probability
1981-03-01
known results of de Finetti (1937, 1972, 1974), Smith (1961), and Savage (1971) and some recent results of Lind- ley (1980) concerning the use of...provides the motivation for de Finettis definition of subjective probabilities as coherent bet prices. From the definition of the probability measure...subjective probability, the probability laws which are traditionally stated as axioms or definitions are obtained instead as theorems. (De Finetti F -7
Unusual case of calculus in floor of mouth: a case report.
Bahadure, Rakesh N; Thosar, Nilima; Jain, Eesha S
2012-09-01
Calculus consists of mineralized bacterial plaque that forms on the surfaces of natural teeth. It is supragingival or subgingival depending upon its relation with gingival margin. The two most common locations for supragingival calculus are the buccal surfaces of maxillary molars and lingual surfaces of mandibular anterior teeth. It is very important to rule out the predisposing factor for calculus formation. In the present case of an 11-year- old female child, 1.2 × 1.5 cm large indurated mass suggestive of calculus in the left side of floor of mouth was observed. After surgical removal, along with indurated mass, an embedded root fragment was seen. Biochemical analysis of the specimen detected the calcium and phosphate ions approximately equals to the level in calculus. Thus, we diagnosed it as a calculus. Oral hygiene instructions and regular follow-up was advised. How to cite this article: Bahadure RN, Thosar N, Jain ES. Unusual Case of Calculus in Floor of Mouth: A Case Report. Int J Clin Pediatr Dent 2012;5(3):223-225.
ERIC Educational Resources Information Center
Lindstrom, Peter A.; And Others
This document consists of four units. The first of these views calculus applications to work, area, and distance problems. It is designed to help students gain experience in: 1) computing limits of Riemann sums; 2) computing definite integrals; and 3) solving elementary area, distance, and work problems by integration. The second module views…
Successful enrichment and recovery of whole mitochondrial genomes from ancient human dental calculus
Ozga, Andrew T.; Nieves‐Colón, Maria A.; Honap, Tanvi P.; Sankaranarayanan, Krithivasan; Hofman, Courtney A.; Milner, George R.; Lewis, Cecil M.; Stone, Anne C.
2016-01-01
ABSTRACT Objectives Archaeological dental calculus is a rich source of host‐associated biomolecules. Importantly, however, dental calculus is more accurately described as a calcified microbial biofilm than a host tissue. As such, concerns regarding destructive analysis of human remains may not apply as strongly to dental calculus, opening the possibility of obtaining human health and ancestry information from dental calculus in cases where destructive analysis of conventional skeletal remains is not permitted. Here we investigate the preservation of human mitochondrial DNA (mtDNA) in archaeological dental calculus and its potential for full mitochondrial genome (mitogenome) reconstruction in maternal lineage ancestry analysis. Materials and Methods Extracted DNA from six individuals at the 700‐year‐old Norris Farms #36 cemetery in Illinois was enriched for mtDNA using in‐solution capture techniques, followed by Illumina high‐throughput sequencing. Results Full mitogenomes (7–34×) were successfully reconstructed from dental calculus for all six individuals, including three individuals who had previously tested negative for DNA preservation in bone using conventional PCR techniques. Mitochondrial haplogroup assignments were consistent with previously published findings, and additional comparative analysis of paired dental calculus and dentine from two individuals yielded equivalent haplotype results. All dental calculus samples exhibited damage patterns consistent with ancient DNA, and mitochondrial sequences were estimated to be 92–100% endogenous. DNA polymerase choice was found to impact error rates in downstream sequence analysis, but these effects can be mitigated by greater sequencing depth. Discussion Dental calculus is a viable alternative source of human DNA that can be used to reconstruct full mitogenomes from archaeological remains. Am J Phys Anthropol 160:220–228, 2016. © 2016 The Authors American Journal of Physical Anthropology
Probabilities of transversions and transitions.
Vol'kenshtein, M V
1976-01-01
The values of the mean relative probabilities of transversions and transitions have been refined on the basis of the data collected by Jukes and found to be equal to 0.34 and 0.66, respectively. Evolutionary factors increase the probability of transversions to 0.44. The relative probabilities of individual substitutions have been determined, and a detailed classification of the nonsense mutations has been given. Such mutations are especially probable in the UGG (Trp) codon. The highest probability of AG, GA transitions correlates with the lowest mean change in the hydrophobic nature of the amino acids coded.
Primary vaginal calculus in a middle-aged woman with mental and physical disabilities.
Ikeda, Yuji; Oda, Katsutoshi; Matsuzawa, Naoki; Shimizu, Ken
2013-07-01
Vaginal calculi are rarely encountered and are often misdiagnosed as bladder calculi because of the difficulty in achieving an appropriate diagnosis. Most vaginal calculi result from the presence of a urethrovaginal fistula; those occurring in the absence of such fistulas are extremely rare. We present a case of a 42-year-old bedridden woman with mental and physical disabilities who had been misdiagnosed for a decade as having a bladder calculus. We removed the calculus nonsurgically and the analyzed the components. Results demonstrated the presence of a primary vaginal calculus. Vaginal calculi may occasionally occur in disabled women, but further investigation of the etiology of such calculi is required.
Factors associated with success in a calculus course: an examination of personal variables
NASA Astrophysics Data System (ADS)
Ubuz, Behiye
2011-01-01
This study examined relationships between students' personal variables (gender, prior achievements, age and academic major) and their success in the first year undergraduate calculus course. The study sample consisted of 59 first year undergraduate students taking Math 154 Calculus II course. A written test about integral, sequence and series including demographic survey items was used to gather data. The test was administered prior to and upon the completion of the calculus course. Multiple regression analysis result indicated that there is relationship between students' personal variables (gender and prior achievements) and their success. Gender differences favouring males typically occurred on Riemann sum and Riemann integral.
Effect of Imperfections on the Collapse of Rectangular Plates Using Variational Calculus.
