The Geometry of Generations

NASA Astrophysics Data System (ADS)

He, Yang-Hui; Jejjala, Vishnu; Matti, Cyril; Nelson, Brent D.; Stillman, Michael

2015-10-01

We present an intriguing and precise interplay between algebraic geometry and the phenomenology of generations of particles. Using the electroweak sector of the MSSM as a testing ground, we compute the moduli space of vacua as an algebraic variety for multiple generations of Standard Model matter and Higgs doublets. The space is shown to have Calabi-Yau, Grassmannian, and toric signatures, which sensitively depend on the number of generations of leptons, as well as inclusion of Majorana mass terms for right-handed neutrinos. We speculate as to why three generations is special.

Remote Symbolic Computation of Loci

ERIC Educational Resources Information Center

Abanades, Miguel A.; Escribano, Jesus; Botana, Francisco

2010-01-01

This article presents a web-based tool designed to compute certified equations and graphs of geometric loci specified using standard Dynamic Geometry Systems (DGS). Complementing the graphing abilities of the considered DGS, the equations of the loci produced by the application are remotely computed using symbolic algebraic techniques from the…

Computers as Cognitive Tools.

ERIC Educational Resources Information Center

Lajoie, Susanne P., Ed.; Derry, Sharon J., Ed.

This book provides exemplars of the types of computer-based learning environments represented by the theoretical camps within the field and the practical applications of the theories. The contributors discuss a variety of computer applications to learning, ranging from school-related topics such as geometry, algebra, biology, history, physics, and…

Secondary Schools Curriculum Guide, Mathematics, Grades 10-12. Revised.

ERIC Educational Resources Information Center

Cranston School Dept., RI.

Behavioral objectives for grades 10 through 12 are specified for plane geometry, algebra, general mathematics, computer mathematics, slide rule mathematics, basic college mathematics, trigonometry, analytic geometry, calculus and probability. Most sections present material in terms of portions of a school year. At least one major objective is…

Secondary Schools Curriculum Guide, Mathematics, Grades 10-12, Levels 87-112.

ERIC Educational Resources Information Center

Rogers, Arnold R., Ed.; And Others

Behavioral objectives for geometry, algebra, computer mathematics, trigonometry, analytic geometry, calculus, and probability are specified for grades 10 through 12. General objectives are stated for major areas under each topic and are followed by a list of specific objectives for that area. This work was prepared under an ESEA Title III…

Duncan F. Gregory, William Walton and the development of British algebra: 'algebraical geometry', 'geometrical algebra', abstraction.

PubMed

Verburgt, Lukas M

2016-01-01

This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.

Space Mathematics: A Resource for Secondary School Teachers

NASA Technical Reports Server (NTRS)

Kastner, Bernice

1985-01-01

A collection of mathematical problems related to NASA space science projects is presented. In developing the examples and problems, attention was given to preserving the authenticity and significance of the original setting while keeping the level of mathematics within the secondary school curriculum. Computation and measurement, algebra, geometry, probability and statistics, exponential and logarithmic functions, trigonometry, matrix algebra, conic sections, and calculus are among the areas addressed.

The 6th International Conference on Computer Science and Computational Mathematics (ICCSCM 2017)

NASA Astrophysics Data System (ADS)

2017-09-01

The ICCSCM 2017 (The 6th International Conference on Computer Science and Computational Mathematics) has aimed to provide a platform to discuss computer science and mathematics related issues including Algebraic Geometry, Algebraic Topology, Approximation Theory, Calculus of Variations, Category Theory; Homological Algebra, Coding Theory, Combinatorics, Control Theory, Cryptology, Geometry, Difference and Functional Equations, Discrete Mathematics, Dynamical Systems and Ergodic Theory, Field Theory and Polynomials, Fluid Mechanics and Solid Mechanics, Fourier Analysis, Functional Analysis, Functions of a Complex Variable, Fuzzy Mathematics, Game Theory, General Algebraic Systems, Graph Theory, Group Theory and Generalizations, Image Processing, Signal Processing and Tomography, Information Fusion, Integral Equations, Lattices, Algebraic Structures, Linear and Multilinear Algebra; Matrix Theory, Mathematical Biology and Other Natural Sciences, Mathematical Economics and Financial Mathematics, Mathematical Physics, Measure Theory and Integration, Neutrosophic Mathematics, Number Theory, Numerical Analysis, Operations Research, Optimization, Operator Theory, Ordinary and Partial Differential Equations, Potential Theory, Real Functions, Rings and Algebras, Statistical Mechanics, Structure Of Matter, Topological Groups, Wavelets and Wavelet Transforms, 3G/4G Network Evolutions, Ad-Hoc, Mobile, Wireless Networks and Mobile Computing, Agent Computing & Multi-Agents Systems, All topics related Image/Signal Processing, Any topics related Computer Networks, Any topics related ISO SC-27 and SC- 17 standards, Any topics related PKI(Public Key Intrastructures), Artifial Intelligences(A.I.) & Pattern/Image Recognitions, Authentication/Authorization Issues, Biometric authentication and algorithms, CDMA/GSM Communication Protocols, Combinatorics, Graph Theory, and Analysis of Algorithms, Cryptography and Foundation of Computer Security, Data Base(D.B.) Management & Information Retrievals, Data Mining, Web Image Mining, & Applications, Defining Spectrum Rights and Open Spectrum Solutions, E-Comerce, Ubiquitous, RFID, Applications, Fingerprint/Hand/Biometrics Recognitions and Technologies, Foundations of High-performance Computing, IC-card Security, OTP, and Key Management Issues, IDS/Firewall, Anti-Spam mail, Anti-virus issues, Mobile Computing for E-Commerce, Network Security Applications, Neural Networks and Biomedical Simulations, Quality of Services and Communication Protocols, Quantum Computing, Coding, and Error Controls, Satellite and Optical Communication Systems, Theory of Parallel Processing and Distributed Computing, Virtual Visions, 3-D Object Retrievals, & Virtual Simulations, Wireless Access Security, etc. The success of ICCSCM 2017 is reflected in the received papers from authors around the world from several countries which allows a highly multinational and multicultural idea and experience exchange. The accepted papers of ICCSCM 2017 are published in this Book. Please check http://www.iccscm.com for further news. A conference such as ICCSCM 2017 can only become successful using a team effort, so herewith we want to thank the International Technical Committee and the Reviewers for their efforts in the review process as well as their valuable advices. We are thankful to all those who contributed to the success of ICCSCM 2017. The Secretary

Veronese geometry and the electroweak vacuum moduli space

NASA Astrophysics Data System (ADS)

He, Yang-Hui; Jejjala, Vishnu; Matti, Cyril; Nelson, Brent D.

2014-09-01

We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F- and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential.

Can a Crescent Mars Ever Be Seen from Earth?

ERIC Educational Resources Information Center

Lamb, John F., Jr.

1990-01-01

Described is an activity that incorporates a computer, geometry, algebra, trigonometry, and calculus to answer questions about the planet Mars. A possible crescent of Mars is compared to those of Venus and Mercury. (KR)

Using Dynamic Geometry and Computer Algebra Systems in Problem Based Courses for Future Engineers

ERIC Educational Resources Information Center

Tomiczková, Svetlana; Lávicka, Miroslav

2015-01-01

It is a modern trend today when formulating the curriculum of a geometric course at the technical universities to start from a real-life problem originated in technical praxis and subsequently to define which geometric theories and which skills are necessary for its solving. Nowadays, interactive and dynamic geometry software plays a more and more…

Entanglement classification with algebraic geometry

NASA Astrophysics Data System (ADS)

Sanz, M.; Braak, D.; Solano, E.; Egusquiza, I. L.

2017-05-01

We approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry, its natural language. We show that the class of symmetric separable states has the structure of a Veronese variety and that its k-secant varieties are SLOCC invariants. Thus SLOCC classes gather naturally into families. This classification presents useful properties such as a linear growth of the number of families with the number of particles, and nesting, i.e. upward consistency of the classification. We attach physical meaning to this classification through the required interaction length of parent Hamiltonians. We show that the states W N and GHZ N are in the same secant family and that, effectively, the former can be obtained in a limit from the latter. This limit is understood in terms of tangents, leading to a refinement of the previous families. We compute explicitly the classification of symmetric states with N≤slant4 qubits in terms of both secant families and its refinement using tangents. This paves the way to further use of projective varieties in algebraic geometry to solve open problems in entanglement theory.

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.)

GeoGebra: A Global Platform for Teaching and Learning Math Together and Using the Synergy of Mathematicians

NASA Astrophysics Data System (ADS)

Kllogjeri, Pellumb

In present age we are witnesses and practioners of computer-based education which is highly speed progressing. The computer-based education allows educators and students to use educational programming language and e-tutors to teach and learn, to interact with one another and share together the results of their work. The computer-based education is done possible by special electronic tools among which the most important are the mathematical programmes. There are many mathematical programmes, but one which is being embraced and used by a daily increasing number of users throughout the world is GeoGebra. The recently published software GeoGebra by Markus Hohenwater (2004) explicitly links geometry and algebra. GeoGebra affords a bidirectional combination of geometry and algebra that differs from earlier software forms. The bidirectional combination means that, for instance, by typing in an equation in the algebra window, the graph of the equation will be shown in the dynamic and graphic window. This programme is so much preferred because of its three main features: the double representation of the mathematical object(geometric and algebraic), there are not strong requirements as to the age and the knowledge in using it(the students of the elementary school can use it as well) and, it is offered free of charge(simply by downloading it). In this paper we are concentrating in the double representation of the mathematical object and its advantages in explaining and forming mathematical concepts and performing operations, in the global opportunities for using GeoGebra and the benefits of using it by cooperating and sharing experiences.

The Development of an Individualized Instructional Program in Beginning College Mathematics Utilizing Computer Based Resource Units. Final Report.

ERIC Educational Resources Information Center

Rockhill, Theron D.

Reported is an attempt to develop and evaluate an individualized instructional program in pre-calculus college mathematics. Four computer based resource units were developed in the areas of set theory, relations and function, algebra, trigonometry, and analytic geometry. Objectives were determined by experienced calculus teachers, and…

Connes distance function on fuzzy sphere and the connection between geometry and statistics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Devi, Yendrembam Chaoba, E-mail: chaoba@bose.res.in; Chakraborty, Biswajit, E-mail: biswajit@bose.res.in; Prajapat, Shivraj, E-mail: shraprajapat@gmail.com

An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the su(2) algebra. This has been computed for both the discrete and the Perelemov’s SU(2) coherent state. Here also, we get a connection between geometry and statistics which ismore » shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by n ∈ ℤ/2.« less

Mathematics. Exceptional Child Education Curriculum K-12.

ERIC Educational Resources Information Center

Jordon, Thelma; And Others

The mathematics curriculum provides a framework of instruction for exceptional child education in grades K-12. Content areas include: numeration, whole numbers, rational numbers, real/complex numbers, calculator literacy, measurement, geometry, statistics, functions/relations, computer literacy, and pre-algebra. The guide is organized by content…

Multiple-block grid adaption for an airplane geometry

NASA Technical Reports Server (NTRS)

Abolhassani, Jamshid Samareh; Smith, Robert E.

1988-01-01

Grid-adaption methods are developed with the capability of moving grid points in accordance with several variables for a three-dimensional multiple-block grid system. These methods are algebraic, and they are implemented for the computation of high-speed flow over an airplane configuration.

Discrimination in a General Algebraic Setting

PubMed Central

Fine, Benjamin; Lipschutz, Seymour; Spellman, Dennis

2015-01-01

Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras. PMID:26171421

RANS modeling of scalar dispersion from localized sources within a simplified urban-area model

NASA Astrophysics Data System (ADS)

Rossi, Riccardo; Capra, Stefano; Iaccarino, Gianluca

2011-11-01

The dispersion of a passive scalar downstream a localized source within a simplified urban-like geometry is examined by means of RANS scalar flux models. The computations are conducted under conditions of neutral stability and for three different incoming wind directions (0°, 45°, 90°) at a roughness Reynolds number of Ret = 391. A Reynolds stress transport model is used to close the flow governing equations whereas both the standard eddy-diffusivity closure and algebraic flux models are employed to close the transport equation for the passive scalar. The comparison with a DNS database shows improved reliability from algebraic scalar flux models towards predicting both the mean concentration and the plume structure. Since algebraic flux models do not increase substantially the computational effort, the results indicate that the use of tensorial-diffusivity can be promising tool for dispersion simulations for the urban environment.

Numerical algebraic geometry for model selection and its application to the life sciences

PubMed Central

Gross, Elizabeth; Davis, Brent; Ho, Kenneth L.; Bates, Daniel J.

2016-01-01

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are available. Here, we consider polynomial models (e.g. mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometrical structures relating models and data, and we demonstrate its utility on examples from cell signalling, synthetic biology and epidemiology. PMID:27733697

Brain activity associated with translation from a visual to a symbolic representation in algebra and geometry.

PubMed

Leikin, Mark; Waisman, Ilana; Shaul, Shelley; Leikin, Roza

2014-03-01

This paper presents a small part of a larger interdisciplinary study that investigates brain activity (using event related potential methodology) of male adolescents when solving mathematical problems of different types. The study design links mathematics education research with neurocognitive studies. In this paper we performed a comparative analysis of brain activity associated with the translation from visual to symbolic representations of mathematical objects in algebra and geometry. Algebraic tasks require translation from graphical to symbolic representation of a function, whereas tasks in geometry require translation from a drawing of a geometric figure to a symbolic representation of its property. The findings demonstrate that electrical activity associated with the performance of geometrical tasks is stronger than that associated with solving algebraic tasks. Additionally, we found different scalp topography of the brain activity associated with algebraic and geometric tasks. Based on these results, we argue that problem solving in algebra and geometry is associated with different patterns of brain activity.

The role of difficulty and gender in numbers, algebra, geometry and mathematics achievement

NASA Astrophysics Data System (ADS)

Rabab'h, Belal Sadiq Hamed; Veloo, Arsaythamby; Perumal, Selvan

2015-05-01

This study aims to identify the role of difficulty and gender in numbers, algebra, geometry and mathematics achievement among secondary schools students in Jordan. The respondent of the study were 337 students from eight public secondary school in Alkoura district by using stratified random sampling. The study comprised of 179 (53%) males and 158 (47%) females students. The mathematics test comprises of 30 items which has eight items for numbers, 14 items for algebra and eight items for geometry. Based on difficulties among male and female students, the findings showed that item 4 (fractions - 0.34) was most difficult for male students and item 6 (square roots - 0.39) for females in numbers. For the algebra, item 11 (inequality - 0.23) was most difficult for male students and item 6 (algebraic expressions - 0.35) for female students. In geometry, item 3 (reflection - 0.34) was most difficult for male students and item 8 (volume - 0.33) for female students. Based on gender differences, female students showed higher achievement in numbers and algebra compare to male students. On the other hand, there was no differences between male and female students achievement in geometry test. This study suggest that teachers need to give more attention on numbers and algebra when teaching mathematics.

Aspects géométriques et intégrables des modèles de matrices aléatoires

NASA Astrophysics Data System (ADS)

Marchal, Olivier

2010-12-01

This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of integrable systems. The variety of these applications shows why matrix models are important from a mathematical point of view. First, the thesis will focus on the study of the merging of two intervals of the eigenvalues density near a singular point. Specifically, we will show why this special limit gives universal equations from the Painlevé II hierarchy of integrable systems theory. Then, following the approach of (bi) orthogonal polynomials introduced by Mehta to compute partition functions, we will find Riemann-Hilbert and isomonodromic problems connected to matrix models, making the link with the theory of Jimbo, Miwa and Ueno. In particular, we will describe how the hermitian two-matrix models provide a degenerate case of Jimbo-Miwa-Ueno's theory that we will generalize in this context. Furthermore, the loop equations method, with its central notions of spectral curve and topological expansion, will lead to the symplectic invariants of algebraic geometry recently proposed by Eynard and Orantin. This last point will be generalized to the case of non-hermitian matrix models (arbitrary beta) paving the way to "quantum algebraic geometry" and to the generalization of symplectic invariants to "quantum curves". Finally, this set up will be applied to combinatorics in the context of topological string theory, with the explicit computation of an hermitian random matrix model enumerating the Gromov-Witten invariants of a toric Calabi-Yau threefold.

BIBLIOGRAPHIES, HIGH SCHOOL MATHEMATICS.

ERIC Educational Resources Information Center

WOODS, PAUL E.

THIS ANNOTATED BIBLIOGRAPHY IS A COMPILATION OF A NUMBER OF HIGHLY REGARDED BOOK LISTS CONSISTING OF LIBRARY BOOKS AND TEXTBOOKS FOR GRADES 7-12. THE BOOKS IN THIS LIST ARE CURRENTLY IN PRINT AND THE CONTENT IS REPRESENTATIVE OF THE FOLLOWING AREAS OF MATHEMATICS--MATHEMATICAL RECREATION, COMPUTERS, ARITHMETIC, ALGEBRA, EUCLIDEAN GEOMETRY,…

Adaptive Technologies for Training and Education

ERIC Educational Resources Information Center

Durlach, Paula J., Ed; Lesgold, Alan M., Ed.

2012-01-01

This edited volume provides an overview of the latest advancements in adaptive training technology. Intelligent tutoring has been deployed for well-defined and relatively static educational domains such as algebra and geometry. However, this adaptive approach to computer-based training has yet to come into wider usage for domains that are less…

Teaching Multistep Equations with Virtual Manipulatives to Secondary Students with Learning Disabilities

ERIC Educational Resources Information Center

Satsangi, Rajiv; Hammer, Rachel; Evmenova, Anya S.

2018-01-01

Students with learning disabilities often struggle with the academic demands presented in secondary mathematics curricula. To combat these students' struggles, researchers have studied various pedagogical practices and classroom technologies for teaching standards covered in subjects such as algebra and geometry. However, as the role of computer-…

Noncommutative Differential Geometry of Generalized Weyl Algebras

NASA Astrophysics Data System (ADS)

Brzeziński, Tomasz

2016-06-01

Elements of noncommutative differential geometry of Z-graded generalized Weyl algebras A(p;q) over the ring of polynomials in two variables and their zero-degree subalgebras B(p;q), which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particular, three classes of skew derivations of A(p;q) are constructed, and three-dimensional first-order differential calculi induced by these derivations are described. The associated integrals are computed and it is shown that the dimension of the integral space coincides with the order of the defining polynomial p(z). It is proven that the restriction of these first-order differential calculi to the calculi on B(p;q) is isomorphic to the direct sum of degree 2 and degree -2 components of A(p;q). A Dirac operator for B(p;q) is constructed from a (strong) connection with respect to this differential calculus on the (free) spinor bimodule defined as the direct sum of degree 1 and degree -1 components of A(p;q). The real structure of KO-dimension two for this Dirac operator is also described.

Calabi-Yau Geometries: Algorithms, Databases and Physics

NASA Astrophysics Data System (ADS)

He, Yang-Hui

2013-08-01

With a bird's-eye view, we survey the landscape of Calabi-Yau threefolds, compact and noncompact, smooth and singular. Emphasis will be placed on the algorithms and databases which have been established over the years, and how they have been useful in the interaction between the physics and the mathematics, especially in string and gauge theories. A skein which runs through this review will be algorithmic and computational algebraic geometry and how, implementing its principles on powerful computers and experimenting with the vast mathematical data, new physics can be learnt. It is hoped that this interdisciplinary glimpse will be of some use to the beginning student.

Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity

NASA Technical Reports Server (NTRS)

Kolecki, Joseph C.

2005-01-01

Tensor analysis is one of the more abstruse, even if one of the more useful, higher math subjects enjoined by students of physics and engineering. It is abstruse because of the intellectual gap that exists between where most physics and engineering mathematics leave off and where tensor analysis traditionally begins. It is useful because of its great generality, computational power, and compact, easy to use, notation. This paper bridges the intellectual gap. It is divided into three parts: algebra, calculus, and relativity. Algebra: In tensor analysis, coordinate independent quantities are sought for applications in physics and engineering. Coordinate independence means that the quantities have such coordinate transformations as to leave them invariant relative to a particular observer s coordinate system. Calculus: Non-zero base vector derivatives contribute terms to dynamical equations that correspond to pseudoaccelerations in accelerated coordinate systems and to curvature or gravity in relativity. These derivatives have a specific general form in tensor analysis. Relativity: Spacetime has an intrinsic geometry. Light is the tool for investigating that geometry. Since the observed geometry of spacetime cannot be made to match the classical geometry of Euclid, Einstein applied another more general geometry differential geometry. The merger of differential geometry and cosmology was accomplished in the theory of relativity. In relativity, gravity is equivalent to curvature.

The Effect of Geogebra on Students' Conceptual and Procedural Knowledge: The Case of Applications of Derivative

ERIC Educational Resources Information Center

Ocal, Mehmet Fatih

2017-01-01

Integrating the properties of computer algebra systems and dynamic geometry environments, Geogebra became an effective and powerful tool for teaching and learning mathematics. One of the reasons that teachers use Geogebra in mathematics classrooms is to make students learn mathematics meaningfully and conceptually. From this perspective, the…

Mathematics, Grade 5, Part 2.

ERIC Educational Resources Information Center

New York City Board of Education, Brooklyn, NY.

This curriculum bulletin is designed to help teachers meet the diverse needs in mathematics of the children in fifth grade classes. In addition to the emphasis that is placed on arithmetic computational skills, the bulletin shows how to include other areas considered important, such as concepts, skills, and ideas from algebra and geometry. The 80…

From "Work-and-Walk-By" to "Sherpa-at-Work"

ERIC Educational Resources Information Center

Drijvers, Paul

2011-01-01

Nowadays, many technological means are available to support teaching, such as the interactive whiteboard, class sets of laptop or netbook computers, and high speed internet access. For mathematics education there are advanced software packages for geometry, algebra, calculus, and statistics, which in many cases are available on line at no cost.…

Pre Service Teachers' Usage of Dynamic Mathematics Software

ERIC Educational Resources Information Center

Bulut, Mehmet; Bulut, Neslihan

2011-01-01

Aim of this study is about mathematics education and dynamic mathematics software. Dynamic mathematics software provides new opportunities for using both computer algebra system and dynamic geometry software. GeoGebra selected as dynamic mathematics software in this research. In this study, it is investigated that what is the usage of pre service…

Calabi's conjecture and some new results in algebraic geometry

PubMed Central

Yau, Shing-Tung

1977-01-01

We announce a proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold and then apply it to prove some new results in algebraic geometry and differential geometry. For example, we prove that the only Kähler structure on a complex projective space is the standard one. PMID:16592394

Calculation of turbulence-driven secondary motion in ducts with arbitrary cross section

NASA Technical Reports Server (NTRS)

Demuren, A. O.

1989-01-01

Calculation methods for turbulent duct flows are generalized for ducts with arbitrary cross-sections. The irregular physical geometry is transformed into a regular one in computational space, and the flow equations are solved with a finite-volume numerical procedure. The turbulent stresses are calculated with an algebraic stress model derived by simplifying model transport equations for the individual Reynolds stresses. Two variants of such a model are considered. These procedures enable the prediction of both the turbulence-driven secondary flow and the anisotropy of the Reynolds stresses, in contrast to some of the earlier calculation methods. Model predictions are compared to experimental data for developed flow in triangular duct, trapezoidal duct and a rod-bundle geometry. The correct trends are predicted, and the quantitative agreement is mostly fair. The simpler variant of the algebraic stress model procured better agreement with the measured data.

Examining Gender DIF on a Multiple-Choice Test of Mathematics: A Confirmatory Approach.

ERIC Educational Resources Information Center

Ryan, Katherine E.; Fan, Meichu

1996-01-01

Results for 3,244 female and 3,033 male junior high school students from the Second International Mathematics Study show that applied items in algebra, geometry, and computation were easier for males but arithmetic items were differentially easier for females. Implications of these findings for assessment and instruction are discussed. (SLD)

GENIE(++): A Multi-Block Structured Grid System

NASA Technical Reports Server (NTRS)

Williams, Tonya; Nadenthiran, Naren; Thornburg, Hugh; Soni, Bharat K.

1996-01-01

The computer code GENIE++ is a continuously evolving grid system containing a multitude of proven geometry/grid techniques. The generation process in GENIE++ is based on an earlier version. The process uses several techniques either separately or in combination to quickly and economically generate sculptured geometry descriptions and grids for arbitrary geometries. The computational mesh is formed by using an appropriate algebraic method. Grid clustering is accomplished with either exponential or hyperbolic tangent routines which allow the user to specify a desired point distribution. Grid smoothing can be accomplished by using an elliptic solver with proper forcing functions. B-spline and Non-Uniform Rational B-splines (NURBS) algorithms are used for surface definition and redistribution. The built in sculptured geometry definition with desired distribution of points, automatic Bezier curve/surface generation for interior boundaries/surfaces, and surface redistribution is based on NURBS. Weighted Lagrance/Hermite transfinite interpolation methods, interactive geometry/grid manipulation modules, and on-line graphical visualization of the generation process are salient features of this system which result in a significant time savings for a given geometry/grid application.

On the computation of steady Hopper flows. II: von Mises materials in various geometries

NASA Astrophysics Data System (ADS)

Gremaud, Pierre A.; Matthews, John V.; O'Malley, Meghan

2004-11-01

Similarity solutions are constructed for the flow of granular materials through hoppers. Unlike previous work, the present approach applies to nonaxisymmetric containers. The model involves ten unknowns (stresses, velocity, and plasticity function) determined by nine nonlinear first order partial differential equations together with a quadratic algebraic constraint (yield condition). A pseudospectral discretization is applied; the resulting problem is solved with a trust region method. The important role of the hopper geometry on the flow is illustrated by several numerical experiments of industrial relevance.

A network-analysis-based comparative study of the throughput behavior of polymer melts in barrier screw geometries

NASA Astrophysics Data System (ADS)

Aigner, M.; Köpplmayr, T.; Kneidinger, C.; Miethlinger, J.

2014-05-01

Barrier screws are widely used in the plastics industry. Due to the extreme diversity of their geometries, describing the flow behavior is difficult and rarely done in practice. We present a systematic approach based on networks that uses tensor algebra and numerical methods to model and calculate selected barrier screw geometries in terms of pressure, mass flow, and residence time. In addition, we report the results of three-dimensional simulations using the commercially available ANSYS Polyflow software. The major drawbacks of three-dimensional finite-element-method (FEM) simulations are that they require vast computational power and, large quantities of memory, and consume considerable time to create a geometric model created by computer-aided design (CAD) and complete a flow calculation. Consequently, a modified 2.5-dimensional finite volume method, termed network analysis is preferable. The results obtained by network analysis and FEM simulations correlated well. Network analysis provides an efficient alternative to complex FEM software in terms of computing power and memory consumption. Furthermore, typical barrier screw geometries can be parameterized and used for flow calculations without timeconsuming CAD-constructions.

Assessment of an Explicit Algebraic Reynolds Stress Model

NASA Technical Reports Server (NTRS)

Carlson, Jan-Renee

2005-01-01

This study assesses an explicit algebraic Reynolds stress turbulence model in the in the three-dimensional Reynolds averaged Navier-Stokes (RANS) solver, ISAAC (Integrated Solution Algorithm for Arbitrary Con gurations). Additionally, it compares solutions for two select configurations between ISAAC and the RANS solver PAB3D. This study compares with either direct numerical simulation data, experimental data, or empirical models for several different geometries with compressible, separated, and high Reynolds number flows. In general, the turbulence model matched data or followed experimental trends well, and for the selected configurations, the computational results of ISAAC closely matched those of PAB3D using the same turbulence model.

Connecting Functions in Geometry and Algebra

ERIC Educational Resources Information Center

Steketee, Scott; Scher, Daniel

2016-01-01

One goal of a mathematics education is that students make significant connections among different branches of mathematics. Connections--such as those between arithmetic and algebra, between two-dimensional and three-dimensional geometry, between compass-and-straight-edge constructions and transformations, and between calculus and analytic…

Nonlinear External Kink Computing with NIMROD

NASA Astrophysics Data System (ADS)

Bunkers, K. J.; Sovinec, C. R.

2016-10-01

Vertical displacement events (VDEs) during disruptions often include non-axisymmetric activity, including external kink modes, which are driven unstable as contact with the wall eats into the q-profile. The NIMROD code is being applied to study external-kink-unstable tokamak profiles in toroidal and cylindrical geometries. Simulations with external kinks show the plasma swallowing a vacuum bubble, similar to. NIMROD reproduces external kinks in both geometries, using an outer vacuum region (modeled as a plasma with a large resistivity), but as the boundary between the vacuum and plasma regions becomes more 3D, the resistivity becomes a 3D function, and it becomes more difficult for algebraic solves to converge. To help allow non-axisymmetric, nonlinear VDE calculations to proceed without restrictively small time-steps, several computational algorithms have been tested. Flexible GMRES, using a Fourier and real space representation for the toroidal angle has shown improvements. Off-diagonal preconditioning and a multigrid approach were tested and showed little improvement. A least squares finite element method (LSQFEM) has also helped improve the algebraic solve. This effort is supported by the U.S. Dept. of Energy, Award Numbers DE-FG02-06ER54850 and DE-FC02-08ER54975.

Comparison of algebraic and analytical approaches to the formulation of the statistical model-based reconstruction problem for X-ray computed tomography.

PubMed

Cierniak, Robert; Lorent, Anna

2016-09-01

The main aim of this paper is to investigate properties of our originally formulated statistical model-based iterative approach applied to the image reconstruction from projections problem which are related to its conditioning, and, in this manner, to prove a superiority of this approach over ones recently used by other authors. The reconstruction algorithm based on this conception uses a maximum likelihood estimation with an objective adjusted to the probability distribution of measured signals obtained from an X-ray computed tomography system with parallel beam geometry. The analysis and experimental results presented here show that our analytical approach outperforms the referential algebraic methodology which is explored widely in the literature and exploited in various commercial implementations. Copyright © 2016 Elsevier Ltd. All rights reserved.

Galois groups of Schubert problems via homotopy computation

NASA Astrophysics Data System (ADS)

Leykin, Anton; Sottile, Frank

2009-09-01

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in mathbb{C}^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_{6006} .

Anytime query-tuned kernel machine classifiers via Cholesky factorization

NASA Technical Reports Server (NTRS)

DeCoste, D.

2002-01-01

We recently demonstrated 2 to 64-fold query-time speedups of Support Vector Machine and Kernel Fisher classifiers via a new computational geometry method for anytime output bounds (DeCoste,2002). This new paper refines our approach in two key ways. First, we introduce a simple linear algebra formulation based on Cholesky factorization, yielding simpler equations and lower computational overhead. Second, this new formulation suggests new methods for achieving additional speedups, including tuning on query samples. We demonstrate effectiveness on benchmark datasets.

Origami, geometry and art

NASA Astrophysics Data System (ADS)

Wares, Arsalan; Elstak, Iwan

2017-02-01

The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra and geometry, like other branches of mathematics, are interrelated.

Misconceptions in Rational Numbers, Probability, Algebra, and Geometry

ERIC Educational Resources Information Center

Rakes, Christopher R.

2010-01-01

In this study, the author examined the relationship of probability misconceptions to algebra, geometry, and rational number misconceptions and investigated the potential of probability instruction as an intervention to address misconceptions in all 4 content areas. Through a review of literature, 5 fundamental concepts were identified that, if…

A new application of algebraic geometry to systems theory

NASA Technical Reports Server (NTRS)

Martin, C. F.; Hermann, R.

1976-01-01

Following an introduction to algebraic geometry, the dominant morphism theorem is stated, and the application of this theorem to systems-theoretic problems, such as the feedback problem, is discussed. The Gaussian elimination method used for solving linear equations is shown to be an example of a dominant morphism.

Case Study Projects for College Mathematics Courses Based on a Particular Function of Two Variables

ERIC Educational Resources Information Center

Shi, Y.

2007-01-01

Based on a sequence of number pairs, a recent paper (Mauch, E. and Shi, Y., 2005, Using a sequence of number pairs as an example in teaching mathematics, "Mathematics and Computer Education," 39(3), 198-205) presented some interesting examples that can be used in teaching high school and college mathematics classes such as algebra, geometry,…

Research issues of geometry-based visual languages and some solutions

NASA Astrophysics Data System (ADS)

Green, Thorn G.

This dissertation addresses the problem of how to design visual language systems that are based upon Geometric Algebra, and provide a visual coupling of algebraic expressions and geometric depictions. This coupling of algebraic expressions and geometric depictions provides a new means for expressing both mathematical and geometric relationships present in mathematics, physics, and Computer-Aided Geometric Design (CAGD). Another significant feature of such a system is that the result of changing a parameter (by dragging the mouse) can be seen immediately in the depiction(s) of all expressions that use that parameter. This greatly aides the cognition of the relationships between variables. Systems for representing such a coupling of algebra and geometry have characteristics of both visual language systems, and systems for scientific visualization. Instead of using a parsing or dataflow paradigm for the visual language representation, the systems instead represent equations as manipulatible constrained diagrams for their visualization. This requires that the design of such a system have (but is not limited to) a means for parsing equations entered by the user, a scheme for producing a visual representation of these equations; techniques for maintaining the coupling between the expressions entered and the diagrams displayed; algorithms for maintaining the consistency of the diagrams; and, indexing capabilities that are efficient enough to allow diagrams to be created, and manipulated in a short enough period of time. The author proposes solutions for how such a design can be realized.

Invariant classification of second-order conformally flat superintegrable systems

NASA Astrophysics Data System (ADS)

Capel, J. J.; Kress, J. M.

2014-12-01

In this paper we continue the work of Kalnins et al in classifying all second-order conformally-superintegrable (Laplace-type) systems over conformally flat spaces, using tools from algebraic geometry and classical invariant theory. The results obtained show, through Stäckel equivalence, that the list of known nondegenerate superintegrable systems over three-dimensional conformally flat spaces is complete. In particular, a seven-dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems. This manifold is foliated by the nonlinear action of the conformal group in three dimensions. Two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation. This foliation is explicitly described using algebraic varieties formed from representations of the conformal group. The proof of these results rely heavily on Gröbner basis calculations using the computer algebra software packages Maple and Singular.

A Comparison of Solver Performance for Complex Gastric Electrophysiology Models

PubMed Central

Sathar, Shameer; Cheng, Leo K.; Trew, Mark L.

2016-01-01

Computational techniques for solving systems of equations arising in gastric electrophysiology have not been studied for efficient solution process. We present a computationally challenging problem of simulating gastric electrophysiology in anatomically realistic stomach geometries with multiple intracellular and extracellular domains. The multiscale nature of the problem and mesh resolution required to capture geometric and functional features necessitates efficient solution methods if the problem is to be tractable. In this study, we investigated and compared several parallel preconditioners for the linear systems arising from tetrahedral discretisation of electrically isotropic and anisotropic problems, with and without stimuli. The results showed that the isotropic problem was computationally less challenging than the anisotropic problem and that the application of extracellular stimuli increased workload considerably. Preconditioning based on block Jacobi and algebraic multigrid solvers were found to have the best overall solution times and least iteration counts, respectively. The algebraic multigrid preconditioner would be expected to perform better on large problems. PMID:26736543

On genera of curves from high-loop generalized unitarity cuts

NASA Astrophysics Data System (ADS)

Huang, Rijun; Zhang, Yang

2013-04-01

Generalized unitarity cut of a Feynman diagram generates an algebraic system of polynomial equations. At high-loop levels, these equations may define a complex curve or a (hyper-)surface with complicated topology. We study the curve cases, i.e., a 4-dimensional L-loop diagram with (4 L-1) cuts. The topology of a complex curve is classified by its genus. Hence in this paper, we use computational algebraic geometry to calculate the genera of curves from two and three-loop unitarity cuts. The global structure of degenerate on-shell equations under some specific kinematic configurations is also sketched. The genus information can also be used to judge if a unitary cut solution could be rationally parameterized.

UCSMP Algebra. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2007

2007-01-01

"University of Chicago School Mathematics Project (UCSMP) Algebra," designed to increase students' skills in algebra, is appropriate for students in grades 7-10, depending on the students' incoming knowledge. This one-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and…

Mathematics: Algebra and Geometry. GED Scoreboost.

ERIC Educational Resources Information Center

Hoyt, Cathy

GED "Scoreboost" materials target exactly the skills one needs to pass the General Educational Development (GED) tests. This book focuses on the GED Mathematics test. To prepare for the test, the test taker needs to learn skills in number and operation sense, data and statistics, geometry and measurement, and algebra. To pass the test,…

Comparison of different models for non-invasive FFR estimation

NASA Astrophysics Data System (ADS)

Mirramezani, Mehran; Shadden, Shawn

2017-11-01

Coronary artery disease is a leading cause of death worldwide. Fractional flow reserve (FFR), derived from invasively measuring the pressure drop across a stenosis, is considered the gold standard to diagnose disease severity and need for treatment. Non-invasive estimation of FFR has gained recent attention for its potential to reduce patient risk and procedural cost versus invasive FFR measurement. Non-invasive FFR can be obtained by using image-based computational fluid dynamics to simulate blood flow and pressure in a patient-specific coronary model. However, 3D simulations require extensive effort for model construction and numerical computation, which limits their routine use. In this study we compare (ordered by increasing computational cost/complexity): reduced-order algebraic models of pressure drop across a stenosis; 1D, 2D (multiring) and 3D CFD models; as well as 3D FSI for the computation of FFR in idealized and patient-specific stenosis geometries. We demonstrate the ability of an appropriate reduced order algebraic model to closely predict FFR when compared to FFR from a full 3D simulation. This work was supported by the NIH, Grant No. R01-HL103419.

A Simple Introduction to Gröbner Basis Methods in String Phenomenology

NASA Astrophysics Data System (ADS)

Gray, James

In this talk I give an elementary introduction to the key algorithm used in recent applications of computational algebraic geometry to the subject of string phenomenology. I begin with a simple description of the algorithm itself and then give 3 examples of its use in physics. I describe how it can be used to obtain constraints on flux parameters, how it can simplify the equations describing vacua in 4d string models and lastly how it can be used to compute the vacuum space of the electroweak sector of the MSSM.

