NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
Learning Activity Package, Algebra-Trigonometry.
ERIC Educational Resources Information Center
Holland, Bill
A series of ten teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, the units cover logic; absolute value, inequalities, exponents, and complex numbers; functions; higher degree equations and the derivative; the trigonometric function; graphs and applications of the trigonometric functions; sequences and…
An Improvement on SSA Congruence for Geometry and Trigonometry.
ERIC Educational Resources Information Center
Yeshurun, Shraga; Kay, David C.
1983-01-01
Three ideas are explored: (1) an improvement of the SSA congruence theorem for trigonometry; (2) a discussion of the failure of SSA in spherical geometry; and (3) an extension of SSA to spherical geometry and hyperbolic geometry. (MNS)
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…
Enhancing Conceptual Understanding of Trigonometry Using Earth Geometry and the Great Circle
ERIC Educational Resources Information Center
Wongapiwatkul, Pimpalak; Laosinchai, Parames; Panijpan, Bhinyo
2011-01-01
Trigonometry is an integral part of the draft for the Senior Secondary Australian National Curriculum for Mathematics, as it is a topic in Unit 2 of both Specialist Mathematics and Mathematics Methods, and a reviewing topic in Unit 1, Topic 3: Measurement and Geometry of General Mathematics. However, learning trigonometric ideas is difficult for…
ERIC Educational Resources Information Center
Vaninsky, Alexander
2011-01-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos--satisfying an axiom sin[superscript 2] + cos[superscript 2] = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two…
Math Sense: Algebra and Geometry.
ERIC Educational Resources Information Center
Howett, Jerry
This book is designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four strands: (1) skill lessons…
Investigating the purpose of trigonometry in the modern sciences
NASA Astrophysics Data System (ADS)
Hertel, Joshua T.
This dissertation reports the results of a qualitative research project that aimed to develop a research-based perspective on the purpose of trigonometry in the modern sciences. The investigation was guided by three objectives. First, the study sought to identify the purpose of trigonometry as described by educators and high school textbooks. Second, the research investigated the perspectives these sources held about definitions of the trigonometric functions. Third, the investigation examined the potential benefits and drawbacks of a line-segment definition of the trigonometric functions. The study followed a grounded theory methodology with data collection and analysis intertwined. Participants included faculty from two large Midwestern research universities, high school teachers, and authors of standards documents. Textbooks were drawn from introductory algebra, geometry, advanced algebra, precalculus, and calculus texts. Data collected included surveys, interviews, and textbook excerpts. Analysis used the constant comparative method (Corbin & Strauss, 2008; Glaser & Strauss, 2006/1967). Analysis resulted in the emergence of a grounded theory, the tensions of trigonometry, which described three interrelated themes within the data: definition, application, and role. Two ideas emerged that connected the tensions of trigonometry, the regions of interaction, which described the interplay between the three tensions, and the idealized dichotomy of trigonometry education, which outlined opposing perspectives on trigonometry: trigonometry for all and trigonometry for some. The grounded theory outlines a range of competing purposes for trigonometry in the modern sciences. It suggests that educators are engaged in a process of continual negotiation that results in the formation of a localized purpose of trigonometry. The benefits and drawbacks of different definitions are not based on mathematical sophistication, but are situational. Furthermore, the theory suggests that
ERIC Educational Resources Information Center
Gur, Hulya
2009-01-01
Background: Trigonometry is an area of mathematics that students believe to be particularly difficult and abstract compared with the other subjects of mathematics. Trigonometry is often introduced early in year 8 with most textbooks traditionally starting with naming sides of right-angled triangles. Students need to see and understand why their…
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…
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…
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.
Computational algebraic geometry of epidemic models
NASA Astrophysics Data System (ADS)
Rodríguez Vega, Martín.
2014-06-01
Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.
ERIC Educational Resources Information Center
Goetz, Albert
2016-01-01
"Media Clips" appears in every issue of "Mathematics Teacher," offering readers contemporary, authentic applications of quantitative reasoning based on print or electronic media. Based on "In All the Light We Cannot See" (2014), by Anthony Doerr, this article provides a brief trigonometry problem that was solved by…
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
Classical versus Computer Algebra Methods in Elementary Geometry
ERIC Educational Resources Information Center
Pech, Pavel
2005-01-01
Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…
Differential geometry on Hopf algebras and quantum groups
Watts, Paul
1994-12-15
The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined.
The Bell states in noncommutative algebraic geometry
NASA Astrophysics Data System (ADS)
Beil, Charlie
2014-10-01
We introduce new mathematical aspects of the Bell states using matrix factorizations, non-noetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial p consists of two matrices ϕ1, ϕ2 such that ϕ1ϕ2 = ϕ2ϕ1 = p id. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on non-noetherian singularities. Moreover, we find that the collapse of a Bell state is intimately related to the representation theory of the noncommutative blowup along its singular support. This presents an exchange in geometry: the nonlocal commutative spacetime of the entangled state emerges from an underlying local noncommutative spacetime.