1979-12-01
UCLASSIFIEO AFI/AE/AA/79D014 ru oIN280 11 AFIT/GAE/AA/7 9D-l 4 EFFECT OF IMPERFECTIONS ON THE COLLAPSE OF RECTANGULAR KLATES USING VARIATIONAL CALCULUS , THESIS...Imperfections on the Collapse of Rectangular Plates Using Variational Calculus THESIS Presented to the Faculty of the School of Engineering of the Air...PLATES USING VARIATIONAL CALCULUS I. Introduction Background The first problems in elastic instabi. ty were solved over 200 years ago by Euler (1
Econometricians, your probabilities are close to their end. Economists, now look for a new world!
Brian, Éric
2015-01-01
The article examines several stochastic layers of epistemic reasoning at work in econometrics and in the current economic methods: (1) the argumentative level; (2) the reasoning based on the analogy with gambling; (3) the models based on analytical calculation of probabilities, where the phenomenon is held centered, its uncertainty being controlled by white noise as to its fluctuation; (4) the axiomatic calculus. Entanglement of these strata is observed. The article calls for reflexion on the topic. Doing so, it introduces the concept of stochastic cognitive artifacts.
Constructor theory of probability
2016-01-01
Unitary quantum theory, having no Born Rule, is non-probabilistic. Hence the notorious problem of reconciling it with the unpredictability and appearance of stochasticity in quantum measurements. Generalizing and improving upon the so-called ‘decision-theoretic approach’, I shall recast that problem in the recently proposed constructor theory of information—where quantum theory is represented as one of a class of superinformation theories, which are local, non-probabilistic theories conforming to certain constructor-theoretic conditions. I prove that the unpredictability of measurement outcomes (to which constructor theory gives an exact meaning) necessarily arises in superinformation theories. Then I explain how the appearance of stochasticity in (finitely many) repeated measurements can arise under superinformation theories. And I establish sufficient conditions for a superinformation theory to inform decisions (made under it) as if it were probabilistic, via a Deutsch–Wallace-type argument—thus defining a class of decision-supporting superinformation theories. This broadens the domain of applicability of that argument to cover constructor-theory compliant theories. In addition, in this version some of the argument's assumptions, previously construed as merely decision-theoretic, follow from physical properties expressed by constructor-theoretic principles. PMID:27616914
Constructor theory of probability.
Marletto, Chiara
2016-08-01
Unitary quantum theory, having no Born Rule, is non-probabilistic. Hence the notorious problem of reconciling it with the unpredictability and appearance of stochasticity in quantum measurements. Generalizing and improving upon the so-called 'decision-theoretic approach', I shall recast that problem in the recently proposed constructor theory of information-where quantum theory is represented as one of a class of superinformation theories, which are local, non-probabilistic theories conforming to certain constructor-theoretic conditions. I prove that the unpredictability of measurement outcomes (to which constructor theory gives an exact meaning) necessarily arises in superinformation theories. Then I explain how the appearance of stochasticity in (finitely many) repeated measurements can arise under superinformation theories. And I establish sufficient conditions for a superinformation theory to inform decisions (made under it) as if it were probabilistic, via a Deutsch-Wallace-type argument-thus defining a class of decision-supporting superinformation theories. This broadens the domain of applicability of that argument to cover constructor-theory compliant theories. In addition, in this version some of the argument's assumptions, previously construed as merely decision-theoretic, follow from physical properties expressed by constructor-theoretic principles.
The chain collocation method: A spectrally accurate calculus of forms
NASA Astrophysics Data System (ADS)
Rufat, Dzhelil; Mason, Gemma; Mullen, Patrick; Desbrun, Mathieu
2014-01-01
Preserving in the discrete realm the underlying geometric, topological, and algebraic structures at stake in partial differential equations has proven to be a fruitful guiding principle for numerical methods in a variety of fields such as elasticity, electromagnetism, or fluid mechanics. However, structure-preserving methods have traditionally used spaces of piecewise polynomial basis functions for differential forms. Yet, in many problems where solutions are smoothly varying in space, a spectral numerical treatment is called for. In an effort to provide structure-preserving numerical tools with spectral accuracy on logically rectangular grids over periodic or bounded domains, we present a spectral extension of the discrete exterior calculus (DEC), with resulting computational tools extending well-known collocation-based spectral methods. Its efficient implementation using fast Fourier transforms is provided as well.
A fractional calculus model of anomalous dispersion of acoustic waves.
Wharmby, Andrew W
2016-09-01
An empirical formula based on viscoelastic analysis techniques that employs concepts from the fractional calculus that was used to model the dielectric behavior of materials exposed to oscillating electromagnetic fields in the radiofrequency, terahertz, and infrared bands. This work adapts and applies the formula to model viscoelastic behavior of materials that show an apparent increase of phase velocity of vibration with an increase in frequency, otherwise known as anomalous dispersion. A fractional order wave equation is derived through the application of the classic elastic-viscoelastic correspondence principle whose analytical solution is used to describe absorption and dispersion of acoustic waves in the viscoelastic material displaying anomalous dispersion in a specific frequency range. A brief discussion and comparison of an alternative fractional order wave equation recently formulated is also included.
Geometric constrained variational calculus. III: The second variation (Part II)
NASA Astrophysics Data System (ADS)
Massa, Enrico; Luria, Gianvittorio; Pagani, Enrico
2016-03-01
The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.
Promoting Metacognition in Introductory Calculus-based Physics Labs
NASA Astrophysics Data System (ADS)
Grennell, Drew; Boudreaux, Andrew
2010-10-01
In the Western Washington University physics department, a project is underway to develop research-based laboratory curriculum for the introductory calculus-based course. Instructional goals not only include supporting students' conceptual understanding and reasoning ability, but also providing students with opportunities to engage in metacognition. For the latter, our approach has been to scaffold reflective thinking with guided questions. Specific instructional strategies include analysis of alternate reasoning presented in fictitious dialogues and comparison of students' initial ideas with their lab group's final, consensus understanding. Assessment of student metacognition includes pre- and post- course data from selected questions on the CLASS survey, analysis of written lab worksheets, and student opinion surveys. CLASS results are similar to a traditional physics course and analysis of lab sheets show that students struggle to engage in a metacognitive process. Future directions include video studies, as well as use of additional written assessments adapted from educational psychology.