Algebra: Level II, Unit 8, Lesson 1; Powers and Roots: Lesson 2; Geometry: Lesson 3; Number Series: Lesson 4. Advanced General Education Program. A High School Self-Study Program.

ERIC Educational Resources Information Center

Manpower Administration (DOL), Washington, DC. Job Corps.

This self-study program for high-school level contains lessons on: Algebra, Powers and Roots, Geometry, and Number Series. Each of the lessons concludes with a Mastery Test to be completed by the student. (DB)

Large calculation of the flow over a hypersonic vehicle using a GPU

NASA Astrophysics Data System (ADS)

Elsen, Erich; LeGresley, Patrick; Darve, Eric

2008-12-01

Graphics processing units are capable of impressive computing performance up to 518 Gflops peak performance. Various groups have been using these processors for general purpose computing; most efforts have focussed on demonstrating relatively basic calculations, e.g. numerical linear algebra, or physical simulations for visualization purposes with limited accuracy. This paper describes the simulation of a hypersonic vehicle configuration with detailed geometry and accurate boundary conditions using the compressible Euler equations. To the authors' knowledge, this is the most sophisticated calculation of this kind in terms of complexity of the geometry, the physical model, the numerical methods employed, and the accuracy of the solution. The Navier-Stokes Stanford University Solver (NSSUS) was used for this purpose. NSSUS is a multi-block structured code with a provably stable and accurate numerical discretization which uses a vertex-based finite-difference method. A multi-grid scheme is used to accelerate the solution of the system. Based on a comparison of the Intel Core 2 Duo and NVIDIA 8800GTX, speed-ups of over 40× were demonstrated for simple test geometries and 20× for complex geometries.

Quantum error-correcting codes from algebraic geometry codes of Castle type

NASA Astrophysics Data System (ADS)

Munuera, Carlos; Tenório, Wanderson; Torres, Fernando

2016-10-01

We study algebraic geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.

Commutative Algebras of Toeplitz Operators in Action

NASA Astrophysics Data System (ADS)

Vasilevski, Nikolai

2011-09-01

We will discuss a quite unexpected phenomenon in the theory of Toeplitz operators on the Bergman space: the existence of a reach family of commutative C*-algebras generated by Toeplitz operators with non-trivial symbols. As it tuns out the smoothness properties of symbols do not play any role in the commutativity, the symbols can be merely measurable. Everything is governed here by the geometry of the underlying manifold, the hyperbolic geometry of the unit disk. We mention as well that the complete characterization of these commutative C*-algebras of Toeplitz operators requires the Berezin quantization procedure. These commutative algebras come with a powerful research tool, the spectral type representation for the operators under study, which permit us to answer to many important questions in the area.

Oleanna Math Program Materials.

ERIC Educational Resources Information Center

Coole, Walter A.

This document is a collection of course outlines, syllabi, and test materials designed for several high school level and lower division mathematics courses taught in an auto-tutorial learning laboratory at Skagit Valley College (Washington). The courses included are: Pre-Algebra, Basic Algebra, Plan Geometry, Intermediate Algebra, Probability and…

Computer algebra and operators

NASA Technical Reports Server (NTRS)

Fateman, Richard; Grossman, Robert

1989-01-01

The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.

Classification of digital affine noncommutative geometries

NASA Astrophysics Data System (ADS)

Majid, Shahn; Pachoł, Anna

2018-03-01

It is known that connected translation invariant n-dimensional noncommutative differentials dxi on the algebra k[x1, …, xn] of polynomials in n-variables over a field k are classified by commutative algebras V on the vector space spanned by the coordinates. These data also apply to construct differentials on the Heisenberg algebra "spacetime" with relations [xμ, xν] = λΘμν, where Θ is an antisymmetric matrix, as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field k =F2 of two elements, in which case translation invariant metrics (i.e., with constant coefficients) are equivalent to making V a Frobenius algebra. We classify all of these and their quantum Levi-Civita bimodule connections for n = 2, 3, with partial results for n = 4. For n = 2, we find 3 inequivalent differential structures admitting 1, 2, and 3 invariant metrics, respectively. For n = 3, we find 6 differential structures admitting 0, 1, 2, 3, 4, 7 invariant metrics, respectively. We give some examples for n = 4 and general n. Surprisingly, not all our geometries for n ≥ 2 have zero quantum Riemann curvature. Quantum gravity is normally seen as a weighted "sum" over all possible metrics but our results are a step towards a deeper approach in which we must also "sum" over differential structures. Over F2 we construct some of our algebras and associated structures by digital gates, opening up the possibility of "digital geometry."

Using Linear Algebra to Introduce Computer Algebra, Numerical Analysis, Data Structures and Algorithms (and To Teach Linear Algebra, Too).

ERIC Educational Resources Information Center

Gonzalez-Vega, Laureano

1999-01-01

Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)

Implementing the Curriculum and Evaluation Standards: First-Year Algebra.

ERIC Educational Resources Information Center

Kysh, Judith

1991-01-01

Described is an alternative first year algebra program developed to bridge the gap between the NCTM's Curriculum and Evaluation Standards and institutional demands of schools. Increased attention is given to graphing as a context for algebra, calculator use, solving "memorable problems," and incorporating geometry concepts, while…

Algorithms for computations of Loday algebras' invariants

NASA Astrophysics Data System (ADS)

Hussain, Sharifah Kartini Said; Rakhimov, I. S.; Basri, W.

2017-04-01

The paper is devoted to applications of some computer programs to study structural determination of Loday algebras. We present how these computer programs can be applied in computations of various invariants of Loday algebras and provide several computer programs in Maple to verify Loday algebras' identities, the isomorphisms between the algebras, as a special case, to describe the automorphism groups, centroids and derivations.

Geometry and Algebra: Glow with the Flow. NASA Connect: Program 2 in the 2000-2001 Series.

ERIC Educational Resources Information Center

National Aeronautics and Space Administration, Hampton, VA. Langley Research Center.

This teaching unit is designed to help students in grades 5 to 8 explore the concepts of geometry and algebra in the context of the force of drag. The units in the series have been developed to enhance and enrich mathematics, science, and technology education and to accommodate different teaching and learning styles. Each unit consists of…

Perceptions of 9th and 10th Grade Students on How Their Environment, Cognition, and Behavior Motivate Them in Algebra and Geometry Courses

ERIC Educational Resources Information Center

Harootunian, Alen

2012-01-01

In this study, relationships were examined between students' perception of their cognition, behavior, environment, and motivation. The purpose of the research study was to explore the extent to which 9th and 10th grade students' perception of environment, cognition, and behavior can predict their motivation in Algebra and Geometry courses. A…

Funny Face Contest: A Formative Assessment

ERIC Educational Resources Information Center

Colen, Yong S.

2010-01-01

Many American students begin their high school mathematics study with the algebra 1-geometry-algebra 2 sequence. After algebra 2, then, students with average or below-average mathematical ability face a dilemma in choosing their next mathematics course. For students to succeed in higher mathematics, understanding the concept of functions is…

University of Chicago School Mathematics Project (UCSMP) Algebra. WWC Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2009

2009-01-01

University of Chicago School Mathematics Project (UCSMP) Algebra is a one-year course covering three primary topics: (1) linear and quadratic expressions, sentences, and functions; (2) exponential expressions and functions; and (3) linear systems. Topics from geometry, probability, and statistics are integrated with the appropriate algebra.…

Application of Navier-Stokes code PAB3D with kappa-epsilon turbulence model to attached and separated flows

NASA Technical Reports Server (NTRS)

Abdol-Hamid, Khaled S.; Lakshmanan, B.; Carlson, John R.

1995-01-01

A three-dimensional Navier-Stokes solver was used to determine how accurately computations can predict local and average skin friction coefficients for attached and separated flows for simple experimental geometries. Algebraic and transport equation closures were used to model turbulence. To simulate anisotropic turbulence, the standard two-equation turbulence model was modified by adding nonlinear terms. The effects of both grid density and the turbulence model on the computed flow fields were also investigated and compared with available experimental data for subsonic and supersonic free-stream conditions.

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2006-10-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2011-03-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Gauge backgrounds and zero-mode counting in F-theory

NASA Astrophysics Data System (ADS)

Bies, Martin; Mayrhofer, Christoph; Weigand, Timo

2017-11-01

Computing the exact spectrum of charged massless matter is a crucial step towards understanding the effective field theory describing F-theory vacua in four dimensions. In this work we further develop a coherent framework to determine the charged massless matter in F-theory compactified on elliptic fourfolds, and demonstrate its application in a concrete example. The gauge background is represented, via duality with M-theory, by algebraic cycles modulo rational equivalence. Intersection theory within the Chow ring allows us to extract coherent sheaves on the base of the elliptic fibration whose cohomology groups encode the charged zero-mode spectrum. The dimensions of these cohomology groups are computed with the help of modern techniques from algebraic geometry, which we implement in the software gap. We exemplify this approach in models with an Abelian and non-Abelian gauge group and observe jumps in the exact massless spectrum as the complex structure moduli are varied. An extended mathematical appendix gives a self-contained introduction to the algebro-geometric concepts underlying our framework.

Carnegie Learning Curricula and Cognitive Tutor™. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2013

2013-01-01

"Carnegie Learning Curricula and Cognitive Tutor"®, published by Carnegie Learning, is a secondary math curricula that offers textbooks and interactive software to provide individualized, self-paced instruction based on student needs. The program includes pre-Algebra, Algebra I, Algebra II, and Geometry, as well as a three-course series…

Quantum correlations are weaved by the spinors of the Euclidean primitives

PubMed Central

2018-01-01

The exceptional Lie group E8 plays a prominent role in both mathematics and theoretical physics. It is the largest symmetry group associated with the most general possible normed division algebra, namely, that of the non-associative real octonions, which—thanks to their non-associativity—form the only possible closed set of spinors (or rotors) that can parallelize the 7-sphere. By contrast, here we show how a similar 7-sphere also arises naturally from the algebraic interplay of the graded Euclidean primitives, such as points, lines, planes and volumes, which characterize the three-dimensional conformal geometry of the ambient physical space, set within its eight-dimensional Clifford-algebraic representation. Remarkably, the resulting algebra remains associative, and allows us to understand the origins and strengths of all quantum correlations locally, in terms of the geometry of the compactified physical space, namely, that of a quaternionic 3-sphere, S3, with S7 being its algebraic representation space. Every quantum correlation can thus be understood as a correlation among a set of points of this S7, computed using manifestly local spinors within S3, thereby extending the stringent bounds of ±2 set by Bell inequalities to the bounds of ±22 on the strengths of all possible strong correlations, in the same quantitatively precise manner as that predicted within quantum mechanics. The resulting geometrical framework thus overcomes Bell’s theorem by producing a strictly deterministic and realistic framework that allows a locally causal understanding of all quantum correlations, without requiring either remote contextuality or backward causation. We demonstrate this by first proving a general theorem concerning the geometrical origins of the correlations predicted by arbitrarily entangled quantum states, and then reproducing the correlations predicted by the EPR-Bohm and the GHZ states. The raison d’être of strong correlations turns out to be the Möbius-like twists in the Hopf bundles of S3 and S7. PMID:29893385

An algebra of reversible computation.

PubMed

Wang, Yong

2016-01-01

We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules: basic reversible processes algebra, algebra of reversible communicating processes, recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.

Experimental Aeroheating Study of Mid-L/D Entry Vehicle Geometries: NASA LaRC 20-Inch Mach 6 Air Tunnel Test 6966

NASA Technical Reports Server (NTRS)

Hollis, Brian R.; Hollingsworth, Kevin E.

2014-01-01

Aeroheating data on mid lift-to-drag ratio entry vehicle configurations has been obtained through hypersonic wind tunnel testing. Vehicles of this class have been proposed for high-mass Mars missions, such as sample return and crewed exploration, for which the conventional sphere-cone entry vehicle geometries of previous Mars missions are insufficient. Several configurations were investigated, including elliptically-blunted cylinders with both circular and elliptical cross sections, biconic geometries based on launch vehicle dual-use shrouds, and parametrically-optimized analytic geometries. Testing was conducted at Mach 6 over a range of Reynolds numbers sufficient to generate laminar, transitional, and turbulent flow. Global aeroheating data were obtained using phosphor thermography. Both stream-wise and cross-flow transition occured on different configurations. Comparisons were made with laminar and turbulent computational predictions generated with an algebraic turbulence model. Predictions were generally in good agreement in regions of laminar or fully-turbulent flow; however for transitional cases, the lack of a transition onset prediction capability produced less accurate comparisons. The data obtained in this study are intended to be used for prelimary mission design studies and the development and validation of computational methods.

Lectures on Kähler Geometry - Series: London Mathematical Society Student Texts (No. 69)

NASA Astrophysics Data System (ADS)

Moroianu, Andrei

2004-03-01

Kähler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. Kähler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous Kähler identities. The final part of the text studies several aspects of compact Kähler manifolds: the Calabi conjecture, Weitzenböck techniques, Calabi Yau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory. The first graduate-level text on Kähler geometry, providing a concise introduction for both mathematicians and physicists with a basic knowledge of calculus in several variables and linear algebra Over 130 exercises and worked examples Self-contained and presents varying viewpoints including Riemannian, complex and algebraic

Prospective Teachers' Views on the Use of Calculators with Computer Algebra System in Algebra Instruction

ERIC Educational Resources Information Center

Ozgun-Koca, S. Ash

2010-01-01

Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This…

Assessing Mathematics Automatically Using Computer Algebra and the Internet

ERIC Educational Resources Information Center

Sangwin, Chris

2004-01-01

This paper reports some recent developments in mathematical computer-aided assessment which employs computer algebra to evaluate students' work using the Internet. Technical and educational issues raised by this use of computer algebra are addressed. Working examples from core calculus and algebra which have been used with first year university…

Packing a Box with Bricks.

ERIC Educational Resources Information Center

Jepsen, Charles H.

1991-01-01

Presented are solutions to variations of a combinatorics problem from a recent International Mathematics Olympiad. In particular, the matrix algebra solution illustrates an interaction among the undergraduate areas of geometry, combinatorics, linear algebra, and group theory. (JJK)

Deriving Laws from Ordering Relations

NASA Technical Reports Server (NTRS)

Knuth, Kevin H.

2003-01-01

It took much effort in the early days of non-Euclidean geometry to break away from the mindset that all spaces are flat and that two distinct parallel lines do not cross. Up to that point, all that was known was Euclidean geometry, and it was difficult to imagine anything else. We have suffered a similar handicap brought on by the enormous relevance of Boolean algebra to the problems of our age-logic and set theory. Previously, I demonstrated that the algebra of questions is not Boolean, but rather is described by the free distributive algebra. To get to this stage took much effort, as many obstacles-most self-placed-had to be overcome. As Boolean algebras were all I had ever known, it was almost impossible for me to imagine working with an algebra where elements do not have complements. With this realization, it became very clear that the sum and product rules of probability theory at the most basic level had absolutely nothing to do with the Boolean algebra of logical statements. Instead, a measure of degree of inclusion can be invented for many different partially ordered sets, and the sum and product rules fall out of the associativity and distributivity of the algebra. To reinforce this very important idea, this paper will go over how these constructions are made, while focusing on the underlying assumptions. I will derive the sum and product rules for a distributive lattice in general and demonstrate how this leads to probability theory on the Boolean lattice and is related to the calculus of quantum mechanical amplitudes on the partially ordered set of experimental setups. I will also discuss the rules that can be derived from modular lattices and their relevance to the cross-ratio of projective geometry.

Matrix De Rham Complex and Quantum A-infinity algebras

NASA Astrophysics Data System (ADS)

Barannikov, S.

2014-04-01

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A ∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin-Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006-48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein-Leites simple associative algebras with odd trace q( N), and gl( N| N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin-Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.

Computations of Internal and External Axisymmetric Nozzle Aerodynamics at Transonic Speeds

NASA Technical Reports Server (NTRS)

Dalbello, Teryn; Georgiadis, Nicholas; Yoder, Dennis; Keith, Theo

2003-01-01

Computational Fluid Dynamics (CFD) analyses of axisymmetric circular-arc boattail nozzles have been completed in support of NASA's Next Generation Launch Technology Program to investigate the effects of high-speed nozzle geometries on the nozzle internal flow and the surrounding boattail regions. These computations span the very difficult transonic flight regime, with shock-induced separations and strong adverse pressure gradients. External afterbody and internal nozzle pressure distributions computed with the Wind code are compared with experimental data. A range of turbulence models were examined in Wind, including an Explicit Algebraic Stress model (EASM). Computations on two nozzle geometries have been completed at freestream Mach numbers ranging from 0.6 to 0.9, driven by nozzle pressure ratios (NPR) ranging from 2.9 to 5. Results obtained on converging-only geometry indicate reasonable agreement to experimental data, with the EASM and Shear Stress Transport (SST) turbulence models providing the best agreement. Calculations completed on a converging-diverging geometry involving large-scale internal flow separation did not converge to a true steady-state solution when run with variable timestepping (steady-state). Calculations obtained using constant timestepping (time-accurate) indicate less variations in flow properties compared with steady-state solutions. This failure to converge to a steady-state solution was found to be the result of difficulties in using variable time-stepping with large-scale separations present in the flow. Nevertheless, time-averaged boattail surface pressure coefficient and internal nozzle pressures show fairly good agreement with experimental data. The SST turbulence model demonstrates the best over-all agreement with experimental data.

Geometric Algebra for Physicists

NASA Astrophysics Data System (ADS)

Doran, Chris; Lasenby, Anthony

2007-11-01

Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.

Finite-dimensional integrable systems: A collection of research problems

NASA Astrophysics Data System (ADS)

Bolsinov, A. V.; Izosimov, A. M.; Tsonev, D. M.

2017-05-01

This article suggests a series of problems related to various algebraic and geometric aspects of integrability. They reflect some recent developments in the theory of finite-dimensional integrable systems such as bi-Poisson linear algebra, Jordan-Kronecker invariants of finite dimensional Lie algebras, the interplay between singularities of Lagrangian fibrations and compatible Poisson brackets, and new techniques in projective geometry.

Generalizing the extensibility of a dynamic geometry software

NASA Astrophysics Data System (ADS)

Herceg, Đorđe; Radaković, Davorka; Herceg, Dejana

2012-09-01

Plug-and-play visual components in a Dynamic Geometry Software (DGS) enable development of visually attractive, rich and highly interactive dynamic drawings. We are developing SLGeometry, a DGS that contains a custom programming language, a computer algebra system (CAS engine) and a graphics subsystem. The basic extensibility framework on SLGeometry supports dynamic addition of new functions from attribute annotated classes that implement runtime metadata registration in code. We present a general plug-in framework for dynamic importing of arbitrary Silverlight user interface (UI) controls into SLGeometry at runtime. The CAS engine maintains a metadata storage that describes each imported visual component and enables two-way communication between the expressions stored in the engine and the UI controls on the screen.

Low Density Parity Check Codes Based on Finite Geometries: A Rediscovery and More

NASA Technical Reports Server (NTRS)

Kou, Yu; Lin, Shu; Fossorier, Marc

1999-01-01

Low density parity check (LDPC) codes with iterative decoding based on belief propagation achieve astonishing error performance close to Shannon limit. No algebraic or geometric method for constructing these codes has been reported and they are largely generated by computer search. As a result, encoding of long LDPC codes is in general very complex. This paper presents two classes of high rate LDPC codes whose constructions are based on finite Euclidean and projective geometries, respectively. These classes of codes a.re cyclic and have good constraint parameters and minimum distances. Cyclic structure adows the use of linear feedback shift registers for encoding. These finite geometry LDPC codes achieve very good error performance with either soft-decision iterative decoding based on belief propagation or Gallager's hard-decision bit flipping algorithm. These codes can be punctured or extended to obtain other good LDPC codes. A generalization of these codes is also presented.

Computing Gröbner Bases within Linear Algebra

NASA Astrophysics Data System (ADS)

Suzuki, Akira

In this paper, we present an alternative algorithm to compute Gröbner bases, which is based on computations on sparse linear algebra. Both of S-polynomial computations and monomial reductions are computed in linear algebra simultaneously in this algorithm. So it can be implemented to any computational system which can handle linear algebra. For a given ideal in a polynomial ring, it calculates a Gröbner basis along with the corresponding term order appropriately.

Dimensional analysis using toric ideals: primitive invariants.

PubMed

Atherton, Mark A; Bates, Ronald A; Wynn, Henry P

2014-01-01

Classical dimensional analysis in its original form starts by expressing the units for derived quantities, such as force, in terms of power products of basic units [Formula: see text] etc. This suggests the use of toric ideal theory from algebraic geometry. Within this the Graver basis provides a unique primitive basis in a well-defined sense, which typically has more terms than the standard Buckingham approach. Some textbook examples are revisited and the full set of primitive invariants found. First, a worked example based on convection is introduced to recall the Buckingham method, but using computer algebra to obtain an integer [Formula: see text] matrix from the initial integer [Formula: see text] matrix holding the exponents for the derived quantities. The [Formula: see text] matrix defines the dimensionless variables. But, rather than this integer linear algebra approach it is shown how, by staying with the power product representation, the full set of invariants (dimensionless groups) is obtained directly from the toric ideal defined by [Formula: see text]. One candidate for the set of invariants is a simple basis of the toric ideal. This, although larger than the rank of [Formula: see text], is typically not unique. However, the alternative Graver basis is unique and defines a maximal set of invariants, which are primitive in a simple sense. In addition to the running example four examples are taken from: a windmill, convection, electrodynamics and the hydrogen atom. The method reveals some named invariants. A selection of computer algebra packages is used to show the considerable ease with which both a simple basis and a Graver basis can be found.

A Mathematics Software Database Update.

ERIC Educational Resources Information Center

Cunningham, R. S.; Smith, David A.

1987-01-01

Contains an update of an earlier listing of software for mathematics instruction at the college level. Topics are: advanced mathematics, algebra, calculus, differential equations, discrete mathematics, equation solving, general mathematics, geometry, linear and matrix algebra, logic, statistics and probability, and trigonometry. (PK)

The 1984 ARI Survey of Army Recruits: Supplementary User’s Manual for October 1984/February 1985 Administration

DTIC Science & Technology

1986-05-01

league baseball playoffs 106. World Series 116. Which of the following mathematics 107. NBA basketball and technical courses, if any, did you take and pass...baseball playoffs 94. World Series (Mark all that apply) 95. NBA bdsketball A. Elementary Algebra B. Plane Geometry e 96. College basketball C...in high school? 108. College basketball (Mark all that apply) 109. NHL hockey A. Elementary Algebra 110. Professional wrestling S. Plane Geometry C

Geometry of quantum state manifolds generated by the Lie algebra operators

NASA Astrophysics Data System (ADS)

Kuzmak, A. R.

2018-03-01

The Fubini-Study metric of quantum state manifold generated by the operators which satisfy the Heisenberg Lie algebra is calculated. The similar problem is studied for the manifold generated by the so(3) Lie algebra operators. Using these results, we calculate the Fubini-Study metrics of state manifolds generated by the position and momentum operators. Also the metrics of quantum state manifolds generated by some spin systems are obtained. Finally, we generalize this problem for operators of an arbitrary Lie algebra.

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

DTIC Science & Technology

2014-11-01

linear hybrid systems by linear algebraic methods. In SAS, volume 6337 of LNCS, pages 373–389. Springer, 2010. [19] E. W. Mayr. Membership in polynomial...383–394, 2009. [31] A. Tarski. A decision method for elementary algebra and geometry. Bull. Amer. Math. Soc., 59, 1951. [32] A. Tiwari. Abstractions...A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 November 2014 CMU

Heat transfer predictions for two turbine nozzle geometries at high Reynolds and Mach numbers

NASA Technical Reports Server (NTRS)

Boyle, R. J.; Jackson, R.

1995-01-01

Predictions of turbine vane and endwall heat transfer and pressure distributions are compared with experimental measurements for two vane geometries. The differences in geometries were due to differences in the hub profile, and both geometries were derived from the design of a high rim speed turbine (HRST). The experiments were conducted in the Isentropic Light Piston Facility (ILPF) at Pyestock at a Reynolds number of 5.3 x 10(exp 6), a Mach number of 1.2, and a wall-to-gas temperature ratio of 0.66. Predictions are given for two different steady-state three-dimensional Navier-Stokes computational analyses. C-type meshes were used, and algebraic models were employed to calculate the turbulent eddy viscosity. The effects of different turbulence modeling assumptions on the predicted results are examined. Comparisons are also given between predicted and measured total pressure distributions behind the vane. The combination of realistic engine geometries and flow conditions proved to be quite demanding in terms of the convergence of the CFD solutions. An appropriate method of grid generation, which resulted in consistently converged CFD solutions, was identified.

Geometry for Pie Lovers.

ERIC Educational Resources Information Center

Fisher, William

1982-01-01

An approach to the instruction of maxima and minima problems that works with tools of geometry and algebra is presented. The focus is on a classic pie-cutting problem, which is viewed as an interesting and instructive task that is an excellent application of transformation geometry. (MP)

Topics for Mathematics Clubs.

ERIC Educational Resources Information Center

Dalton, LeRoy C., Ed.; Snyder, Henry D., Ed.

The ten chapters in this booklet cover topics not ordinarily discussed in the classroom: Fibonacci sequences, projective geometry, groups, infinity and transfinite numbers, Pascal's Triangle, topology, experiments with natural numbers, non-Euclidean geometries, Boolean algebras, and the imaginary and the infinite in geometry. Each chapter is…

DOE Office of Scientific and Technical Information (OSTI.GOV)

Agarwala, Susama; Delaney, Colleen

This paper defines a generalization of the Connes-Moscovici Hopf algebra, H(1), that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in non-commutative geometry, and the latter, a much studied object in perturbative quantum field theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.

Transonic Navier-Stokes computations of strake-generated vortex interactions for a fighter-like configuration

NASA Technical Reports Server (NTRS)

Reznick, Steve

1988-01-01

Transonic Euler/Navier-Stokes computations are accomplished for wing-body flow fields using a computer program called Transonic Navier-Stokes (TNS). The wing-body grids are generated using a program called ZONER, which subdivides a coarse grid about a fighter-like aircraft configuration into smaller zones, which are tailored to local grid requirements. These zones can be either finely clustered for capture of viscous effects, or coarsely clustered for inviscid portions of the flow field. Different equation sets may be solved in the different zone types. This modular approach also affords the opportunity to modify a local region of the grid without recomputing the global grid. This capability speeds up the design optimization process when quick modifications to the geometry definition are desired. The solution algorithm embodied in TNS is implicit, and is capable of capturing pressure gradients associated with shocks. The algebraic turbulence model employed has proven adequate for viscous interactions with moderate separation. Results confirm that the TNS program can successfully be used to simulate transonic viscous flows about complicated 3-D geometries.

An Experience of the Czechoslovakian Experimental Center

ERIC Educational Resources Information Center

Vysin, J.

1975-01-01

The Czechoslovakian Academy of Sciences is sponsoring an experimental approach to the modernization of the geometry curriculum. Geometry is viewed as ancillary to other parts of the curriculum and is taught as appropriate to other subjects (e.g., algebra). Combinatorial geometry is taught formally. (SD)

From geometry to algebra and vice versa: Realistic mathematics education principles for analyzing geometry tasks

NASA Astrophysics Data System (ADS)

Jupri, Al

2017-04-01

In this article we address how Realistic Mathematics Education (RME) principles, including the intertwinement and the reality principles, are used to analyze geometry tasks. To do so, we carried out three phases of a small-scale study. First we analyzed four geometry problems - considered as tasks inviting the use of problem solving and reasoning skills - theoretically in the light of the RME principles. Second, we tested two problems to 31 undergraduate students of mathematics education program and other two problems to 16 master students of primary mathematics education program. Finally, we analyzed student written work and compared these empirical to the theoretical results. We found that there are discrepancies between what we expected theoretically and what occurred empirically in terms of mathematization and of intertwinement of mathematical concepts from geometry to algebra and vice versa. We conclude that the RME principles provide a fruitful framework for analyzing geometry tasks that, for instance, are intended for assessing student problem solving and reasoning skills.

The Standard Model in noncommutative geometry: fundamental fermions as internal forms

NASA Astrophysics Data System (ADS)

Dąbrowski, Ludwik; D'Andrea, Francesco; Sitarz, Andrzej

2018-05-01

Given the algebra, Hilbert space H, grading and real structure of the finite spectral triple of the Standard Model, we classify all possible Dirac operators such that H is a self-Morita equivalence bimodule for the associated Clifford algebra.

Solving Geometric Problems by Using Algebraic Representation for Junior High School Level 3 in Van Hiele at Geometric Thinking Level

ERIC Educational Resources Information Center

Suwito, Abi; Yuwono, Ipung; Parta, I. Nengah; Irawati, Santi; Oktavianingtyas, Ervin

2016-01-01

This study aims to determine the ability of algebra students who have 3 levels van Hiele levels. Follow its framework Dindyal framework (2007). Students are required to do 10 algebra shaped multiple choice, then students work 15 about the geometry of the van Hiele level in the form of multiple choice questions. The question has been tested levels…

Linking Computer Algebra Systems and Paper-and-Pencil Techniques To Support the Teaching of Mathematics.

ERIC Educational Resources Information Center

van Herwaarden, Onno A.; Gielen, Joseph L. W.

2002-01-01

Focuses on students showing a lack of conceptual insight while using computer algebra systems (CAS) in the setting of an elementary calculus and linear algebra course for first year university students in social sciences. The use of a computer algebra environment has been incorporated into a more traditional course but with special attention on…

Software Reviews.

ERIC Educational Resources Information Center

Bitter, Gary G., Ed.

1989-01-01

Describes three software packages: (1) "MacMendeleev"--database/graphic display for chemistry, grades 10-12, Macintosh; (2) "Geometry One: Foundations"--geometry tutorial, grades 7-12, IBM; (3) "Mathematics Exploration Toolkit"--algebra and calculus tutorial, grades 8-12, IBM. (MVL)

Secondary School Mathematics Curriculum Improvement Study Information Bulletin 7.

ERIC Educational Resources Information Center

Secondary School Mathematics Curriculum Improvement Study, New York, NY.

The background, objectives, and design of Secondary School Mathematics Curriculum Improvement Study (SSMCIS) are summarized. Details are given of the content of the text series, "Unified Modern Mathematics," in the areas of algebra, geometry, linear algebra, probability and statistics, analysis (calculus), logic, and computer…

Mathematics Unit Plans. PACE '94.

ERIC Educational Resources Information Center

Wiles, Clyde A., Ed.; Schoon, Kenneth J., Ed.

This booklet contains mathematics unit plans for Algebra 1, Geometry, Math for Technology, Mathematical Problem Solving, and Pre-Algebra developed by PACE (Promoting Academic Excellence In Mathematics, Science & Technology for Workers of the 21st Century). Each unit plan contains suggested timing, objectives, skills to be acquired, workplace…

Local algebraic analysis of differential systems

NASA Astrophysics Data System (ADS)

Kaptsov, O. V.

2015-06-01

We propose a new approach for studying the compatibility of partial differential equations. This approach is a synthesis of the Riquier method, Gröbner basis theory, and elements of algebraic geometry. As applications, we consider systems including the wave equation and the sine-Gordon equation.

Strategies Toward Automation of Overset Structured Surface Grid Generation

NASA Technical Reports Server (NTRS)

Chan, William M.

2017-01-01

An outline of a strategy for automation of overset structured surface grid generation on complex geometries is described. The starting point of the process consists of an unstructured surface triangulation representation of the geometry derived from a native CAD, STEP, or IGES definition, and a set of discretized surface curves that captures all geometric features of interest. The procedure for surface grid generation is decomposed into an algebraic meshing step, a hyperbolic meshing step, and a gap-filling step. This paper will focus primarily on the high-level plan with details on the algebraic step. The algorithmic procedure for the algebraic step involves analyzing the topology of the network of surface curves, distributing grid points appropriately on these curves, identifying domains bounded by four curves that can be meshed algebraically, concatenating the resulting grids into fewer patches, and extending appropriate boundaries of the concatenated grids to provide proper overlap. Results are presented for grids created on various aerospace vehicle components.

Combinatorial Formulas for Characteristic Classes, and Localization of Secondary Topological Invariants.

NASA Astrophysics Data System (ADS)

Smirnov, Mikhail

1995-01-01

The problems solved in this thesis originated from combinatorial formulas for characteristic classes. This thesis deals with Chern-Simons classes, their generalizations and related algebraic and analytic problems. (1) In this thesis, I describe a new class of algebras whose elements contain Chern and generalized Chern -Simons classes. There is a Poisson bracket in these algebras, similar to the bracket in Kontsevich's noncommutative symplectic geometry (Kon). I prove that the Poisson bracket gives rise to a graded Lie algebra containing differential forms representing Chern and Chern-Simons classes. This is a new result. I describe algebraic analogs of the dilogarithm and higher polylogarithms in the algebra corresponding to Chern-Simons classes. (2) I study the properties of this bracket. It is possible to write the exterior differential and other operations in the algebra using this bracket. The bracket of any two Chern classes is zero and the bracket of a Chern class and a Chern-Simons class is d-closed. The construction developed here easily gives explicit formulas for known secondary classes and makes it possible to construct new ones. (3) I develop an algebraic model for the action of the gauge group and describe how elements of algebra corresponding to the secondary characteristic classes change under this action (see theorem 3 page xi). (4) It is possible give new explicit formulas for cocycles on a gauge group of a bundle and for the corresponding cocycles on the Lie algebra of the gauge group. I use formulas for secondary characteristic classes and an algebraic approach developed in chapter 1. I also use the work of Faddeev, Reiman and Semyonov-Tian-Shanskii (FRS) on cocycles as quantum anomalies. (5) I apply the methods of differential geometry of formal power series to construct universal characteristic and secondary characteristic classes. Given a pair of gauge equivalent connections using local formulas I obtain dilogarithmic and trilogarithmic analogs of Chern-Simons classes.

Curriculum Guide for Baccalaureate Oriented Courses in Mathematics.

ERIC Educational Resources Information Center

Darnes, G. Robert, Ed.

A mathematics curriculum guide is presented for the purpose of offering statewide guidelines to colleges for determining the content of those courses which might be considered standard courses in the first two years of the college curriculum. Courses covered include: intermediate algebra, college algebra, trigonometry, analytic geometry,…

Graphs and Zero-Divisors

ERIC Educational Resources Information Center

Axtell, M.; Stickles, J.

2010-01-01

The last ten years have seen an explosion of research in the zero-divisor graphs of commutative rings--by professional mathematicians "and" undergraduates. The objective is to find algebraic information within the geometry of these graphs. This topic is approachable by anyone with one or two semesters of abstract algebra. This article gives the…

Studies in Mathematics, Volume X. Applied Mathematics in the High School.

ERIC Educational Resources Information Center

Schiffer, Max M.

This publication contains a sequence of lectures given to high school mathematics teachers by the author. Applications of mathematics emphasized are elementary algebra, geometry, and matrix algebra. Included are: (1) an introduction concerning teaching applications of mathematics; (2) Chapter 1: Mechanics for the High School Student; (3) Chapter…

DOE Office of Scientific and Technical Information (OSTI.GOV)

Buchstaber, V M; Ustinov, A V

We describe the coefficient rings of universal formal group laws which arise in algebraic geometry, algebraic topology and their application to mathematical physics. We also describe the homomorphisms of these coefficient rings coming from reductions of one formal group law to another. The proofs are based on the number-theoretic properties of binomial coefficients. Bibliography: 37 titles.

Towards Cohomology of Renormalization: Bigrading the Combinatorial Hopf Algebra of Rooted Trees

NASA Astrophysics Data System (ADS)

Broadhurst, D. J.; Kreimer, D.

The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, ℌR, generated by a single primitive divergence, solves a universal problem in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the cocommutative subalgebra ℌladder of pure ladder diagrams and the Connes-Moscovici noncocommutative subalgebra ℌCM of noncommutative geometry. These three Hopf algebras admit a bigrading by n, the number of nodes, and an index k that specifies the degree of primitivity. In each case, we use iterations of the relevant coproduct to compute the dimensions of subspaces with modest values of n and k and infer a simple generating procedure for the remainder. The results for ℌladder are familiar from the theory of partitions, while those for ℌCM involve novel transforms of partitions. Most beautiful is the bigrading of ℌR, the largest of the three. Thanks to Sloane's superseeker, we discovered that it saturates all possible inequalities. We prove this by using the universal Hochschild-closed one-cocycle B+, which plugs one set of divergences into another, and by generalizing the concept of natural growth beyond that entailed by the Connes-Moscovici case. We emphasize the yet greater challenge of handling the infinite set of decorations of realistic quantum field theory.

Implementing Computer Algebra Enabled Questions for the Assessment and Learning of Mathematics

ERIC Educational Resources Information Center

Sangwin, Christopher J.; Naismith, Laura

2008-01-01

We present principles for the design of an online system to support computer algebra enabled questions for use within the teaching and learning of mathematics in higher education. The introduction of a computer algebra system (CAS) into a computer aided assessment (CAA) system affords sophisticated response processing of student provided answers.…

Geometry and physics

PubMed Central

Atiyah, Michael; Dijkgraaf, Robbert; Hitchin, Nigel

2010-01-01

We review the remarkably fruitful interactions between mathematics and quantum physics in the past decades, pointing out some general trends and highlighting several examples, such as the counting of curves in algebraic geometry, invariants of knots and four-dimensional topology. PMID:20123740

Fractal Patterns and Chaos Games

ERIC Educational Resources Information Center

Devaney, Robert L.

2004-01-01

Teachers incorporate the chaos game and the concept of a fractal into various areas of the algebra and geometry curriculum. The chaos game approach to fractals provides teachers with an opportunity to help students comprehend the geometry of affine transformations.

Mathematical biology modules based on modern molecular biology and modern discrete mathematics.