Algebraic geometry realization of quantum Hall soliton
NASA Astrophysics Data System (ADS)
Abounasr, R.; Ait Ben Haddou, M.; El Rhalami, A.; Saidi, E. H.
2005-02-01
Using the Iqbal-Netzike-Vafa dictionary giving the correspondence between the H2 homology of del Pezzo surfaces and p-branes, we develop a way to approach the system of brane bounds in M-theory on S1. We first review the structure of 10-dimensional quantum Hall soliton (QHS) from the view of M-theory on S1. Then, we show how the D0 dissolution in D2-brane is realized in M-theory language and derive the p-brane constraint equations used to define appropriately the QHS. Finally, we build an algebraic geometry realization of the QHS in type IIA superstring and show how to get its type IIB dual. Other aspects are also discussed.
From geometry to algebra: the Euclidean way with technology
NASA Astrophysics Data System (ADS)
Ferrarello, Daniela; Flavia Mammana, Maria; Pennisi, Mario
2016-05-01
In this paper, we present the results of an experimental classroom activity, history-based with a phylogenetic approach, to achieve algebra properties through geometry. In particular, we used Euclidean propositions, processed them by a dynamic geometry system and translate them into algebraic special products.
Elements of Mathematics, Book 9: Linear Algebra and Geometry with Trigonometry.
ERIC Educational Resources Information Center
Exner, Robert; And Others
One of 12 books developed for use with the core material (Book O) of the Elements of Mathematics Program, this text covers material well beyond the scope of the usual secondary mathematics sequences. These materials are designed for highly motivated students with strong verbal abilities; mathematical theories and ideas are developed through…
Algebraic grid generation for complex geometries
NASA Technical Reports Server (NTRS)
Shih, T. I.-P.; Bailey, R. T.; Nguyen, H. L.; Roelke, R. J.
1991-01-01
An efficient computer program called GRID2D/3D has been developed to generate single and composite grid systems within geometrically complex two- and three-dimensional (2D and 3D) spatial domains that can deform with time. GRID2D/3D generates single grid systems by using algebraic grid generation methods based on transfinite interpolation. The distribution of grid points within the spatial domain is controlled by stretching functions and grid lines can intersect boundaries of the spatial domain orthogonally. GRID2D/3D generates composite grid systems by patching together two or more single grid systems. The patching can be discontinuous or continuous. For 2D spatial domains the boundary curves are constructed by using either cubic or tension spline interpolation. For 3D spatial domains the boundary surfaces are constructed by using a new technique, developed in this study, referred to as 3D bidirectional Hermite interpolation.
From string theory to algebraic geometry and back
Brinzanescu, Vasile
2011-02-10
We describe some facts in physics which go up to the modern string theory and the related concepts in algebraic geometry. Then we present some recent results on moduli-spaces of vector bundles on non-Kaehler Calabi-Yau 3-folds and their consequences for heterotic string theory.
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.
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…
Math Sense: Algebra and Geometry. Teacher's Resource Guide.
ERIC Educational Resources Information Center
Phillips, Jan; Osmus, Kathy
This book is a teacher's resource guide designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four…
Multilinear Computing and Multilinear Algebraic Geometry
2016-08-10
satisfying a polynomial growth condi- tion: for such a given function, computation of its Fenchel dual/conjugate is polynomial-time reducible to...computation of the given function. Hence the computation of a norm or a convex function of polynomial growth is NP-hard if and only if the computation of... growth and that of its Fenchel dual. This paper has been submitted and is available as a preprint (see [19] in Section 4) Semialgebraic geometry of
Reverse engineering: algebraic boundary representations to constructive solid geometry.
Buchele, S. F.; Ellingson, W. A.
1997-12-17
Recent advances in reverse engineering have focused on recovering a boundary representation (b-rep) of an object, often for integration with rapid prototyping. This boundary representation may be a 3-D point cloud, a triangulation of points, or piecewise algebraic or parametric surfaces. This paper presents work in progress to develop an algorithm to extend the current state of the art in reverse engineering of mechanical parts. This algorithm will take algebraic surface representations as input and will produce a constructive solid geometry (CSG) description that uses solid primitives such as rectangular block, pyramid, sphere, cylinder, and cone. The proposed algorithm will automatically generate a CSG solid model of a part given its algebraic b-rep, thus allowing direct input into a CAD system and subsequent CSG model generation.