Stokes integral of economic growth. Calculus and the Solow model
NASA Astrophysics Data System (ADS)
Mimkes, Jürgen
2010-04-01
Economic growth depends on capital and labor and two-dimensional calculus has been applied to economic theory. This leads to Riemann and Stokes integrals and to the first and second laws of production and growth. The mathematical structure is the same as in thermodynamics, economic properties may be related to physical terms: capital to energy, production to physical work, GDP per capita to temperature, production function to entropy. This is called econophysics. Production, trade and banking may be compared to motors, heat pumps or refrigerators. The Carnot process of the first law creates two levels in each system: cold and hot in physics; buyer and seller, investor and saver, rich and poor in economics. The efficiency rises with the income difference of rich and poor. The results of econophysics are compared to neoclassical theory.
A new class of problems in the calculus of variations
NASA Astrophysics Data System (ADS)
Ekeland, Ivar; Long, Yiming; Zhou, Qinglong
2013-11-01
This paper investigates an infinite-horizon problem in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Following Chichilnisky, we introduce an additional term, which models concern for the well-being of future generations. We show that there are no optimal solutions, but that there are equilibrium strateges, i.e. Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thom. Our analysis extends earlier work by Ekeland and Lazrak.
Site occupancy models with heterogeneous detection probabilities
Royle, J. Andrew
2006-01-01
Models for estimating the probability of occurrence of a species in the presence of imperfect detection are important in many ecological disciplines. In these ?site occupancy? models, the possibility of heterogeneity in detection probabilities among sites must be considered because variation in abundance (and other factors) among sampled sites induces variation in detection probability (p). In this article, I develop occurrence probability models that allow for heterogeneous detection probabilities by considering several common classes of mixture distributions for p. For any mixing distribution, the likelihood has the general form of a zero-inflated binomial mixture for which inference based upon integrated likelihood is straightforward. A recent paper by Link (2003, Biometrics 59, 1123?1130) demonstrates that in closed population models used for estimating population size, different classes of mixture distributions are indistinguishable from data, yet can produce very different inferences about population size. I demonstrate that this problem can also arise in models for estimating site occupancy in the presence of heterogeneous detection probabilities. The implications of this are discussed in the context of an application to avian survey data and the development of animal monitoring programs.
Prevention of retrograde calculus migration with the Stone Cone.
Pardalidis, N P; Papatsoris, A G; Kosmaoglou, E V
2005-02-01
Retrograde calculus migration during ureteroscopic lithotripsy remains a problem in 5-40% of cases. We assessed the safety and efficacy of the Stone Cone device, in comparison with the standard flat wire basket. A total of 56 consecutive patients with ureteral calculi, suitable for ureteroscopic extraction and/or lithotripsy, where included in this prospective study. Patients were randomly allocated into two groups. In group A (30 patients), we used the Stone Cone, while in group B (26 patients) we used the standard flat wire basket. The Stone Cone was placed through a cystoscope under fluoroscopic guidance, or when necessary under direct ureteroscopic control. Whenever necessary, intracorporeal electrohydraulic lithotripsy took place in both groups. Statistical significance was assessed by the paired t-test. The mean operative time was 48.5 min in group A, and 42.4 min in group B. Intact calculus extraction was possible in 16.6% in group A, and in 7.6% in group B (P < 0.01). Retrograde stone migration was revealed in 23% in group B only (P < 0.001). Also, residual fragments > 3 mm were recorded in 30.7% in group B only (P < 0.001). None of the patients in group A required auxiliary procedures, in contrary to 23% in group B (P < 0.001). No major complications were recorded in group A, while in group B a case of major ureteral mucosal abrasion was recorded. The Stone Cone is safe and efficient in preventing retrograde stone migration and in minimizing residual fragments during ureteroscopic lithotripsy in comparison with the flat wire basket.
CR-Calculus and adaptive array theory applied to MIMO random vibration control tests
NASA Astrophysics Data System (ADS)
Musella, U.; Manzato, S.; Peeters, B.; Guillaume, P.
2016-09-01
Performing Multiple-Input Multiple-Output (MIMO) tests to reproduce the vibration environment in a user-defined number of control points of a unit under test is necessary in applications where a realistic environment replication has to be achieved. MIMO tests require vibration control strategies to calculate the required drive signal vector that gives an acceptable replication of the target. This target is a (complex) vector with magnitude and phase information at the control points for MIMO Sine Control tests while in MIMO Random Control tests, in the most general case, the target is a complete spectral density matrix. The idea behind this work is to tailor a MIMO random vibration control approach that can be generalized to other MIMO tests, e.g. MIMO Sine and MIMO Time Waveform Replication. In this work the approach is to use gradient-based procedures over the complex space, applying the so called CR-Calculus and the adaptive array theory. With this approach it is possible to better control the process performances allowing the step-by-step Jacobian Matrix update. The theoretical bases behind the work are followed by an application of the developed method to a two-exciter two-axis system and by performance comparisons with standard methods.
Probability workshop to be better in probability topic
NASA Astrophysics Data System (ADS)
Asmat, Aszila; Ujang, Suriyati; Wahid, Sharifah Norhuda Syed
2015-02-01
The purpose of the present study was to examine whether statistics anxiety and attitudes towards probability topic among students in higher education level have an effect on their performance. 62 fourth semester science students were given statistics anxiety questionnaires about their perception towards probability topic. Result indicated that students' performance in probability topic is not related to anxiety level, which means that the higher level in statistics anxiety will not cause lower score in probability topic performance. The study also revealed that motivated students gained from probability workshop ensure that their performance in probability topic shows a positive improvement compared before the workshop. In addition there exists a significance difference in students' performance between genders with better achievement among female students compared to male students. Thus, more initiatives in learning programs with different teaching approaches is needed to provide useful information in improving student learning outcome in higher learning institution.
NASA Astrophysics Data System (ADS)
Sevimli, Eyup
2016-08-01
This study aims to evaluate the consistency of teaching content with teaching approaches in calculus on the basis of lecturers' views. In this sense, the structures of the examples given in two commonly used calculus textbooks, both in traditional and reform classrooms, are compared. The content analysis findings show that the examples in both textbooks are presented in a rather formal language and generally highlight procedural knowledge. And, even though the examples in the reform book chosen are structured using multiple representations, only a small number of them incorporated the usage of instructional technology. The lecturers' views which were gathered indicated that, although, on the one hand, the example structures of the traditional textbook largely overlapped with the characteristics of the traditional approach, the example structures of the reform textbook, on the other hand, were found to be inconsistent with the characteristics of the reform approach, especially with regard to its environment and knowledge components. At the end of the paper, some suggestions for further studies are provided for book authors and researchers.