PubMed

Robeva, Raina; Davies, Robin; Hodge, Terrell; Enyedi, Alexander

2010-01-01

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to introduce students to mathematical methods beyond the interface of biology with calculus. Based on ongoing research, and designed to use the project-based-learning approach, the modules highlight applications of modern discrete mathematics and algebraic statistics to pressing problems in molecular biology. For the majority of projects, calculus is not a required prerequisite and, due to the modest amount of mathematical background needed for some of the modules, the materials can be used for an early introduction to mathematical modeling. At the same time, most modules are connected with topics in linear and abstract algebra, algebraic geometry, and probability, and they can be used as meaningful applied introductions into the relevant advanced-level mathematics courses. Open-source software is used to facilitate the relevant computations. As a detailed example, we outline a module that focuses on Boolean models of the lac operon network.

Mathematical Biology Modules Based on Modern Molecular Biology and Modern Discrete Mathematics

PubMed Central

Davies, Robin; Hodge, Terrell; Enyedi, Alexander

2010-01-01

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to introduce students to mathematical methods beyond the interface of biology with calculus. Based on ongoing research, and designed to use the project-based-learning approach, the modules highlight applications of modern discrete mathematics and algebraic statistics to pressing problems in molecular biology. For the majority of projects, calculus is not a required prerequisite and, due to the modest amount of mathematical background needed for some of the modules, the materials can be used for an early introduction to mathematical modeling. At the same time, most modules are connected with topics in linear and abstract algebra, algebraic geometry, and probability, and they can be used as meaningful applied introductions into the relevant advanced-level mathematics courses. Open-source software is used to facilitate the relevant computations. As a detailed example, we outline a module that focuses on Boolean models of the lac operon network. PMID:20810955

Some applications of mathematics in theoretical physics - A review

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bora, Kalpana

2016-06-21

Mathematics is a very beautiful subject−very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like−differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical toolsmore » are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.« less

Higher symmetries of the Schrödinger operator in Newton-Cartan geometry

NASA Astrophysics Data System (ADS)

Gundry, James

2017-03-01

We establish several relationships between the non-relativistic conformal symmetries of Newton-Cartan geometry and the Schrödinger equation. In particular we discuss the algebra sch(d) of vector fields conformally-preserving a flat Newton-Cartan spacetime, and we prove that its curved generalisation generates the symmetry group of the covariant Schrödinger equation coupled to a Newtonian potential and generalised Coriolis force. We provide intrinsic Newton-Cartan definitions of Killing tensors and conformal Schrödinger-Killing tensors, and we discuss their respective links to conserved quantities and to the higher symmetries of the Schrödinger equation. Finally we consider the role of conformal symmetries in Newtonian twistor theory, where the infinite-dimensional algebra of holomorphic vector fields on twistor space corresponds to the symmetry algebra cnc(3) on the Newton-Cartan spacetime.

Some applications of mathematics in theoretical physics - A review

NASA Astrophysics Data System (ADS)

Bora, Kalpana

2016-06-01

Mathematics is a very beautiful subject-very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like-differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical tools are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.

Category-theoretic models of algebraic computer systems

NASA Astrophysics Data System (ADS)

Kovalyov, S. P.

2016-01-01

A computer system is said to be algebraic if it contains nodes that implement unconventional computation paradigms based on universal algebra. A category-based approach to modeling such systems that provides a theoretical basis for mapping tasks to these systems' architecture is proposed. The construction of algebraic models of general-purpose computations involving conditional statements and overflow control is formally described by a reflector in an appropriate category of algebras. It is proved that this reflector takes the modulo ring whose operations are implemented in the conventional arithmetic processors to the Łukasiewicz logic matrix. Enrichments of the set of ring operations that form bases in the Łukasiewicz logic matrix are found.

Sound Off! Don't Sacrifice Geometry on the Common Core Altar

ERIC Educational Resources Information Center

Nirode, Wayne

2013-01-01

Although high school geometry could be a meaningful course in exploring, reasoning, proving, and communicating, it often lacks authentic proof and has become just another course in algebra. This article examines why geometry is important to learn and provides an outline of what that learning experience should be.

The Pontryagin class for pre-Courant algebroids

NASA Astrophysics Data System (ADS)

Liu, Zhangju; Sheng, Yunhe; Xu, Xiaomeng

2016-06-01

In this paper, we show that the Jacobiator J of a pre-Courant algebroid is closed naturally. The corresponding equivalence class [J♭ ] is defined as the Pontryagin class, which is the obstruction of a pre-Courant algebroid to be deformed into a Courant algebroid. We construct a Leibniz 2-algebra and a Lie 2-algebra associated to a pre-Courant algebroid and prove that these algebraic structures are isomorphic under deformations. Finally, we introduce the twisted action of a Lie algebra on a manifold to give more examples of pre-Courant algebroids, which include the Cartan geometry.

Labeled trees and the efficient computation of derivations

NASA Technical Reports Server (NTRS)

Grossman, Robert; Larson, Richard G.

1989-01-01

The effective parallel symbolic computation of operators under composition is discussed. Examples include differential operators under composition and vector fields under the Lie bracket. Data structures consisting of formal linear combinations of rooted labeled trees are discussed. A multiplication on rooted labeled trees is defined, thereby making the set of these data structures into an associative algebra. An algebra homomorphism is defined from the original algebra of operators into this algebra of trees. An algebra homomorphism from the algebra of trees into the algebra of differential operators is then described. The cancellation which occurs when noncommuting operators are expressed in terms of commuting ones occurs naturally when the operators are represented using this data structure. This leads to an algorithm which, for operators which are derivations, speeds up the computation exponentially in the degree of the operator. It is shown that the algebra of trees leads naturally to a parallel version of the algorithm.

Application of Computer Graphics to Graphing in Algebra and Trigonometry. Final Report.

ERIC Educational Resources Information Center

Morris, J. Richard

This project was designed to improve the graphing competency of students in elementary algebra, intermediate algebra, and trigonometry courses at Virginia Commonwealth University. Computer graphics programs were designed using an Apple II Plus computer and implemented using Pascal. The software package is interactive and gives students control…

Assessing Elementary Algebra with STACK

ERIC Educational Resources Information Center

Sangwin, Christopher J.

2007-01-01

This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…

RIACS

NASA Technical Reports Server (NTRS)

Oliger, Joseph

1997-01-01

Topics considered include: high-performance computing; cognitive and perceptual prostheses (computational aids designed to leverage human abilities); autonomous systems. Also included: development of a 3D unstructured grid code based on a finite volume formulation and applied to the Navier-stokes equations; Cartesian grid methods for complex geometry; multigrid methods for solving elliptic problems on unstructured grids; algebraic non-overlapping domain decomposition methods for compressible fluid flow problems on unstructured meshes; numerical methods for the compressible navier-stokes equations with application to aerodynamic flows; research in aerodynamic shape optimization; S-HARP: a parallel dynamic spectral partitioner; numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains; application of high-order shock capturing schemes to direct simulation of turbulence; multicast technology; network testbeds; supercomputer consolidation project.

Weaving Geometry and Algebra Together

ERIC Educational Resources Information Center

Cetner, Michelle

2015-01-01

When thinking about student reasoning and sense making, teachers must consider the nature of tasks given to students along with how to plan to use the tasks in the classroom. Students should be presented with tasks in a way that encourages them to draw connections between algebraic and geometric concepts. This article focuses on the idea that it…

A Subject Classification of Math Lab Activities from School Science and Mathematics 1974-1980.

ERIC Educational Resources Information Center

Grinstein, Louise S.

1982-01-01

Presented here is an index which indicates the title and location of each activity by volume and page numbers. The majority of items relate to arithmetic, elementary algebra, and plane geometry, but material also covers such topics as statistics, probability, trigonometry set theory, topology, and modern algebra. (MP)

The Symmetry Group of the Permutahedron

ERIC Educational Resources Information Center

Crisman, Karl-Dieter

2011-01-01

Although it can be visualized fairly easily and its symmetry group is easy to calculate, the permutahedron is a somewhat neglected combinatorial object. We propose it as a useful case study in abstract algebra. It supplies concrete examples of group actions, the difference between right and left actions, and how geometry and algebra can work…

University of Chicago School Mathematics Project 6-12 Curriculum. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2011

2011-01-01

The "University of Chicago School Mathematics Project ("UCSMP") 6-12 Curriculum" is a series of yearlong courses--(1) Transition Mathematics; (2) Algebra; (3) Geometry; (4) Advanced Algebra; (5) Functions, Statistics, and Trigonometry; and (6) Precalculus and Discrete Mathematics--emphasizing problem solving, real-world applications, and the use…

Enhancing Mathematical Communication for Virtual Math Teams

ERIC Educational Resources Information Center

Stahl, Gerry; Çakir, Murat Perit; Weimar, Stephen; Weusijana, Baba Kofi; Ou, Jimmy Xiantong

2010-01-01

The Math Forum is an online resource center for pre-algebra, algebra, geometry and pre-calculus. Its Virtual Math Teams (VMT) service provides an integrated web-based environment for small teams of people to discuss math and to work collaboratively on math problems or explore interesting mathematical micro-worlds together. The VMT Project studies…

Focus in High School Mathematics: Reasoning and Sense Making in Algebra

ERIC Educational Resources Information Center

Graham, Karen; Cuoco, Albert; Zimmermann, Gwendolyn

2010-01-01

This book examines the five key elements (meaningful use of symbols, mindful manipulation, reasoned solving, connection algebra with geometry, and linking expressions and functions) identified in "Focus in High School Mathematics: Reasoning and Sense Making" in more detail and elaborates on the associated reasoning habits. This volume is one of a…

Computer Algebra Systems in Undergraduate Instruction.

ERIC Educational Resources Information Center

Small, Don; And Others

1986-01-01

Computer algebra systems (such as MACSYMA and muMath) can carry out many of the operations of calculus, linear algebra, and differential equations. Use of them with sketching graphs of rational functions and with other topics is discussed. (MNS)

Metaphorical motion in mathematical reasoning: further evidence for pre-motor implementation of structure mapping in abstract domains.

PubMed

Fields, Chris

2013-08-01

The theory of computation and category theory both employ arrow-based notations that suggest that the basic metaphor "state changes are like motions" plays a fundamental role in all mathematical reasoning involving formal manipulations. If this is correct, structure-mapping inferences implemented by the pre-motor action planning system can be expected to be involved in solving any mathematics problems not solvable by table lookups and number line manipulations alone. Available functional imaging studies of multi-digit arithmetic, algebra, geometry and calculus problem solving are consistent with this expectation.

Transforming Middle School Geometry: Designing Professional Development Materials that Support the Teaching and Learning of Similarity

ERIC Educational Resources Information Center

Seago, Nanette; Jacobs, Jennifer; Driscoll, Mark

2010-01-01

Although there are increasing numbers of professional development (PD) materials intended to foster teachers' mathematical knowledge for teaching within the topics of number and algebra, little attention has been given to geometry. In this article we describe the Learning and Teaching Geometry project's approach to the development of PD materials…

PREFACE: Algebra, Geometry, and Mathematical Physics 2010

NASA Astrophysics Data System (ADS)

Stolin, A.; Abramov, V.; Fuchs, J.; Paal, E.; Shestopalov, Y.; Silvestrov, S.

2012-02-01

This proceedings volume presents results obtained by the participants of the 6th Baltic-Nordic workshop 'Algebra, Geometry, and Mathematical Physics (AGMP-6)' held at the Sven Lovén Centre for Marine Sciences in Tjärnö, Sweden on October 25-30, 2010. The Baltic-Nordic Network AGMP 'Algebra, Geometry, and Mathematical Physics' http://www.agmp.eu was created in 2005 on the initiative of two Estonian universities and two Swedish universities: Tallinn University of Technology represented by Eugen Paal (coordinator of the network), Tartu University represented by Viktor Abramov, Lund University represented by Sergei Silvestrov, and Chalmers University of Technology and the University of Gothenburg represented by Alexander Stolin. The goal was to promote international and interdisciplinary cooperation between scientists and research groups in the countries of the Baltic-Nordic region in mathematics and mathematical physics, with special emphasis on the important role played by algebra and geometry in modern physics, engineering and technologies. The main activities of the AGMP network consist of a series of regular annual international workshops, conferences and research schools. The AGMP network also constitutes an important educational forum for scientific exchange and dissimilation of research results for PhD students and Postdocs. The network has expanded since its creation, and nowadays its activities extend beyond countries in the Baltic-Nordic region to universities in other European countries and participants from elsewhere in the world. As one of the important research-dissimilation outcomes of its activities, the network has a tradition of producing high-quality research proceedings volumes after network events, publishing them with various international publishers. The PDF also contains the following: List of AGMP workshops and other AGMP activities Main topics discussed at AGMP-6 Review of AGMP-6 proceedings Acknowledgments List of Conference Participants

Horizon fluffs: In the context of generalized minimal massive gravity

NASA Astrophysics Data System (ADS)

Setare, Mohammad Reza; Adami, Hamed

2018-02-01

We consider a metric which describes Bañados geometries and show that the considered metric is a solution of the generalized minimal massive gravity (GMMG) model. We consider the Killing vector field which preserves the form of the considered metric. Using the off-shell quasi-local approach we obtain the asymptotic conserved charges of the given solution. Similar to the Einstein gravity in the presence of negative cosmological constant, for the GMMG model, we also show that the algebra among the asymptotic conserved charges is isomorphic to two copies of the Virasoro algebra. Eventually, we find a relation between the algebra of the near-horizon and the asymptotic conserved charges. This relation shows that the main part of the horizon fluffs proposed by Afshar et al., Sheikh-Jabbari and Yavartanoo appear for generic black holes in the class of Bañados geometries in the context of the GMMG model.

Descriptions of Free and Freeware Software in the Mathematics Teaching

NASA Astrophysics Data System (ADS)

Antunes de Macedo, Josue; Neves de Almeida, Samara; Voelzke, Marcos Rincon

2016-05-01

This paper presents the analysis and the cataloging of free and freeware mathematical software available on the internet, a brief explanation of them, and types of licenses for use in teaching and learning. The methodology is based on the qualitative research. Among the different types of software found, it stands out in algebra, the Winmat, that works with linear algebra, matrices and linear systems. In geometry, the GeoGebra, which can be used in the study of functions, plan and spatial geometry, algebra and calculus. For graphing, can quote the Graph and Graphequation. With Graphmatica software, it is possible to build various graphs of mathematical equations on the same screen, representing cartesian equations, inequalities, parametric among other functions. The Winplot allows the user to build graphics in two and three dimensions functions and mathematical equations. Thus, this work aims to present the teachers some free math software able to be used in the classroom.

SD-CAS: Spin Dynamics by Computer Algebra System.

PubMed

Filip, Xenia; Filip, Claudiu

2010-11-01

A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples. Copyright © 2010 Elsevier Inc. All rights reserved.

Foundation Mathematics for the Physical Sciences

NASA Astrophysics Data System (ADS)

Riley, K. F.; Hobson, M. P.

2011-03-01

1. Arithmetic and geometry; 2. Preliminary algebra; 3. Differential calculus; 4. Integral calculus; 5. Complex numbers and hyperbolic functions; 6. Series and limits; 7. Partial differentiation; 8. Multiple integrals; 9. Vector algebra; 10. Matrices and vector spaces; 11. Vector calculus; 12. Line, surface and volume integrals; 13. Laplace transforms; 14. Ordinary differential equations; 15. Elementary probability; Appendices; Index.

Student Solution Manual for Foundation Mathematics for the Physical Sciences

NASA Astrophysics Data System (ADS)

Riley, K. F.; Hobson, M. P.

2011-03-01

1. Arithmetic and geometry; 2. Preliminary algebra; 3. Differential calculus; 4. Integral calculus; 5. Complex numbers and hyperbolic functions; 6. Series and limits; 7. Partial differentiation; 8. Multiple integrals; 9. Vector algebra; 10. Matrices and vector spaces; 11. Vector calculus; 12. Line, surface and volume integrals; 13. Laplace transforms; 14. Ordinary differential equations; 15. Elementary probability; Appendix.

Evaluation of Mathematics Teacher Candidates' the Ellipse Knowledge According to the Revised Bloom's Taxonomy

ERIC Educational Resources Information Center

Kurtulus, Aytaç; Ada, Aytaç

2017-01-01

In this study, the teacher candidates who learnt to find the algebraic equation corresponding to geometric structure of the ellipse in analytic geometry classes were requested to find the algebraic representations corresponding to the structures that contained ellipses in different positions. Thus, it would be possible to determine higher order…

The Koslowski-Sahlmann representation: quantum configuration space

NASA Astrophysics Data System (ADS)

Campiglia, Miguel; Varadarajan, Madhavan

2014-09-01

The Koslowski-Sahlmann (KS) representation is a generalization of the representation underlying the discrete spatial geometry of loop quantum gravity (LQG), to accommodate states labelled by smooth spatial geometries. As shown recently, the KS representation supports, in addition to the action of the holonomy and flux operators, the action of operators which are the quantum counterparts of certain connection dependent functions known as ‘background exponentials’. Here we show that the KS representation displays the following properties which are the exact counterparts of LQG ones: (i) the abelian * algebra of SU(2) holonomies and ‘U(1)’ background exponentials can be completed to a C* algebra, (ii) the space of semianalytic SU(2) connections is topologically dense in the spectrum of this algebra, (iii) there exists a measure on this spectrum for which the KS Hilbert space is realized as the space of square integrable functions on the spectrum, (iv) the spectrum admits a characterization as a projective limit of finite numbers of copies of SU(2) and U(1), (v) the algebra underlying the KS representation is constructed from cylindrical functions and their derivations in exactly the same way as the LQG (holonomy-flux) algebra except that the KS cylindrical functions depend on the holonomies and the background exponentials, this extra dependence being responsible for the differences between the KS and LQG algebras. While these results are obtained for compact spaces, they are expected to be of use for the construction of the KS representation in the asymptotically flat case.

Tropical geometry of statistical models.

PubMed

Pachter, Lior; Sturmfels, Bernd

2004-11-16

This article presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algorithm is an efficient tool for evaluating specific coordinates. Here, we address the question of how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. The Newton polytope of a statistical model plays a key role. Our results are applied to the hidden Markov model and the general Markov model on a binary tree.

Extended Riemannian geometry II: local heterotic double field theory

NASA Astrophysics Data System (ADS)

Deser, Andreas; Heller, Marc Andre; Sämann, Christian

2018-04-01

We continue our exploration of local Double Field Theory (DFT) in terms of symplectic graded manifolds carrying compatible derivations and study the case of heterotic DFT. We start by developing in detail the differential graded manifold that captures heterotic Generalized Geometry which leads to new observations on the generalized metric and its twists. We then give a symplectic pre-N Q-manifold that captures the symmetries and the geometry of local heterotic DFT. We derive a weakened form of the section condition, which arises algebraically from consistency of the symmetry Lie 2-algebra and its action on extended tensors. We also give appropriate notions of twists — which are required for global formulations — and of the torsion and Riemann tensors. Finally, we show how the observed α'-corrections are interpreted naturally in our framework.

Asymptotic symmetries and geometry on the boundary in the first order formalism

NASA Astrophysics Data System (ADS)

Korovin, Yegor

2018-03-01

Proper understanding of the geometry on the boundary of a spacetime is a critical step on the way to extending holography to spaces with non-AdS asymptotics. In general the boundary cannot be described in terms of the Riemannian geometry and the first order formalism is more appropriate as we show. We analyze the asymptotic symmetries in the first order formalism for large classes of theories on AdS, Lifshitz or flat space. In all cases the asymptotic symmetry algebra is realized on the first order variables as a gauged symmetry algebra. First order formalism geometrizes and simplifies the analysis. We apply our framework to the issue of scale versus conformal invariance in AdS/CFT and obtain new perspective on the structure of asymptotic expansions for AdS and flat spaces.

Gauss-Manin Connection in Disguise: Calabi-Yau Threefolds

NASA Astrophysics Data System (ADS)

Alim, Murad; Movasati, Hossein; Scheidegger, Emanuel; Yau, Shing-Tung

2016-06-01

We describe a Lie Algebra on the moduli space of non-rigid compact Calabi-Yau threefolds enhanced with differential forms and its relation to the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation. In particular, we describe algebraic topological string partition functions {{F}g^alg, g ≥ 1}, which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck's algebraic de Rham cohomology and on the algebraic Gauss-Manin connection. In this way, we recover a result of Yamaguchi-Yau and Alim-Länge in an algebraic context. Our proofs use the fact that the special polynomial generators defined using the special geometry of deformation spaces of Calabi-Yau threefolds correspond to coordinates on such a moduli space. We discuss the mirror quintic as an example.

A comparative study of turbulence models for overset grids

NASA Technical Reports Server (NTRS)

Renze, Kevin J.; Buning, Pieter G.; Rajagopalan, R. G.

1992-01-01

The implementation of two different types of turbulence models for a flow solver using the Chimera overset grid method is examined. Various turbulence model characteristics, such as length scale determination and transition modeling, are found to have a significant impact on the computed pressure distribution for a multielement airfoil case. No inherent problem is found with using either algebraic or one-equation turbulence models with an overset grid scheme, but simulation of turbulence for multiple-body or complex geometry flows is very difficult regardless of the gridding method. For complex geometry flowfields, modification of the Baldwin-Lomax turbulence model is necessary to select the appropriate length scale in wall-bounded regions. The overset grid approach presents no obstacle to use of a one- or two-equation turbulence model. Both Baldwin-Lomax and Baldwin-Barth models have problems providing accurate eddy viscosity levels for complex multiple-body flowfields such as those involving the Space Shuttle.

Classical integrable many-body systems disconnected with semi-simple Lie algebras

NASA Astrophysics Data System (ADS)

Inozemtsev, V. I.

2017-05-01

The review of the results in the theory of integrable many-body systems disconnected with semisimple Lie algebras is done. The one-dimensional systems of light Calogero-Sutherland-Moser particles interacting with one particle of infinite mass located at the origin are described in detail. In some cases the exact solutions of the equations of motion are obtained. The general theory of integration of the equations of motion needs the methods of algebraic geometry. The Lax pairs with spectral parameter are constructed for this purpose. The theory still contains many unsolved problems.

Calculus of Elementary Functions, Part I. Teacher's Commentary. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics including algebra, axiomatic geometry, trigonometry, and analytic geometry. It does not assume they have acquired a background of elementary functions. This teacher's guide contains background information, suggested instructional procedures, and…

Does Watching "Do the Math" Affect Self-Efficacy and Achievement in Mathematics?

ERIC Educational Resources Information Center

Cavazos, Blanca Guadalupe

2014-01-01

"Do The Math," a 1-hour, live, educational television program provides on-air instruction in general math, geometry, pre-algebra and algebra to a target audience of 4th-12th graders. A team of math teachers also provides tutoring to students who call in for help with homework. The purpose of this study was to investigate whether watching…

Math Ties: Problem Solving, Logic Teasers, and Math Puzzles All "Tied" To the Math Curriculum. Book B1.

ERIC Educational Resources Information Center

Santi, Terri

This book contains a classroom-tested approach to the teaching of problem solving to all students in Grades 6-8, regardless of ability. Information on problem solving in general is provided, then mathematical problems on logic, exponents, fractions, pre-algebra, algebra, geometry, number theory, set theory, ratio, proportion, percent, probability,…

Are Parents Ready for New High School Curriculum Requirements? Policy Report 28

ERIC Educational Resources Information Center

Landauer-Menchik, Bettie

2006-01-01

The State Board of Education has recommended the implementation of a new, more rigorous curriculum for Michigan high schools. All students would be required to take four years of English; one year each of Algebra I, Geometry, Algebra II, and an additional math class in the senior year; one year each of Biology, Physics or Chemistry, and one…

Marriages of mathematics and physics: A challenge for biology.

PubMed

Islami, Arezoo; Longo, Giuseppe

2017-12-01

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of "geometric judgments" from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) "space" should be revisited for the purposes of life sciences. Copyright © 2017. Published by Elsevier Ltd.

On Fock-space representations of quantized enveloping algebras related to noncommutative differential geometry

NASA Astrophysics Data System (ADS)

Jurčo, B.; Schlieker, M.

1995-07-01

In this paper explicitly natural (from the geometrical point of view) Fock-space representations (contragradient Verma modules) of the quantized enveloping algebras are constructed. In order to do so, one starts from the Gauss decomposition of the quantum group and introduces the differential operators on the corresponding q-deformed flag manifold (assumed as a left comodule for the quantum group) by a projection to it of the right action of the quantized enveloping algebra on the quantum group. Finally, the representatives of the elements of the quantized enveloping algebra corresponding to the left-invariant vector fields on the quantum group are expressed as first-order differential operators on the q-deformed flag manifold.

Moving frames and prolongation algebras

NASA Technical Reports Server (NTRS)

Estabrook, F. B.

1982-01-01

Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

NASA Astrophysics Data System (ADS)

Nazarov, Anton

2012-11-01

In this paper we present Affine.m-a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Catalogue identifier: AENA_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, UK Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 24 844 No. of bytes in distributed program, including test data, etc.: 1 045 908 Distribution format: tar.gz Programming language: Mathematica. Computer: i386-i686, x86_64. Operating system: Linux, Windows, Mac OS, Solaris. RAM: 5-500 Mb Classification: 4.2, 5. Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems. Solution method: We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases. Restrictions: Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical. Unusual features: We offer the possibility of using a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface. Running time: From seconds to days depending on the rank of the algebra and the complexity of the representation.

On Some Algebraic and Combinatorial Properties of Dunkl Elements

NASA Astrophysics Data System (ADS)

Kirillov, Anatol N.

2013-06-01

We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements and Schubert calculus, in Advances in Geometry (eds. J.-S. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, Boston, 1995), pp. 147-182, A. Postnikov, On a quantum version of Pieri's formula, in Advances in Geometry (eds. J.-S. Brylinski, R. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, 1995), pp. 371-383 and A. N. Kirillov and T. Maenor, A Note on Quantum K-Theory of Flag Varieties, preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [E. Mukhin, V. Tarasov and A. Varchenko, Bethe Subalgebras of the Group Algebra of the Symmetric Group, preprint arXiv:1004.4248]. Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra in a connection with the values of the β-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan-Robbins polytope.

On Some Algebraic and Combinatorial Properties of Dunkl Elements

NASA Astrophysics Data System (ADS)

Kirillov, Anatol N.

2012-11-01

We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements and Schubert calculus, in Advances in Geometry (eds. J.-S. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, Boston, 1995), pp. 147-182, A. Postnikov, On a quantum version of Pieri's formula, in Advances in Geometry (eds. J.-S. Brylinski, R. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, 1995), pp. 371-383 and A. N. Kirillov and T. Maenor, A Note on Quantum K-Theory of Flag Varieties, preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [E. Mukhin, V. Tarasov and A. Varchenko, Bethe Subalgebras of the Group Algebra of the Symmetric Group, preprint arXiv:1004.4248]. Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra in a connection with the values of the β-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan-Robbins polytope.

Advanced Algebra and Calculus. High School Mathematics Curricula. Instructor's Guide.

ERIC Educational Resources Information Center

Natour, Denise M.

This manual is an instructor's guide for the utilization of the "CCA High School Mathematics Curricula: Advanced Algebra and Calculus" courseware developed by the Computer-based Education Research Laboratory (CERL). The curriculum comprises 34 algebra lessons within 12 units and 15 calculus lessons that are computer-based and require…

Changes in Pre-Service Teachers' Algebraic Misconceptions by Using Computer-Assisted Instruction

ERIC Educational Resources Information Center

Lin, ByCheng-Yao; Ko, Yi-Yin; Kuo, Yu-Chun

2014-01-01

In order to carry out current reforms regarding algebra and technology in elementary school mathematics successfully, pre-service elementary mathematics teachers must be equipped with adequate understandings of algebraic concepts and self-confidence in using computers for their future teaching. This paper examines the differences in preservice…

Some Applications Of Semigroups And Computer Algebra In Discrete Structures

NASA Astrophysics Data System (ADS)

Bijev, G.

2009-11-01

An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. A map (p) is defined, which is similar to an orthogonal projection in linear vector spaces. Some other important maps with properties similar to those of the generalized inverses (pseudoinverses) of linear transformations and matrices corresponding to them are also defined and investigated. Let Ax = b be an equation with matrix A and vectors x and b Boolean. Stochastic experiments for solving the equation, which involves the maps defined and use computer algebra methods, have been made. As a result, the Hamming distance between vectors Ax = p(b) and b is equal or close to the least possible. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra and research. Some examples for computations with binary relations using Maple are given.

Gauss Elimination: Workhorse of Linear Algebra.

DTIC Science & Technology

1995-08-05

linear algebra computation for solving systems, computing determinants and determining the rank of matrix. All of these are discussed in varying contexts. These include different arithmetic or algebraic setting such as integer arithmetic or polynomial rings as well as conventional real (floating-point) arithmetic. These have effects on both accuracy and complexity analyses of the algorithm. These, too, are covered here. The impact of modern parallel computer architecture on GE is also

Parallel Algorithms for Least Squares and Related Computations.

DTIC Science & Technology

1991-03-22

for dense computations in linear algebra . The work has recently been published in a general reference book on parallel algorithms by SIAM. AFO SR...written his Ph.D. dissertation with the principal investigator. (See publication 6.) • Parallel Algorithms for Dense Linear Algebra Computations. Our...and describe and to put into perspective a selection of the more important parallel algorithms for numerical linear algebra . We give a major new

Generic, Type-Safe and Object Oriented Computer Algebra Software

NASA Astrophysics Data System (ADS)

Kredel, Heinz; Jolly, Raphael

Advances in computer science, in particular object oriented programming, and software engineering have had little practical impact on computer algebra systems in the last 30 years. The software design of existing systems is still dominated by ad-hoc memory management, weakly typed algorithm libraries and proprietary domain specific interactive expression interpreters. We discuss a modular approach to computer algebra software: usage of state-of-the-art memory management and run-time systems (e.g. JVM) usage of strongly typed, generic, object oriented programming languages (e.g. Java) and usage of general purpose, dynamic interactive expression interpreters (e.g. Python) To illustrate the workability of this approach, we have implemented and studied computer algebra systems in Java and Scala. In this paper we report on the current state of this work by presenting new examples.

Redesigning the Quantum Mechanics Curriculum to Incorporate Problem Solving Using a Computer Algebra System

NASA Astrophysics Data System (ADS)

Roussel, Marc R.

1999-10-01

One of the traditional obstacles to learning quantum mechanics is the relatively high level of mathematical proficiency required to solve even routine problems. Modern computer algebra systems are now sufficiently reliable that they can be used as mathematical assistants to alleviate this difficulty. In the quantum mechanics course at the University of Lethbridge, the traditional three lecture hours per week have been replaced by two lecture hours and a one-hour computer-aided problem solving session using a computer algebra system (Maple). While this somewhat reduces the number of topics that can be tackled during the term, students have a better opportunity to familiarize themselves with the underlying theory with this course design. Maple is also available to students during examinations. The use of a computer algebra system expands the class of feasible problems during a time-limited exercise such as a midterm or final examination. A modern computer algebra system is a complex piece of software, so some time needs to be devoted to teaching the students its proper use. However, the advantages to the teaching of quantum mechanics appear to outweigh the disadvantages.

Holography for a De Sitter-Esque geometry

NASA Astrophysics Data System (ADS)

Anninos, Dionysios; de Buyl, Sophie; Detournay, Stéphane

2011-05-01

Warped dS3 arises as a solution to topologically massive gravity (TMG) with positive cosmological constant +1/ ℓ 2 and Chern-Simons coefficient 1/ μ in the region μ 2 ℓ 2 < 27. It is given by a real line fibration over two-dimensional de Sitter space and is equivalent to the rotating Nariai geometry at fixed polar angle. We study the thermodynamic and asymptotic structure of a family of geometries with warped dS3 asymptotics. Interestingly, these solutions have both a cosmological horizon and an internal one, and their entropy is unbounded from above unlike black holes in regular de Sitter space. The asymptotic symmetry group resides at future infinity and is given by a semi-direct product of a Virasoro algebra and a current algebra. The right moving central charge vanishes when μ 2 ℓ 2 = 27/5. We discuss the possible holographic interpretation of these de Sitter-esque spacetimes.

Discrete Surface Evolution and Mesh Deformation for Aircraft Icing Applications

NASA Technical Reports Server (NTRS)

Thompson, David; Tong, Xiaoling; Arnoldus, Qiuhan; Collins, Eric; McLaurin, David; Luke, Edward; Bidwell, Colin S.

2013-01-01

Robust, automated mesh generation for problems with deforming geometries, such as ice accreting on aerodynamic surfaces, remains a challenging problem. Here we describe a technique to deform a discrete surface as it evolves due to the accretion of ice. The surface evolution algorithm is based on a smoothed, face-offsetting approach. We also describe a fast algebraic technique to propagate the computed surface deformations into the surrounding volume mesh while maintaining geometric mesh quality. Preliminary results presented here demonstrate the ecacy of the approach for a sphere with a prescribed accretion rate, a rime ice accretion, and a more complex glaze ice accretion.

Transition mixing study

NASA Technical Reports Server (NTRS)

Reynolds, R.; White, C.

1986-01-01

A computer model capable of analyzing the flow field in the transition liner of small gas turbine engines is developed. A FORTRAN code has been assembled from existing codes and physical submodels and used to predict the flow in several test geometries which contain characteristics similar to transition liners, and for which experimental data was available. Comparisons between the predictions and measurements indicate that the code produces qualitative results but that the turbulence models, both K-E and algebraic Reynolds Stress, underestimate the cross-stream diffusion. The code has also been used to perform a numerical experiment to examine the effect of a variety of parameters on the mixing process in transition liners. Comparisons illustrate that geometries with significant curvature show a drift of the jet trajectory toward the convex wall and weaker wake region vortices and decreased penetration for jets located on the convex wall of the liner, when compared to jets located on concave walls. Also shown were the approximate equivalency of angled slots and round holes and a technique by which jet mixing correlations developed for rectangular channels can be used for can geometries.

Learning coefficient of generalization error in Bayesian estimation and vandermonde matrix-type singularity.

PubMed

Aoyagi, Miki; Nagata, Kenji

2012-06-01

The term algebraic statistics arises from the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry (Sturmfels, 2009 ). The purpose of our study is to consider the generalization error and stochastic complexity in learning theory by using the log-canonical threshold in algebraic geometry. Such thresholds correspond to the main term of the generalization error in Bayesian estimation, which is called a learning coefficient (Watanabe, 2001a , 2001b ). The learning coefficient serves to measure the learning efficiencies in hierarchical learning models. In this letter, we consider learning coefficients for Vandermonde matrix-type singularities, by using a new approach: focusing on the generators of the ideal, which defines singularities. We give tight new bound values of learning coefficients for the Vandermonde matrix-type singularities and the explicit values with certain conditions. By applying our results, we can show the learning coefficients of three-layered neural networks and normal mixture models.

Equivariant Verlinde Algebra from Superconformal Index and Argyres-Seiberg Duality

NASA Astrophysics Data System (ADS)

Gukov, Sergei; Pei, Du; Yan, Wenbin; Ye, Ke

2018-02-01

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory {T[Σ,G]} on {L(k,1) × S^1}, the other is the {^L G} "equivariant Verlinde formula", or equivalently partition function of {^L G_C} complex Chern-Simons theory on {Σ× S^1}. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally G and its Langlands dual {^L G}. When G is not simply-connected, we provide a recipe of computing the index of {T[Σ,G]} as summation over the indices of T[Σ,\\tilde{G}] with non-trivial background 't Hooft fluxes, where \\tilde{G} is the universal cover of G. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for {G=SU(2)} or SO(3). In the end, as an application of this newly found relation, we consider the more general case where G is SU( N) or PSU( N) and show that equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg duality. We also attach a Mathematica notebook that can be used to compute the SU(3) equivariant Verlinde coefficients.

A Computer Algebra Approach to Solving Chemical Equilibria in General Chemistry

ERIC Educational Resources Information Center

Kalainoff, Melinda; Lachance, Russ; Riegner, Dawn; Biaglow, Andrew

2012-01-01

In this article, we report on a semester-long study of the incorporation into our general chemistry course, of advanced algebraic and computer algebra techniques for solving chemical equilibrium problems. The method presented here is an alternative to the commonly used concentration table method for describing chemical equilibria in general…

Maple (Computer Algebra System) in Teaching Pre-Calculus: Example of Absolute Value Function

ERIC Educational Resources Information Center

Tuluk, Güler

2014-01-01

Modules in Computer Algebra Systems (CAS) make Mathematics interesting and easy to understand. The present study focused on the implementation of the algebraic, tabular (numerical), and graphical approaches used for the construction of the concept of absolute value function in teaching mathematical content knowledge along with Maple 9. The study…

Computing with impure numbers - Automatic consistency checking and units conversion using computer algebra

NASA Technical Reports Server (NTRS)

Stoutemyer, D. R.

1977-01-01

The computer algebra language MACSYMA enables the programmer to include symbolic physical units in computer calculations, and features automatic detection of dimensionally-inhomogeneous formulas and conversion of inconsistent units in a dimensionally homogeneous formula. Some examples illustrate these features.

Two and three dimensional grid generation by an algebraic homotopy procedure

NASA Technical Reports Server (NTRS)

Moitra, Anutosh

1990-01-01

An algebraic method for generating two- and three-dimensional grid systems for aerospace vehicles is presented. The method is based on algebraic procedures derived from homotopic relations for blending between inner and outer boundaries of any given configuration. Stable properties of homotopic maps have been exploited to provide near-orthogonality and specified constant spacing at the inner boundary. The method has been successfully applied to analytically generated blended wing-body configurations as well as discretely defined geometries such as the High-Speed Civil Transport Aircraft. Grid examples representative of the capabilities of the method are presented.

Weak Lie symmetry and extended Lie algebra

DOE Office of Scientific and Technical Information (OSTI.GOV)

Goenner, Hubert

2013-04-15

The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found ('extended Lie algebras') which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).

Linear-Algebra Programs

NASA Technical Reports Server (NTRS)

Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.

1982-01-01

The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.

Bridging Algebra & Geometry with "n"-Gram Proofs

ERIC Educational Resources Information Center

Craven, Joshua D.