Prime factorization using quantum annealing and computational algebraic geometry
Dridi, Raouf; Alghassi, Hedayat
2017-01-01
We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gröbner bases. We present a novel autonomous algorithm which combines the two approaches and leads to the factorization of all bi-primes up to just over 200000, the largest number factored to date using a quantum processor. We also explain how Gröbner bases can be used to reduce the degree of Hamiltonians. PMID:28220854
Prime factorization using quantum annealing and computational algebraic geometry
NASA Astrophysics Data System (ADS)
Dridi, Raouf; Alghassi, Hedayat
2017-02-01
We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gröbner bases. We present a novel autonomous algorithm which combines the two approaches and leads to the factorization of all bi-primes up to just over 200000, the largest number factored to date using a quantum processor. We also explain how Gröbner bases can be used to reduce the degree of Hamiltonians.
Trigonometry, Including Snell's Theorem.
ERIC Educational Resources Information Center
Kent, David
1980-01-01
Aspects of the instruction of trigonometry in secondary school mathematics are reviewed. Portions of this document cover basic introductions, a student-developed theorem, the cosine rule, inverse functions, and a sample outdoor activity. (MP)
Trigonometry with Year 8: Part 3
ERIC Educational Resources Information Center
Steer, Jessica; de Vila, Maria Antonieta; Eaton, James
2009-01-01
This final article focuses in particular on the engagement of year 8 students who were taught trigonometry using dynamic geometry software ("Geometer's SketchPad"), as outlined in "MT214" and with resources from the ATM website. The project was implemented in three different classrooms in two different, multiracial,…
Historical Reflections on Teaching Trigonometry
ERIC Educational Resources Information Center
Bressoud, David M.
2010-01-01
The study of trigonometry suffers from a basic dichotomy that presents a serious obstacle to many students. On the one hand, there is triangle trigonometry, in which angles are commonly measured in degrees and trigonometric functions are defined as ratios of sides of a right-angled triangle. On the other hand, there is circle trigonometry, in…
ERIC Educational Resources Information Center
Shafiq, Hasan
2013-01-01
Over the last few decades school accountability for student performance has become an issue at the forefront of education. The federal No Child Left Behind Act of 2001 (NCLB) and various regulations by individual states have set standards for student performance at both the district and individual public and charter school levels, and certain…
The Path from Algebra I to Geometry to Algebra II to Developmental Studies Mathematics.
ERIC Educational Resources Information Center
Blackburn, Katie
Nearly 48 percent of the students in Developmental Studies Mathematics courses at a major southern university during the fall of 1983 had taken college preparatory mathematics courses in high school. Some had even taken trigonometry, pre-calculus, or calculus. Yet, they were placed in university developmental studies mathematics courses because…
Geometric algebra and information geometry for quantum computational software
NASA Astrophysics Data System (ADS)
Cafaro, Carlo
2017-03-01
The art of quantum algorithm design is highly nontrivial. Grover's search algorithm constitutes a masterpiece of quantum computational software. In this article, we use methods of geometric algebra (GA) and information geometry (IG) to enhance the algebraic efficiency and the geometrical significance of the digital and analog representations of Grover's algorithm, respectively. Specifically, GA is used to describe the Grover iterate and the discretized iterative procedure that exploits quantum interference to amplify the probability amplitude of the target-state before measuring the query register. The transition from digital to analog descriptions occurs via Stone's theorem which relates the (unitary) Grover iterate to a suitable (Hermitian) Hamiltonian that controls Schrodinger's quantum mechanical evolution of a quantum state towards the target state. Once the discrete-to-continuos transition is completed, IG is used to interpret Grover's iterative procedure as a geodesic path on the manifold of the parametric density operators of pure quantum states constructed from the continuous approximation of the parametric quantum output state in Grover's algorithm. Finally, we discuss the dissipationless nature of quantum computing, recover the quadratic speedup relation, and identify the superfluity of the Walsh-Hadamard operation from an IG perspective with emphasis on statistical mechanical considerations.
Integrating the Study of Trigonometry, Vectors, and Force Through Modeling.
ERIC Educational Resources Information Center
Doerr, Helen M.
1996-01-01
Investigates the construction of understanding of the motion of an object down an inclined plane which takes place through the process of model building in an integrated algebra, trigonometry, and physics class. Discusses four major themes related to student learning through modeling that emerged from the results. Discusses implications for…
Integrand Reduction Reloaded: Algebraic Geometry and Finite Fields
NASA Astrophysics Data System (ADS)
Sameshima, Ray D.; Ferroglia, Andrea; Ossola, Giovanni
2017-01-01
The evaluation of scattering amplitudes in quantum field theory allows us to compare the phenomenological prediction of particle theory with the measurement at collider experiments. The study of scattering amplitudes, in terms of their symmetries and analytic properties, provides a theoretical framework to develop techniques and efficient algorithms for the evaluation of physical cross sections and differential distributions. Tree-level calculations have been known for a long time. Loop amplitudes, which are needed to reduce the theoretical uncertainty, are more challenging since they involve a large number of Feynman diagrams, expressed as integrals of rational functions. At one-loop, the problem has been solved thanks to the combined effect of integrand reduction, such as the OPP method, and unitarity. However, plenty of work is still needed at higher orders, starting with the two-loop case. Recently, integrand reduction has been revisited using algebraic geometry. In this presentation, we review the salient features of integrand reduction for dimensionally regulated Feynman integrals, and describe an interesting technique for their reduction based on multivariate polynomial division. We also show a novel approach to improve its efficiency by introducing finite fields. Supported in part by the National Science Foundation under Grant PHY-1417354.