Intrinsic Probability of a Multifractal Set
NASA Astrophysics Data System (ADS)
Hosokawa, Iwao
1991-12-01
It is shown that a self-similar measure isotropically distributed in a d-dimensional set should have its own intermittency exponents equivalent to its own generalized dimensions (in the sense of Hentschel and Procaccia), and that the intermittency exponents are completely designated by an intrinsic probability which governs the spatial distribution of the measure. Based on this, it is proven that the intrinsic probability uniquely determines the spatial distribution of the scaling index α of the measure as well as the so-called f-α spectrum of the multifractal set.
Quantum probability and quantum decision-making.
Yukalov, V I; Sornette, D
2016-01-13
A rigorous general definition of quantum probability is given, which is valid not only for elementary events but also for composite events, for operationally testable measurements as well as for inconclusive measurements, and also for non-commuting observables in addition to commutative observables. Our proposed definition of quantum probability makes it possible to describe quantum measurements and quantum decision-making on the same common mathematical footing. Conditions are formulated for the case when quantum decision theory reduces to its classical counterpart and for the situation where the use of quantum decision theory is necessary.
NASA Astrophysics Data System (ADS)
Paneva-Konovska, Jordanka
2013-10-01
In this paper we consider a family of 3 m-indices generalizations of the classical Mittag-Leffler function, called multi-index (3 m-parametric) Mittag-Leffler functions. We survey the basic properties of these entire functions, find their order and type, and new representations by means of Mellin-Barnes type contour integrals, Wright p Ψ q -functions and Fox H-functions, asymptotic estimates. Formulas for integer and fractional order integration and differentiations are found, and these are extended also for the operators of the generalized fractional calculus (multiple Erdélyi-Kober operators). Some interesting particular cases of the multi-index Mittag-Leffler functions are discussed. The convergence of series of such type functions in the complex plane is considered, and analogues of the Cauchy-Hadamard, Abel, Tauber and Littlewood theorems are provided.
NASA Astrophysics Data System (ADS)
Paneva-Konovska, Jordanka
2013-10-01
In this paper we consider a family of 3m-indices generalizations of the classical Mittag-Leffler function, called multi-index (3m-parametric) Mittag-Leffler functions. We survey the basic properties of these entire functions, find their order and type, and new representations by means of Mellin-Barnes type contour integrals, Wright pΨq-functions and Fox H-functions, asymptotic estimates. Formulas for integer and fractional order integration and differentiations are found, and these are extended also for the operators of the generalized fractional calculus (multiple Erdélyi-Kober operators). Some interesting particular cases of the multi-index Mittag-Leffler functions are discussed. The convergence of series of such type functions in the complex plane is considered, and analogues of the Cauchy-Hadamard, Abel, Tauber and Littlewood theorems are provided.
Propensity, Probability, and Quantum Theory
NASA Astrophysics Data System (ADS)
Ballentine, Leslie E.
2016-08-01
Quantum mechanics and probability theory share one peculiarity. Both have well established mathematical formalisms, yet both are subject to controversy about the meaning and interpretation of their basic concepts. Since probability plays a fundamental role in QM, the conceptual problems of one theory can affect the other. We first classify the interpretations of probability into three major classes: (a) inferential probability, (b) ensemble probability, and (c) propensity. Class (a) is the basis of inductive logic; (b) deals with the frequencies of events in repeatable experiments; (c) describes a form of causality that is weaker than determinism. An important, but neglected, paper by P. Humphreys demonstrated that propensity must differ mathematically, as well as conceptually, from probability, but he did not develop a theory of propensity. Such a theory is developed in this paper. Propensity theory shares many, but not all, of the axioms of probability theory. As a consequence, propensity supports the Law of Large Numbers from probability theory, but does not support Bayes theorem. Although there are particular problems within QM to which any of the classes of probability may be applied, it is argued that the intrinsic quantum probabilities (calculated from a state vector or density matrix) are most naturally interpreted as quantum propensities. This does not alter the familiar statistical interpretation of QM. But the interpretation of quantum states as representing knowledge is untenable. Examples show that a density matrix fails to represent knowledge.
Trajectory versus probability density entropy.
Bologna, M; Grigolini, P; Karagiorgis, M; Rosa, A
2001-07-01
We show that the widely accepted conviction that a connection can be established between the probability density entropy and the Kolmogorov-Sinai (KS) entropy is questionable. We adopt the definition of density entropy as a functional of a distribution density whose time evolution is determined by a transport equation, conceived as the only prescription to use for the calculation. Although the transport equation is built up for the purpose of affording a picture equivalent to that stemming from trajectory dynamics, no direct use of trajectory time evolution is allowed, once the transport equation is defined. With this definition in mind we prove that the detection of a time regime of increase of the density entropy with a rate identical to the KS entropy is possible only in a limited number of cases. The proposals made by some authors to establish a connection between the two entropies in general, violate our definition of density entropy and imply the concept of trajectory, which is foreign to that of density entropy.
Rare calcium oxalate monohydrate calculus attached to the wall of the renal pelvis.
Grases, Felix; Costa-Bauza, Antonia; Prieto, Rafael M; Saus, Carlos; Servera, Antonio; García-Miralles, Reyes; Benejam, Joan
2011-04-01
Most renal calculi can be classified using well-established criteria in a manner that reflects both composition and fine structure under specific pathophysiological conditions. However, when a large patient population is considered, rare renal calculi invariably appear, some of which have never been classified; careful study is required to establish stone etiology in such cases. The patient in the present case report formed two types of calculi. One was attached on the wall of the renal pelvis near the ureter and part of the calculus was embedded inside pelvic renal tissue. The calculus developed on an ossified calcification located in the pelvis tissue. Current knowledge on the development of calcification in soft tissues suggests a pre-existing injury as an inducer of its development. A mechanism of calculus formation is proposed. The second stone was a typical jack-stone calculus.
History of the Infinitely Small and the Infinitely Large in Calculus.