2010-01-01

For many students, geometry is the first course in which mathematical proof takes center stage. To help ease students into writing proofs, the author tries to create lessons and activities throughout the year that challenge students to prove their own conjectures by using tools learned in previous mathematics courses. Teachers cannot get all…

Calculus of Elementary Functions, Part II. Teacher's Commentary. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. This teacher's guide is for Part II of the course. It is designed to follow Part I of the text. The guide contains background information, suggested instructional…

Calculus of Elementary Functions, Part I. Student Text. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. This text, Part I, contains the first five chapters of the course and two appendices. Chapters included are: (1) Polynomial Functions; (2) The Derivative of a Polynomial…

Calculus of Elementary Functions, Part II. Student Text. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. This text, Part II, contains material designed to follow Part I. Chapters included in this text are: (6) Derivatives of Exponential and Related Functions; (7) Area and…

Origami, Geometry and Art

ERIC Educational Resources Information Center

Wares, Arsalan; Elstak, Iwan

2017-01-01

The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra…

Computers and the Multiplicity of Polynomial Roots.

ERIC Educational Resources Information Center

Wavrik, John J.

1982-01-01

Described are stages in the development of a computer program to solve a particular algebra problem and the nature of algebraic computation is presented. A program in BASIC is provided to give ideas to others for developing their own programs. (MP)

Derive Workshop Matrix Algebra and Linear Algebra.

ERIC Educational Resources Information Center

Townsley Kulich, Lisa; Victor, Barbara

This document presents the course content for a workshop that integrates the use of the computer algebra system Derive with topics in matrix and linear algebra. The first section is a guide to using Derive that provides information on how to write algebraic expressions, make graphs, save files, edit, define functions, differentiate expressions,…

Computational approach to compact Riemann surfaces

NASA Astrophysics Data System (ADS)

Frauendiener, Jörg; Klein, Christian

2017-01-01

A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw-Curtis algorithm and contour integrals. As an application of the code, solutions to the Kadomtsev-Petviashvili equation are computed on non-hyperelliptic Riemann surfaces.

The Relationship between School-Facilitated Parental Involvement and Academic Math Achievement of High School Students in Virginia Who Receive Special Education Services

ERIC Educational Resources Information Center

Stein, Allison

2017-01-01

This study examined how school-facilitated parental involvement affects Standards of Learning (SOL) end-of-course exams for high school students in Virginia who are receiving special education services. This study examined test results from the 2012-2013, 2013-2014, and 2014-2015 school years for the Algebra I, Geometry, and Algebra II SOL exams,…

Generalized -deformed correlation functions as spectral functions of hyperbolic geometry

NASA Astrophysics Data System (ADS)

Bonora, L.; Bytsenko, A. A.; Guimarães, M. E. X.

2014-08-01

We analyze the role of vertex operator algebra and 2d amplitudes from the point of view of the representation theory of infinite-dimensional Lie algebras, MacMahon and Ruelle functions. By definition p-dimensional MacMahon function, with , is the generating function of p-dimensional partitions of integers. These functions can be represented as amplitudes of a two-dimensional c = 1 CFT, and, as such, they can be generalized to . With some abuse of language we call the latter amplitudes generalized MacMahon functions. In this paper we show that generalized p-dimensional MacMahon functions can be rewritten in terms of Ruelle spectral functions, whose spectrum is encoded in the Patterson-Selberg function of three-dimensional hyperbolic geometry.

Numeric invariants from multidimensional persistence

DOE PAGES

Skryzalin, Jacek; Carlsson, Gunnar

2017-05-19

Topological data analysis is the study of data using techniques from algebraic topology. Often, one begins with a finite set of points representing data and a “filter” function which assigns a real number to each datum. Using both the data and the filter function, one can construct a filtered complex for further analysis. For example, applying the homology functor to the filtered complex produces an algebraic object known as a “one-dimensional persistence module”, which can often be interpreted as a finite set of intervals representing various geometric features in the data. If one runs the above process incorporating multiple filtermore » functions simultaneously, one instead obtains a multidimensional persistence module. Unfortunately, these are much more difficult to interpret. In this article, we analyze the space of multidimensional persistence modules from the perspective of algebraic geometry. First we build a moduli space of a certain subclass of easily analyzed multidimensional persistence modules, which we construct specifically to capture much of the information which can be gained by using multidimensional persistence instead of one-dimensional persistence. Fruthermore, we argue that the global sections of this space provide interesting numeric invariants when evaluated against our subclass of multidimensional persistence modules. Finally, we extend these global sections to the space of all multidimensional persistence modules and discuss how the resulting numeric invariants might be used to study data. This paper extends the results of Adcock et al. (Homol Homotopy Appl 18(1), 381–402, 2016) by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson et al. (J Comput Geom 1(1), 72–100, 2010).« less

Development process of in-service training intended for teachers to perform teaching of mathematics with computer algebra systems

NASA Astrophysics Data System (ADS)

Ardıç, Mehmet Alper; Işleyen, Tevfik

2018-01-01

In this study, we deal with the development process of in-service training activities designed in order for mathematics teachers of secondary education to realize teaching of mathematics, utilizing computer algebra systems. In addition, the results obtained from the researches carried out during and after the in-service training were summarized. Last section focuses on suggestions any teacher can use to carry out activities aimed at using computer algebra systems in teaching environments.

A Comparison of Success and Failure Rates between Computer-Assisted and Traditional College Algebra Sections

ERIC Educational Resources Information Center

Herron, Sherry; Gandy, Rex; Ye, Ningjun; Syed, Nasser

2012-01-01

A unique aspect of the implementation of a computer algebra system (CAS) at a comprehensive university in the U.S. allowed us to compare the student success and failure rates to the traditional method of teaching college algebra. Due to space limitations, the university offered sections of both CAS and traditional simultaneously and, upon…

A scalable nonlinear fluid-structure interaction solver based on a Schwarz preconditioner with isogeometric unstructured coarse spaces in 3D

NASA Astrophysics Data System (ADS)

Kong, Fande; Cai, Xiao-Chuan

2017-07-01

Nonlinear fluid-structure interaction (FSI) problems on unstructured meshes in 3D appear in many applications in science and engineering, such as vibration analysis of aircrafts and patient-specific diagnosis of cardiovascular diseases. In this work, we develop a highly scalable, parallel algorithmic and software framework for FSI problems consisting of a nonlinear fluid system and a nonlinear solid system, that are coupled monolithically. The FSI system is discretized by a stabilized finite element method in space and a fully implicit backward difference scheme in time. To solve the large, sparse system of nonlinear algebraic equations at each time step, we propose an inexact Newton-Krylov method together with a multilevel, smoothed Schwarz preconditioner with isogeometric coarse meshes generated by a geometry preserving coarsening algorithm. Here "geometry" includes the boundary of the computational domain and the wet interface between the fluid and the solid. We show numerically that the proposed algorithm and implementation are highly scalable in terms of the number of linear and nonlinear iterations and the total compute time on a supercomputer with more than 10,000 processor cores for several problems with hundreds of millions of unknowns.

A scalable nonlinear fluid–structure interaction solver based on a Schwarz preconditioner with isogeometric unstructured coarse spaces in 3D

DOE PAGES

Kong, Fande; Cai, Xiao-Chuan

2017-03-24

Nonlinear fluid-structure interaction (FSI) problems on unstructured meshes in 3D appear many applications in science and engineering, such as vibration analysis of aircrafts and patient-specific diagnosis of cardiovascular diseases. In this work, we develop a highly scalable, parallel algorithmic and software framework for FSI problems consisting of a nonlinear fluid system and a nonlinear solid system, that are coupled monolithically. The FSI system is discretized by a stabilized finite element method in space and a fully implicit backward difference scheme in time. To solve the large, sparse system of nonlinear algebraic equations at each time step, we propose an inexactmore » Newton-Krylov method together with a multilevel, smoothed Schwarz preconditioner with isogeometric coarse meshes generated by a geometry preserving coarsening algorithm. Here ''geometry'' includes the boundary of the computational domain and the wet interface between the fluid and the solid. We show numerically that the proposed algorithm and implementation are highly scalable in terms of the number of linear and nonlinear iterations and the total compute time on a supercomputer with more than 10,000 processor cores for several problems with hundreds of millions of unknowns.« less

Generalized Kähler geometry and current algebras in classical N=2 superconformal WZW model

NASA Astrophysics Data System (ADS)

Parkhomenko, S. E.

2018-04-01

I examine the Generalized Kähler (GK) geometry of classical N = (2, 2) superconformal WZW model on a compact group and relate the right-moving and left-moving Kac-Moody superalgebra currents to the GK geometry data using biholomorphic gerbe formulation and Hamiltonian formalism. It is shown that the canonical Poisson homogeneous space structure induced by the GK geometry of the group manifold is crucial to provide N = (2, 2) superconformal σ-model with the Kac-Moody superalgebra symmetries. Then, the biholomorphic gerbe geometry is used to prove that Kac-Moody superalgebra currents are globally defined.

Oasis: A high-level/high-performance open source Navier-Stokes solver

NASA Astrophysics Data System (ADS)

Mortensen, Mikael; Valen-Sendstad, Kristian

2015-03-01

Oasis is a high-level/high-performance finite element Navier-Stokes solver written from scratch in Python using building blocks from the FEniCS project (fenicsproject.org). The solver is unstructured and targets large-scale applications in complex geometries on massively parallel clusters. Oasis utilizes MPI and interfaces, through FEniCS, to the linear algebra backend PETSc. Oasis advocates a high-level, programmable user interface through the creation of highly flexible Python modules for new problems. Through the high-level Python interface the user is placed in complete control of every aspect of the solver. A version of the solver, that is using piecewise linear elements for both velocity and pressure, is shown to reproduce very well the classical, spectral, turbulent channel simulations of Moser et al. (1999). The computational speed is strongly dominated by the iterative solvers provided by the linear algebra backend, which is arguably the best performance any similar implicit solver using PETSc may hope for. Higher order accuracy is also demonstrated and new solvers may be easily added within the same framework.

Stability Analysis of Finite Difference Schemes for Hyperbolic Systems, and Problems in Applied and Computational Linear Algebra.

DTIC Science & Technology

FINITE DIFFERENCE THEORY, * LINEAR ALGEBRA , APPLIED MATHEMATICS, APPROXIMATION(MATHEMATICS), BOUNDARY VALUE PROBLEMS, COMPUTATIONS, HYPERBOLAS, MATHEMATICAL MODELS, NUMERICAL ANALYSIS, PARTIAL DIFFERENTIAL EQUATIONS, STABILITY.

Libraries for Software Use on Peregrine | High-Performance Computing | NREL

Science.gov Websites

-specific libraries. Libraries List Name Description BLAS Basic Linear Algebra Subroutines, libraries only managing hierarchically structured data. LAPACK Standard Netlib offering for computational linear algebra

On superintegrable monopole systems

NASA Astrophysics Data System (ADS)

Fazlul Hoque, Md; Marquette, Ian; Zhang, Yao-Zhong

2018-02-01

Superintegrable systems with monopole interactions in flat and curved spaces have attracted much attention. For example, models in spaces with a Taub-NUT metric are well-known to admit the Kepler-type symmetries and provide non-trivial generalizations of the usual Kepler problems. In this paper, we overview new families of superintegrable Kepler, MIC-harmonic oscillator and deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and curved Taub-NUT spaces. We present their higher-order, algebraically independent integrals of motion via the direct and constructive approaches which prove the superintegrability of the models. The integrals form symmetry polynomial algebras of the systems with structure constants involving Casimir operators of certain Lie algebras. Such algebraic approaches provide a deeper understanding to the degeneracies of the energy spectra and connection between wave functions and differential equations and geometry.

Computer Program For Linear Algebra

NASA Technical Reports Server (NTRS)

Krogh, F. T.; Hanson, R. J.

1987-01-01

Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.

Gauge Gravity and Electroweak Theory

NASA Astrophysics Data System (ADS)

Hestenes, David

2008-09-01

Reformulation of the Dirac equation in terms of the real Spacetime Algebra (STA) reveals hidden geometric structure, including a geometric role for the unit imaginary as generator of rotations in a spacelike plane. The STA and the real Dirac equation play essential roles in a new Gauge Theory Gravity (GTG) version of General Relativity (GR). Besides clarifying the conceptual foundations of GR and facilitating complex computations, GTG opens up new possibilities for a unified gauge theory of gravity and quantum mechanics, including spacetime geometry of electroweak interactions. The Weinberg-Salam model fits perfectly into this geometric framework, and a promising variant that replaces chiral states with Majorana states is formulated to incorporate zitterbewegung in electron states.

Basic exploration geophysics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Robinson, E.S.

1988-01-01

An introduction to geophysical methods used to explore for natural resources and to survey earth's geology is presented in this volume. It is suitable for second-and third-year undergraduate students majoring in geology or engineering and for professional engineering and for professional engineers and earth scientists without formal instruction in geophysics. The author assumes the reader is familiar with geometry, algebra, and trigonometry. Geophysical exploration includes seismic refraction and reflection surveying, electrical resistivity and electromagnetic field surveying, and geophysical well logging. Surveying operations are described in step-by-step procedures and are illustrated by practical examples. Computer-based methods of processing and interpreting datamore » as well as geographical methods are introduced.« less

Examining Opportunity-to-Learn and Success in High School Mathematics Performance in California under NCLB

ERIC Educational Resources Information Center

Gavrilovic, Daniel Miodrag

2013-01-01

The No Child Left Behind Act of 2001 has put many schools under a lot of pressure to meet its high demands. In this quantitative study, the effects that the NCLB act has had on students' opportunity to learn (OTL) and Subject Level Success (SS) from 2004 to 2012 in 9th, 10th, and 11th grade math coursework (Algebra 1, Geometry, Algebra 2, and…

Spectral geometry of {kappa}-Minkowski space

DOE Office of Scientific and Technical Information (OSTI.GOV)

D'Andrea, Francesco

After recalling Snyder's idea [Phys. Rev. 71, 38 (1947)] of using vector fields over a smooth manifold as 'coordinates on a noncommutative space', we discuss a two-dimensional toy-model whose 'dual' noncommutative coordinates form a Lie algebra: this is the well-known {kappa}-Minkowski space [Phys. Lett. B 334, 348 (1994)]. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of {kappa}-Minkowski as linear operators on an Hilbert space (a major problem in the construction of a physical theory), study its 'spectral properties', and discuss how tomore » obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of Dimitrijevic et al. [Eur. Phys. J. C 31, 129 (2003)] can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.« less

Imagination, Intuition, and Computing in School Algebra.

ERIC Educational Resources Information Center

Kieren, Thomas E.; Olson, Alton T.

1989-01-01

Two incidents involving novice teachers with classes in grades 7 and 10 are presented. Then considered are the nature of intuitive mathematics and contributions computers can make to such intuitive mathematics, particularly in Algebra. (MNS)

Towards Student Instrumentation of Computer-Based Algebra Systems in University Courses

ERIC Educational Resources Information Center

Stewart, Sepideh; Thomas, Michael O. J.; Hannah, John

2005-01-01

There are many perceived benefits of using technology, such as computer algebra systems, in undergraduate mathematics courses. However, attaining these benefits sometimes proves elusive. Some of the key variables are the teaching approach and the student instrumentation of the technology. This paper considers the instrumentation of computer-based…

Digital Maps, Matrices and Computer Algebra

ERIC Educational Resources Information Center

Knight, D. G.

2005-01-01

The way in which computer algebra systems, such as Maple, have made the study of complex problems accessible to undergraduate mathematicians with modest computational skills is illustrated by some large matrix calculations, which arise from representing the Earth's surface by digital elevation models. Such problems are often considered to lie in…

The Simulation of an Oxidation-Reduction Titration Curve with Computer Algebra

ERIC Educational Resources Information Center

Whiteley, Richard V., Jr.

2015-01-01

Although the simulation of an oxidation/reduction titration curve is an important exercise in an undergraduate course in quantitative analysis, that exercise is frequently simplified to accommodate computational limitations. With the use of readily available computer algebra systems, however, such curves for complicated systems can be generated…

Computing the Moore-Penrose Inverse of a Matrix with a Computer Algebra System

ERIC Educational Resources Information Center

Schmidt, Karsten

2008-01-01

In this paper "Derive" functions are provided for the computation of the Moore-Penrose inverse of a matrix, as well as for solving systems of linear equations by means of the Moore-Penrose inverse. Making it possible to compute the Moore-Penrose inverse easily with one of the most commonly used Computer Algebra Systems--and to have the blueprint…

Linear systems with structure group and their feedback invariants

NASA Technical Reports Server (NTRS)

Martin, C.; Hermann, R.

1977-01-01

A general method described by Hermann and Martin (1976) for the study of the feedback invariants of linear systems is considered. It is shown that this method, which makes use of ideas of topology and algebraic geometry, is very useful in the investigation of feedback problems for which the classical methods are not suitable. The transfer function as a curve in the Grassmanian is examined. The general concepts studied in the context of specific systems and applications are organized in terms of the theory of Lie groups and algebraic geometry. Attention is given to linear systems which have a structure group, linear mechanical systems, and feedback invariants. The investigation shows that Lie group techniques are powerful and useful tools for analysis of the feedback structure of linear systems.

Adhesive-bonded scarf and stepped-lap joints

NASA Technical Reports Server (NTRS)

Hart-Smith, L. J.

1973-01-01

Continuum mechanics solutions are derived for the static load-carrying capacity of scarf and stepped-lap adhesive-bonded joints. The analyses account for adhesive plasticity and adherend stiffness imbalance and thermal mismatch. The scarf joint solutions include a simple algebraic formula which serves as a close lower bound, within a small fraction of a per cent of the true answer for most practical geometries and materials. Digital computer programs were developed and, for the stepped-lap joints, the critical adherend and adhesive stresses are computed for each step. The scarf joint solutions exhibit grossly different behavior from that for double-lap joints for long overlaps inasmuch as that the potential bond shear strength continues to increase with indefinitely long overlaps on the scarf joints. The stepped-lap joint solutions exhibit some characteristics of both the scarf and double-lap joints. The stepped-lap computer program handles arbitrary (different) step lengths and thickness and the solutions obtained have clarified potentially weak design details and the remedies. The program has been used effectively to optimize the joint proportions.

Deformation Theory and Physics Model Building

NASA Astrophysics Data System (ADS)

Sternheimer, Daniel

2006-08-01

The mathematical theory of deformations has proved to be a powerful tool in modeling physical reality. We start with a short historical and philosophical review of the context and concentrate this rapid presentation on a few interrelated directions where deformation theory is essential in bringing a new framework - which has then to be developed using adapted tools, some of which come from the deformation aspect. Minkowskian space-time can be deformed into Anti de Sitter, where massless particles become composite (also dynamically): this opens new perspectives in particle physics, at least at the electroweak level, including prediction of new mesons. Nonlinear group representations and covariant field equations, coming from interactions, can be viewed as some deformation of their linear (free) part: recognizing this fact can provide a good framework for treating problems in this area, in particular global solutions. Last but not least, (algebras associated with) classical mechanics (and field theory) on a Poisson phase space can be deformed to (algebras associated with) quantum mechanics (and quantum field theory). That is now a frontier domain in mathematics and theoretical physics called deformation quantization, with multiple ramifications, avatars and connections in both mathematics and physics. These include representation theory, quantum groups (when considering Hopf algebras instead of associative or Lie algebras), noncommutative geometry and manifolds, algebraic geometry, number theory, and of course what is regrouped under the name of M-theory. We shall here look at these from the unifying point of view of deformation theory and refer to a limited number of papers as a starting point for further study.

An Algebraic Approach to Guarantee Harmonic Balance Method Using Gröbner Base

NASA Astrophysics Data System (ADS)

Yagi, Masakazu; Hisakado, Takashi; Okumura, Kohshi

Harmonic balance (HB) method is well known principle for analyzing periodic oscillations on nonlinear networks and systems. Because the HB method has a truncation error, approximated solutions have been guaranteed by error bounds. However, its numerical computation is very time-consuming compared with solving the HB equation. This paper proposes an algebraic representation of the error bound using Gröbner base. The algebraic representation enables to decrease the computational cost of the error bound considerably. Moreover, using singular points of the algebraic representation, we can obtain accurate break points of the error bound by collisions.

Teaching of Real Numbers by Using the Archimedes-Cantor Approach and Computer Algebra Systems

ERIC Educational Resources Information Center

Vorob'ev, Evgenii M.

2015-01-01

Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of…

Pre-Service Elementary Teachers Make Connections between Geometry and Algebra through the Use of Technology

ERIC Educational Resources Information Center

Mohr, Doris J.

2008-01-01

In a geometry content course for pre-service elementary teachers, technology was utilized to assist students in making sense of shapes. They learned to write simple procedures in Logo that would program a turtle to draw various quadrilaterals. In the context of writing these procedures, the pre-service teachers used variables to represent the…

PAB3D Simulations of a Nozzle with Fluidic Injection for Yaw Thrust-Vector Control

NASA Technical Reports Server (NTRS)

Deere, Karen A.

1998-01-01

An experimental and computational study was conducted on an exhaust nozzle with fluidic injection for yaw thrust-vector control. The nozzle concept was tested experimentally in the NASA Langley Jet Exit Test Facility (JETF) at nozzle pressure ratios up to 4 and secondary fluidic injection flow rates up to 15 percent of the primary flow rate. Although many injection-port geometries and two nozzle planforms (symmetric and asymmetric) were tested experimentally, this paper focuses on the computational results of the more successful asymmetric planform with a slot injection port. This nozzle concept was simulated with the Navier-Stokes flow solver, PAB3D, invoking the Shih, Zhu, and Lumley algebraic Reynolds stress turbulence model (ASM) at nozzle pressure ratios (NPRs) of 2,3, and 4 with secondary to primary injection flow rates (w(sub s)/w(sub p)) of 0, 2, 7 and 10 percent.

Anatomical medial surfaces with efficient resolution of branches singularities.

PubMed

Gil, Debora; Vera, Sergio; Borràs, Agnés; Andaluz, Albert; González Ballester, Miguel A

2017-01-01

Medial surfaces are powerful tools for shape description, but their use has been limited due to the sensibility of existing methods to branching artifacts. Medial branching artifacts are associated to perturbations of the object boundary rather than to geometric features. Such instability is a main obstacle for a confident application in shape recognition and description. Medial branches correspond to singularities of the medial surface and, thus, they are problematic for existing morphological and energy-based algorithms. In this paper, we use algebraic geometry concepts in an energy-based approach to compute a medial surface presenting a stable branching topology. We also present an efficient GPU-CPU implementation using standard image processing tools. We show the method computational efficiency and quality on a custom made synthetic database. Finally, we present some results on a medical imaging application for localization of abdominal pathologies. Copyright © 2016 Elsevier B.V. All rights reserved.

The Overgrid Interface for Computational Simulations on Overset Grids

NASA Technical Reports Server (NTRS)

Chan, William M.; Kwak, Dochan (Technical Monitor)

2002-01-01

Computational simulations using overset grids typically involve multiple steps and a variety of software modules. A graphical interface called OVERGRID has been specially designed for such purposes. Data required and created by the different steps include geometry, grids, domain connectivity information and flow solver input parameters. The interface provides a unified environment for the visualization, processing, generation and diagnosis of such data. General modules are available for the manipulation of structured grids and unstructured surface triangulations. Modules more specific for the overset approach include surface curve generators, hyperbolic and algebraic surface grid generators, a hyperbolic volume grid generator, Cartesian box grid generators, and domain connectivity: pre-processing tools. An interface provides automatic selection and viewing of flow solver boundary conditions, and various other flow solver inputs. For problems involving multiple components in relative motion, a module is available to build the component/grid relationships and to prescribe and animate the dynamics of the different components.

Multi-Mounted X-Ray Computed Tomography.

PubMed

Fu, Jian; Liu, Zhenzhong; Wang, Jingzheng

2016-01-01

Most existing X-ray computed tomography (CT) techniques work in single-mounted mode and need to scan the inspected objects one by one. It is time-consuming and not acceptable for the inspection in a large scale. In this paper, we report a multi-mounted CT method and its first engineering implementation. It consists of a multi-mounted scanning geometry and the corresponding algebraic iterative reconstruction algorithm. This approach permits the CT rotation scanning of multiple objects simultaneously without the increase of penetration thickness and the signal crosstalk. Compared with the conventional single-mounted methods, it has the potential to improve the imaging efficiency and suppress the artifacts from the beam hardening and the scatter. This work comprises a numerical study of the method and its experimental verification using a dataset measured with a developed multi-mounted X-ray CT prototype system. We believe that this technique is of particular interest for pushing the engineering applications of X-ray CT.

Improved finite-difference computation of the van der Waals force: One-dimensional case

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pinto, Fabrizio

2009-10-15

We present an improved demonstration of the calculation of Casimir forces in one-dimensional systems based on the recently proposed numerical imaginary frequency Green's function computation approach. The dispersion force on two thick lossy dielectric slabs separated by an empty gap and placed within a perfectly conducting cavity is obtained from the Green's function of the modified Helmholtz equation by means of an ordinary finite-difference method. In order to demonstrate the possibility to develop algorithms to explore complex geometries in two and three dimensions to higher order in the mesh spacing, we generalize existing classical electromagnetism algebraic methods to generate themore » difference equations for dielectric boundaries not coinciding with any grid points. Diagnostic tests are presented to monitor the accuracy of our implementation of the method and follow-up applications in higher dimensions are introduced.« less

Exceptional quantum geometry and particle physics

NASA Astrophysics Data System (ADS)

Dubois-Violette, Michel

2016-11-01

Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group SU (3) and on the existence of 3 generations, we develop an argumentation suggesting that the "finite quantum space" corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space C ⊕C3 is associated to the quark-lepton symmetry (one complex for the lepton and 3 for the corresponding quark). More generally it is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of "the algebra of real functions" on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.

Matematicas Para El Primer Ciclo Secundario, Volumen I (Parte 1). Traduccion Preliminar de la Edicion Inglesa Revisada. (Mathematics for Junior High School, Volume I, Part 1, Preliminary Translation of the Revised English Edition).

ERIC Educational Resources Information Center

Anderson, R. D.; And Others

This is part one of a two-part SMSG mathematics text for junior high school students. Key ideas emphasized are structure of arithmetic from an algebraic viewpoint, the real number system, and metric and non-metric relations in geometry. Topics included are numbers; cardinal numbers; geometry of lines, points, and planes; geometry of angles,…

Image Algebra Matlab language version 2.3 for image processing and compression research

NASA Astrophysics Data System (ADS)

Schmalz, Mark S.; Ritter, Gerhard X.; Hayden, Eric

2010-08-01

Image algebra is a rigorous, concise notation that unifies linear and nonlinear mathematics in the image domain. Image algebra was developed under DARPA and US Air Force sponsorship at University of Florida for over 15 years beginning in 1984. Image algebra has been implemented in a variety of programming languages designed specifically to support the development of image processing and computer vision algorithms and software. The University of Florida has been associated with development of the languages FORTRAN, Ada, Lisp, and C++. The latter implementation involved a class library, iac++, that supported image algebra programming in C++. Since image processing and computer vision are generally performed with operands that are array-based, the Matlab™ programming language is ideal for implementing the common subset of image algebra. Objects include sets and set operations, images and operations on images, as well as templates and image-template convolution operations. This implementation, called Image Algebra Matlab (IAM), has been found to be useful for research in data, image, and video compression, as described herein. Due to the widespread acceptance of the Matlab programming language in the computing community, IAM offers exciting possibilities for supporting a large group of users. The control over an object's computational resources provided to the algorithm designer by Matlab means that IAM programs can employ versatile representations for the operands and operations of the algebra, which are supported by the underlying libraries written in Matlab. In a previous publication, we showed how the functionality of IAC++ could be carried forth into a Matlab implementation, and provided practical details of a prototype implementation called IAM Version 1. In this paper, we further elaborate the purpose and structure of image algebra, then present a maturing implementation of Image Algebra Matlab called IAM Version 2.3, which extends the previous implementation of IAM to include polymorphic operations over different point sets, as well as recursive convolution operations and functional composition. We also show how image algebra and IAM can be employed in image processing and compression research, as well as algorithm development and analysis.

Generalized Cartan Calculus in general dimension

DOE PAGES

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.

Computer-Aided College Algebra: Learning Components that Students Find Beneficial

ERIC Educational Resources Information Center

Aichele, Douglas B.; Francisco, Cynthia; Utley, Juliana; Wescoatt, Benjamin

2011-01-01

A mixed-method study was conducted during the Fall 2008 semester to better understand the experiences of students participating in computer-aided instruction of College Algebra using the software MyMathLab. The learning environment included a computer learning system for the majority of the instruction, a support system via focus groups (weekly…

Computer Algebra Systems in Education Newsletter[s].

ERIC Educational Resources Information Center

Computer Algebra Systems in Education Newsletter, 1990

1990-01-01

Computer Algebra Systems (CAS) are computer systems for the exact solution of problems in symbolic form. The newspaper is designed to serve as a conduit for information and ideas on the use of CAS in education, especially in lower division college and university courses. Articles included are about CAS programs in several colleges, experiences…

Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nataf, J.M.; Winkelmann, F.

We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK's symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of thesemore » methods to solving the partial differential equations for two-dimensional heat flow is illustrated.« less

Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nataf, J.M.; Winkelmann, F.

We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK`s symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of thesemore » methods to solving the partial differential equations for two-dimensional heat flow is illustrated.« less

Sixth SIAM conference on applied linear algebra: Final program and abstracts. Final technical report

DOE Office of Scientific and Technical Information (OSTI.GOV)

NONE

1997-12-31

Linear algebra plays a central role in mathematics and applications. The analysis and solution of problems from an amazingly wide variety of disciplines depend on the theory and computational techniques of linear algebra. In turn, the diversity of disciplines depending on linear algebra also serves to focus and shape its development. Some problems have special properties (numerical, structural) that can be exploited. Some are simply so large that conventional approaches are impractical. New computer architectures motivate new algorithms, and fresh ways to look at old ones. The pervasive nature of linear algebra in analyzing and solving problems means that peoplemore » from a wide spectrum--universities, industrial and government laboratories, financial institutions, and many others--share an interest in current developments in linear algebra. This conference aims to bring them together for their mutual benefit. Abstracts of papers presented are included.« less

Schwarz maps of algebraic linear ordinary differential equations

NASA Astrophysics Data System (ADS)

Sanabria Malagón, Camilo

2017-12-01

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.

A computer program for calculating the perfect gas inviscid flow field about blunt axisymmetric bodies at an angle of attack of 0 deg

NASA Technical Reports Server (NTRS)

Zoby, E. V.; Graves, R. A., Jr.

1973-01-01

A method for the rapid calculation of the inviscid shock layer about blunt axisymmetric bodies at an angle of attack of 0 deg has been developed. The procedure is of an inverse nature, that is, a shock wave is assumed and calculations proceed along rays normal to the shock. The solution is iterated until the given body is computed. The flow field solution procedure is programed at the Langley Research Center for the Control Data 6600 computer. The geometries specified in the program are sphores, ellipsoids, paraboloids, and hyperboloids which may conical afterbodies. The normal momentum equation is replaced with an approximate algebraic expression. This simplification significantly reduces machine computation time. Comparisons of the present results with shock shapes and surface pressure distributions obtained by the more exact methods indicate that the program provides reasonably accurate results for smooth bodies in axisymmetric flow. However, further research is required to establish the proper approximate form of the normal momentum equation for the two-dimensional case.

Survey of Turbulence Models for the Computation of Turbulent Jet Flow and Noise

NASA Technical Reports Server (NTRS)

Nallasamy, N.

1999-01-01

The report presents an overview of jet noise computation utilizing the computational fluid dynamic solution of the turbulent jet flow field. The jet flow solution obtained with an appropriate turbulence model provides the turbulence characteristics needed for the computation of jet mixing noise. A brief account of turbulence models that are relevant for the jet noise computation is presented. The jet flow solutions that have been directly used to calculate jet noise are first reviewed. Then, the turbulent jet flow studies that compute the turbulence characteristics that may be used for noise calculations are summarized. In particular, flow solutions obtained with the k-e model, algebraic Reynolds stress model, and Reynolds stress transport equation model are reviewed. Since, the small scale jet mixing noise predictions can be improved by utilizing anisotropic turbulence characteristics, turbulence models that can provide the Reynolds stress components must now be considered for jet flow computations. In this regard, algebraic stress models and Reynolds stress transport models are good candidates. Reynolds stress transport models involve more modeling and computational effort and time compared to algebraic stress models. Hence, it is recommended that an algebraic Reynolds stress model (ASM) be implemented in flow solvers to compute the Reynolds stress components.

VNAP2: A Computer Program for Computation of Two-dimensional, Time-dependent, Compressible, Turbulent Flow

NASA Technical Reports Server (NTRS)

Cline, M. C.

1981-01-01

A computer program, VNAP2, for calculating turbulent (as well as laminar and inviscid), steady, and unsteady flow is presented. It solves the two dimensional, time dependent, compressible Navier-Stokes equations. The turbulence is modeled with either an algebraic mixing length model, a one equation model, or the Jones-Launder two equation model. The geometry may be a single or a dual flowing stream. The interior grid points are computed using the unsplit MacCormack scheme. Two options to speed up the calculations for high Reynolds number flows are included. The boundary grid points are computed using a reference plane characteristic scheme with the viscous terms treated as source functions. An explicit artificial viscosity is included for shock computations. The fluid is assumed to be a perfect gas. The flow boundaries may be arbitrary curved solid walls, inflow/outflow boundaries, or free jet envelopes. Typical problems that can be solved concern nozzles, inlets, jet powered afterbodies, airfoils, and free jet expansions. The accuracy and efficiency of the program are shown by calculations of several inviscid and turbulent flows. The program and its use are described completely, and six sample cases and a code listing are included.

Asymptotic identity in min-plus algebra: a report on CPNS.

PubMed

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions.

Asymptotic Identity in Min-Plus Algebra: A Report on CPNS

PubMed Central

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions. PMID:21822446

FAST TRACK COMMUNICATION Algebraic classification of the Weyl tensor in higher dimensions based on its 'superenergy' tensor

NASA Astrophysics Data System (ADS)

Senovilla, José M. M.

2010-11-01

The algebraic classification of the Weyl tensor in the arbitrary dimension n is recovered by means of the principal directions of its 'superenergy' tensor. This point of view can be helpful in order to compute the Weyl aligned null directions explicitly, and permits one to obtain the algebraic type of the Weyl tensor by computing the principal eigenvalue of rank-2 symmetric future tensors. The algebraic types compatible with states of intrinsic gravitational radiation can then be explored. The underlying ideas are general, so that a classification of arbitrary tensors in the general dimension can be achieved.

Computers in the Classroom: Teacher's Resource Manual for Algebra.

ERIC Educational Resources Information Center

Koetke, Walter

Demonstration programs, possible assignments for students (with solutions), and remedial drill programs for students to use are presented to aid teachers using a computer or a computer terminal in the teaching of algebra. The text can be followed page by page or used as a well-indexed reference work, and specific suggestions are made on how and…

A Comparison of Equality in Computer Algebra and Correctness in Mathematical Pedagogy (II)

ERIC Educational Resources Information Center

Bradford, Russell; Davenport, James H.; Sangwin, Chris

2010-01-01

A perennial problem in computer-aided assessment is that "a right answer", pedagogically speaking, is not the same thing as "a mathematically correct expression", as verified by a computer algebra system, or indeed other techniques such as random evaluation. Paper I in this series considered the difference in cases where there was "the right…

The Spectral Shift Function and Spectral Flow

NASA Astrophysics Data System (ADS)

Azamov, N. A.; Carey, A. L.; Sukochev, F. A.

2007-11-01

At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general semifinite von Neumann algebra {mathcal{N}} and the perturbation V in {mathcal{N}} . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples where D has compact resolvent belonging to {mathcal{N}} and V is any bounded self-adjoint operator in {mathcal{N}} . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8-10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].

Newton-Cartan gravity and torsion

NASA Astrophysics Data System (ADS)

Bergshoeff, Eric; Chatzistavrakidis, Athanasios; Romano, Luca; Rosseel, Jan

2017-10-01

We compare the gauging of the Bargmann algebra, for the case of arbitrary torsion, with the result that one obtains from a null-reduction of General Relativity. Whereas the two procedures lead to the same result for Newton-Cartan geometry with arbitrary torsion, the null-reduction of the Einstein equations necessarily leads to Newton-Cartan gravity with zero torsion. We show, for three space-time dimensions, how Newton-Cartan gravity with arbitrary torsion can be obtained by starting from a Schrödinger field theory with dynamical exponent z = 2 for a complex compensating scalar and next coupling this field theory to a z = 2 Schrödinger geometry with arbitrary torsion. The latter theory can be obtained from either a gauging of the Schrödinger algebra, for arbitrary torsion, or from a null-reduction of conformal gravity.

Using computer algebra and SMT-solvers to analyze a mathematical model of cholera propagation

NASA Astrophysics Data System (ADS)

Trujillo Arredondo, Mariana

2014-06-01

We analyze a mathematical model for the transmission of cholera. The model is already defined and involves variables such as the pathogen agent, which in this case is the bacterium Vibrio cholera, and the human population. The human population is divided into three classes: susceptible, infectious and removed. Using Computer Algebra, specifically Maple we obtain two equilibrium states: the disease free state and the endemic state. Using Maple it is possible to prove that the disease free state is locally asymptotically stable if and only if R0 < 1. Using Maple it is possible to prove that the endemic equilibrium state is locally stable when it exists, it is to say when R0 > 1. Using the package Red-Log of the Computer algebra system Reduce and the SMT-Solver Z3Py it is possible to obtain numerical conditions for the model. The formula for the basic reproductive number makes a synthesis with all epidemic parameters in the model. Also it is possible to make numerical simulations which are very illustrative about the epidemic patters that are expected to be observed in real situations. We claim that these kinds of software are very useful in the analysis of epidemic models given that the symbolic computation provides algebraic formulas for the basic reproductive number and such algebraic formulas are very useful to derive control measures. For other side, computer algebra software is a powerful tool to make the stability analysis for epidemic models given that the all steps in the stability analysis can be made automatically: finding the equilibrium points, computing the jacobian, computing the characteristic polynomial for the jacobian, and applying the Routh-Hurwitz theorem to the characteristic polynomial. Finally, using SMT-Solvers is possible to make automatically checks of satisfiability, validity and quantifiers elimination being these computations very useful to analyse complicated epidemic models.