ERIC Educational Resources Information Center
Kleiner, Israel
2001-01-01
Considers examples of aspects of the infinitely small and large as they unfolded in the history of calculus from the 17th through the 20th centuries. Presents didactic observations at relevant places in the historical account. (Author/MM)
Calculus Technique of Integration by Parts, Correlated with a Geometric Picture
ERIC Educational Resources Information Center
Fromhold, Albert T., Jr.
2005-01-01
The method of integration by parts is one of the most useful in integral calculus. Among the most important applications is the integration of differentials involving products, differentials in involving logarithms, and differentials involving inverse circular functions.
The Probabilities of Unique Events
2012-08-30
probabilities into quantum mechanics, and some psychologists have argued that they have a role to play in accounting for errors in judgment [30]. But, in...Discussion The mechanisms underlying naive estimates of the probabilities of unique events are largely inaccessible to consciousness , but they...Can quantum probability provide a new direc- tion for cognitive modeling? Behavioral and Brain Sciences (in press). 31. Paolacci G, Chandler J
Probability Surveys, Conditional Probability, and Ecological Risk Assessment
We show that probability-based environmental resource monitoring programs, such as the U.S. Environmental Protection Agency’s (U.S. EPA) Environmental Monitoring and Assessment Program, and conditional probability analysis can serve as a basis for estimating ecological risk over ...
PROBABILITY SURVEYS, CONDITIONAL PROBABILITIES, AND ECOLOGICAL RISK ASSESSMENT
We show that probability-based environmental resource monitoring programs, such as U.S. Environmental Protection Agency's (U.S. EPA) Environmental Monitoring and Asscssment Program EMAP) can be analyzed with a conditional probability analysis (CPA) to conduct quantitative probabi...
PROBABILITY SURVEYS , CONDITIONAL PROBABILITIES AND ECOLOGICAL RISK ASSESSMENT
We show that probability-based environmental resource monitoring programs, such as the U.S. Environmental Protection Agency's (U.S. EPA) Environmental Monitoring and Assessment Program, and conditional probability analysis can serve as a basis for estimating ecological risk over ...
Vaginal Calculus in a Woman With Mixed Urinary Incontinence and Vaginal Mesh Exposure.
Winkelman, William D; Rabban, Joseph T; Korn, Abner P
2016-01-01
Vaginal calculi are extremely rare and are most commonly encountered in the setting of an urethrovaginal or vesicovaginal fistula. We present a case of a 72-year-old woman with mixed urinary incontinence and vaginal mesh exposure incidentally found to have a large vaginal calculus. We removed the calculus surgically and analyzed the components. Results demonstrated the presence of ammonium-magnesium phosphate hexahydrate and carbonate apatite.
Composition and distribution of elements and ultrastructural topography of a human cardiac calculus.
Cheng, Ching-Li; Chang, Hsiao-Huang; Huang, Pei-Jung; Chu, Yu-Ting; Lin, Shan-Yang
2013-04-01
Trace elements (TEs) may contribute to the formation of calculi or stones or be involved in the aetiopathogenesis of stone diseases. The compositions and spatial distribution of elements from the inner nucleus to outer crust of the cardiac calculus were investigated by energy-dispersive X-ray fluorescence (EDXRF) spectrometer. The surface topograph, distribution map of elements, elemental and chemical compositions were also determined by environmental scanning electron microscope (ESEM)-energy-dispersive X-ray (EDX) analysis. Twenty-five elements were identifiable from 18 positions on the cardiac calculus by EDXRF spectrometer, in which the highest concentrations of toxic TEs (Ni, Pt, Hg, Sn, Pb, W, Au, Al, Si) and higher levels of essential TEs (Ca, Sr, Cr, P) were detected. A moderate positive Pearson's correlation between TEs concentrations of Mg, Ca or P and location differences from centre to periphery in the cardiac calculus was observed. A positive correlation was also found for Ca/Zn and Ca/Cu, indicating the gradual increase of calcium concentration from inner nucleus to outer crust of cardiac calculus. The drop-like nodules/crystals on the surface of petrous part of cardiac calculus were observed from ESEM analysis. ESEM-EDX analysis determined the calculus to be predominantly composed of calcium hydroxyapatite and cholesterol, as indicated by the petrous surface and drop-like nodules/crystals, respectively. This composition was confirmed using a portable Raman analyser. The spatial distribution analysis indicated a gradual increase in Mg, P and Ca concentrations from the inner nucleus to the outer crust of the cardiac calculus. The major chemical compositions of calcium hydroxyapatite and cholesterol were detected on this cardiac calculus.
An intrahepatic calculus superimposed over the right renal shadow: a case of mistaken identity.
Learney, Robert M; Shrotri, Nitin
2010-08-01
A 36-year-old Caucasian British woman presented with a classic case of right renal colic. Initial plain abdominal radiography and intravenous urography identified an 8 x 5 mm calculus apparently lying within a right lower pole calyx. Following failed extracorporeal lithotripsy and flexible ureterorenoscopy, cross-sectional imaging revealed a misdiagnosis by superposition of an intrahepatic calculus over the right renal shadow. This case serves to support cross-sectional imaging in the diagnosis of renal calculi.
Calculus structure on the Lie conformal algebra complex and the variational complex
NASA Astrophysics Data System (ADS)
De Sole, Alberto; Hekmati, Pedram; Kac, Victor G.
2011-05-01
We construct a calculus structure on the Lie conformal algebra cochain complex. By restricting to degree one chains, we recover the structure of a {mathfrak g}-complex introduced in [A. De Sole and V. G. Kac, Commun. Math. Phys. 292, 667 (2009), 10.1007/s00220-009-0886-1]. A special case of this construction is the variational calculus, for which we provide explicit formulas.
Information Processing Using Quantum Probability
NASA Astrophysics Data System (ADS)
Behera, Laxmidhar
2006-11-01
This paper presents an information processing paradigm that introduces collective response of multiple agents (computational units) while the level of intelligence associated with the information processing has been increased manifold. It is shown that if the potential field of the Schroedinger wave equation is modulated using a self-organized learning scheme, then the probability density function associated with the stochastic data is transferred to the probability amplitude function which is the response of the Schroedinger wave equation. This approach illustrates that information processing of data with stochastic behavior can be efficiently done using quantum probability instead of classical probability. The proposed scheme has been demonstrated through two applications: denoising and adaptive control.
A discussion on the origin of quantum probabilities
Holik, Federico; Sáenz, Manuel; Plastino, Angel
2014-01-15
We study the origin of quantum probabilities as arising from non-Boolean propositional-operational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorovian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case). -- Highlights: •Several recent works use a derivation similar to that of R.T. Cox to obtain quantum probabilities. •We apply Cox’s method to the lattice of subspaces of the Hilbert space. •We obtain a derivation of quantum probabilities which includes mixed states. •The method presented in this work is susceptible to generalization. •It includes quantum mechanics and classical mechanics as particular cases.
Drinking water composition and incidence of urinary calculus: introducing a new index.
Basiri, Abbas; Shakhssalim, Nasser; Khoshdel, Ali Reza; Pakmanesh, Hamid; Radfar, Mohammad Hadi
2011-01-01
INTRODUCTION. We searched for a pathophysiologically based feature of major water electrolytes, which may define water quality better than the water hardness, respecting urinary calculus formation. MATERIALS AND METHODS. Utilizing a multistage stratified sampling, 2310 patients were diagnosed in the imaging centers of the provincial capitals in Iran between 2007 and 2008. These were composed of 1755 patients who were settled residents of 24 provincial capitals. Data on the regional drinking water composition, obtained from an accredited registry, and their relationships with the region's incidence of urinary calculi were evaluated by metaregression models. The stone risk index (defined as the ratio of calcium to magnesium-bicarbonate product in drinking water) was used to assess the risk of calculus formation. RESULTS. No correlation was found between the urinary calculus incidence and the amount of calcium, bicarbonate, or the total hardness of the drinking water. In contrast, water magnesium had a marginally significant nonlinear inverse relationship with the incidence of the disease in the capitals (R(2) = 26%, P = .05 for a power model). The stone risk index was associated nonlinearly with the calculus incidence (R(2) = 28.4%, P = .04). CONCLUSIONS. Urinary calculus incidence was inversely related with drinking water magnesium content. We introduced a new index constructed on the foundation of a pathophysiologically based formula; the stone risk index had a strong positive association with calculus incidence. This index can have therapeutic and preventive applications, yet to be confirmed by clinical trials.
NASA Astrophysics Data System (ADS)
Cahaya, Cindy; Masulili, Sri Lelyati C.; Lessang, Robert; Radi, Basuni
2017-02-01
Coronary Artery Disease (CAD) or Coronary Heart Disease (CHD) is a disease that happened because of blood flow being blocked by atherosclerosis. Atherosclerosis is a process of hardening of the arteries which characterized by thickening and loss of elasticity of the intimal layer of vascular wall, by lipid deposit. Periodontitis is a chronic multifactorial inflammatory disease caused by microorganism and characterized by progressive destruction of the tooth supporting apparatus leading to tooth loss. Many studies use saliva as a valuable source for clinically information, as an asset for early diagnosis, prognostic and reviewer for pascatherapy status. Dental calculus had happened as a consequence of saliva supersaturation by calcium and phosphate. Salivary flow rate and its composition influence the formation of calculus. Increasing salivary calcium levels is characteristic of periodontitis patients. An important hipotesis in Cardiology is chronic infections contribute in atherosclerosis. Objective: To analyse the correlation between calcium and phosphate levels in saliva to calculus accumulation on CHD patients. Result: Correlation analysis between salivary calcium levels with calculus accumulation in patients with CHD and non-CHD showed no significant p value, p=0.59 and p=0.518. Correlation analysis between salivary phosphate levels and calculus accumulation showed no significant p value, p=0.836 for CHD patients and p=0.484 for non-CHD patients. Conclusion: There are no correlation between calcium levels and phosphate levels with calculus accumulation in CHD patients. Further research need to be done.
NASA Astrophysics Data System (ADS)
Tenreiro Machado, J.; Galhano, Alexandra M.
2017-01-01
The development of models constitutes a fundamental step in the study of natural and artificial systems. Present day science aims to address broader and more complex areas of application requiring, therefore, new concepts and models. This paper explores the concept of generalized two-port network by embedding the ideas of fractional calculus, memristor, transformer and gyrator. Each element represents separately one possible direction for generalizing the classical elements, but the cross-fertilization of the distinct topics has been overlooked. In this line of thought, the proposal of a novel element is a logical conjecture for obeying the symmetries that have been discovered in nature.
Causal inference, probability theory, and graphical insights.
Baker, Stuart G
2013-11-10
Causal inference from observational studies is a fundamental topic in biostatistics. The causal graph literature typically views probability theory as insufficient to express causal concepts in observational studies. In contrast, the view here is that probability theory is a desirable and sufficient basis for many topics in causal inference for the following two reasons. First, probability theory is generally more flexible than causal graphs: Besides explaining such causal graph topics as M-bias (adjusting for a collider) and bias amplification and attenuation (when adjusting for instrumental variable), probability theory is also the foundation of the paired availability design for historical controls, which does not fit into a causal graph framework. Second, probability theory is the basis for insightful graphical displays including the BK-Plot for understanding Simpson's paradox with a binary confounder, the BK2-Plot for understanding bias amplification and attenuation in the presence of an unobserved binary confounder, and the PAD-Plot for understanding the principal stratification component of the paired availability design.
Capture probabilities for secondary resonances
NASA Technical Reports Server (NTRS)
Malhotra, Renu
1990-01-01
A perturbed pendulum model is used to analyze secondary resonances, and it is shown that a self-similarity between secondary and primary resonances exists. Henrard's (1982) theory is used to obtain formulas for the capture probability into secondary resonances. The tidal evolution of Miranda and Umbriel is considered as an example, and significant probabilities of capture into secondary resonances are found.
Calculus of P-Acquisition versus P-Sparing by Plankton in the Oligotrophic Ocean (Invited)
NASA Astrophysics Data System (ADS)
van Mooy, B.; Dyhrman, S. T.; Lomas, M. W.