Algebraic properties of automata associated to Petri nets and applications to computation in biological systems.

PubMed

Egri-Nagy, Attila; Nehaniv, Chrystopher L

2008-01-01

Biochemical and genetic regulatory networks are often modeled by Petri nets. We study the algebraic structure of the computations carried out by Petri nets from the viewpoint of algebraic automata theory. Petri nets comprise a formalized graphical modeling language, often used to describe computation occurring within biochemical and genetic regulatory networks, but the semantics may be interpreted in different ways in the realm of automata. Therefore, there are several different ways to turn a Petri net into a state-transition automaton. Here, we systematically investigate different conversion methods and describe cases where they may yield radically different algebraic structures. We focus on the existence of group components of the corresponding transformation semigroups, as these reflect symmetries of the computation occurring within the biological system under study. Results are illustrated by applications to the Petri net modelling of intermediary metabolism. Petri nets with inhibition are shown to be computationally rich, regardless of the particular interpretation method. Along these lines we provide a mathematical argument suggesting a reason for the apparent all-pervasiveness of inhibitory connections in living systems.

Transverse Laplacians for Substitution Tilings

NASA Astrophysics Data System (ADS)

Julien, Antoine; Savinien, Jean

2011-01-01

Pearson and Bellissard recently built a spectral triple - the data of Riemannian noncommutative geometry - for ultrametric Cantor sets. They derived a family of Laplace-Beltrami like operators on those sets. Motivated by the applications to specific examples, we revisit their work for the transversals of tiling spaces, which are particular self-similar Cantor sets. We use Bratteli diagrams to encode the self-similarity, and Cuntz-Krieger algebras to implement it. We show that the abscissa of convergence of the ζ-function of the spectral triple gives indications on the exponent of complexity of the tiling. We determine completely the spectrum of the Laplace-Beltrami operators, give an explicit method of calculation for their eigenvalues, compute their Weyl asymptotics, and a Seeley equivalent for their heat kernels.

It's a Wonderful Life: Using Public Domain Cinema Clips To Teach Affective Objectives and Illustrate Real-World Algebra Applications.

ERIC Educational Resources Information Center

Palmer, Loretta

A basic algebra unit was developed at Utah Valley State College to emphasize applications of mathematical concepts in the work world, using video and computer-generated graphics to integrate textual material. The course was implemented in three introductory algebra sections involving 80 students and taught algebraic concepts using such areas as…

Using trees to compute approximate solutions to ordinary differential equations exactly

NASA Technical Reports Server (NTRS)

Grossman, Robert

1991-01-01

Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.

Reducing Communication in Algebraic Multigrid Using Additive Variants

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vassilevski, Panayot S.; Yang, Ulrike Meier

Algebraic multigrid (AMG) has proven to be an effective scalable solver on many high performance computers. However, its increasing communication complexity on coarser levels has shown to seriously impact its performance on computers with high communication cost. Moreover, additive AMG variants provide not only increased parallelism as well as decreased numbers of messages per cycle but also generally exhibit slower convergence. Here we present various new additive variants with convergence rates that are significantly improved compared to the classical additive algebraic multigrid method and investigate their potential for decreased communication, and improved communication-computation overlap, features that are essential for goodmore » performance on future exascale architectures.« less

Reducing Communication in Algebraic Multigrid Using Additive Variants

DOE PAGES

Vassilevski, Panayot S.; Yang, Ulrike Meier

2014-02-12

Algebraic multigrid (AMG) has proven to be an effective scalable solver on many high performance computers. However, its increasing communication complexity on coarser levels has shown to seriously impact its performance on computers with high communication cost. Moreover, additive AMG variants provide not only increased parallelism as well as decreased numbers of messages per cycle but also generally exhibit slower convergence. Here we present various new additive variants with convergence rates that are significantly improved compared to the classical additive algebraic multigrid method and investigate their potential for decreased communication, and improved communication-computation overlap, features that are essential for goodmore » performance on future exascale architectures.« less

Arithmetic Circuit Verification Based on Symbolic Computer Algebra

NASA Astrophysics Data System (ADS)

Watanabe, Yuki; Homma, Naofumi; Aoki, Takafumi; Higuchi, Tatsuo

This paper presents a formal approach to verify arithmetic circuits using symbolic computer algebra. Our method describes arithmetic circuits directly with high-level mathematical objects based on weighted number systems and arithmetic formulae. Such circuit description can be effectively verified by polynomial reduction techniques using Gröbner Bases. In this paper, we describe how the symbolic computer algebra can be used to describe and verify arithmetic circuits. The advantageous effects of the proposed approach are demonstrated through experimental verification of some arithmetic circuits such as multiply-accumulator and FIR filter. The result shows that the proposed approach has a definite possibility of verifying practical arithmetic circuits.

Soft hairy warped black hole entropy

NASA Astrophysics Data System (ADS)

Grumiller, Daniel; Hacker, Philip; Merbis, Wout

2018-02-01

We reconsider warped black hole solutions in topologically massive gravity and find novel boundary conditions that allow for soft hairy excitations on the horizon. To compute the associated symmetry algebra we develop a general framework to compute asymptotic symmetries in any Chern-Simons-like theory of gravity. We use this to show that the near horizon symmetry algebra consists of two u (1) current algebras and recover the surprisingly simple entropy formula S = 2 π( J 0 + + J 0 - ), where J 0 ± are zero mode charges of the current algebras. This provides the first example of a locally non-maximally symmetric configuration exhibiting this entropy law and thus non-trivial evidence for its universality.

Duct flow nonuniformities for Space Shuttle Main Engine (SSME)

NASA Technical Reports Server (NTRS)

1987-01-01

A three-duct Space Shuttle Main Engine (SSME) Hot Gas Manifold geometry code was developed for use. The methodology of the program is described, recommendations on its implementation made, and an input guide, input deck listing, and a source code listing provided. The code listing is strewn with an abundance of comments to assist the user in following its development and logic. A working source deck will be provided. A thorough analysis was made of the proper boundary conditions and chemistry kinetics necessary for an accurate computational analysis of the flow environment in the SSME fuel side preburner chamber during the initial startup transient. Pertinent results were presented to facilitate incorporation of these findings into an appropriate CFD code. The computation must be a turbulent computation, since the flow field turbulent mixing will have a profound effect on the chemistry. Because of the additional equations demanded by the chemistry model it is recommended that for expediency a simple algebraic mixing length model be adopted. Performing this computation for all or selected time intervals of the startup time will require an abundance of computer CPU time regardless of the specific CFD code selected.

Numerical methods on some structured matrix algebra problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jessup, E.R.

1996-06-01

This proposal concerned the design, analysis, and implementation of serial and parallel algorithms for certain structured matrix algebra problems. It emphasized large order problems and so focused on methods that can be implemented efficiently on distributed-memory MIMD multiprocessors. Such machines supply the computing power and extensive memory demanded by the large order problems. We proposed to examine three classes of matrix algebra problems: the symmetric and nonsymmetric eigenvalue problems (especially the tridiagonal cases) and the solution of linear systems with specially structured coefficient matrices. As all of these are of practical interest, a major goal of this work was tomore » translate our research in linear algebra into useful tools for use by the computational scientists interested in these and related applications. Thus, in addition to software specific to the linear algebra problems, we proposed to produce a programming paradigm and library to aid in the design and implementation of programs for distributed-memory MIMD computers. We now report on our progress on each of the problems and on the programming tools.« less

Zooming in on AdS3/CFT2 near a BPS bound

NASA Astrophysics Data System (ADS)

Hartong, Jelle; Lei, Yang; Obers, Niels; Oling, Gerben

2018-05-01

Any ( d + 1)-dimensional CFT with a U(1) flavor symmetry, a BPS bound and an exactly marginal coupling admits a decoupling limit in which one zooms in on the spectrum close to the bound. This limit is an Inönü-Wigner contraction of so(2 , d+1)⊕ u(1) that leads to a relativistic algebra with a scaling generator but no conformal generators. In 2D CFTs, Lorentz boosts are abelian and by adding a second u(1) we find a contraction of two copies of sl(2, ℝ) ⊕ u(1) to two copies of P 2 c , the 2-dimensional centrally extended Poincaré algebra. We show that the bulk is described by a novel non-Lorentzian geometry that we refer to as pseudo-Newton-Cartan geometry. Both the Chern-Simons action on sl(2, ℝ) ⊕ u(1) and the entire phase space of asymptotically AdS3 spacetimes are well-behaved in the corresponding limit if we fix the radial component for the u(1) connection. With this choice, the resulting Newton-Cartan foliation structure is now associated not with time, but with the emerging holographic direction. Since the leaves of this foliation do not mix, the emergence of the holographic direction is much simpler than in AdS3 holography. Furthermore, we show that the asymptotic symmetry algebra of the limit theory consists of a left- and a right-moving warped Virasoro algebra.

Analyzing and Comparing the Two Grade-Ten Mathematics Textbooks Versions (Arabic and English) Used in Abu Dhabi Schools

ERIC Educational Resources Information Center

Abosalem, Yousef

2016-01-01

This study was conducted to compare two grade-ten mathematics textbooks according to Bloom's Taxonomies. In the Arabic version, 37 out of 70 periods (55.29%) were given to plane geometry and trigonometry, whereas 29 out of 70 periods (41.41%) were allocated for geometry and trigonometry. Also, 12 periods (17.14%) were allocated for algebra in the…

Geometric model of topological insulators from the Maxwell algebra

NASA Astrophysics Data System (ADS)

Palumbo, Giandomenico

2017-11-01

We propose a novel geometric model of time-reversal-invariant topological insulators in three dimensions in presence of an external electromagnetic field. Their gapped boundary supports relativistic quantum Hall states and is described by a Chern-Simons theory, where the gauge connection takes values in the Maxwell algebra. This represents a non-central extension of the Poincaré algebra and takes into account both the Lorentz and magnetic-translation symmetries of the surface states. In this way, we derive a relativistic version of the Wen-Zee term and we show that the non-minimal coupling between the background geometry and the electromagnetic field in the model is in agreement with the main properties of the relativistic quantum Hall states in the flat space.

In Praise of Numerical Computation

NASA Astrophysics Data System (ADS)

Yap, Chee K.

Theoretical Computer Science has developed an almost exclusively discrete/algebraic persona. We have effectively shut ourselves off from half of the world of computing: a host of problems in Computational Science & Engineering (CS&E) are defined on the continuum, and, for them, the discrete viewpoint is inadequate. The computational techniques in such problems are well-known to numerical analysis and applied mathematics, but are rarely discussed in theoretical algorithms: iteration, subdivision and approximation. By various case studies, I will indicate how our discrete/algebraic view of computing has many shortcomings in CS&E. We want embrace the continuous/analytic view, but in a new synthesis with the discrete/algebraic view. I will suggest a pathway, by way of an exact numerical model of computation, that allows us to incorporate iteration and approximation into our algorithms’ design. Some recent results give a peek into how this view of algorithmic development might look like, and its distinctive form suggests the name “numerical computational geometry” for such activities.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Suh, Uhi Rinn, E-mail: uhrisu1@math.snu.ac.kr

We introduce a classical BRST complex (See Definition 3.2.) and show that one can construct a classical affine W-algebra via the complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible λ-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we describe explicit forms ofmore » free generators and compute λ-brackets between them. Provided some assumptions on a classical affine fractional W-algebra, we find an infinite sequence of integrable systems related to the algebra, using the generalized Drinfel’d and Sokolov reduction.« less

Effects of Computer Algebra System (CAS) with Metacognitive Training on Mathematical Reasoning.

ERIC Educational Resources Information Center

Kramarski, Bracha; Hirsch, Chaya

2003-01-01

Describes a study that investigated the differential effects of Computer Algebra Systems (CAS) and metacognitive training (META) on mathematical reasoning. Participants were 83 Israeli eighth-grade students. Results showed that CAS embedded within META significantly outperformed the META and CAS alone conditions, which in turn significantly…

Reflections on John Monaghan's "Computer Algebra, Instrumentation, and the Anthropological Approach"

ERIC Educational Resources Information Center

Blume, Glen

2007-01-01

Reactions to John Monaghan's "Computer Algebra, Instrumentation and the Anthropological Approach" focus on a variety of issues related to the ergonomic approach (instrumentation) and anthropological approach to mathematical activity and practice. These include uses of the term technique; several possibilities for integration of the two approaches;…

A Flexible, Extensible Online Testing System for Mathematics

ERIC Educational Resources Information Center

Passmore, Tim; Brookshaw, Leigh; Butler, Harry

2011-01-01

An online testing system developed for entry-skills testing of first-year university students in algebra and calculus is described. The system combines the open-source computer algebra system "Maxima" with computer scripts to parse student answers, which are entered using standard mathematical notation and conventions. The answers can…

Computer Algebra Systems and Theorems on Real Roots of Polynomials

ERIC Educational Resources Information Center

Aidoo, Anthony Y.; Manthey, Joseph L.; Ward, Kim Y.

2010-01-01

A computer algebra system is used to derive a theorem on the existence of roots of a quadratic equation on any bounded real interval. This is extended to a cubic polynomial. We discuss how students could be led to derive and prove these theorems. (Contains 1 figure.)

Inequalities, Assessment and Computer Algebra

ERIC Educational Resources Information Center

Sangwin, Christopher J.

2015-01-01

The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary…

Computer Algebra System Calculators: Gender Issues and Teachers' Expectations

ERIC Educational Resources Information Center

Forgasz, Helen J.; Griffith, Shirly

2006-01-01

In this paper we present findings from two studies focusing on computer algebra system (CAS) calculators. In Victoria, Australia, it is currently mandatory for students to use graphics calculators in some grade 12 mathematics examinations. Since 2001, a pilot study has been conducted involving Victorian Certificate of Education (VCE) students…

Using Computer Symbolic Algebra to Solve Differential Equations.

ERIC Educational Resources Information Center

Mathews, John H.

1989-01-01

This article illustrates that mathematical theory can be incorporated into the process to solve differential equations by a computer algebra system, muMATH. After an introduction to functions of muMATH, several short programs for enhancing the capabilities of the system are discussed. Listed are six references. (YP)

Color Algebras

NASA Technical Reports Server (NTRS)

Mulligan, Jeffrey B.

2017-01-01

A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.

Class and Homework Problems: The Break-Even Radius of Insulation Computed Using Excel Solver and WolframAlpha

ERIC Educational Resources Information Center

Foley, Greg

2014-01-01

A problem that illustrates two ways of computing the break-even radius of insulation is outlined. The problem is suitable for students who are taking an introductory module in heat transfer or transport phenomena and who have some previous knowledge of the numerical solution of non- linear algebraic equations. The potential for computer algebra,…

How to Orbit the Earth.

ERIC Educational Resources Information Center

Quimby, Donald J.

1984-01-01

Discusses the geometry, algebra, and logic involved in the solution of a "Mindbenders" problem in "Discover" magazine and applies it to calculations of satellite orbital velocity. Extends the solution of this probe to other applications of falling objects. (JM)

Reflective Properties of a Parabolic Mirror.

ERIC Educational Resources Information Center

Ramsey, Gordon P.

1991-01-01

An incident light ray parallel to the optical axis of a parabolic mirror will be reflected at the focal point and vice versa. Presents a mathematical proof that uses calculus, algebra, and geometry to prove this reflective property. (MDH)

42 CFR Appendix A to Part 75 - Standards for Accreditation of Educational Programs for Radiographers

Code of Federal Regulations, 2011 CFR

2011-10-01

... film evaluation; (k) Methods of patient care; (l) Pathology; (m) Radiologic physics; and (n) Radiation.... Courses in physics, chemistry, biology, algebra, and geometry are strongly recommended. (b) The number of...

42 CFR Appendix A to Part 75 - Standards for Accreditation of Educational Programs for Radiographers

Code of Federal Regulations, 2014 CFR

2014-10-01

... film evaluation; (k) Methods of patient care; (l) Pathology; (m) Radiologic physics; and (n) Radiation.... Courses in physics, chemistry, biology, algebra, and geometry are strongly recommended. (b) The number of...

42 CFR Appendix A to Part 75 - Standards for Accreditation of Educational Programs for Radiographers

Code of Federal Regulations, 2012 CFR

2012-10-01

... film evaluation; (k) Methods of patient care; (l) Pathology; (m) Radiologic physics; and (n) Radiation.... Courses in physics, chemistry, biology, algebra, and geometry are strongly recommended. (b) The number of...

42 CFR Appendix A to Part 75 - Standards for Accreditation of Educational Programs for Radiographers

Code of Federal Regulations, 2013 CFR

2013-10-01

... film evaluation; (k) Methods of patient care; (l) Pathology; (m) Radiologic physics; and (n) Radiation.... Courses in physics, chemistry, biology, algebra, and geometry are strongly recommended. (b) The number of...

The History of Mathematics and Mathematical Education

ERIC Educational Resources Information Center

Grattan-Guinness, I.

1977-01-01

Answers to questions which were asked after the author's various lectures in Australia are gathered here. Topics touched upon include "new" mathematics, unknown constants and free variables, propositional functions, linear algebra, arithmetic and geometry, and student assessment. (MN)

Lightning Radio Source Retrieval Using Advanced Lightning Direction Finder (ALDF) Networks

NASA Technical Reports Server (NTRS)

Koshak, William J.; Blakeslee, Richard J.; Bailey, J. C.

1998-01-01

A linear algebraic solution is provided for the problem of retrieving the location and time of occurrence of lightning ground strikes from an Advanced Lightning Direction Finder (ALDF) network. The ALDF network measures field strength, magnetic bearing and arrival time of lightning radio emissions. Solutions for the plane (i.e., no Earth curvature) are provided that implement all of tile measurements mentioned above. Tests of the retrieval method are provided using computer-simulated data sets. We also introduce a quadratic planar solution that is useful when only three arrival time measurements are available. The algebra of the quadratic root results are examined in detail to clarify what portions of the analysis region lead to fundamental ambiguities in source location. Complex root results are shown to be associated with the presence of measurement errors when the lightning source lies near an outer sensor baseline of the ALDF network. In the absence of measurement errors, quadratic root degeneracy (no source location ambiguity) is shown to exist exactly on the outer sensor baselines for arbitrary non-collinear network geometries. The accuracy of the quadratic planar method is tested with computer generated data sets. The results are generally better than those obtained from the three station linear planar method when bearing errors are about 2 deg. We also note some of the advantages and disadvantages of these methods over the nonlinear method of chi(sup 2) minimization employed by the National Lightning Detection Network (NLDN) and discussed in Cummins et al.(1993, 1995, 1998).

A Discrete Global Grid System Programming Language Using MapReduce

NASA Astrophysics Data System (ADS)

Peterson, P.; Shatz, I.

2016-12-01

A discrete global grid system (DGGS) is a powerful mechanism for storing and integrating geospatial information. As a "pixelization" of the Earth, many image processing techniques lend themselves to the transformation of data values referenced to the DGGS cells. It has been shown that image algebra, as an example, and advanced algebra, like Fast Fourier Transformation, can be used on the DGGS tiling structure for geoprocessing and spatial analysis. MapReduce has been shown to provide advantages for processing and generating large data sets within distributed and parallel computing. The DGGS structure is ideally suited for big distributed Earth data. We proposed that basic expressions could be created to form the atoms of a generalized DGGS language using the MapReduce programming model. We created three very efficient expressions: Selectors (aka filter) - A selection function that generate a set of cells, cell collections, or geometries; Calculators (aka map) - A computational function (including quantization of raw measurements and data sources) that generate values in a DGGS cell; and Aggregators (aka reduce) - A function that generate spatial statistics from cell values within a cell. We found that these three basic MapReduce operations along with a forth function, the Iterator, for horizontal and vertical traversing of any DGGS structure, provided simple building block resulting in very efficient operations and processes that could be used with any DGGS. We provide examples and a demonstration of their effectiveness using the ISEA3H DGGS on the PYXIS Studio.

Data Retrieval Algorithms for Validating the Optical Transient Detector and the Lightning Imaging Sensor

NASA Technical Reports Server (NTRS)

Koshak, W. J.; Blakeslee, R. J.; Bailey, J. C.

2000-01-01

A linear algebraic solution is provided for the problem of retrieving the location and time of occurrence of lightning ground strikes from an Advanced Lightning Direction Finder (ALDF) network. The ALDF network measures field strength, magnetic bearing, and arrival time of lightning radio emissions. Solutions for the plane (i.e., no earth curvature) are provided that implement all of these measurements. The accuracy of the retrieval method is tested using computer-simulated datasets, and the relative influence of bearing and arrival time data an the outcome of the final solution is formally demonstrated. The algorithm is sufficiently accurate to validate NASA:s Optical Transient Detector and Lightning Imaging Sensor. A quadratic planar solution that is useful when only three arrival time measurements are available is also introduced. The algebra of the quadratic root results are examined in detail to clarify what portions of the analysis region lead to fundamental ambiguities in sc)iirce location, Complex root results are shown to be associated with the presence of measurement errors when the lightning source lies near an outer sensor baseline of the ALDF network. For arbitrary noncollinear network geometries and in the absence of measurement errors, it is shown that the two quadratic roots are equivalent (no source location ambiguity) on the outer sensor baselines. The accuracy of the quadratic planar method is tested with computer-generated datasets, and the results are generally better than those obtained from the three-station linear planar method when bearing errors are about 2 deg.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Slattery, Stuart R.

In this study we analyze and extend mesh-free algorithms for three-dimensional data transfer problems in partitioned multiphysics simulations. We first provide a direct comparison between a mesh-based weighted residual method using the common-refinement scheme and two mesh-free algorithms leveraging compactly supported radial basis functions: one using a spline interpolation and one using a moving least square reconstruction. Through the comparison we assess both the conservation and accuracy of the data transfer obtained from each of the methods. We do so for a varying set of geometries with and without curvature and sharp features and for functions with and without smoothnessmore » and with varying gradients. Our results show that the mesh-based and mesh-free algorithms are complementary with cases where each was demonstrated to perform better than the other. We then focus on the mesh-free methods by developing a set of algorithms to parallelize them based on sparse linear algebra techniques. This includes a discussion of fast parallel radius searching in point clouds and restructuring the interpolation algorithms to leverage data structures and linear algebra services designed for large distributed computing environments. The scalability of our new algorithms is demonstrated on a leadership class computing facility using a set of basic scaling studies. Finally, these scaling studies show that for problems with reasonable load balance, our new algorithms for both spline interpolation and moving least square reconstruction demonstrate both strong and weak scalability using more than 100,000 MPI processes with billions of degrees of freedom in the data transfer operation.« less

Comparison of methods for developing the dynamics of rigid-body systems

NASA Technical Reports Server (NTRS)

Ju, M. S.; Mansour, J. M.

1989-01-01

Several approaches for developing the equations of motion for a three-degree-of-freedom PUMA robot were compared on the basis of computational efficiency (i.e., the number of additions, subtractions, multiplications, and divisions). Of particular interest was the investigation of the use of computer algebra as a tool for developing the equations of motion. Three approaches were implemented algebraically: Lagrange's method, Kane's method, and Wittenburg's method. Each formulation was developed in absolute and relative coordinates. These six cases were compared to each other and to a recursive numerical formulation. The results showed that all of the formulations implemented algebraically required fewer calculations than the recursive numerical algorithm. The algebraic formulations required fewer calculations in absolute coordinates than in relative coordinates. Each of the algebraic formulations could be simplified, using patterns from Kane's method, to yield the same number of calculations in a given coordinate system.

Generalized Quantum Field Theory Based on a Nonlinear Deformed Heisenberg Algebra

NASA Astrophysics Data System (ADS)

Ribeiro-Silva, C. I.; Oliveira-Neto, N. M.

We consider a quantum field theory based on a nonlinear Heisenberg algebra which describes phenomenologically a composite particle. Perturbative computation, considering the λϕ4 interaction was done and we also performed some comparison with a quantum field theory based on the q-oscillator algebra.

Some Applications of Algebraic System Solving

ERIC Educational Resources Information Center

Roanes-Lozano, Eugenio

2011-01-01

Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact…

Astro Algebra [CD-ROM].

ERIC Educational Resources Information Center

1997

Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…

A Structural Model of Algebra Achievement: Computational Fluency and Spatial Visualisation as Mediators of the Effect of Working Memory on Algebra Achievement

ERIC Educational Resources Information Center

Tolar, Tammy Daun; Lederberg, Amy R.; Fletcher, Jack M.

2009-01-01

The goal of this study was to develop and evaluate a structural model of the relations among cognitive abilities and arithmetic skills and college students' algebra achievement. The model of algebra achievement was compared to a model of performance on the Scholastic Assessment in Mathematics (SAT-M) to determine whether the pattern of relations…

Decomposition of algebraic sets and applications to weak centers of cubic systems

NASA Astrophysics Data System (ADS)

Chen, Xingwu; Zhang, Weinian

2009-10-01

There are many methods such as Gröbner basis, characteristic set and resultant, in computing an algebraic set of a system of multivariate polynomials. The common difficulties come from the complexity of computation, singularity of the corresponding matrices and some unnecessary factors in successive computation. In this paper, we decompose algebraic sets, stratum by stratum, into a union of constructible sets with Sylvester resultants, so as to simplify the procedure of elimination. Applying this decomposition to systems of multivariate polynomials resulted from period constants of reversible cubic differential systems which possess a quadratic isochronous center, we determine the order of weak centers and discuss the bifurcation of critical periods.

A path model for Whittaker vectors

NASA Astrophysics Data System (ADS)

Di Francesco, Philippe; Kedem, Rinat; Turmunkh, Bolor

2017-06-01

In this paper we construct weighted path models to compute Whittaker vectors in the completion of Verma modules, as well as Whittaker functions of fundamental type, for all finite-dimensional simple Lie algebras, affine Lie algebras, and the quantum algebra U_q(slr+1) . This leads to series expressions for the Whittaker functions. We show how this construction leads directly to the quantum Toda equations satisfied by these functions, and to the q-difference equations in the quantum case. We investigate the critical limit of affine Whittaker functions computed in this way.

"MONSTROUS MOONSHINE" and Physics

NASA Astrophysics Data System (ADS)

Pushkin, A. V.

The report presents some results obtained by the author on the quantum gravitation theory. Algebraic structure of this theory proves to be related to the commutative nonassociative Griess algebra. The theory symmetry is the automorphism group of Griess algebra: "Monster" simple group. Knowledge of the theory symmetry allows to compute observed physical values in the `zero' approximation. The report presents such computed results for values {m_{p}}/{m_{c}} and α, for the latter the `zero' approximation accuracy, controlled by the theory, being one order of magnitude higher than the accuracy of modern measurements.

Quantum Koszul formula on quantum spacetime

NASA Astrophysics Data System (ADS)

Majid, Shahn; Williams, Liam

2018-07-01

Noncommutative or quantum Riemannian geometry has been proposed as an effective theory for aspects of quantum gravity. Here the metric is an invertible bimodule map Ω1⊗AΩ1 → A where A is a possibly noncommutative or 'quantum' spacetime coordinate algebra and (Ω1 , d) is a specified bimodule of 1-forms or 'differential calculus' over it. In this paper we explore the proposal of a 'quantum Koszul formula' in Majid [12] with initial data a degree - 2 bilinear map ⊥ on the full exterior algebra Ω obeying the 4-term relations

A versatile embedded boundary adaptive mesh method for compressible flow in complex geometry

NASA Astrophysics Data System (ADS)

Al-Marouf, M.; Samtaney, R.

2017-05-01

We present an embedded ghost fluid method for numerical solutions of the compressible Navier Stokes (CNS) equations in arbitrary complex domains. A PDE multidimensional extrapolation approach is used to reconstruct the solution in the ghost fluid regions and imposing boundary conditions on the fluid-solid interface, coupled with a multi-dimensional algebraic interpolation for freshly cleared cells. The CNS equations are numerically solved by the second order multidimensional upwind method. Block-structured adaptive mesh refinement, implemented with the Chombo framework, is utilized to reduce the computational cost while keeping high resolution mesh around the embedded boundary and regions of high gradient solutions. The versatility of the method is demonstrated via several numerical examples, in both static and moving geometry, ranging from low Mach number nearly incompressible flows to supersonic flows. Our simulation results are extensively verified against other numerical results and validated against available experimental results where applicable. The significance and advantages of our implementation, which revolve around balancing between the solution accuracy and implementation difficulties, are briefly discussed as well.

High speed imaging of dynamic processes with a switched source x-ray CT system

NASA Astrophysics Data System (ADS)

Thompson, William M.; Lionheart, William R. B.; Morton, Edward J.; Cunningham, Mike; Luggar, Russell D.

2015-05-01

Conventional x-ray computed tomography (CT) scanners are limited in their scanning speed by the mechanical constraints of their rotating gantries and as such do not provide the necessary temporal resolution for imaging of fast-moving dynamic processes, such as moving fluid flows. The Real Time Tomography (RTT) system is a family of fast cone beam CT scanners which instead use multiple fixed discrete sources and complete rings of detectors in an offset geometry. We demonstrate the potential of this system for use in the imaging of such high speed dynamic processes and give results using simulated and real experimental data. The unusual scanning geometry results in some challenges in image reconstruction, which are overcome using algebraic iterative reconstruction techniques and explicit regularisation. Through the use of a simple temporal regularisation term and by optimising the source firing pattern, we show that temporal resolution of the system may be increased at the expense of spatial resolution, which may be advantageous in some situations. Results are given showing temporal resolution of approximately 500 µs with simulated data and 3 ms with real experimental data.

Students' Use of Computational Thinking in Linear Algebra

ERIC Educational Resources Information Center

Bagley, Spencer; Rabin, Jeffrey M.

2016-01-01

In this work, we examine students' ways of thinking when presented with a novel linear algebra problem. Our intent was to explore how students employ and coordinate three modes of thinking, which we call computational, abstract, and geometric, following similar frameworks proposed by Hillel (2000) and Sierpinska (2000). However, the undergraduate…

High Level Technology in a Low Level Mathematics Course.

ERIC Educational Resources Information Center

Schultz, James E.; Noguera, Norma

2000-01-01

Describes a teaching experiment in which spreadsheets and computer algebra systems were used to teach a low-level college consumer mathematics course. Students were successful in using different types of functions to solve a variety of problems drawn from real-world situations. Provides an existence proof that computer algebra systems can assist…

Revisiting Newtonian and Non-Newtonian Fluid Mechanics Using Computer Algebra

ERIC Educational Resources Information Center

Knight, D. G.

2006-01-01

This article illustrates how a computer algebra system, such as Maple[R], can assist in the study of theoretical fluid mechanics, for both Newtonian and non-Newtonian fluids. The continuity equation, the stress equations of motion, the Navier-Stokes equations, and various constitutive equations are treated, using a full, but straightforward,…

Algebraic Functions, Computer Programming, and the Challenge of Transfer

ERIC Educational Resources Information Center

Schanzer, Emmanuel Tanenbaum

2015-01-01

Students' struggles with algebra are well documented. Prior to the introduction of functions, mathematics is typically focused on applying a set of arithmetic operations to compute an answer. The introduction of functions, however, marks the point at which mathematics begins to focus on building up abstractions as a way to solve complex problems.…

Some Unexpected Results Using Computer Algebra Systems.

ERIC Educational Resources Information Center

Alonso, Felix; Garcia, Alfonsa; Garcia, Francisco; Hoya, Sara; Rodriguez, Gerardo; de la Villa, Agustin

2001-01-01

Shows how teachers can often use unexpected outputs from Computer Algebra Systems (CAS) to reinforce concepts and to show students the importance of thinking about how they use the software and reflecting on their results. Presents different examples where DERIVE, MAPLE, or Mathematica does not work as expected and suggests how to use them as a…

Exploring Students' Understanding of Ordinary Differential Equations Using Computer Algebraic System (CAS)

ERIC Educational Resources Information Center

Maat, Siti Mistima; Zakaria, Effandi

2011-01-01

Ordinary differential equations (ODEs) are one of the important topics in engineering mathematics that lead to the understanding of technical concepts among students. This study was conducted to explore the students' understanding of ODEs when they solve ODE questions using a traditional method as well as a computer algebraic system, particularly…

Analysis of Computer Algebra System Tutorials Using Cognitive Load Theory

ERIC Educational Resources Information Center

May, Patricia

2004-01-01

Most research in the area of Computer Algebra Systems (CAS) has been designed to compare the effectiveness of instructional technology to traditional lecture-based formats. While results are promising, research also indicates evidence of the steep learning curve imposed by the technology. Yet no studies have been conducted to investigate this…

A Study of the Use of a Handheld Computer Algebra System in Discrete Mathematics

ERIC Educational Resources Information Center

Powers, Robert A.; Allison, Dean E.; Grassl, Richard M.

2005-01-01

This study investigated the impact of the TI-92 handheld Computer Algebra System (CAS) on student achievement in a discrete mathematics course. Specifically, the researchers examined the differences between a CAS section and a control section of discrete mathematics on students' in-class examinations. Additionally, they analysed student approaches…

Developing a TI-92 Manual Generator Based on Computer Algebra Systems

ERIC Educational Resources Information Center

Jun, Youngcook

2004-01-01

The electronic medium suitable for mathematics learning and teaching is often designed with a notebook interface provided in a computer algebra system. Such a notebook interface facilitates a workspace for mathematical activities along with an online help system. In this paper, the proposed feature is implemented in the Mathematica's notebook…

Daubechies wavelets for linear scaling density functional theory.

PubMed

Mohr, Stephan; Ratcliff, Laura E; Boulanger, Paul; Genovese, Luigi; Caliste, Damien; Deutsch, Thierry; Goedecker, Stefan

2014-05-28

We demonstrate that Daubechies wavelets can be used to construct a minimal set of optimized localized adaptively contracted basis functions in which the Kohn-Sham orbitals can be represented with an arbitrarily high, controllable precision. Ground state energies and the forces acting on the ions can be calculated in this basis with the same accuracy as if they were calculated directly in a Daubechies wavelets basis, provided that the amplitude of these adaptively contracted basis functions is sufficiently small on the surface of the localization region, which is guaranteed by the optimization procedure described in this work. This approach reduces the computational costs of density functional theory calculations, and can be combined with sparse matrix algebra to obtain linear scaling with respect to the number of electrons in the system. Calculations on systems of 10,000 atoms or more thus become feasible in a systematic basis set with moderate computational resources. Further computational savings can be achieved by exploiting the similarity of the adaptively contracted basis functions for closely related environments, e.g., in geometry optimizations or combined calculations of neutral and charged systems.

PROTEUS two-dimensional Navier-Stokes computer code, version 1.0. Volume 3: Programmer's reference

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Benson, Thomas J.; Suresh, Ambady

1990-01-01

A new computer code was developed to solve the 2-D or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating-direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 3 is the Programmer's Reference, and describes the program structure, the FORTRAN variables stored in common blocks, and the details of each subprogram.

High School Students with Learning Disabilities: Mathematics Instruction, Study Skills, and High Stakes Tests

ERIC Educational Resources Information Center

Steele, Marcee M.

2010-01-01

This article reviews characteristics of high school students with learning disabilities and presents instructional modifications and study skills to help them succeed in algebra and geometry courses and on high stakes mathematics assessments.

Literal algebra for satellite dynamics. [perturbation analysis

NASA Technical Reports Server (NTRS)

Gaposchkin, E. M.

1975-01-01

A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.

Designing Virtual Worlds for Use in Mathematics Education.

ERIC Educational Resources Information Center

Winn, William; Bricken, William

Virtual Reality (VR) is a computer generated, multi-dimensional, inclusive environment that can build axioms of algebra into the behavior of the world. This paper discusses the use of VR to represent part of the algebra curriculum in order to improve students' classroom experiences in learning algebra. Students learn to construct their knowledge…

Linear Algebra and Image Processing

ERIC Educational Resources Information Center

Allali, Mohamed

2010-01-01

We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)

Teaching the "Diagonalization Concept" in Linear Algebra with Technology: A Case Study at Galatasaray University

ERIC Educational Resources Information Center

Yildiz Ulus, Aysegul

2013-01-01

This paper examines experimental and algorithmic contributions of advanced calculators (graphing and computer algebra system, CAS) in teaching the concept of "diagonalization," one of the key topics in Linear Algebra courses taught at the undergraduate level. Specifically, the proposed hypothesis of this study is to assess the effective…

Computational algebraic geometry for statistical modeling FY09Q2 progress.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Thompson, David C.; Rojas, Joseph Maurice; Pebay, Philippe Pierre

2009-03-01

This is a progress report on polynomial system solving for statistical modeling. This is a progress report on polynomial system solving for statistical modeling. This quarter we have developed our first model of shock response data and an algorithm for identifying the chamber cone containing a polynomial system in n variables with n+k terms within polynomial time - a significant improvement over previous algorithms, all having exponential worst-case complexity. We have implemented and verified the chamber cone algorithm for n+3 and are working to extend the implementation to handle arbitrary k. Later sections of this report explain chamber cones inmore » more detail; the next section provides an overview of the project and how the current progress fits into it.« less

Holography and the Coleman-Mermin-Wagner theorem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Anninos, Dionysios; Hartnoll, Sean A.; Iqbal, Nabil

2010-09-15

In 2+1 dimensions at finite temperature, spontaneous symmetry breaking of global symmetries is precluded by large thermal fluctuations of the order parameter. The holographic correspondence implies that analogous effects must also occur in 3+1 dimensional theories with gauged symmetries in certain curved spacetimes with horizon. By performing a one loop computation in the background of a holographic superconductor, we show that bulk quantum fluctuations wash out the classical order parameter at sufficiently large distance scales. The low temperature phase is seen to exhibit algebraic long-range order. Beyond the specific example we study, holography suggests that IR singular quantum fluctuations ofmore » the fields and geometry will play an interesting role for many 3+1 dimensional asymptotically anti-de Sitter spacetimes with planar horizon.« less

Editors' preface for the topical issue on Seven papers on Noncommutative Geometry and Operator Algebras

NASA Astrophysics Data System (ADS)

Guido, Daniele; Landi, Giovanni; Vassout, Stéphane

2016-07-01

This topical issue grew out of the International Conference ;Noncommutative Geometry and Applications; held 16-21 June 2014 at Villa Mondragone, Frascati (Roma). The main purpose of the conference was to have a unified view of different incarnations of noncommutative geometry and its applications. The seven papers collected in the present topical issue represent a good sample of the topics covered at the workshop. The conference itself was one of the climaxes of the Franco-Italian project GREFI-GENCO, which was initiated in 2007 by CNRS and INDAM to promote and enhance collaboration and exchanges between French and Italian researchers in the area of noncommutative geometry.

Network-Physics(NP) Bec DIGITAL(#)-VULNERABILITY Versus Fault-Tolerant Analog

NASA Astrophysics Data System (ADS)

Alexander, G. K.; Hathaway, M.; Schmidt, H. E.; Siegel, E.

2011-03-01

Siegel[AMS Joint Mtg.(2002)-Abs.973-60-124] digits logarithmic-(Newcomb(1881)-Weyl(1914; 1916)-Benford(1938)-"NeWBe"/"OLDbe")-law algebraic-inversion to ONLY BEQS BEC:Quanta/Bosons= digits: Synthesis reveals EMP-like SEVERE VULNERABILITY of ONLY DIGITAL-networks(VS. FAULT-TOLERANT ANALOG INvulnerability) via Barabasi "Network-Physics" relative-``statics''(VS.dynamics-[Willinger-Alderson-Doyle(Not.AMS(5/09)]-]critique); (so called)"Quantum-computing is simple-arithmetic(sans division/ factorization); algorithmic-complexities: INtractibility/ UNdecidability/ INefficiency/NONcomputability / HARDNESS(so MIScalled) "noise"-induced-phase-transitions(NITS) ACCELERATION: Cook-Levin theorem Reducibility is Renormalization-(Semi)-Group fixed-points; number-Randomness DEFINITION via WHAT? Query(VS. Goldreich[Not.AMS(02)] How? mea culpa)can ONLY be MBCS "hot-plasma" versus digit-clumping NON-random BEC; Modular-arithmetic Congruences= Signal X Noise PRODUCTS = clock-model; NON-Shor[Physica A,341,586(04)] BEC logarithmic-law inversion factorization:Watkins number-thy. U stat.-phys.); P=/=NP TRIVIAL Proof: Euclid!!! [(So Miscalled) computational-complexity J-O obviation via geometry.

A Formally-Verified Decision Procedure for Univariate Polynomial Computation Based on Sturm's Theorem

NASA Technical Reports Server (NTRS)

Narkawicz, Anthony J.; Munoz, Cesar A.

2014-01-01

Sturm's Theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semiopen interval. This paper presents a formalization of this theorem in the PVS theorem prover, as well as a decision procedure that checks whether a polynomial is always positive, nonnegative, nonzero, negative, or nonpositive on any input interval. The soundness and completeness of the decision procedure is proven in PVS. The procedure and its correctness properties enable the implementation of a PVS strategy for automatically proving existential and universal univariate polynomial inequalities. Since the decision procedure is formally verified in PVS, the soundness of the strategy depends solely on the internal logic of PVS rather than on an external oracle. The procedure itself uses a combination of Sturm's Theorem, an interval bisection procedure, and the fact that a polynomial with exactly one root in a bounded interval is always nonnegative on that interval if and only if it is nonnegative at both endpoints.

Multi-Mounted X-Ray Computed Tomography

PubMed Central

Fu, Jian; Liu, Zhenzhong; Wang, Jingzheng

2016-01-01

Most existing X-ray computed tomography (CT) techniques work in single-mounted mode and need to scan the inspected objects one by one. It is time-consuming and not acceptable for the inspection in a large scale. In this paper, we report a multi-mounted CT method and its first engineering implementation. It consists of a multi-mounted scanning geometry and the corresponding algebraic iterative reconstruction algorithm. This approach permits the CT rotation scanning of multiple objects simultaneously without the increase of penetration thickness and the signal crosstalk. Compared with the conventional single-mounted methods, it has the potential to improve the imaging efficiency and suppress the artifacts from the beam hardening and the scatter. This work comprises a numerical study of the method and its experimental verification using a dataset measured with a developed multi-mounted X-ray CT prototype system. We believe that this technique is of particular interest for pushing the engineering applications of X-ray CT. PMID:27073911

Astronomy Education using the Web and a Computer Algebra System

NASA Astrophysics Data System (ADS)

Flurchick, K. M.; Culver, Roger B.; Griego, Ben

2013-04-01

The combination of a web server and a Computer Algebra System to provide students the ability to explore and investigate astronomical concepts presented in a class can help student understanding. This combination of technologies provides a framework to extend the classroom experience with independent student exploration. In this presentation we report on the developmen of this web based material and some initial results of students making use of the computational tools using webMathematica^TM. The material developed allow the student toanalyze and investigate a variety of astronomical phenomena, including topics such as the Runge-Lenz vector, descriptions of the orbits of some of the exo-planets, Bode' law and other topics related to celestial mechanics. The server based Computer Algebra System system allows for computations without installing software on the student's computer but provides a powerful environment to explore the various concepts. The current system is installed at North Carolina A&T State University and has been used in several undergraduate classes.

The language of mathematics: investigating the ways language counts for children's mathematical development.

PubMed

Vukovic, Rose K; Lesaux, Nonie K

2013-06-01

This longitudinal study examined how language ability relates to mathematical development in a linguistically and ethnically diverse sample of children from 6 to 9 years of age. Study participants were 75 native English speakers and 92 language minority learners followed from first to fourth grades. Autoregression in a structural equation modeling (SEM) framework was used to evaluate the relation between children's language ability and gains in different domains of mathematical cognition (i.e., arithmetic, data analysis/probability, algebra, and geometry). The results showed that language ability predicts gains in data analysis/probability and geometry, but not in arithmetic or algebra, after controlling for visual-spatial working memory, reading ability, and sex. The effect of language on gains in mathematical cognition did not differ between language minority learners and native English speakers. These findings suggest that language influences how children make meaning of mathematics but is not involved in complex arithmetical procedures whether presented with Arabic symbols as in arithmetic or with abstract symbols as in algebraic reasoning. The findings further indicate that early language experiences are important for later mathematical development regardless of language background, denoting the need for intensive and targeted language opportunities for language minority and native English learners to develop mathematical concepts and representations. Copyright © 2013. Published by Elsevier Inc.

Analysis of the Harrier forebody/inlet design using computational techniques

NASA Technical Reports Server (NTRS)

Chow, Chuen-Yen

1993-01-01

Under the support of this Cooperative Agreement, computations of transonic flow past the complex forebody/inlet configuration of the AV-8B Harrier II have been performed. The actual aircraft configuration was measured and its surface and surrounding domain were defined using computational structured grids. The thin-layer Navier-Stokes equations were used to model the flow along with the Chimera embedded multi-grid technique. A fully conservative, alternating direction implicit (ADI), approximately-factored, partially flux-split algorithm was employed to perform the computation. An existing code was altered to conform with the needs of the study, and some special engine face boundary conditions were developed. The algorithm incorporated the Chimera technique and an algebraic turbulence model in order to deal with the embedded multi-grids and viscous governing equations. Comparison with experimental data has yielded good agreement for the simplifications incorporated into the analysis. The aim of the present research was to provide a methodology for the numerical solution of complex, combined external/internal flows. This is the first time-dependent Navier-Stokes solution for a geometry in which the fuselage and inlet share a wall. The results indicate the methodology used here is a viable tool for transonic aircraft modeling.

PROTEUS two-dimensional Navier-Stokes computer code, version 1.0. Volume 2: User's guide

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Benson, Thomas J.; Suresh, Ambady

1990-01-01

A new computer code was developed to solve the two-dimensional or axisymmetric, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 2 is the User's Guide, and describes the program's general features, the input and output, the procedure for setting up initial conditions, the computer resource requirements, the diagnostic messages that may be generated, the job control language used to run the program, and several test cases.

Mimicking Natural Laminar to Turbulent Flow Transition: A Systematic CFD Study Using PAB3D

NASA Technical Reports Server (NTRS)

Pao, S. Paul; Abdol-Hamid, Khaled S.

2005-01-01

For applied aerodynamic computations using a general purpose Navier-Stokes code, the common practice of treating laminar to turbulent flow transition over a non-slip surface is somewhat arbitrary by either treating the entire flow as turbulent or forcing the flow to undergo transition at given trip locations in the computational domain. In this study, the possibility of using the PAB3D code, standard k-epsilon turbulence model, and the Girimaji explicit algebraic stresses model to mimic natural laminar to turbulent flow transition was explored. The sensitivity of flow transition with respect to two limiters in the standard k-epsilon turbulence model was examined using a flat plate and a 6:1 aspect ratio prolate spheroid for our computations. For the flat plate, a systematic dependence of transition Reynolds number on background turbulence intensity was found. For the prolate spheroid, the transition patterns in the three-dimensional boundary layer at different flow conditions were sensitive to the free stream turbulence viscosity limit, the reference Reynolds number and the angle of attack, but not to background turbulence intensity below a certain threshold value. The computed results showed encouraging agreements with the experimental measurements at the corresponding geometry and flow conditions.

Learning Nonadjacent Dependencies: No Need for Algebraic-Like Computations

ERIC Educational Resources Information Center

Perruchet, Pierre; Tyler, Michael D.; Galland, Nadine; Peereman, Ronald

2004-01-01

Is it possible to learn the relation between 2 nonadjacent events? M. Pena, L. L. Bonatti, M. Nespor, and J. Mehler (2002) claimed this to be possible, but only in conditions suggesting the involvement of algebraic-like computations. The present article reports simulation studies and experimental data showing that the observations on which Pe?a et…

Higher Inductive Types as Homotopy-Initial Algebras

DTIC Science & Technology

2016-08-01

Higher Inductive Types as Homotopy-Initial Algebras Kristina Sojakova CMU-CS-16-125 August 2016 School of Computer Science Carnegie Mellon University...talk at the Workshop on Logic, Language, Information and Computation (WoLLIC 2011). 1, 2.1 [38] M. Warren. Homotopy-Theoretic Aspects of Constructive Type Theory. PhD thesis, Carnegie Mellon University, 2008. 1 143

A Computer Program To Increase Comprehension of the Cartesian Rectangular Coordinate System in High School Pre-Algebra Students.

ERIC Educational Resources Information Center

Exley, I. Sheck

The high percentage of high school pre-algebra students having difficulty learning the abstract concept of graphing ordered pairs on the Cartesian rectangular coordinate system was addressed by the creation and implementation of a computer-managed instructional program. Modules consisted of a pretest, instruction, two practice sessions, and a…

Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper-and-Pencil

ERIC Educational Resources Information Center

Fonger, Nicole L.; Davis, Jon D.; Rohwer, Mary Lou

2018-01-01

This research addresses the issue of how to support students' representational fluency--the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is…

Computer Algebra Systems: Permitted but Are They Used?

ERIC Educational Resources Information Center

Pierce, Robyn; Bardini, Caroline

2015-01-01

Since the 1990s, computer algebra systems (CAS) have been available in Australia as hand-held devices designed for students with the expectation that they will be used in the mathematics classroom. The data discussed in this paper was collected as part of a pilot study that investigated first year university mathematics and statistics students'…

Do Mathematicians Integrate Computer Algebra Systems in University Teaching? Comparing a Literature Review to an International Survey Study

ERIC Educational Resources Information Center

Marshall, Neil; Buteau, Chantal; Jarvis, Daniel H.; Lavicza, Zsolt

2012-01-01

We present a comparative study of a literature review of 326 selected contributions (Buteau, Marshall, Jarvis & Lavicza, 2010) to an international (US, UK, Hungary) survey of mathematicians (Lavicza, 2008) regarding the use of Computer Algebra Systems (CAS) in post-secondary mathematics education. The comparison results are organized with respect…

Examining the Use of Computer Algebra Systems in University-Level Mathematics Teaching

ERIC Educational Resources Information Center

Lavicza, Zsolt

2009-01-01

The use of Computer Algebra Systems (CAS) is becoming increasingly important and widespread in mathematics research and teaching. In this paper, I will report on a questionnaire study enquiring about mathematicians' use of CAS in mathematics teaching in three countries; the United States, the United Kingdom, and Hungary. Based on the responses…

Experiences in Evaluating Outcomes in Tool-Based, Competence Building Education in Dynamical Systems Using Symbolic Computer Algebra

ERIC Educational Resources Information Center

Perram, John W.; Andersen, Morten; Ellekilde, Lars-Peter; Hjorth, Poul G.

2004-01-01

This paper discusses experience with alternative assessment strategies for an introductory course in dynamical systems, where the use of computer algebra and calculus is fully integrated into the learning process, so that the standard written examination would not be appropriate. Instead, students' competence was assessed by grading three large…

Solution of the Schrodinger Equation for a Diatomic Oscillator Using Linear Algebra: An Undergraduate Computational Experiment

ERIC Educational Resources Information Center

Gasyna, Zbigniew L.

2008-01-01

Computational experiment is proposed in which a linear algebra method is applied to the solution of the Schrodinger equation for a diatomic oscillator. Calculations of the vibration-rotation spectrum for the HCl molecule are presented and the results show excellent agreement with experimental data. (Contains 1 table and 1 figure.)

Color Algebras

NASA Technical Reports Server (NTRS)

Mulligan, Jeffrey B.

2017-01-01

A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. The difficulty addressed here is the fact that, because of metamerism, we cannot know with certainty the spectrum that produced a particular color solely on the basis of sensory data. Knowledge of the spectrum is not required to compute additive mixture of colors, but is critical for subtractive (multiplicative) mixture. Therefore, we cannot predict with certainty the multiplicative interactions between colors based solely on sensory data. There are two potential applications of a color algebra: first, to aid modeling phenomena of human visual perception, such as color constancy and transparency; and, second, to provide better models of the interactions of lights and surfaces for computer graphics rendering.

An algebraic interpretation of PSP composition.

PubMed

Vaucher, G

1998-01-01

The introduction of time in artificial neurons is a delicate problem on which many groups are working. Our approach combines some properties of biological models and the algebraic properties of McCulloch and Pitts artificial neuron (AN) (McCulloch and Pitts, 1943) to produce a new model which links both characteristics. In this extended artificial neuron, postsynaptic potentials (PSPs) are considered as numerical elements, having two degrees of freedom, on which the neuron computes operations. Modelled in this manner, a group of neurons can be seen as a computer with an asynchronous architecture. To formalize the functioning of this computer, we propose an algebra of impulses. This approach might also be interesting in the modelling of the passive electrical properties in some biological neurons.

FINAL REPORT: GEOMETRY AND ELEMENTARY PARTICLE PHYSICS

DOE Office of Scientific and Technical Information (OSTI.GOV)

Singer, Isadore M.

2008-03-04

The effect on mathematics of collaborations between high-energy theoretical physics and modern mathematics has been remarkable. Mirror symmetry has revolutionized enumerative geometry, and Seiberg-Witten invariants have greatly simplified the study of four manifolds. And because of their application to string theory, physicists now need to know cohomology theory, characteristic classes, index theory, K-theory, algebraic geometry, differential geometry, and non-commutative geometry. Much more is coming. We are experiencing a deeper contact between the two sciences, which will stimulate new mathematics essential to the physicists’ quest for the unification of quantum mechanics and relativity. Our grant, supported by the Department of Energymore » for twelve years, has been instrumental in promoting an effective interaction between geometry and string theory, by supporting the Mathematical Physics seminar, postdoc research, collaborations, graduate students and several research papers.« less

Optical systolic solutions of linear algebraic equations

NASA Technical Reports Server (NTRS)

Neuman, C. P.; Casasent, D.

1984-01-01

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Exact solution of some linear matrix equations using algebraic methods

NASA Technical Reports Server (NTRS)

Djaferis, T. E.; Mitter, S. K.

1979-01-01

Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.

JOURNAL SCOPE GUIDELINES: Paper classification scheme

NASA Astrophysics Data System (ADS)

2005-06-01

This scheme is used to clarify the journal's scope and enable authors and readers to more easily locate the appropriate section for their work. For each of the sections listed in the scope statement we suggest some more detailed subject areas which help define that subject area. These lists are by no means exhaustive and are intended only as a guide to the type of papers we envisage appearing in each section. We acknowledge that no classification scheme can be perfect and that there are some papers which might be placed in more than one section. We are happy to provide further advice on paper classification to authors upon request (please email jphysa@iop.org). 1. Statistical physics numerical and computational methods statistical mechanics, phase transitions and critical phenomena quantum condensed matter theory Bose-Einstein condensation strongly correlated electron systems exactly solvable models in statistical mechanics lattice models, random walks and combinatorics field-theoretical models in statistical mechanics disordered systems, spin glasses and neural networks nonequilibrium systems network theory 2. Chaotic and complex systems nonlinear dynamics and classical chaos fractals and multifractals quantum chaos classical and quantum transport cellular automata granular systems and self-organization pattern formation biophysical models 3. Mathematical physics combinatorics algebraic structures and number theory matrix theory classical and quantum groups, symmetry and representation theory Lie algebras, special functions and orthogonal polynomials ordinary and partial differential equations difference and functional equations integrable systems soliton theory functional analysis and operator theory inverse problems geometry, differential geometry and topology numerical approximation and analysis geometric integration computational methods 4. Quantum mechanics and quantum information theory coherent states eigenvalue problems supersymmetric quantum mechanics scattering theory relativistic quantum mechanics semiclassical approximations foundations of quantum mechanics and measurement theory entanglement and quantum nonlocality geometric phases and quantum tomography quantum tunnelling decoherence and open systems quantum cryptography, communication and computation theoretical quantum optics 5. Classical and quantum field theory quantum field theory gauge and conformal field theory quantum electrodynamics and quantum chromodynamics Casimir effect integrable field theory random matrix theory applications in field theory string theory and its developments classical field theory and electromagnetism metamaterials 6. Fluid and plasma theory turbulence fundamental plasma physics kinetic theory magnetohydrodynamics and multifluid descriptions strongly coupled plasmas one-component plasmas non-neutral plasmas astrophysical and dusty plasmas

Characterizing and Supporting Change in Algebra Students' Representational Fluency in a CAS/Paper-and-Pencil Environment

ERIC Educational Resources Information Center

Fonger, Nicole L.

2012-01-01

Representational fluency (RF) includes an ability to interpret, create, move within and among, and connect tool-based representations of mathematical objects. Taken as an indicator of conceptual understanding, there is a need to better support school algebra students' RF in learning environments that utilize both computer algebra systems…

A Comparison Study between a Traditional and Experimental Program.

ERIC Educational Resources Information Center

Dogan, Hamide

This paper is part of a dissertation defended in January 2001 as part of the author's Ph.D. requirement. The study investigated the effects of use of Mathematica, a computer algebra system, in learning basic linear algebra concepts, It was done by means of comparing two first year linear algebra classes, one traditional and one Mathematica…

Spatial-Operator Algebra For Flexible-Link Manipulators

NASA Technical Reports Server (NTRS)

Jain, Abhinandan; Rodriguez, Guillermo

1994-01-01

Method of computing dynamics of multiple-flexible-link robotic manipulators based on spatial-operator algebra, which originally applied to rigid-link manipulators. Aspects of spatial-operator-algebra approach described in several previous articles in NASA Tech Briefs-most recently "Robot Control Based on Spatial-Operator Algebra" (NPO-17918). In extension of spatial-operator algebra to manipulators with flexible links, each link represented by finite-element model: mass of flexible link apportioned among smaller, lumped-mass rigid bodies, coupling of motions expressed in terms of vibrational modes. This leads to operator expression for modal-mass matrix of link.

Topics in string theory

NASA Astrophysics Data System (ADS)

Jejjala, Vishnumohan

2002-01-01

This Thesis explores aspects of superstring theory on orbifold spaces and applies some of the intuition gleaned from the study of the non-commutative geometry of space-time to understanding the fractional quantum Hall effect. The moduli space of vacua of marginal and relevant deformations of N = 4 super-Yang-Mills gauge theory in four dimensions is interpreted in terms of non-commutative geometry. A formalism for thinking about the algebraic geometry of the moduli space is developed. Within this framework, the representation theory of the algebras studied provides a natural exposition of D-brane fractionation. The non-commutative moduli space of deformations preserving N = 1 supersymmetry is examined in detail through various examples. In string theory, by the AdS/CFT correspondence, deformations of the N = 4 field theory are dual to the near-horizon geometries of D-branes on orbifolds of AdS5 x S 5. The physics of D-branes on the dual AdS backgrounds is explored. Quivers encapsulate the matter content of supersymmetric field theories on the worldvolumes of D-branes at orbifold singularities. New techniques for constructing quivers are presented here. When N is a normal subgroup of a finite group G, the quiver corresponding to fixed points of the orbifold M/G is computed from a G/N action on the quiver corresponding to M/G . These techniques prove useful for constructing non-Abelian quivers and for examining discrete torsion orbifolds. Quivers obtained through our constructions contain interesting low-energy phenomenology. The matter content on a brane at an isolated singularity of the Delta27 orbifold embeds the Standard Model. The symmetries of the quiver require exactly three generations of fields in the particle spectrum. Lepton masses are suppressed relative to quark masses because lepton Yukawa couplings do not appear in the superpotential. Lepton masses are generated through the Kahler potential and are related to the supersymmetry breaking scale. The model makes falsifiable predictions about TeV scale physics. Susskind has proposed that the fractional quantum Hall system can be realized through an Abelian Chern-Simons theory with a Moyal product. Susskind's Chern-Simons field is a hydrodynamical quantity. Lopez and Fradkin have an alternate Chern-Simons description couched in terms of a statistical gauge field. We show that this statistical Chern-Simons theory also possesses a non-commutative structure and develop the dictionary between the two Chern-Simons pictures.

Applications: Using Algebra in an Accounting Practice.

ERIC Educational Resources Information Center

Eisner, Gail A.

1994-01-01

Presents examples of algebra from the field of accounting including proportional ownership of stock, separation of a loan payment into principal and interest portions, depreciation methods, and salary withholdings computations. (MKR)

Students' Comparison of Their Trigonometric Answers with the Answers of a Computer Algebra System in Terms of Equivalence and Correctness

ERIC Educational Resources Information Center

Tonisson, Eno; Lepp, Marina

2015-01-01

The answers offered by computer algebra systems (CAS) can sometimes differ from those expected by the students or teachers. The comparison of the students' answers and CAS answers could provide ground for discussion about equivalence and correctness. Investigating the students' comparison of the answers gives the possibility to study different…

Differences between Expected Answers and the Answers Given by Computer Algebra Systems to School Equations

ERIC Educational Resources Information Center

Tonisson, Eno

2015-01-01

Sometimes Computer Algebra Systems (CAS) offer an answer that is somewhat different from the answer that is probably expected by the student or teacher. These (somewhat unexpected) answers could serve as a catalyst for rich mathematical discussion. In this study, over 120 equations from school mathematics were solved using 8 different CAS. Many…

The Effects of Computer Algebra System on Undergraduate Students' Spatial Visualization Skills in a Calculus Course

ERIC Educational Resources Information Center

Karakus, Fatih; Aydin, Bünyamin

2017-01-01

This study aimed at determining the effects of using a computer algebra system (CAS) on undergraduate students' spatial visualization skills in a calculus course. This study used an experimental design. The "one group pretest-posttest design" was the research model. The participants were 41 sophomore students (26 female and 15 male)…

Introducing Computer Algebra to School Teachers of Mathematics

ERIC Educational Resources Information Center

Man, Yiu-Kwong

2007-01-01

Since the last decade, the use of computer algebra systems at the Hong Kong school level is still very limited. Among various reasons behind, the lack of exposure of this kind of software to local school teachers should be taken into account. In this article, we describe how to introduce MAPLE in a BEd module of a local teacher-training programme.…

The Automation of Stochastization Algorithm with Use of SymPy Computer Algebra Library

NASA Astrophysics Data System (ADS)

Demidova, Anastasya; Gevorkyan, Migran; Kulyabov, Dmitry; Korolkova, Anna; Sevastianov, Leonid

2018-02-01

SymPy computer algebra library is used for automatic generation of ordinary and stochastic systems of differential equations from the schemes of kinetic interaction. Schemes of this type are used not only in chemical kinetics but also in biological, ecological and technical models. This paper describes the automatic generation algorithm with an emphasis on application details.

On the Integration of Computer Algebra Systems (CAS) by Canadian Mathematicians: Results of a National Survey

ERIC Educational Resources Information Center

Buteau, Chantal; Jarvis, Daniel H.; Lavicza, Zsolt

2014-01-01

In this article, we outline the findings of a Canadian survey study (N = 302) that focused on the extent of computer algebra systems (CAS)-based technology use in postsecondary mathematics instruction. Results suggest that a considerable number of Canadian mathematicians use CAS in research and teaching. CAS use in research was found to be the…

CENTER CONDITIONS AND CYCLICITY FOR A FAMILY OF CUBIC SYSTEMS: COMPUTER ALGEBRA APPROACH.

PubMed

Ferčec, Brigita; Mahdi, Adam

2013-01-01

Using methods of computational algebra we obtain an upper bound for the cyclicity of a family of cubic systems. We overcame the problem of nonradicality of the associated Bautin ideal by moving from the ring of polynomials to a coordinate ring. Finally, we determine the number of limit cycles bifurcating from each component of the center variety.

Novel symmetries in Christ-Lee model

NASA Astrophysics Data System (ADS)

Kumar, R.; Shukla, A.

2016-07-01

We demonstrate that the gauge-fixed Lagrangian of the Christ-Lee model respects four fermionic symmetries, namely; (anti-)BRST symmetries, (anti-)co-BRST symmetries within the framework of BRST formalism. The appropriate anticommutators amongst the fermionic symmetries lead to a unique bosonic symmetry. It turns out that the algebra obeyed by the symmetry transformations (and their corresponding conserved charges) is reminiscent of the algebra satisfied by the de Rham cohomological operators of differential geometry. We also provide the physical realizations of the cohomological operators in terms of the symmetry properties. Thus, the present model provides a simple model for the Hodge theory.

A tour about existence and uniqueness of dg enhancements and lifts

NASA Astrophysics Data System (ADS)

Canonaco, Alberto; Stellari, Paolo

2017-12-01

This paper surveys the recent advances concerning the relations between triangulated (or derived) categories and their dg enhancements. We explain when some interesting triangulated categories arising in algebraic geometry have a unique dg enhancement. This is the case, for example, for the unbounded derived category of quasi-coherent sheaves on an algebraic stack or for its full triangulated subcategory of perfect complexes. Moreover we give an account of the recent results about the possibility to lift exact functors between the bounded derived categories of coherent sheaves on smooth schemes to dg (quasi-)functors.

Pointless strings

DOE Office of Scientific and Technical Information (OSTI.GOV)

Periwal, V.

1988-01-01

The author proves that bosonic string perturbation theory diverges and is not Borel summable. This is an indication of a non-perturbative instability of the bosonic string vacuum. He formulates two-dimensional sigma models in terms of algebras of functions. He extends this formulation to general C* algebras. He illustrates the utility of these algebraic notions by calculating some determinants of interest in the study of string propagation in orbifold backgrounds. He studies the geometry of spaces of field theories and show that the vanishing of the curvature of the natural Gel'fand-Naimark-Segal metric on such spaces is exactly the strong associativity conditionmore » of the operator product expansion.He shows that string scattering amplitudes arise as invariants of renormalization, when he formulates renormalization in terms of rescalings of the metric on the string world-sheet.« less

Ada Linear-Algebra Program

NASA Technical Reports Server (NTRS)

Klumpp, A. R.; Lawson, C. L.

1988-01-01

Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.

General Algebraic Modeling System Tutorial | High-Performance Computing |

Science.gov Websites

power generation from two different fuels. The goal is to minimize the cost for one of the fuels while Here's a basic tutorial for modeling optimization problems with the General Algebraic Modeling System (GAMS). Overview The GAMS (General Algebraic Modeling System) package is essentially a compiler for a

The Effect of an Intelligent Tutoring System (ITS) on Student Achievement in Algebraic Expression

ERIC Educational Resources Information Center

Chien, Tsai Chen; Md. Yunus, Aida Suraya; Ali, Wan Zah Wan; Bakar, Ab. Rahim

2008-01-01

In this experimental study, use of Computer Assisted Instruction (CAI) followed by use of an Intelligent Tutoring System (CAI+ITS) was compared to the use of CAI (CAI only) in tutoring students on the topic of Algebraic Expression. Two groups of students participated in the study. One group of 32 students studied algebraic expression in a CAI…

ALDF Data Retrieval Algorithms for Validating the Optical Transient Detector (OTD) and the Lightning Imaging Sensor (LIS)

NASA Technical Reports Server (NTRS)

Koshak, W. J.; Blakeslee, R. J.; Bailey, J. C.

1997-01-01

A linear algebraic solution is provided for the problem of retrieving the location and time of occurrence of lightning ground strikes from in Advanced Lightning Direction Finder (ALDF) network. The ALDF network measures field strength, magnetic bearing, and arrival time of lightning radio emissions and solutions for the plane (i.e.. no Earth curvature) are provided that implement all of these measurements. The accuracy of the retrieval method is tested using computer-simulated data sets and the relative influence of bearing and arrival time data on the outcome of the final solution is formally demonstrated. The algorithm is sufficiently accurate to validate NASA's Optical Transient Detector (OTD) and Lightning Imaging System (LIS). We also introduce a quadratic planar solution that is useful when only three arrival time measurements are available. The algebra of the quadratic root results are examined in detail to clarify what portions of the analysis region lead to fundamental ambiguities in source location. Complex root results are shown to be associated with the presence of measurement errors when the lightning source lies near an outer sensor baseline of the ALDF network. For arbitrary noncollinear network geometries and in the absence of measurement errors, it is shown that the two quadratic roots are equivalent (no source location ambiguity) on the outer sensor baselines. The accuracy of the quadratic planar method is tested with computer-generated data sets and the results are generally better than those obtained from the three station linear planar method when bearing errors are about 2 degrees.

Mesh-free data transfer algorithms for partitioned multiphysics problems: Conservation, accuracy, and parallelism

DOE PAGES

Slattery, Stuart R.

2015-12-02

In this study we analyze and extend mesh-free algorithms for three-dimensional data transfer problems in partitioned multiphysics simulations. We first provide a direct comparison between a mesh-based weighted residual method using the common-refinement scheme and two mesh-free algorithms leveraging compactly supported radial basis functions: one using a spline interpolation and one using a moving least square reconstruction. Through the comparison we assess both the conservation and accuracy of the data transfer obtained from each of the methods. We do so for a varying set of geometries with and without curvature and sharp features and for functions with and without smoothnessmore » and with varying gradients. Our results show that the mesh-based and mesh-free algorithms are complementary with cases where each was demonstrated to perform better than the other. We then focus on the mesh-free methods by developing a set of algorithms to parallelize them based on sparse linear algebra techniques. This includes a discussion of fast parallel radius searching in point clouds and restructuring the interpolation algorithms to leverage data structures and linear algebra services designed for large distributed computing environments. The scalability of our new algorithms is demonstrated on a leadership class computing facility using a set of basic scaling studies. Finally, these scaling studies show that for problems with reasonable load balance, our new algorithms for both spline interpolation and moving least square reconstruction demonstrate both strong and weak scalability using more than 100,000 MPI processes with billions of degrees of freedom in the data transfer operation.« less

Computational analysis of forebody tangential slot blowing on the high alpha research vehicle

NASA Technical Reports Server (NTRS)

Gee, Ken

1995-01-01

A numerical analysis of forebody tangential slot blowing as a means of generating side force and yawing moment is conducted using an aircraft geometry. The Reynolds-averaged, thin-layer, Navier-Stokes equations are solved using a partially flux-split, approximately-factored algorithm. An algebraic turbulence model is used to determine the turbulent eddy viscosity values. Solutions are obtained using both patched and overset grid systems. In the patched grid model, and actuator plane is used to introduce jet variables into the flow field. The overset grid model is used to model the physical slot geometry and facilitate modeling of the full aircraft configuration. A slot optimization study indicates that a short slot located close to the nose of the aircraft provided the most side force and yawing moment per unit blowing coefficient. Comparison of computed surface pressure with that obtained in full-scale wind tunnel tests produce good agreement, indicating the numerical method and grid system used in the study are valid. Full aircraft computations resolve the changes in vortex burst point due to blowing. A time-accurate full-aircraft solution shows the effect of blowing on the changes in the frequency of the aerodynamic loads over the vertical tails. A study of the effects of freestream Mach number and various jet parameters indicates blowing remains effective through the transonic Mach range. An investigation of the force onset time lag associated with forebody blowing shows the lag to be minimal. The knowledge obtained in this study may be applied to the design of a forebody tangential slot blowing system for use on flight aircraft.

Genetic algorithms in teaching artificial intelligence (automated generation of specific algebras)

NASA Astrophysics Data System (ADS)

Habiballa, Hashim; Jendryscik, Radek

2017-11-01

The problem of teaching essential Artificial Intelligence (AI) methods is an important task for an educator in the branch of soft-computing. The key focus is often given to proper understanding of the principle of AI methods in two essential points - why we use soft-computing methods at all and how we apply these methods to generate reasonable results in sensible time. We present one interesting problem solved in the non-educational research concerning automated generation of specific algebras in the huge search space. We emphasize above mentioned points as an educational case study of an interesting problem in automated generation of specific algebras.

A Brief Historical Introduction to Determinants with Applications

ERIC Educational Resources Information Center

Debnath, L.

2013-01-01

This article deals with a short historical introduction to determinants with applications to the theory of equations, geometry, multiple integrals, differential equations and linear algebra. Included are some properties of determinants with proofs, eigenvalues, eigenvectors and characteristic equations with examples of applications to simple…

Teaching Environmental Awareness in Mathematics

ERIC Educational Resources Information Center

Jianguo, Mao

2004-01-01

This article is all about the integration of environmental education to the middle school mathematics in China. To raise environmental awareness and improve environmental quality, environmental education is a must. Environment-related materials can be found in middle school algebra and geometry textbooks. In order to provide environmental…

Introducing Group Theory through Music

ERIC Educational Resources Information Center

Johnson, Craig M.

2009-01-01

The central ideas of postcalculus mathematics courses offered in college are difficult to introduce in middle and secondary schools, especially through the engineering and sciences examples traditionally used in algebra, geometry, and trigonometry textbooks. However, certain concepts in music theory can be used to expose students to interesting…

Comparison of Virginia's College and Career Ready Mathematics Performance Expectations with the Common Core State Standards for Mathematics

ERIC Educational Resources Information Center

Virginia Department of Education, 2010

2010-01-01

This paper presents a comparison of Virginia's mathematics performance expectations with the common core state standards for mathematics. The comparison focuses on number and quantity, algebra, functions, geometry, and statistics and probability. (Contains 1 footnote.)

The International Mathematical Olympiad Training Session.

ERIC Educational Resources Information Center

Rousseau, Cecil; Patruno, Gregg

1985-01-01

The Mathematical Olympiad Training Session is designed to give United States students a problem-oriented exposure to subject areas (algebra, geometry, number theory, combinatorics, and inequalities) through an intensive three-week course. Techniques used during the session, with three sample problems and their solutions, are presented. (JN)

Technology Tips: Investigating Extrema with GeoGebra

ERIC Educational Resources Information Center

Cullen, Craig J.; Hertel, Joshua T.; John, Sheryl

2013-01-01

The NCTM Algebra Standard suggests that students use technology to explore the effects of varying the parameters in y = ax2 + bx + c. This article discusses an extension of this task that incorporates dynamic geometry software to engage students in generating, testing, and proving mathematical conjectures.

Curriculum Change in Secondary School Mathematics

ERIC Educational Resources Information Center

Alspaugh, John W.; and others

1970-01-01

Discusses six major trends in mathematics curriculum development: lowering of grade placement, teaching methods from memorization to discovery, introduction and deletion of content, integration of plane and solid geometry, algebra, and trigonometry, emphasis upon needs and characteristics of student, and increasing rate of curriculum change.…

Two-spectral Yang-Baxter operators in topological quantum computation

NASA Astrophysics Data System (ADS)

Sanchez, William F.

2011-05-01

One of the current trends in quantum computing is the application of algebraic topological methods in the design of new algorithms and quantum computers, giving rise to topological quantum computing. One of the tools used in it is the Yang-Baxter equation whose solutions are interpreted as universal quantum gates. Lately, more general Yang-Baxter equations have been investigated, making progress as two-spectral equations and Yang-Baxter systems. This paper intends to apply these new findings to the field of topological quantum computation, more specifically, the proposition of the two-spectral Yang-Baxter operators as universal quantum gates for 2 qubits and 2 qutrits systems, obtaining 4x4 and 9x9 matrices respectively, and further elaboration of the corresponding Hamiltonian by the use of computer algebra software Mathematica® and its Qucalc package. In addition, possible physical systems to which the Yang-Baxter operators obtained can be applied are considered. In the present work it is demonstrated the utility of the Yang-Baxter equation to generate universal quantum gates and the power of computer algebra to design them; it is expected that these mathematical studies contribute to the further development of quantum computers

FAST TRACK COMMUNICATION: On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

NASA Astrophysics Data System (ADS)

Man, Yiu-Kwong

2010-10-01

In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.