2010-12-01
Phosphate concentrations in the Sargasso Sea are among the lowest observed in the open oceans; at times concentrations can become sub-nanomolar. It has been suggested that the scarcity of phosphate in this region plays a key role in limiting rates of primary production, bacterial respiration, and nitrogen fixation. The myriad physiological responses of plankton to phosphate scarcity fall into two basic categories: 1) expending energy and resources in an attempt to acquire P from dissolved organic matter, or 2) adjusting the biochemical composition of the cell in an attempt to spare P and continue to grow with less P. In theory, these two types of responses should have opposing impacts on the partitioning of P between the particulate and dissolved pools. In a transect across the north-western boundary of the Sargasso Sea, we measured examples of these two types of responses: rates of alkaline phosphatase activity and relative abundance of non-phosphorus membrane lipids. Although, both parameters generally increased with decreasing phosphate concentrations, there were distinct differences in theses responses. Analysis of these differences has provided key insights on the calculus of P-acquisition vs. P-sparing by members of the planktonic community.
Jain, PK; Kumra, Madhumani; Rehani, Shweta; Mathias, Yulia; Gupta, Ramakant; Mehendiratta, Monica; Chander, Anil
2016-01-01
Introduction Chronic inflammatory periodontal diseases i.e. gingivitis and periodontitis are one of the most common afflictions faced by human beings. Dental plaque, which is a pool of pathogenic microorganisms, remains to be current mainstay in etiopathogenesis. Dental calculus, which is a mineralized product of this plaque remains ignored and is considered merely as an ash heap of minor significance. However, the intriguing array in disease etiopathogenesis bulldozed researchers to suspect the role of calculus in disease chrysalis but still the viability of bacteria inside calculus and thus its pathogenicity remains an intricacy; the answer to which lies in the Pandora’s Box. Aim The present study was undertaken to investigate the viability of bacteria within dental calculus along with their identification. Also, to classify dental calculus on the basis of mineralization and to observe the variation of viable microflora found in dental calculus with the extent of mineralization and disease severity. Materials and Methods A total of 60 samples were obtained, by harvesting two samples of supragingival calculus from each patient having chronic inflammatory periodontal disease. These samples were divided into two groups (Group A and Group B). Samples of Group A were kept non-irradiated and samples of Group B were exposed to UV radiation. The samples were categorized into less, moderately and highly mineralized according to the force required for crushing them. All the crushed calculus samples were then divided into three parts. These were used for dark-field microscopy, gram staining and bacterial cultures. Bacterial identification of the cultures obtained was also carried out by performing various biochemical assays. Results The present study revealed the presence of motile spirochaetes within the samples under dark-field microscope. Gram staining revealed presence of numerous gram positive cocci and gram negative bacilli. Bacterial cultures showed growth of
Failure probability under parameter uncertainty.
Gerrard, R; Tsanakas, A
2011-05-01
In many problems of risk analysis, failure is equivalent to the event of a random risk factor exceeding a given threshold. Failure probabilities can be controlled if a decisionmaker is able to set the threshold at an appropriate level. This abstract situation applies, for example, to environmental risks with infrastructure controls; to supply chain risks with inventory controls; and to insurance solvency risks with capital controls. However, uncertainty around the distribution of the risk factor implies that parameter error will be present and the measures taken to control failure probabilities may not be effective. We show that parameter uncertainty increases the probability (understood as expected frequency) of failures. For a large class of loss distributions, arising from increasing transformations of location-scale families (including the log-normal, Weibull, and Pareto distributions), the article shows that failure probabilities can be exactly calculated, as they are independent of the true (but unknown) parameters. Hence it is possible to obtain an explicit measure of the effect of parameter uncertainty on failure probability. Failure probability can be controlled in two different ways: (1) by reducing the nominal required failure probability, depending on the size of the available data set, and (2) by modifying of the distribution itself that is used to calculate the risk control. Approach (1) corresponds to a frequentist/regulatory view of probability, while approach (2) is consistent with a Bayesian/personalistic view. We furthermore show that the two approaches are consistent in achieving the required failure probability. Finally, we briefly discuss the effects of data pooling and its systemic risk implications.
Cluster membership probability: polarimetric approach
NASA Astrophysics Data System (ADS)
Medhi, Biman J.; Tamura, Motohide
2013-04-01
Interstellar polarimetric data of the six open clusters Hogg 15, NGC 6611, NGC 5606, NGC 6231, NGC 5749 and NGC 6250 have been used to estimate the membership probability for the stars within them. For proper-motion member stars, the membership probability estimated using the polarimetric data is in good agreement with the proper-motion cluster membership probability. However, for proper-motion non-member stars, the membership probability estimated by the polarimetric method is in total disagreement with the proper-motion cluster membership probability. The inconsistencies in the determined memberships may be because of the fundamental differences between the two methods of determination: one is based on stellar proper motion in space and the other is based on selective extinction of the stellar output by the asymmetric aligned dust grains present in the interstellar medium. The results and analysis suggest that the scatter of the Stokes vectors q (per cent) and u (per cent) for the proper-motion member stars depends on the interstellar and intracluster differential reddening in the open cluster. It is found that this method could be used to estimate the cluster membership probability if we have additional polarimetric and photometric information for a star to identify it as a probable member/non-member of a particular cluster, such as the maximum wavelength value (λmax), the unit weight error of the fit (σ1), the dispersion in the polarimetric position angles (overline{ɛ }), reddening (E(B - V)) or the differential intracluster reddening (ΔE(B - V)). This method could also be used to estimate the membership probability of known member stars having no membership probability as well as to resolve disagreements about membership among different proper-motion surveys.
Holographic Probabilities in Eternal Inflation
NASA Astrophysics Data System (ADS)
Bousso, Raphael
2006-11-01
In the global description of eternal inflation, probabilities for vacua are notoriously ambiguous. The local point of view is preferred by holography and naturally picks out a simple probability measure. It is insensitive to large expansion factors or lifetimes and so resolves a recently noted paradox. Any cosmological measure must be complemented with the probability for observers to emerge in a given vacuum. In lieu of anthropic criteria, I propose to estimate this by the entropy that can be produced in a local patch. This allows for prior-free predictions.