Effects of Using a Computer Algebra System (CAS) on Junior College Students' Attitudes towards CAS and Achievement in Mathematics

ERIC Educational Resources Information Center

Leng, Ng Wee; Choo, Kwee Tiow; Soon, Lau Hock; Yi-Huak, Koh; Sun, Yap Yew

2005-01-01

This study examines the effects of using Texas Instruments' Voyage 200 calculator (V200), a graphing calculator with a built-in computer algebra system (CAS), on attitudes towards CAS and achievement in mathematics of junior college students (17 year olds). Students' attitudes towards CAS were examined using a 40-item Likert-type instrument…

Communication Avoiding and Overlapping for Numerical Linear Algebra

DTIC Science & Technology

2012-05-08

future exascale systems, communication cost must be avoided or overlapped. Communication-avoiding 2.5D algorithms improve scalability by reducing...linear algebra problems to future exascale systems, communication cost must be avoided or overlapped. Communication-avoiding 2.5D algorithms improve...will continue to grow relative to the cost of computation. With exascale computing as the long-term goal, the community needs to develop techniques

Mixing Microworld and CAS Features in Building Computer Systems that Help Students Learn Algebra

ERIC Educational Resources Information Center

Nicaud, Jean-Francois; Bouhineau, Denis; Chaachoua, Hamid

2004-01-01

We present the design principles for a new kind of computer system that helps students learn algebra. The fundamental idea is to have a system based on the microworld paradigm that allows students to make their own calculations, as they do with paper and pencil, without being obliged to use commands, and to verify the correctness of these…

Exploring Interactive and Dynamic Simulations Using a Computer Algebra System in an Advanced Placement Chemistry Course

ERIC Educational Resources Information Center

Matsumoto, Paul S.

2014-01-01

The article describes the use of Mathematica, a computer algebra system (CAS), in a high school chemistry course. Mathematica was used to generate a graph, where a slider controls the value of parameter(s) in the equation; thus, students can visualize the effect of the parameter(s) on the behavior of the system. Also, Mathematica can show the…

Computer Aided Instruction for a Course in Boolean Algebra and Logic Design. Final Report (Revised).

ERIC Educational Resources Information Center

Roy, Rob

The use of computers to prepare deficient college and graduate students for courses that build upon previously acquired information would solve the growing problem of professors who must spend up to one third of their class time in review of material. But examination of students who were taught Boolean Algebra and Logic Design by means of Computer…

Signal Processing for Radar Target Tracking and Identification

DTIC Science & Technology

1996-12-01

Computes the likelihood for various potential jump moves. 12. matrix_mult.m: Parallel implementation of linear algebra ... Elementary Lineary Algebra with Applications, John Wiley k Sons, Inc., New York, 1987. [9] A. K. Bhattacharyya, and D. L. Sengupta, Radar Cross...Miller, ’Target Tracking and Recognition Using Jump-Diffusion Processes," ARO’s 11th Army Conf. on Applied Mathemat- ics and Computing, June 8-11

Algebras Generated by Geometric Scalar Forms and their Applications in Physics and Social Sciences

DOE Office of Scientific and Technical Information (OSTI.GOV)

Keller, Jaime

2008-09-17

The present paper analyzes the consequences of defining that the geometric scalar form is not necessarily quadratic, but in general K-atic, that is obtained from the K{sup th} power of the linear form, requiring {l_brace}e{sub i};i = 1,...,N;(e{sub i}){sup K} = 1{r_brace} and d-vector {sigma}{sub i}x{sub i}e{sub i}. We consider the algebras which are thus generated, for positive integer K, a generalization of the geometric algebras we know under the names of Clifford or Grassmann algebras. We then obtain a set of geometric K-algebras. We also consider the generalization of special functions of geometry which corresponds to the K-order scalarmore » forms (as trigonometric functions and other related geometric functions which are based on the use of quadratic forms). We present an overview of the use of quadratic forms in physics as in our general theory, we have called START. And, in order to give an introduction to the use of the more general K-algebras and to the possible application to sciences other than physics, the application to social sciences is considered.For the applications to physics we show that quadratic spaces are a fundamental clue to understand the structure of theoretical physics (see, for example, Keller in ICNAAM 2005 and 2006)« less

Software for Training in Pre-College Mathematics

NASA Technical Reports Server (NTRS)

Shelton, Robert O.; Moebes, Travis A.; VanAlstine, Scot

2003-01-01

The Intelligent Math Tutor (IMT) is a computer program for training students in pre-college and college-level mathematics courses, including fundamentals, intermediate algebra, college algebra, and trigonometry. The IMT can be executed on a server computer for access by students via the Internet; alternatively, it can be executed on students computers equipped with compact- disk/read-only-memory (CD-ROM) drives. The IMT provides interactive exercises, assessment, tracking, and an on-line graphing calculator with algebraic-manipulation capabilities. The IMT provides an innovative combination of content, delivery mechanism, and artificial intelligence. Careful organization and presentation of the content make it possible to provide intelligent feedback to the student based on performance on exercises and tests. The tracking and feedback mechanisms are implemented within the capabilities of a commercial off-the-shelf development software tool and are written in the Unified Modeling Language to maximize reuse and minimize development cost. The graphical calculator is a standard feature of most college and pre-college algebra and trigonometry courses. Placing this functionality in a Java applet decreases the cost, provides greater capabilities, and provides an opportunity to integrate the calculator with the lessons.

A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian decomposition method.

PubMed

Benhammouda, Brahim

2016-01-01

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

The Effect of a Math Emporium Course Redesign in Developmental and Introductory Mathematics Courses on Student Achievement and Students' Attitudes toward Mathematics at a Two-Year College

ERIC Educational Resources Information Center

Bishop, Amy Renee

2010-01-01

The purpose of this research was to determine the effect of computer-based instruction on student mathematics achievement and students' attitudes toward mathematics in developmental and introductory mathematics courses, namely Elementary Algebra, Intermediate Algebra, and College Algebra, at a community college. The researcher also examined the…

Boolean Minimization and Algebraic Factorization Procedures for Fully Testable Sequential Machines

DTIC Science & Technology

1989-09-01

Boolean Minimization and Algebraic Factorization Procedures for Fully Testable Sequential Machines Srinivas Devadas and Kurt Keutzer F ( Abstract In this...Projects Agency under contract number N00014-87-K-0825. Author Information Devadas : Department of Electrical Engineering and Computer Science, Room 36...MA 02139; (617) 253-0292. 0 * Boolean Minimization and Algebraic Factorization Procedures for Fully Testable Sequential Machines Siivas Devadas

An Example of Competence-Based Learning: Use of Maxima in Linear Algebra for Engineers

ERIC Educational Resources Information Center

Diaz, Ana; Garcia, Alfonsa; de la Villa, Agustin

2011-01-01

This paper analyses the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is…

Math Wonders to Inspire Teachers and Students.

ERIC Educational Resources Information Center

Posamentier, Alfred S.

This book offers ideas to enrich instruction and help teachers explore the intrinsic beauty of math. Through dozens of examples from arithmetic, algebra, geometry, and probability, the symmetries, patterns, processes, paradoxes, and surprises that have facilitated generations of great thinkers are revealed. Activities include: (1) The Beauty in…

Connected Mathematics Project (CMP). What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2010

2010-01-01

The "Connected Mathematics Project" ("CMP") is a mathematics curriculum designed for students in grades 6-8. Each grade level of the curriculum is a full-year program and covers numbers, algebra, geometry/measurement, probability, and statistics. The curriculum uses an investigative approach, and students utilize interactive…

Two Essays in Economic Education

ERIC Educational Resources Information Center

Evans, Brent A.

2013-01-01

Prior researchers (Anderson et al. 1994; Ballard & Johnson 2004; Hoag & Benedict 2010) have shown that different math abilities do not equally correlate with success in economics, yet no research has specifically compared algebra and geometry skills as predictors of economics success. In the first essay, I find that students' standardized…

46 CFR 310.55 - Scholastic requirements.

Code of Federal Regulations, 2010 CFR

2010-10-01

... SAT. A candidate electing to use the ACT, shall take all the tests, namely, English, Mathematics... Mathematics (from algebra, geometry and trigonometry); (B) 3 units of English; and (C) 1 unit of Physics or... science; (B) Foreign language; (C) Economics; and, (D) Social science. (2) Evidence of academic work...

Cultivating Deductive Thinking with Angle Chasing

ERIC Educational Resources Information Center

Edwards, Michael todd; Quinlan, James; Harper, Suzanne R.; Cox, Dana C.; Phelps, Steve

2014-01-01

Despite Common Core State Standards for Mathematics (CCSSI 2010) recommendations, too often students' introduction to proof consists of the study of formal axiomatic systems--for example, triangle congruence proofs--typically in an introductory geometry course with no connection back to previous work in earlier algebra courses. Van Hiele…

Platonic Symmetry and Geometric Thinking

ERIC Educational Resources Information Center

Zsombor-Murray, Paul

2007-01-01

Cubic symmetry is used to build the other four Platonic solids and some formalism from classical geometry is introduced. Initially, the approach is via geometric construction, e.g., the "golden ratio" is necessary to construct an icosahedron with pentagonal faces. Then conventional elementary vector algebra is used to extract quantitative…

Some Mathematics and Physics of Ball Games.

ERIC Educational Resources Information Center

Hughes, D. E.

1985-01-01

Gives examples on the applications of arithmetic, geometry, and some calculus, vector algebra, and mechanics to ball games. Suggestions for further interesting investigations are provided together with references to other articles and books on applications of mathematics and physics to ball games and sports in general. (JN)

Mathematics Assessment Sampler 3-5

ERIC Educational Resources Information Center

National Council of Teachers of Mathematics, 2005

2005-01-01

The sample assessment items in this volume are sorted according to the strands of number and operations, algebra, geometry, measurement, and data analysis and probability. Because one goal of assessment is to determine students' abilities to communicate mathematically, the writing team suggests ways to extend or modify multiple-choice and…

Possibility of Engineering Education That Makes Use of Algebraic Calculators by Various Scenes

NASA Astrophysics Data System (ADS)

Umeno, Yoshio

Algebraic calculators are graphing calculators with a feature of computer algebra system. It can be said that we can solve mathematics only by pushing some keys of these calculators in technical colleges or universities. They also possess another feature, so we can make extensive use in engineering education. For example, we can use them for a basic education, a programming education, English education, and creative thinking tools for excellent students. In this paper, we will introduce the summary of algebraic calculators, then, consider how we utilize them in engineer education.

Algebraic model checking for Boolean gene regulatory networks.

PubMed

Tran, Quoc-Nam

2011-01-01

We present a computational method in which modular and Groebner bases (GB) computation in Boolean rings are used for solving problems in Boolean gene regulatory networks (BN). In contrast to other known algebraic approaches, the degree of intermediate polynomials during the calculation of Groebner bases using our method will never grow resulting in a significant improvement in running time and memory space consumption. We also show how calculation in temporal logic for model checking can be done by means of our direct and efficient Groebner basis computation in Boolean rings. We present our experimental results in finding attractors and control strategies of Boolean networks to illustrate our theoretical arguments. The results are promising. Our algebraic approach is more efficient than the state-of-the-art model checker NuSMV on BNs. More importantly, our approach finds all solutions for the BN problems.

Hawking fluxes, fermionic currents, W{sub 1+{infinity}} algebra, and anomalies

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bonora, L.; Cvitan, M.; Theoretical Physics Department, Faculty of Science, University of Zagreb Bijenicka cesta 32, HR-10002 Zagreb

2009-10-15

We complete the analysis carried out in previous papers by studying the Hawking radiation for a Kerr black hole carried to infinity by fermionic currents of any spin. We find agreement with the thermal spectrum of the Hawking radiation for fermionic degrees of freedom. We start by showing that the near-horizon physics for a Kerr black hole is approximated by an effective two-dimensional field theory of fermionic fields. Then, starting from two-dimensional currents of any spin that form a W{sub 1+{infinity}} algebra, we construct an infinite set of covariant currents, each of which carries the corresponding moment of the Hawkingmore » radiation. All together they agree with the thermal spectrum of the latter. We show that the predictive power of this method is based not on the anomalies of the higher-spin currents (which are trivial) but on the underlying W{sub 1+{infinity}} structure. Our results point toward the existence in the near-horizon geometry of a symmetry larger than the Virasoro algebra, which very likely takes the form of a W{sub {infinity}} algebra.« less

Quantum gravity from noncommutative spacetime

NASA Astrophysics Data System (ADS)

Lee, Jungjai; Yang, Hyun Seok

2014-12-01

We review a novel and authentic way to quantize gravity. This novel approach is based on the fact that Einstein gravity can be formulated in terms of a symplectic geometry rather than a Riemannian geometry in the context of emergent gravity. An essential step for emergent gravity is to realize the equivalence principle, the most important property in the theory of gravity (general relativity), from U(1) gauge theory on a symplectic or Poisson manifold. Through the realization of the equivalence principle, which is an intrinsic property in symplectic geometry known as the Darboux theorem or the Moser lemma, one can understand how diffeomorphism symmetry arises from noncommutative U(1) gauge theory; thus, gravity can emerge from the noncommutative electromagnetism, which is also an interacting theory. As a consequence, a background-independent quantum gravity in which the prior existence of any spacetime structure is not a priori assumed but is defined by using the fundamental ingredients in quantum gravity theory can be formulated. This scheme for quantum gravity can be used to resolve many notorious problems in theoretical physics, such as the cosmological constant problem, to understand the nature of dark energy, and to explain why gravity is so weak compared to other forces. In particular, it leads to a remarkable picture of what matter is. A matter field, such as leptons and quarks, simply arises as a stable localized geometry, which is a topological object in the defining algebra (noncommutative ★-algebra) of quantum gravity.

Geometric Model of Topological Insulators from the Maxwell Algebra

NASA Astrophysics Data System (ADS)

Palumbo, Giandomenico

I propose a novel geometric model of time-reversal-invariant topological insulators in three dimensions in presence of an external electromagnetic field. Their gapped boundary supports relativistic quantum Hall states and is described by a Chern-Simons theory, where the gauge connection takes values in the Maxwell algebra. This represents a non-central extension of the Poincare' algebra and takes into account both the Lorentz and magnetic-translation symmetries of the surface states. In this way, I derive a relativistic version of the Wen-Zee term and I show that the non-minimal coupling between the background geometry and the electromagnetic field in the model is in agreement with the main properties of the relativistic quantum Hall states in the flat space. This work is part of the DITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

NASA Astrophysics Data System (ADS)

Lehrer, G. I.; Zhang, R. B.

2017-01-01

We give an elementary and explicit proof of the first fundamental theorem of invariant theory for the orthosymplectic supergroup by generalising the geometric method of Atiyah, Bott and Patodi to the supergroup context. We use methods from super-algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space. In this way, super Schur-Weyl-Brauer duality is established between the orthosymplectic supergroup of superdimension ( m|2 n) and the Brauer algebra with parameter m - 2 n. The result may be interpreted either in terms of the group scheme OSp( V) over C, where V is a finite dimensional super space, or as a statement about the orthosymplectic Lie supergroup over the infinite dimensional Grassmann algebra {Λ}. We take the latter point of view here, and also state a corresponding theorem for the orthosymplectic Lie superalgebra, which involves an extra invariant generator, the super-Pfaffian.

Geometric descriptions of entangled states by auxiliary varieties

NASA Astrophysics Data System (ADS)

Holweck, Frédéric; Luque, Jean-Gabriel; Thibon, Jean-Yves

2012-10-01

The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting, we describe well-known classifications of multipartite entanglement such as 2 × 2 × (n + 1), for n ⩾ 1, quantum systems and a new description with the 2 × 3 × 3 quantum system. Our results complete the approach of Miyake and make stronger connections with recent work of algebraic geometers. Moreover, for the quantum systems detailed in this paper, we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.

Topological String Theory and Enumerative Geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Song, Y. S

In this thesis we investigate several problems which have their roots in both topological string theory and enumerative geometry. In the former case, underlying theories are topological field theories, whereas the latter case is concerned with intersection theories on moduli spaces. A permeating theme in this thesis is to examine the close interplay between these two complementary fields of study. The main problems addressed are as follows: In considering the Hurwitz enumeration problem of branched covers of compact connected Riemann surfaces, we completely solve the problem in the case of simple Hurwitz numbers. In addition, utilizing the connection between Hurwitzmore » numbers and Hodge integrals, we derive a generating function for the latter on the moduli space {bar M}{sub g,2} of 2-pointed, genus-g Deligne-Mumford stable curves. We also investigate Givental's recent conjecture regarding semisimple Frobenius structures and Gromov-Witten invariants, both of which are closely related to topological field theories; we consider the case of a complex projective line P{sup 1} as a specific example and verify his conjecture at low genera. In the last chapter, we demonstrate that certain topological open string amplitudes can be computed via relative stable morphisms in the algebraic category.« less

Methodology and application of high performance electrostatic field simulation in the KATRIN experiment

NASA Astrophysics Data System (ADS)

Corona, Thomas

The Karlsruhe Tritium Neutrino (KATRIN) experiment is a tritium beta decay experiment designed to make a direct, model independent measurement of the electron neutrino mass. The experimental apparatus employs strong ( O[T]) magnetostatic and (O[10 5 V/m]) electrostatic fields in regions of ultra high (O[10-11 mbar]) vacuum in order to obtain precise measurements of the electron energy spectrum near the endpoint of tritium beta-decay. The electrostatic fields in KATRIN are formed by multiscale electrode geometries, necessitating the development of high performance field simulation software. To this end, we present a Boundary Element Method (BEM) with analytic boundary integral terms in conjunction with the Robin Hood linear algebraic solver, a nonstationary successive subspace correction (SSC) method. We describe an implementation of these techniques for high performance computing environments in the software KEMField, along with the geometry modeling and discretization software KGeoBag. We detail the application of KEMField and KGeoBag to KATRIN's spectrometer and detector sections, and demonstrate its use in furthering several of KATRIN's scientific goals. Finally, we present the results of a measurement designed to probe the electrostatic profile of KATRIN's main spectrometer in comparison to simulated results.

PROTEUS two-dimensional Navier-Stokes computer code, version 1.0. Volume 1: Analysis description

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Benson, Thomas J.; Suresh, Ambady

1990-01-01

A new computer code was developed to solve the two-dimensional or axisymmetric, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 1 is the Analysis Description, and describes in detail the governing equations, the turbulence model, the linearization of the equations and boundary conditions, the time and space differencing formulas, the ADI solution procedure, and the artificial viscosity models.

ADPAC v1.0: User's Manual

NASA Technical Reports Server (NTRS)

Hall, Edward J.; Heidegger, Nathan J.; Delaney, Robert A.

1999-01-01

The overall objective of this study was to evaluate the effects of turbulence models in a 3-D numerical analysis on the wake prediction capability. The current version of the computer code resulting from this study is referred to as ADPAC v7 (Advanced Ducted Propfan Analysis Codes -Version 7). This report is intended to serve as a computer program user's manual for the ADPAC code used and modified under Task 15 of NASA Contract NAS3-27394. The ADPAC program is based on a flexible multiple-block and discretization scheme permitting coupled 2-D/3-D mesh block solutions with application to a wide variety of geometries. Aerodynamic calculations are based on a four-stage Runge-Kutta time-marching finite volume solution technique with added numerical dissipation. Steady flow predictions are accelerated by a multigrid procedure. Turbulence models now available in the ADPAC code are: a simple mixing-length model, the algebraic Baldwin-Lomax model with user defined coefficients, the one-equation Spalart-Allmaras model, and a two-equation k-R model. The consolidated ADPAC code is capable of executing in either a serial or parallel computing mode from a single source code.

A finite element computation of turbulent boundary layer flows with an algebraic stress turbulence model

NASA Technical Reports Server (NTRS)

Kim, Sang-Wook; Chen, Yen-Sen

1988-01-01

An algebraic stress turbulence model and a computational procedure for turbulent boundary layer flows which is based on the semidiscrete Galerkin FEM are discussed. In the algebraic stress turbulence model, the eddy viscosity expression is obtained from the Reynolds stress turbulence model, and the turbulent kinetic energy dissipation rate equation is improved by including a production range time scale. Good agreement with experimental data is found for the examples of a fully developed channel flow, a fully developed pipe flow, a flat plate boundary layer flow, a plane jet exhausting into a moving stream, a circular jet exhausting into a moving stream, and a wall jet flow.

Combinatorial quantisation of the Euclidean torus universe

NASA Astrophysics Data System (ADS)

Meusburger, C.; Noui, K.

2010-12-01

We quantise the Euclidean torus universe via a combinatorial quantisation formalism based on its formulation as a Chern-Simons gauge theory and on the representation theory of the Drinfel'd double DSU(2). The resulting quantum algebra of observables is given by two commuting copies of the Heisenberg algebra, and the associated Hilbert space can be identified with the space of square integrable functions on the torus. We show that this Hilbert space carries a unitary representation of the modular group and discuss the role of modular invariance in the theory. We derive the classical limit of the theory and relate the quantum observables to the geometry of the torus universe.

Weak completions, bornologies and rigid cohomology

NASA Astrophysics Data System (ADS)

Cortiñas, Guillermo; Cuntz, Joachim; Meyer, Ralf; Tamme, Georg

2018-07-01

Let V be a complete discrete valuation ring with residue field k of positive characteristic and with fraction field K of characteristic 0. We clarify the analysis behind the Monsky-Washnitzer completion of a commutative V-algebra using completions of bornological V-algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field k that computes its rigid cohomology in the sense of Berthelot.

The Chess and Mathematics Connection: More than Just a Game

ERIC Educational Resources Information Center

Berkman, Robert M.

2004-01-01

This article describes connections between chess and mathematics, including examples of activities that connect chess with set theory, patterns, algebra, geometry, combinatorics, and Pascal's triangle. The author observes that competitive games play a dual purpose in advancing the work of mathematics educators: to reinforce a specific skill and to…

A Brief History of the Most Remarkable Numbers "e," "i" and "?" in Mathematical Sciences with Applications

ERIC Educational Resources Information Center

Debnath, Lokenath

2015-01-01

This paper deals with a brief history of the most remarkable Euler numbers "e,"?"i"?and?"?" in mathematical sciences. Included are many properties of the constants "e,"?"i"?and?"?" and their applications in algebra, geometry, physics, chemistry, ecology, business and industry. Special…

Chattanooga Math Trail: Community Mathematics Modules, Volume 1.

ERIC Educational Resources Information Center

McAllister, Deborah A.; Mealer, Adrian; Moyer, Peggy S.; McDonald, Shirley A.; Peoples, John B.

This collection of community mathematics modules, or "math trail", is appropriate for middle grades and high school students (grades 5-12). Collectively, the modules pay attention to all 10 of the National Council of Teachers of Mathematics (NCTM) standards which include five content standards (Number and Operations, Algebra, Geometry,…

New Trends in Mathematics Teaching, Volume III.

ERIC Educational Resources Information Center

United Nations Educational, Scientific, and Cultural Organization, Paris (France).

Each of the ten chapters in this volume is intended to present an objective analysis of the trends of some important subtopic in mathematics education and each includes a bibliography for fuller study. The chapters cover primary school mathematics, algebra, geometry, probability and statistics, analysis, logic, applications of mathematics, methods…

Core-Plus Mathematics. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2010

2010-01-01

"Core-Plus Mathematics" is a four-year curriculum that replaces the traditional sequence with courses that each feature interwoven strands of algebra and functions, statistics and probability, geometry and trigonometry, and discrete mathematics. The first three courses in the series provide a common core of broadly useful mathematics,…

STEM Picks Up Speed

ERIC Educational Resources Information Center

Demski, Jennifer

2009-01-01

Algebra, geometry, earth science, physics--these require patience and perseverance to master. That kind of academic stamina is hard to advertise to kids nurtured on the instant engagement and gratification of modern digital technology. And there's little hope they'll be sustained by an intrinsic interest in math and science; they have to be shown…

Teaching with New Technology: Four "Early Majority" Teachers

ERIC Educational Resources Information Center

Pierce, Robyn; Stacey, Kaye

2013-01-01

This paper explores how four good teachers, who do not have a special interest in technology, meet the challenge of introducing the rapidly developing mathematics analysis software (e.g. spreadsheets, function graphers, symbolic algebra manipulation and dynamic geometry) into their classrooms. These teachers' practice is viewed through the…

Transition Mathematics. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2007

2007-01-01

"Transition Mathematics" aims to increase 7th- through 12th-grade students' skills in applied arithmetic, pre-algebra, and pre-geometry. This one-year curriculum also addresses general application to different wordings of problems, types of numbers, and contexts for problems and aims to promote mathematical reading skills. The curriculum…

Prospective Secondary Mathematics Teachers' Perspectives and Mathematical Knowledge for Teaching

ERIC Educational Resources Information Center

Karagöz-Akar, Gülseren

2016-01-01

This study investigated the relationship between prospective secondary mathematics teachers' perspectives and their mathematical knowledge for teaching in action. Data from two prospective teachers' practice-teachings, one in geometry and one in algebra, their lesson plans and self-reflections were analyzed with Teacher Perspectives and Knowledge…

Everyday Mathematics. Revised. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2007

2007-01-01

"Everyday Mathematics," published by Wright Group/McGraw-Hill, is a core curriculum for students in kindergarten through grade 6 covering numeration and order, operations, functions and sequences, data and chance, algebra, geometry and spatial sense, measures and measurement, reference frames, and patterns. At each grade level, the…

Advanced Placement Mathematics Calculus, Grade 12 Curriculum Guide.

ERIC Educational Resources Information Center

Scharf, John; And Others

This document is a guide to the advanced placement program in calculus for grade 12 in the city schools in Warren, Ohio. The program covers analytic geometry, differential and integral calculus of algebraic functions, elementary transcendental functions, and applications of differentiation and integration. The philosophy and aims of the program…

Exploring the Effects of Project-Based Learning in Secondary Mathematics Education

ERIC Educational Resources Information Center

Holmes, Vicki-Lynn; Hwang, Yooyeun

2016-01-01

This mixed-method, longitudinal study investigated the benefits of project-based learning (PBL) on secondary-mathematics students' academic skill development and motivated strategies for learning (i.e., cognitive, social, and motivational). The focus of this study was academic skill development (algebra- and geometry-assessment scores) and other…

Everyday Mathematics. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2006

2006-01-01

"Everyday Mathematics," published by Wright Group/McGraw-Hill, is a core curriculum for students in kindergarten through grade 6 covering numeration and order, operations, functions and sequences, data and chance, algebra, geometry and spatial sense, measures and measurement, reference frames, and patterns. At each grade level, the "Everyday…

Connected Mathematics Project (CMP). What Works Clearinghouse Intervention Report. Updated

ERIC Educational Resources Information Center

What Works Clearinghouse, 2017

2017-01-01

"Connected Mathematics Project" (CMP) is a math curriculum for students in grades 6-8. It uses interactive problems and everyday situations to explore mathematical ideas, with a goal of fostering a problem-centered, inquiry-based learning environment. At each grade level, the curriculum covers numbers, algebra, geometry/measurement,…

Using Technology to Promote Mathematical Discourse Concerning Women in Mathematics

ERIC Educational Resources Information Center

Phy, Lyn

2008-01-01

This paper discusses uses of technology to facilitate mathematical discourse concerning women in mathematics. Such a topic can be introduced in various traditional courses such as algebra, geometry, trigonometry, probability and statistics, or calculus, but it is not included in traditional textbooks. Through the ideas presented here, you can…

Stretching Probability Explorations with Geoboards

ERIC Educational Resources Information Center

Wheeler, Ann; Champion, Joe

2016-01-01

Students are faced with many transitions in their middle school mathematics classes. To build knowledge, skills, and confidence in the key areas of algebra and geometry, students often need to practice using numbers and polygons in a variety of contexts. Teachers also want students to explore ideas from probability and statistics. Teachers know…

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…

Teachers' Reactions to Pre-Differentiated and Enriched Mathematics Curricula

ERIC Educational Resources Information Center

Rubenstein, Lisa DaVia; Gilson, Cindy M.; Bruce-Davis, Micah N.; Gubbins, E. Jean

2015-01-01

Modern classrooms are often comprised of a heterogeneous student population with varying abilities. To address this variance, third-grade teachers implemented researcher-designed, pre-differentiated, and enriched math curricula in algebra, geometry and measurement, and graphing and data analysis. The goal of the curricula was to provide academic…

Teaching Third-Degree Price Discrimination

ERIC Educational Resources Information Center

Round, David K.; McIver, Ron P.

2006-01-01

Third-degree price discrimination is taught in almost every intermediate microeconomics class. The theory, geometry, and the algebra behind the concept are simple, and the phenomenon is commonly associated with the sale of many of the goods and services used frequently by students. Classroom discussion is usually vibrant as students can relate…

Hermann-Bernoulli-Laplace-Hamilton-Runge-Lenz Vector.

ERIC Educational Resources Information Center

Subramanian, P. R.; And Others

1991-01-01

A way for students to refresh and use their knowledge in both mathematics and physics is presented. By the study of the properties of the "Runge-Lenz" vector the subjects of algebra, analytical geometry, calculus, classical mechanics, differential equations, matrices, quantum mechanics, trigonometry, and vector analysis can be reviewed. (KR)

Physics for Water and Wastewater Operators.

ERIC Educational Resources Information Center

Koundakjian, Philip

This physics course covers the following main subject areas: (1) liquids; (2) pressure; (3) liquid flow; (4) temperature and heat; and (5) electric currents. The prerequisites for understanding this material are basic algebra and geometry. The lessons are composed mostly of sample problems and calculations that water and wastewater operators have…

A Geometric Solution of a Cournot Ogilopoly with Nonidentical Firms.

ERIC Educational Resources Information Center

Sarkar, Jyotirmoy; Gupta, Barnali; Pal, Debashis

1998-01-01

Maintains that a proper understanding of the Augustin Cournot model of imperfect competition and strategic interactions among firms in various contexts is essential for economics education. Although most models rely on complicated algebra, this one requires nothing more than high school level geometry. Includes a graphical analysis. (MJP)

Construction of Orthonormal Wavelets Using Symbolic Algebraic Methods

NASA Astrophysics Data System (ADS)

Černá, Dana; Finěk, Václav

2009-09-01

Our contribution is concerned with the solution of nonlinear algebraic equations systems arising from the computation of scaling coefficients of orthonormal wavelets with compact support. Specifically Daubechies wavelets, symmlets, coiflets, and generalized coiflets. These wavelets are defined as a solution of equation systems which are partly linear and partly nonlinear. The idea of presented methods consists in replacing those equations for scaling coefficients by equations for scaling moments. It enables us to eliminate some quadratic conditions in the original system and then simplify it. The simplified system is solved with the aid of the Gröbner basis method. The advantage of our approach is that in some cases, it provides all possible solutions and these solutions can be computed to arbitrary precision. For small systems, we are even able to find explicit solutions. The computation was carried out by symbolic algebra software Maple.

Quanta of geometry and unification

NASA Astrophysics Data System (ADS)

Chamseddine, Ali H.

2016-11-01

This is a tribute to Abdus Salam’s memory whose insight and creative thinking set for me a role model to follow. In this contribution I show that the simple requirement of volume quantization in spacetime (with Euclidean signature) uniquely determines the geometry to be that of a noncommutative space whose finite part is based on an algebra that leads to Pati-Salam grand unified models. The Standard Model corresponds to a special case where a mathematical constraint (order one condition) is satisfied. This provides evidence that Salam was a visionary who was generations ahead of his time.

Global differential geometry: An introduction for control engineers

NASA Technical Reports Server (NTRS)

Doolin, B. F.; Martin, C. F.

1982-01-01

The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.

Quanta of Geometry and Unification

NASA Astrophysics Data System (ADS)

Chamseddine, Ali H.

This is a tribute to Abdus Salam's memory whose insight and creative thinking set for me a role model to follow. In this contribution I show that the simple requirement of volume quantization in space-time (with Euclidean signature) uniquely determines the geometry to be that of a noncommutative space whose finite part is based on an algebra that leads to Pati-Salam grand unified models. The Standard Model corresponds to a special case where a mathematical constraint (order one condition) is satisfied. This provides evidence that Salam was a visionary who was generations ahead of his time.

Noncommutative Geometry of the Moyal Plane: Translation Isometries, Connes' Distance on Coherent States, Pythagoras Equality

NASA Astrophysics Data System (ADS)

Martinetti, Pierre; Tomassini, Luca

2013-10-01

We study the metric aspect of the Moyal plane from Connes' noncommutative geometry point of view. First, we compute Connes' spectral distance associated with the natural isometric action of on the algebra of the Moyal plane . We show that the distance between any state of and any of its translated states is precisely the amplitude of the translation. As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane. We investigate the classical limit, showing that the set of coherent states equipped with Connes' spectral distance tends towards the Euclidean plane as the parameter of deformation goes to zero. The extension of these results to the action of the symplectic group is also discussed, with particular emphasis on the orbits of coherent states under rotations. Second, we compute the spectral distance in the double Moyal plane, intended as the product of (the minimal unitization of) by . We show that on the set of states obtained by translation of an arbitrary state of , this distance is given by the Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital and non-degenerate spectral triples. Applied to the Doplicher- Fredenhagen-Roberts model of quantum spacetime [DFR], these two theorems show that Connes' spectral distance and the DFR quantum length coincide on the set of states of optimal localization.

Reliability, Risk and Cost Trade-Offs for Composite Designs

NASA Technical Reports Server (NTRS)

Shiao, Michael C.; Singhal, Surendra N.; Chamis, Christos C.

1996-01-01

Risk and cost trade-offs have been simulated using a probabilistic method. The probabilistic method accounts for all naturally-occurring uncertainties including those in constituent material properties, fabrication variables, structure geometry and loading conditions. The probability density function of first buckling load for a set of uncertain variables is computed. The probabilistic sensitivity factors of uncertain variables to the first buckling load is calculated. The reliability-based cost for a composite fuselage panel is defined and minimized with respect to requisite design parameters. The optimization is achieved by solving a system of nonlinear algebraic equations whose coefficients are functions of probabilistic sensitivity factors. With optimum design parameters such as the mean and coefficient of variation (representing range of scatter) of uncertain variables, the most efficient and economical manufacturing procedure can be selected. In this paper, optimum values of the requisite design parameters for a predetermined cost due to failure occurrence are computationally determined. The results for the fuselage panel analysis show that the higher the cost due to failure occurrence, the smaller the optimum coefficient of variation of fiber modulus (design parameter) in longitudinal direction.

Numerical linear algebra in data mining

NASA Astrophysics Data System (ADS)

Eldén, Lars

Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.

Assessment of polytechnic students' understanding of basic algebra

NASA Astrophysics Data System (ADS)

Mokmin, Nur Azlina Mohamed; Masood, Mona

2015-12-01

It is important for engineering students to excel in algebra. Previous studies show that the algebraic fraction is a subtopic of algebra that was found to be the most challenging for engineering students. This study is done with 191 first semester engineering students who have enrolled in engineering programs in Malaysian polytechnic. The respondents are divided into Group 1 (Distinction) and Group 2 (Credit) based on their Mathematics SPM result. A computer application is developed for this study to assess student information and understanding of the algebraic fraction topic. The result is analyzed using SPSS and Microsoft Excel. The test results show that there are significant differences between Group 1 and Group 2 and that most of the students scored below the minimum requirement.

Integration of a CAS/DGS as a CAD system in the mathematics curriculum for architecture students

NASA Astrophysics Data System (ADS)

Falcón, R. M.

2011-09-01

Students of Architecture and Building Engineering Degrees work with Computer Aided Design systems daily in order to design and model architectonic constructions. Since this kind of software is based on the creation and transformation of geometrical objects, it seems to be a useful tool in Maths classes in order to capture the attention of the students. However, users of these systems cannot display the set of formulas and equations which constitute the basis of their studio. Moreover, if they want to represent curves or surfaces starting from its corresponding equations, they have to define specific macros which require the knowledge of some computer language or they have to create a table of points in order to convert a set of nodes into polylines, polysolids or splines. More specific concepts, like, for instance, those related to differential geometry, are not implemented in this kind of software, although they are taught in our Maths classes. In a very similar virtual environment, Computer Algebra and Dynamic Geometry Systems offer the possibility of implementing several concepts which can be found in the usual mathematics curriculum for Building Engineering: curves, surfaces and calculus. Specifically, the use of sliders related to the Euler's angles and the generation of tools which project 3D into 2D, facilitate the design and model of curves and rigid objects in space, by starting from their parametric equations. In this article, we show the experience carried out in an experimental and control group in the context of the Maths classes of the Building Engineering Degree of the University of Seville, where students have created their own building models by understanding and testing the usefulness of the mathematical concepts.

Elementary Particle Spectroscopy in Regular Solid Rewrite

DOE Office of Scientific and Technical Information (OSTI.GOV)

Trell, Erik

The Nilpotent Universal Computer Rewrite System (NUCRS) has operationalized the radical ontological dilemma of Nothing at All versus Anything at All down to the ground recursive syntax and principal mathematical realisation of this categorical dichotomy as such and so governing all its sui generis modalities, leading to fulfilment of their individual terms and compass when the respective choice sequence operations are brought to closure. Focussing on the general grammar, NUCRS by pure logic and its algebraic notations hence bootstraps Quantum Mechanics, aware that it ''is the likely keystone of a fundamental computational foundation'' also for e.g. physics, molecular biology andmore » neuroscience. The present work deals with classical geometry where morphology is the modality, and ventures that the ancient regular solids are its specific rewrite system, in effect extensively anticipating the detailed elementary particle spectroscopy, and further on to essential structures at large both over the inorganic and organic realms. The geodetic antipode to Nothing is extension, with natural eigenvector the endless straight line which when deployed according to the NUCRS as well as Plotelemeian topographic prescriptions forms a real three-dimensional eigenspace with cubical eigenelements where observed quark-skewed quantum-chromodynamical particle events self-generate as an Aristotelean phase transition between the straight and round extremes of absolute endlessness under the symmetry- and gauge-preserving, canonical coset decomposition SO(3)xO(5) of Lie algebra SU(3). The cubical eigen-space and eigen-elements are the parental state and frame, and the other solids are a range of transition matrix elements and portions adapting to the spherical root vector symmetries and so reproducibly reproducing the elementary particle spectroscopy, including a modular, truncated octahedron nano-composition of the Electron which piecemeal enter into molecular structures or compressed to each other fuse into atomic honeycombs of periodic table signature.« less

Instructors' use of technology in post-secondary undergraduate mathematics teaching: a local study

NASA Astrophysics Data System (ADS)

Jesso, A. T.; Kondratieva, M. F.

2016-02-01

In this study, instructors of undergraduate mathematics from post-secondary institutions in Newfoundland were surveyed (N = 13) and interviewed (N = 8) about their use of, experiences with, and views on, technologically assisted teaching. It was found that the majority of them regularly use technologies for organizational and communication purposes. However, the use of math-specific technology such as computer algebra systems, or dynamic geometry software for instructional, exploratory, and creative activities with students takes place mostly on an individual basis, only occasionally, and is very much topic specific. This was even the case for those instructors who use technology proficiently in their research. The data also suggested that familiarity with and discussions of examples of technology implementation in teaching at regular and field-oriented professional development seminars within mathematics departments could potentially increase the use of math-specific technology by instructors.

(q,{mu}) and (p,q,{zeta})-exponential functions: Rogers-Szego'' polynomials and Fourier-Gauss transform

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hounkonnou, Mahouton Norbert; Nkouankam, Elvis Benzo Ngompe

2010-10-15

From the realization of q-oscillator algebra in terms of generalized derivative, we compute the matrix elements from deformed exponential functions and deduce generating functions associated with Rogers-Szego polynomials as well as their relevant properties. We also compute the matrix elements associated with the (p,q)-oscillator algebra (a generalization of the q-one) and perform the Fourier-Gauss transform of a generalization of the deformed exponential functions.

A Electro-Optical Image Algebra Processing System for Automatic Target Recognition

NASA Astrophysics Data System (ADS)

Coffield, Patrick Cyrus

The proposed electro-optical image algebra processing system is designed specifically for image processing and other related computations. The design is a hybridization of an optical correlator and a massively paralleled, single instruction multiple data processor. The architecture of the design consists of three tightly coupled components: a spatial configuration processor (the optical analog portion), a weighting processor (digital), and an accumulation processor (digital). The systolic flow of data and image processing operations are directed by a control buffer and pipelined to each of the three processing components. The image processing operations are defined in terms of basic operations of an image algebra developed by the University of Florida. The algebra is capable of describing all common image-to-image transformations. The merit of this architectural design is how it implements the natural decomposition of algebraic functions into spatially distributed, point use operations. The effect of this particular decomposition allows convolution type operations to be computed strictly as a function of the number of elements in the template (mask, filter, etc.) instead of the number of picture elements in the image. Thus, a substantial increase in throughput is realized. The implementation of the proposed design may be accomplished in many ways. While a hybrid electro-optical implementation is of primary interest, the benefits and design issues of an all digital implementation are also discussed. The potential utility of this architectural design lies in its ability to control a large variety of the arithmetic and logic operations of the image algebra's generalized matrix product. The generalized matrix product is the most powerful fundamental operation in the algebra, thus allowing a wide range of applications. No other known device or design has made this claim of processing speed and general implementation of a heterogeneous image algebra.

Efficient solvers for coupled models in respiratory mechanics.

PubMed

Verdugo, Francesc; Roth, Christian J; Yoshihara, Lena; Wall, Wolfgang A

2017-02-01

We present efficient preconditioners for one of the most physiologically relevant pulmonary models currently available. Our underlying motivation is to enable the efficient simulation of such a lung model on high-performance computing platforms in order to assess mechanical ventilation strategies and contributing to design more protective patient-specific ventilation treatments. The system of linear equations to be solved using the proposed preconditioners is essentially the monolithic system arising in fluid-structure interaction (FSI) extended by additional algebraic constraints. The introduction of these constraints leads to a saddle point problem that cannot be solved with usual FSI preconditioners available in the literature. The key ingredient in this work is to use the idea of the semi-implicit method for pressure-linked equations (SIMPLE) for getting rid of the saddle point structure, resulting in a standard FSI problem that can be treated with available techniques. The numerical examples show that the resulting preconditioners approach the optimal performance of multigrid methods, even though the lung model is a complex multiphysics problem. Moreover, the preconditioners are robust enough to deal with physiologically relevant simulations involving complex real-world patient-specific lung geometries. The same approach is applicable to other challenging biomedical applications where coupling between flow and tissue deformations is modeled with additional algebraic constraints. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

Modeling Simple Telescope Optics in Secondary Mathematics Classrooms

NASA Astrophysics Data System (ADS)

Siegel, Lauren; Dickinson, G.; Hooper, E. J.; Daniels, M.

2007-12-01

This presentation describes the results of collaboration between instructors in the UTeach teacher preparation program at the University of Texas at Austin, and an astronomer teaching at the university as part of a National Science Foundation Astronomy and Astrophysics Postdoctoral Fellowship. The astronomer provided training to give pre-service teachers an authentic understanding of the principles of telescope optics. This made it possible for the preservice teachers to include real design constraints and optical properties into lessons developed as part of a collaborative field experience to teach astronomical telescope design and construction to high school Algebra II students. One result is a sequence of investigations designed to explore how and why the physical and mathematical properties of parabolic mirrors both enable and constrain our ability to build and use telescopes to focus light from distant objects. Various approaches, including generating and exploring computer models, traditional proofs, even making paper models, are all woven together into a coherent set of eleven investigations for use in mathematics and science classrooms. The presentation will include a description of the suite of investigations, as well as a discussion of the collaborative process which generated the work and resulted in an article submission to a preeminent teaching journal. Teaching Algebra and Geometry Concepts by Modeling Telescope Optics, 2008, Mathematics Teacher is currently in press. Many thanks to the University of Texas UTeach Program for sponsorship of this submission.

Radiation dose reduction in medical x-ray CT via Fourier-based iterative reconstruction

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fahimian, Benjamin P.; Zhao Yunzhe; Huang Zhifeng

Purpose: A Fourier-based iterative reconstruction technique, termed Equally Sloped Tomography (EST), is developed in conjunction with advanced mathematical regularization to investigate radiation dose reduction in x-ray CT. The method is experimentally implemented on fan-beam CT and evaluated as a function of imaging dose on a series of image quality phantoms and anonymous pediatric patient data sets. Numerical simulation experiments are also performed to explore the extension of EST to helical cone-beam geometry. Methods: EST is a Fourier based iterative algorithm, which iterates back and forth between real and Fourier space utilizing the algebraically exact pseudopolar fast Fourier transform (PPFFT). Inmore » each iteration, physical constraints and mathematical regularization are applied in real space, while the measured data are enforced in Fourier space. The algorithm is automatically terminated when a proposed termination criterion is met. Experimentally, fan-beam projections were acquired by the Siemens z-flying focal spot technology, and subsequently interleaved and rebinned to a pseudopolar grid. Image quality phantoms were scanned at systematically varied mAs settings, reconstructed by EST and conventional reconstruction methods such as filtered back projection (FBP), and quantified using metrics including resolution, signal-to-noise ratios (SNRs), and contrast-to-noise ratios (CNRs). Pediatric data sets were reconstructed at their original acquisition settings and additionally simulated to lower dose settings for comparison and evaluation of the potential for radiation dose reduction. Numerical experiments were conducted to quantify EST and other iterative methods in terms of image quality and computation time. The extension of EST to helical cone-beam CT was implemented by using the advanced single-slice rebinning (ASSR) method. Results: Based on the phantom and pediatric patient fan-beam CT data, it is demonstrated that EST reconstructions with the lowest scanner flux setting of 39 mAs produce comparable image quality, resolution, and contrast relative to FBP with the 140 mAs flux setting. Compared to the algebraic reconstruction technique and the expectation maximization statistical reconstruction algorithm, a significant reduction in computation time is achieved with EST. Finally, numerical experiments on helical cone-beam CT data suggest that the combination of EST and ASSR produces reconstructions with higher image quality and lower noise than the Feldkamp Davis and Kress (FDK) method and the conventional ASSR approach. Conclusions: A Fourier-based iterative method has been applied to the reconstruction of fan-bean CT data with reduced x-ray fluence. This method incorporates advantageous features in both real and Fourier space iterative schemes: using a fast and algebraically exact method to calculate forward projection, enforcing the measured data in Fourier space, and applying physical constraints and flexible regularization in real space. Our results suggest that EST can be utilized for radiation dose reduction in x-ray CT via the readily implementable technique of lowering mAs settings. Numerical experiments further indicate that EST requires less computation time than several other iterative algorithms and can, in principle, be extended to helical cone-beam geometry in combination with the ASSR method.« less

Radiation dose reduction in medical x-ray CT via Fourier-based iterative reconstruction.

PubMed

Fahimian, Benjamin P; Zhao, Yunzhe; Huang, Zhifeng; Fung, Russell; Mao, Yu; Zhu, Chun; Khatonabadi, Maryam; DeMarco, John J; Osher, Stanley J; McNitt-Gray, Michael F; Miao, Jianwei

2013-03-01

A Fourier-based iterative reconstruction technique, termed Equally Sloped Tomography (EST), is developed in conjunction with advanced mathematical regularization to investigate radiation dose reduction in x-ray CT. The method is experimentally implemented on fan-beam CT and evaluated as a function of imaging dose on a series of image quality phantoms and anonymous pediatric patient data sets. Numerical simulation experiments are also performed to explore the extension of EST to helical cone-beam geometry. EST is a Fourier based iterative algorithm, which iterates back and forth between real and Fourier space utilizing the algebraically exact pseudopolar fast Fourier transform (PPFFT). In each iteration, physical constraints and mathematical regularization are applied in real space, while the measured data are enforced in Fourier space. The algorithm is automatically terminated when a proposed termination criterion is met. Experimentally, fan-beam projections were acquired by the Siemens z-flying focal spot technology, and subsequently interleaved and rebinned to a pseudopolar grid. Image quality phantoms were scanned at systematically varied mAs settings, reconstructed by EST and conventional reconstruction methods such as filtered back projection (FBP), and quantified using metrics including resolution, signal-to-noise ratios (SNRs), and contrast-to-noise ratios (CNRs). Pediatric data sets were reconstructed at their original acquisition settings and additionally simulated to lower dose settings for comparison and evaluation of the potential for radiation dose reduction. Numerical experiments were conducted to quantify EST and other iterative methods in terms of image quality and computation time. The extension of EST to helical cone-beam CT was implemented by using the advanced single-slice rebinning (ASSR) method. Based on the phantom and pediatric patient fan-beam CT data, it is demonstrated that EST reconstructions with the lowest scanner flux setting of 39 mAs produce comparable image quality, resolution, and contrast relative to FBP with the 140 mAs flux setting. Compared to the algebraic reconstruction technique and the expectation maximization statistical reconstruction algorithm, a significant reduction in computation time is achieved with EST. Finally, numerical experiments on helical cone-beam CT data suggest that the combination of EST and ASSR produces reconstructions with higher image quality and lower noise than the Feldkamp Davis and Kress (FDK) method and the conventional ASSR approach. A Fourier-based iterative method has been applied to the reconstruction of fan-bean CT data with reduced x-ray fluence. This method incorporates advantageous features in both real and Fourier space iterative schemes: using a fast and algebraically exact method to calculate forward projection, enforcing the measured data in Fourier space, and applying physical constraints and flexible regularization in real space. Our results suggest that EST can be utilized for radiation dose reduction in x-ray CT via the readily implementable technique of lowering mAs settings. Numerical experiments further indicate that EST requires less computation time than several other iterative algorithms and can, in principle, be extended to helical cone-beam geometry in combination with the ASSR method.

Radiation dose reduction in medical x-ray CT via Fourier-based iterative reconstruction

PubMed Central

Fahimian, Benjamin P.; Zhao, Yunzhe; Huang, Zhifeng; Fung, Russell; Mao, Yu; Zhu, Chun; Khatonabadi, Maryam; DeMarco, John J.; Osher, Stanley J.; McNitt-Gray, Michael F.; Miao, Jianwei

2013-01-01

Purpose: A Fourier-based iterative reconstruction technique, termed Equally Sloped Tomography (EST), is developed in conjunction with advanced mathematical regularization to investigate radiation dose reduction in x-ray CT. The method is experimentally implemented on fan-beam CT and evaluated as a function of imaging dose on a series of image quality phantoms and anonymous pediatric patient data sets. Numerical simulation experiments are also performed to explore the extension of EST to helical cone-beam geometry. Methods: EST is a Fourier based iterative algorithm, which iterates back and forth between real and Fourier space utilizing the algebraically exact pseudopolar fast Fourier transform (PPFFT). In each iteration, physical constraints and mathematical regularization are applied in real space, while the measured data are enforced in Fourier space. The algorithm is automatically terminated when a proposed termination criterion is met. Experimentally, fan-beam projections were acquired by the Siemens z-flying focal spot technology, and subsequently interleaved and rebinned to a pseudopolar grid. Image quality phantoms were scanned at systematically varied mAs settings, reconstructed by EST and conventional reconstruction methods such as filtered back projection (FBP), and quantified using metrics including resolution, signal-to-noise ratios (SNRs), and contrast-to-noise ratios (CNRs). Pediatric data sets were reconstructed at their original acquisition settings and additionally simulated to lower dose settings for comparison and evaluation of the potential for radiation dose reduction. Numerical experiments were conducted to quantify EST and other iterative methods in terms of image quality and computation time. The extension of EST to helical cone-beam CT was implemented by using the advanced single-slice rebinning (ASSR) method. Results: Based on the phantom and pediatric patient fan-beam CT data, it is demonstrated that EST reconstructions with the lowest scanner flux setting of 39 mAs produce comparable image quality, resolution, and contrast relative to FBP with the 140 mAs flux setting. Compared to the algebraic reconstruction technique and the expectation maximization statistical reconstruction algorithm, a significant reduction in computation time is achieved with EST. Finally, numerical experiments on helical cone-beam CT data suggest that the combination of EST and ASSR produces reconstructions with higher image quality and lower noise than the Feldkamp Davis and Kress (FDK) method and the conventional ASSR approach. Conclusions: A Fourier-based iterative method has been applied to the reconstruction of fan-bean CT data with reduced x-ray fluence. This method incorporates advantageous features in both real and Fourier space iterative schemes: using a fast and algebraically exact method to calculate forward projection, enforcing the measured data in Fourier space, and applying physical constraints and flexible regularization in real space. Our results suggest that EST can be utilized for radiation dose reduction in x-ray CT via the readily implementable technique of lowering mAs settings. Numerical experiments further indicate that EST requires less computation time than several other iterative algorithms and can, in principle, be extended to helical cone-beam geometry in combination with the ASSR method. PMID:23464329

Pre-Algebra Groups. Concepts & Applications.

ERIC Educational Resources Information Center

Montgomery County Public Schools, Rockville, MD.

Discussion material and exercises related to pre-algebra groups are provided in this five chapter manual. Chapter 1 (mappings) focuses on restricted domains, order of operations (parentheses and exponents), rules of assignment, and computer extensions. Chapter 2 considers finite number systems, including binary operations, clock arithmetic,…

A ∞-Algebra of an Elliptic Curve and Eisenstein Series

NASA Astrophysics Data System (ADS)

Polishchuk, Alexander

2011-02-01

We compute explicitly the A ∞-structure on the algebra {Ext^*(mathcal{O}_C oplus L, mathcal{O}_C oplus L)} , where L is a line bundle of degree 1 on an elliptic curve C. The answer involves higher derivatives of Eisenstein series.

The Golden Ratio: A Golden Opportunity to Investigate Multiple Representations of a Problem.

ERIC Educational Resources Information Center

Dickey, Edwin M.

1993-01-01

This article explores the multiple representations (verbal, algebraic, graphical, and numerical) that can be used to study the golden ratio. Emphasis is placed on using technology (both calculators and computers) to investigate the algebraic, graphical, and numerical representations. (JAF)

Heterogeneous Software System Interoperability Through Computer-Aided Resolution of Modeling Differences

DTIC Science & Technology

2002-06-01

techniques for addressing the software component retrieval problem. Steigerwald [Ste91] introduced the use of algebraic specifications for defining the...provided in terms of a specification written using Luqi’s Prototype Specification Description Language (PSDL) [LBY88] augmented with an algebraic

A Cohomological Perspective on Algebraic Quantum Field Theory

NASA Astrophysics Data System (ADS)

Hawkins, Eli

2018-05-01

Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model. This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited. To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory.

An Ada Linear-Algebra Software Package Modeled After HAL/S

NASA Technical Reports Server (NTRS)

Klumpp, Allan R.; Lawson, Charles L.

1990-01-01

New avionics software written more easily. Software package extends Ada programming language to include linear-algebra capabilities similar to those of HAL/S programming language. Designed for such avionics applications as Space Station flight software. In addition to built-in functions of HAL/S, package incorporates quaternion functions used in Space Shuttle and Galileo projects and routines from LINPAK solving systems of equations involving general square matrices. Contains two generic programs: one for floating-point computations and one for integer computations. Written on IBM/AT personal computer running under PC DOS, v.3.1.

Developing Fair Tests for Mathematics Curriculum Comparison Studies: The Role of Content Analyses

ERIC Educational Resources Information Center

Chavez, Oscar; Papick, Ira; Ross, Daniel J.; Grouws, Douglas A.

2011-01-01

This article describes the process of development of assessment instruments for a three-year longitudinal comparative study that focused on evaluating American high school students' mathematics learning from two distinct approaches to content organization: curriculum built around a sequence of three full-year courses (Algebra 1, Geometry, and…

The Mathematics of Skateboarding: A Relevant Application of the 5Es of Constructivism

ERIC Educational Resources Information Center

Robertson, William H.; Meyer, Rachelle D.; Wilkerson, Trena L.

2012-01-01

Getting high school students to enjoy mathematics and to connect concepts to their daily lives is a challenge for many educators. The Mathematics of Skateboarding demonstrated innovative and creative ways to engage students in content and skills mapped to state requirements for high school students in Algebra and Geometry.

The Path to College Calculus: The Impact of High School Mathematics Coursework

ERIC Educational Resources Information Center

Sadler, Philip; Sonnert, Gerhard

2018-01-01

This study addresses a longstanding question among high school mathematics teachers and college mathematics professors: Which is the best preparation for college calculus-- (a) a high level of mastery of mathematics considered preparatory for calculus (algebra, geometry, precalculus) or (b) taking calculus itself in high school? We used a data set…

Steps Forward and Back in Adult Numeracy Teacher Professional Development: A Reflection on a Teacher Workshop Experience

ERIC Educational Resources Information Center

Saliga, Linda Marie; Daviso, Al; Stuart, Denise; Pachnowski, Lynne

2015-01-01

In this project, a university team of teacher education and mathematics professors conducted eight professional development sessions for General Educational Development (GED) teachers in the area of mathematics teaching. Topics included concretely modeling mathematics concepts in algebra, number sense, geometry, and differentiating instruction in…

Geometric and Applied Optics, Science (Experimental): 5318.04.

ERIC Educational Resources Information Center

Sanderson, Robert C.

This unit of instruction presents a laboratory-oriented course which relates the sources and behaviors of light to man's control and uses of light. Successful completion of Algebra I and Plane Geometry is strongly recommended as indicators of success. The course is recommended if the student plans further studies in science, optical technology, or…

Ideas for the Classroom

ERIC Educational Resources Information Center

Mathematics Teaching Incorporating Micromath, 2006

2006-01-01

In this article, the author shares some of the activities for the classroom invented by Gill Hatch. One of those activities is the activity for older students, which is for the five-year-olds through to post-graduates. Card-sorting game, geometry games, algebra games, and loop games are also some of those activities for the classroom invented by…

Using Nonlinear Programming in International Trade Theory: The Factor-Proportions Model

ERIC Educational Resources Information Center

Gilbert, John

2004-01-01

Students at all levels benefit from a multi-faceted approach to learning abstract material. The most commonly used technique in teaching the pure theory of international trade is a combination of geometry and algebraic derivations. Numerical simulation can provide a valuable third support to these approaches. The author describes a simple…

A Comparison between Mathematics Textbook Content and a Statewide Mathematics Proficiency Test.

ERIC Educational Resources Information Center

Chandler, Donald G.; Brosnan, Patricia A.

1995-01-01

Percentages of mathematics content for 7 text series, grades 1-8, were compared with percentages on the Ohio Ninth Grade Proficiency Test. Ratios of text:test percentages were arithmetic (63:30), measurement (10:25), geometry (12:15), data analysis (11:15), and algebra (4:15). Implications are discussed. (MSD)

Mathematics for Junior High School, Volume II (Part 2).

ERIC Educational Resources Information Center

Anderson, R. D.; And Others

This is part two of a two-part SMSG mathematics text for junior high school students. Key ideas emphasized are structure of arithmetic from an algebraic viewpoint, the real number system as a progressing development, and metric and non-metric relations in geometry. Chapter topics include real numbers, similar triangles, variation, non-metric…

Student and Teacher Perceptions of Teacher Oral Communication Behavior in Algebra and Geometry Classrooms

ERIC Educational Resources Information Center

Assuah, Charles K.

2010-01-01

Oral communication in mathematics classroom plays an essential role in the mathematics learning process, because it allows students to share ideas, refine their thoughts, reflect on their methods, and clarify their understanding (NCTM, 2000). Knowledge about teacher oral communication behaviors allows researchers and policy makers to identify and…

Australian Curriculum Linked Lessons. Fluency

ERIC Educational Resources Information Center

Hurrell, Derek

2014-01-01

In this article, Derek Hurrell, points out that while it's easy to fall into the impression that the proficiency strand "Fluency" is all about knowing basic number facts in all its many and splendid ways. He add it is easy to overlook, that within Fluency there are requirements that are based in Algebra; Measurement and Geometry; and…

State College- and Career-Ready High School Graduation Requirements. Updated

ERIC Educational Resources Information Center

Achieve, Inc., 2013

2013-01-01

Research by Achieve, ACT, and others suggests that for high school graduates to be prepared for success in a wide range of postsecondary settings, they need to take four years of challenging mathematics--covering Advanced Algebra; Geometry; and data, probability, and statistics content--and four years of rigorous English aligned with college- and…

Math in the Box

ERIC Educational Resources Information Center

DeYoung, Mary J.

2009-01-01

This article describes how to make an origami paper box and explores the algebra, geometry, and other mathematics that unfolds. A set of origami steps that transforms the paper into an open box can hold mathematical surprises for both students and teachers. An origami lesson can engage students in an open-ended exploration of the relationship…

On the Axiomatization of Mathematical Understanding: Continuous Functions in the Transition to Topology

ERIC Educational Resources Information Center

Cheshire, Daniel C.

2017-01-01

The introduction to general topology represents a challenging transition for students of advanced mathematics. It requires the generalization of their previous understanding of ideas from fields like geometry, linear algebra, and real or complex analysis to fit within a more abstract conceptual system. Students must adopt a new lexicon of…

A brief survey of constrained mechanics and variational problems in terms of differential forms

NASA Technical Reports Server (NTRS)

Hermann, Robert

1994-01-01

There has been considerable interest recently in constrained mechanics and variational problems. This is in part due to applied interests (such as 'non-holonomic mechanics in robotics') and in other part due to the fact that several schools of 'pure' mathematics have found that this classical subject is of importance for what they are trying to do. I have made various attempts at developing these subjects since my Lincoln lab days of the late 1950's. In this Chapter, I will sketch a Unified point of view, using Cartan's approach with differential forms. This has the advantage from the C-O-R viewpoint being developed in this Volume that the extension from 'smooth' to 'generalized' data is very systematic and algebraic. (I will only deal with the 'smooth' point of view in this Chapter; I will develop the 'generalized function' material at a later point.) The material presented briefly here about Variational Calculus and Constrained Mechanics can be found in more detail in my books, 'Differential Geometry and the Calculus of Variations', 'Lie Algebras and Quantum Mechanics', and 'Geometry, Physics and Systems'.

Quantum coherence generating power, maximally abelian subalgebras, and Grassmannian geometry

NASA Astrophysics Data System (ADS)

Zanardi, Paolo; Campos Venuti, Lorenzo

2018-01-01

We establish a direct connection between the power of a unitary map in d-dimensions (d < ∞) to generate quantum coherence and the geometry of the set Md of maximally abelian subalgebras (of the quantum system full operator algebra). This set can be seen as a topologically non-trivial subset of the Grassmannian over linear operators. The natural distance over the Grassmannian induces a metric structure on Md, which quantifies the lack of commutativity between the pairs of subalgebras. Given a maximally abelian subalgebra, one can define, on physical grounds, an associated measure of quantum coherence. We show that the average quantum coherence generated by a unitary map acting on a uniform ensemble of quantum states in the algebra (the so-called coherence generating power of the map) is proportional to the distance between a pair of maximally abelian subalgebras in Md connected by the unitary transformation itself. By embedding the Grassmannian into a projective space, one can pull-back the standard Fubini-Study metric on Md and define in this way novel geometrical measures of quantum coherence generating power. We also briefly discuss the associated differential metric structures.

Many-core graph analytics using accelerated sparse linear algebra routines

NASA Astrophysics Data System (ADS)

Kozacik, Stephen; Paolini, Aaron L.; Fox, Paul; Kelmelis, Eric

2016-05-01

Graph analytics is a key component in identifying emerging trends and threats in many real-world applications. Largescale graph analytics frameworks provide a convenient and highly-scalable platform for developing algorithms to analyze large datasets. Although conceptually scalable, these techniques exhibit poor performance on modern computational hardware. Another model of graph computation has emerged that promises improved performance and scalability by using abstract linear algebra operations as the basis for graph analysis as laid out by the GraphBLAS standard. By using sparse linear algebra as the basis, existing highly efficient algorithms can be adapted to perform computations on the graph. This approach, however, is often less intuitive to graph analytics experts, who are accustomed to vertex-centric APIs such as Giraph, GraphX, and Tinkerpop. We are developing an implementation of the high-level operations supported by these APIs in terms of linear algebra operations. This implementation is be backed by many-core implementations of the fundamental GraphBLAS operations required, and offers the advantages of both the intuitive programming model of a vertex-centric API and the performance of a sparse linear algebra implementation. This technology can reduce the number of nodes required, as well as the run-time for a graph analysis problem, enabling customers to perform more complex analysis with less hardware at lower cost. All of this can be accomplished without the requirement for the customer to make any changes to their analytics code, thanks to the compatibility with existing graph APIs.

Classification of quantum groups and Belavin–Drinfeld cohomologies for orthogonal and symplectic Lie algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kadets, Boris; Karolinsky, Eugene; Pop, Iulia

2016-05-15

In this paper we continue to study Belavin–Drinfeld cohomology introduced in Kadets et al., Commun. Math. Phys. 344(1), 1-24 (2016) and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra #Mathematical Fraktur Small G#. Here we compute Belavin–Drinfeld cohomology for all non-skewsymmetric r-matrices on the Belavin–Drinfeld list for simple Lie algebras of type B, C, and D.

Asymptotic symmetries of Rindler space at the horizon and null infinity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chung, Hyeyoun

2010-08-15

We investigate the asymptotic symmetries of Rindler space at null infinity and at the event horizon using both systematic and ad hoc methods. We find that the approaches that yield infinite-dimensional asymptotic symmetry algebras in the case of anti-de Sitter and flat spaces only give a finite-dimensional algebra for Rindler space at null infinity. We calculate the charges corresponding to these symmetries and confirm that they are finite, conserved, and integrable, and that the algebra of charges gives a representation of the asymptotic symmetry algebra. We also use relaxed boundary conditions to find infinite-dimensional asymptotic symmetry algebras for Rindler spacemore » at null infinity and at the event horizon. We compute the charges corresponding to these symmetries and confirm that they are finite and integrable. We also determine sufficient conditions for the charges to be conserved on-shell, and for the charge algebra to give a representation of the asymptotic symmetry algebra. In all cases, we find that the central extension of the charge algebra is trivial.« less

Quantum teleportation and Birman-Murakami-Wenzl algebra

NASA Astrophysics Data System (ADS)

Zhang, Kun; Zhang, Yong

2017-02-01

In this paper, we investigate the relationship of quantum teleportation in quantum information science and the Birman-Murakami-Wenzl (BMW) algebra in low-dimensional topology. For simplicity, we focus on the two spin-1/2 representation of the BMW algebra, which is generated by both the Temperley-Lieb projector and the Yang-Baxter gate. We describe quantum teleportation using the Temperley-Lieb projector and the Yang-Baxter gate, respectively, and study teleportation-based quantum computation using the Yang-Baxter gate. On the other hand, we exploit the extended Temperley-Lieb diagrammatical approach to clearly show that the tangle relations of the BMW algebra have a natural interpretation of quantum teleportation. Inspired by this interpretation, we construct a general representation of the tangle relations of the BMW algebra and obtain interesting representations of the BMW algebra. Therefore, our research sheds a light on a link between quantum information science and low-dimensional topology.

Line defect Schur indices, Verlinde algebras and U(1) r fixed points

NASA Astrophysics Data System (ADS)

Neitzke, Andrew; Yan, Fei

2017-11-01

Given an N=2 superconformal field theory, we reconsider the Schur index ℐ L ( q) in the presence of a half line defect L. Recently Cordova-Gaiotto-Shao found that ℐ L ( q) admits an expansion in terms of characters of the chiral algebra A introduced by Beem et al., with simple coefficients υ L, β ( q). We report a puzzling new feature of this expansion: the q → 1 limit of the coefficients υ L, β ( q) is linearly related to the vacuum expectation values 〈 L〉 in U(1) r -invariant vacua of the theory compactified on S 1. This relation can be expressed algebraically as a commutative diagram involving three algebras: the algebra generated by line defects, the algebra of functions on U(1) r -invariant vacua, and a Verlindelike algebra associated to A . Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type ( A 1, A 2), ( A 1, A 4), ( A 1, A 6), ( A 1, D 3) and ( A 1, D 5). In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.

Quantum dressing orbits on compact groups

NASA Astrophysics Data System (ADS)

Jurčo, Branislav; Šťovíček, Pavel

1993-02-01

The quantum double is shown to imply the dressing transformation on quantum compact groups and the quantum Iwasawa decompositon in the general case. Quantum dressing orbits are described explicitly as *-algebras. The dual coalgebras consisting of differential operators are related to the quantum Weyl elements. Besides, the differential geometry on a quantum leaf allows a remarkably simple construction of irreducible *-representations of the algebras of quantum functions. Representation spaces then consist of analytic functions on classical phase spaces. These representations are also interpreted in the framework of quantization in the spirit of Berezin applied to symplectic leaves on classical compact groups. Convenient “coherent states” are introduced and a correspondence between classical and quantum observables is given.

Analysis on singular spaces: Lie manifolds and operator algebras

NASA Astrophysics Data System (ADS)

Nistor, Victor

2016-07-01

We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference Noncommutative geometry and applications, Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds-called ;Lie manifolds; -that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here-work that spans over close to two decades-was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.

Generalizations of the classical Yang-Baxter equation and O-operators

NASA Astrophysics Data System (ADS)

Bai, Chengming; Guo, Li; Ni, Xiang

2011-06-01

Tensor solutions (r-matrices) of the classical Yang-Baxter equation (CYBE) in a Lie algebra, obtained as the classical limit of the R-matrix solution of the quantum Yang-Baxter equation, is an important structure appearing in different areas such as integrable systems, symplectic geometry, quantum groups, and quantum field theory. Further study of CYBE led to its interpretation as certain operators, giving rise to the concept of {O}-operators. The O-operators were in turn interpreted as tensor solutions of CYBE by enlarging the Lie algebra [Bai, C., "A unified algebraic approach to the classical Yang-Baxter equation," J. Phys. A: Math. Theor. 40, 11073 (2007)], 10.1088/1751-8113/40/36/007. The purpose of this paper is to extend this study to a more general class of operators that were recently introduced [Bai, C., Guo, L., and Ni, X., "Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras," Commun. Math. Phys. 297, 553 (2010)], 10.1007/s00220-010-0998-7 in the study of Lax pairs in integrable systems. Relations between O-operators, relative differential operators, and Rota-Baxter operators are also discussed.

Non-geometric fluxes, quasi-Hopf twist deformations, and nonassociative quantum mechanics

NASA Astrophysics Data System (ADS)

Mylonas, Dionysios; Schupp, Peter; Szabo, Richard J.

2014-12-01

We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting, nonassociativity is characterised by the associator 3-cocycle which controls non-coassociativity of the quasi-Hopf algebra. We use abelian 2-cocycle twists to construct maps between the dynamical nonassociative star product and a family of associative star products parametrized by constant momentum surfaces in phase space. We define a suitable integration on these nonassociative spaces and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. Using this star product quantization on phase space together with 3-cyclicity, we formulate a consistent version of nonassociative quantum mechanics, in which we calculate the expectation values of area and volume operators, and find coarse-graining of the string background due to the R-flux.

A computing method for sound propagation through a nonuniform jet stream

NASA Technical Reports Server (NTRS)

Padula, S. L.; Liu, C. H.

1974-01-01

The classical formulation of sound propagation through a jet flow was found to be inadequate for computer solutions. Previous investigations selected the phase and amplitude of the acoustic pressure as dependent variables requiring the solution of a system of nonlinear algebraic equations. The nonlinearities complicated both the analysis and the computation. A reformulation of the convective wave equation in terms of a new set of dependent variables is developed with a special emphasis on its suitability for numerical solutions on fast computers. The technique is very attractive because the resulting equations are linear in nonwaving variables. The computer solution to such a linear system of algebraic equations may be obtained by well-defined and direct means which are conservative of computer time and storage space. Typical examples are illustrated and computational results are compared with available numerical and experimental data.

The ALARM Experiment

ERIC Educational Resources Information Center

Gerhardt, Ira

2015-01-01

An experiment was conducted over three recent semesters of an introductory calculus course to test whether it was possible to quantify the effect that difficulty with basic algebraic and arithmetic computation had on individual performance. Points lost during the term were classified as being due to either algebraic and arithmetic mistakes…

Applications of Maple To Algebraic Cryptography.

ERIC Educational Resources Information Center

Sigmon, Neil P.

1997-01-01

Demonstrates the use of technology to enhance the appreciation of applications involving abstract algebra. The symbolic manipulator Maple can perform computations required for a linear cryptosystem. One major benefit of this process is that students can encipher and decipher messages using a linear cryptosystem without becoming confused and…

Teaching of real numbers by using the Archimedes-Cantor approach and computer algebra systems

NASA Astrophysics Data System (ADS)

Vorob'ev, Evgenii M.

2015-11-01

Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of CAS. In the case of real numbers, the Archimedes-Cantor approach satisfies this requirement. The name of Archimedes brings back the exhaustion method. Cantor's name reminds us of the use of Cauchy rational sequences to represent real numbers. The usage of CAS with the Archimedes-Cantor approach enables the discussion of various representations of real numbers such as graphical, decimal, approximate decimal with precision estimates, and representation as points on a straight line. Exercises with numbers such as e, π, the golden ratio ϕ, and algebraic irrational numbers can help students better understand the real numbers. The Archimedes-Cantor approach also reveals a deep and close relationship between real numbers and continuity, in particular the continuity of functions.

Non-local geometry inside Lifshitz horizon

NASA Astrophysics Data System (ADS)

Hu, Qi; Lee, Sung-Sik

2017-07-01

Based on the quantum renormalization group, we derive the bulk geometry that emerges in the holographic dual of the fermionic U( N ) vector model at a nonzero charge density. The obstruction that prohibits the metallic state from being smoothly deformable to the direct product state under the renormalization group flow gives rise to a horizon at a finite radial coordinate in the bulk. The region outside the horizon is described by the Lifshitz geometry with a higher-spin hair determined by microscopic details of the boundary theory. On the other hand, the interior of the horizon is not described by any Riemannian manifold, as it exhibits an algebraic non-locality. The non-local structure inside the horizon carries the information on the shape of the filled Fermi sea.

Cryptographic Properties of Monotone Boolean Functions

DTIC Science & Technology

2016-01-01

Algebraic attacks on stream ciphers with linear feedback, in: Advances in Cryptology (Eurocrypt 2003), Lecture Notes in Comput. Sci. 2656, Springer, Berlin...spectrum, algebraic immu- nity MSC 2010: 06E30, 94C10, 94A60, 11T71, 05E99 || Communicated by: Carlo Blundo 1 Introduction Let F 2 be the prime eld of...7]. For the reader’s convenience, we recall some basic notions below. Any f ∈ Bn can be expressed in algebraic normal form (ANF) as f(x 1 , x 2

Bisimulation equivalence of differential-algebraic systems

NASA Astrophysics Data System (ADS)

Megawati, Noorma Yulia; Schaft, Arjan van der

2018-01-01

In this paper, the notion of bisimulation relation for linear input-state-output systems is extended to general linear differential-algebraic (DAE) systems. Geometric control theory is used to derive a linear-algebraic characterisation of bisimulation relations, and an algorithm for computing the maximal bisimulation relation between two linear DAE systems. The general definition is specialised to the case where the matrix pencil sE - A is regular. Furthermore, by developing a one-sided version of bisimulation, characterisations of simulation and abstraction are obtained.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chung, Won Sang, E-mail: mimip4444@hanmail.net; Hounkonnou, Mahouton Norbert, E-mail: norbert.hounkonnou@cipma.uac.bj; Arjika, Sama, E-mail: rjksama2008@gmail.com

In this paper, we propose a full characterization of a generalized q-deformed Tamm-Dancoff oscillator algebra and investigate its main mathematical and physical properties. Specifically, we study its various representations and find the condition satisfied by the deformed q-number to define the algebra structure function. Particular Fock spaces involving finite and infinite dimensions are examined. A deformed calculus is performed as well as a coordinate realization for this algebra. A relevant example is exhibited. Associated coherent states are constructed. Finally, some thermodynamics aspects are computed and discussed.

Motivating the Concept of Eigenvectors via Cryptography

ERIC Educational Resources Information Center

Siap, Irfan

2008-01-01

New methods of teaching linear algebra in the undergraduate curriculum have attracted much interest lately. Most of this work is focused on evaluating and discussing the integration of special computer software into the Linear Algebra curriculum. In this article, I discuss my approach on introducing the concept of eigenvectors and eigenvalues,…

Solving Optimization Problems with Spreadsheets

ERIC Educational Resources Information Center

Beigie, Darin

2017-01-01

Spreadsheets provide a rich setting for first-year algebra students to solve problems. Individual spreadsheet cells play the role of variables, and creating algebraic expressions for a spreadsheet to perform a task allows students to achieve a glimpse of how mathematics is used to program a computer and solve problems. Classic optimization…