A Complete Graphical Calculus for Spekkens' Toy Bit Theory
NASA Astrophysics Data System (ADS)
Backens, Miriam; Duman, Ali Nabi
2016-01-01
While quantum theory cannot be described by a local hidden variable model, it is nevertheless possible to construct such models that exhibit features commonly associated with quantum mechanics. These models are also used to explore the question of ψ -ontic versus ψ -epistemic theories for quantum mechanics. Spekkens' toy theory is one such model. It arises from classical probabilistic mechanics via a limit on the knowledge an observer may have about the state of a system. The toy theory for the simplest possible underlying system closely resembles stabilizer quantum mechanics, a fragment of quantum theory which is efficiently classically simulable but also non-local. Further analysis of the similarities and differences between those two theories can thus yield new insights into what distinguishes quantum theory from classical theories, and ψ -ontic from ψ -epistemic theories. In this paper, we develop a graphical language for Spekkens' toy theory. Graphical languages offer intuitive and rigorous formalisms for the analysis of quantum mechanics and similar theories. To compare quantum mechanics and a toy model, it is useful to have similar formalisms for both. We show that our language fully describes Spekkens' toy theory and in particular, that it is complete: meaning any equality that can be derived using other formalisms can also be derived entirely graphically. Our language is inspired by a similar graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens' toy bit theory and stabilizer quantum mechanics can be analysed and compared using analogous graphical formalisms.
A physically based connection between fractional calculus and fractal geometry
NASA Astrophysics Data System (ADS)
Butera, Salvatore; Di Paola, Mario
2014-11-01
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.
Motivating Calculus-Based Kinematics Instruction with Super Mario Bros
NASA Astrophysics Data System (ADS)
Nordine, Jeffrey C.
2011-09-01
High-quality physics instruction is contextualized, motivates students to learn, and represents the discipline as a way of investigating the world rather than as a collection of facts and equations. Inquiry-oriented pedagogy, such as problem-based instruction, holds great promise for both teaching physics content and representing the process of doing real science.2 A challenge for physics teachers is to find instructional contexts that are meaningful, accessible, and motivating for students. Today's students are spending a growing fraction of their lives interacting with virtual environments, and these environments—physically realistic or not—can provide valuable contexts for physics explorations3-5 and lead to thoughtful discussions about decisions that programmers make when designing virtual environments. In this article, I describe a problem-based approach to calculus-based kinematics instruction that contextualizes students' learning within the Super Mario Bros. video game—a game that is more than 20 years old, but still remarkably popular with today's high school and college students.
A physically based connection between fractional calculus and fractal geometry
Butera, Salvatore; Di Paola, Mario
2014-11-15
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.
Exploring the solar coronal magnetic field using variational calculus
NASA Astrophysics Data System (ADS)
Bhattacharyya, Ramit; Kumar, Dinesh
2012-07-01
The topological properties of magnetic field has an ubiquitous influence on various energetic processes in the solar atmosphere. For example, the preservation of magnetic topology in a near-ideal magnetofluid characterized by high magnetic Reynolds number is believed to produce discontinuities in an otherwise continuous magnetic field. The subsequent reconnection of magnetic field lines across the corresponding surface of discontinuity is believed to be one of the possible mechanisms to heat the solar corona to its million degree temperature. Any understanding of coronal magnetic field then should involve appropriate topological constraints in their proper physical context. The calculus of variations provides a powerful mathematical tool to include preservation of these constraints in the framework of the problem under consideration. In this work we pose a variational problem based on two-fluid magnetohydrodynamics and relevant to the solar corona. The Euler-Lagrange equations obtained are of double curl in nature and support non-zero plasma-beta along with mass flow of the magnetofluid. These equations are solved in Cartesian coordinates utilizing a geometry compatible to the solar atmosphere, and a basic comparative study of the non force-free, force-free, and potential magnetic field obtained as solutions of the same Euler-Lagrange equations is presented.
Fast free-form deformable registration via calculus of variations.
Lu, Weiguo; Chen, Ming-Li; Olivera, Gustavo H; Ruchala, Kenneth J; Mackie, Thomas R
2004-07-21
In this paper, we present a fully automatic, fast and accurate deformable registration technique. This technique deals with free-form deformation. It minimizes an energy functional that combines both similarity and smoothness measures. By using calculus of variations, the minimization problem was represented as a set of nonlinear elliptic partial differential equations (PDEs). A Gauss-Seidel finite difference scheme is used to iteratively solve the PDE. The registration is refined by a multi-resolution approach. The whole process is fully automatic. It takes less than 3 min to register two three-dimensional (3D) image sets of size 256 x 256 x 61 using a single 933 MHz personal computer. Extensive experiments are presented. These experiments include simulations, phantom studies and clinical image studies. Experimental results show that our model and algorithm are suited for registration of temporal images of a deformable body. The registration of inspiration and expiration phases of the lung images shows that the method is able to deal with large deformations. When applied to the daily CT images of a prostate patient, the results show that registration based on iterative refinement of displacement field is appropriate to describe the local deformations in the prostate and the rectum. Similarity measures improved significantly after the registration. The target application of this paper is for radiotherapy treatment planning and evaluation that incorporates internal organ deformation throughout the course of radiation therapy. The registration method could also be equally applied in diagnostic radiology.
Calculus and design of discrete velocity models using computer algebra
NASA Astrophysics Data System (ADS)
Babovsky, Hans; Grabmeier, Johannes
2016-11-01
In [2, 3], a framework for a calculus with Discrete Velocity Models (DVM) has been derived. The rotatonal symmetry of the discrete velocities can be modelled algebraically by the action of the cyclic group C4 - or including reflections of the dihedral group D4. Taking this point of view, the linearized collision operator can be represented in a compact form as a matrix of elements in the group algebra. Or in other words, by choosing a special numbering it exhibits a certain block structure which lets it appear as a matrix with entries in a certain polynomial ring. A convenient way for approaching such a structure is the use of a computer algebra system able to treat these (predefined) algebraic structures. We used the computer algebra system FriCAS/AXIOM [4, 5] for the generation of the velocity and the collision sets and for the analysis of the structure of the collision operator. Concerning the fluid dynamic limit, the system provides the characterization of sets of collisions and their contribution to the flow parameters. It allows the design of rotationally invariant symmetric models for prescribed Prandtl numbers. The implementation in FriCAS/AXIOM is explained and its results for a 25-velocity model are presented.