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.
Fibonacci Numbers and an Area Puzzle: Connecting Geometry and Algebra in the Mathematics Classroom.
ERIC Educational Resources Information Center
Sullivan, Mary M.; Panasuk, Regina M.
1997-01-01
Presents a mathematical puzzle that asks about "missing" area and leads to an exploration of the Fibonacci sequence as well as genuine inquiry in plane geometry connected to algebra. Discusses the inquiry, the concepts, the solution, and an extension that deepens all students' understanding of the connections between algebra and…
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.
[Thomas Fincke and trigonometry].
Schönbeck, Jürgen
2004-01-01
Thomas Fincke (January 6th, 1561 - April 24th, 1650), born in Flensburg (Germany), was one of the very most important and significant scientists in Denmark during the seventeenth century, a mathematician and astrologer and physician in the beginning of modern science, a representative of humanism and an influentual academic organizer. He studied in Strasbourg (since 1577) and Padua (since 1583) and received his M.D. in Basel (1587), he practised as a physician throughtout his life (since 1587 or 1590) and became a professor at Copenhagen (1591). But he was best known because of his Geometriae rotundi libri XIIII (1583), a famous book on plane and spherical trigonometry, based not on Euclid but on Petrus Ramus. In this influentual work, in which Fincke introduced the terms tangent and secant and probable first noticed the Law of Tangents and the so-called Newton-Oppel-Mauduit-Simpson-Mollweide-Gauss-formula, he showed himself to be ,,abreast of the mathematics of his time".
Spectral properties of sums of Hermitian matrices and algebraic geometry
NASA Astrophysics Data System (ADS)
Chau Huu-Tai, P.; Van Isacker, P.
2016-04-01
It is shown that all the eigenvectors of a sum of Hermitian matrices belong to the same algebraic variety. A polynomial system characterizing this variety is given and a set of nonlinear equations is derived which allows the construction of the variety. Moreover, in some specific cases, explicit expressions for the eigenvectors and eigenvalues can be obtained. Explicit solutions of selected models are also derived.
Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry
NASA Astrophysics Data System (ADS)
Eynard, B.; Marchal, O.
2009-03-01
In this article, we define a non-commutative deformation of the ``symplectic invariants'' (introduced in [13]) of an algebraic hyperelliptic plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some algebraic geometry quantities. In particular our non-commutative Bergman kernel satisfies a Rauch variational formula. Those non-commutative invariants are inspired from the large N expansion of formal non-hermitian matrix models. Thus they are expected to be related to the enumeration problem of discrete non-orientable surfaces of arbitrary topologies.
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.
The Circle Approach to Trigonometry
ERIC Educational Resources Information Center
Moore, Kevin c.; LaForest, Kevin R.
2014-01-01
How do students think about an angle measure of ninety degrees? How do they think about ratios and values on the unit circle? How might angle measure be used to connect right-triangle trigonometry and circular functions? And why might asking these questions be important when introducing trigonometric functions to students? When teaching…
Mollweide's Formula in Teaching Trigonometry
ERIC Educational Resources Information Center
Karjanto, Natanael
2011-01-01
Trigonometry is one of the topics in mathematics that the students in both high school and pre-undergraduate levels need to learn. Generally, the topic covers trigonometric functions, trigonometric equations, trigonometric identities and solving oblique triangles using the Laws of Sines and Cosines. However, when solving the oblique triangles,…
NASA Astrophysics Data System (ADS)
Orantin, N.
2007-09-01
The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links and extend them beyond the matrix models, following my work's evolution. First, I take care to define properly the hermitian 2 matrix model which gives rise to generating functions of discrete surfaces equipped with a spin structure. Then, I show how to compute all the terms in the topological expansion of any observable by using algebraic geometry tools. They are obtained as differential forms on an algebraic curve associated to the model: the spectral curve. In a second part, I show how to define such differentials on any algebraic curve even if it does not come from a matrix model. I then study their numerous symmetry properties under deformations of the algebraic curve. In particular, I show that these objects coincide with the topological expansion of the observable of a matrix model if the algebraic curve is the spectral curve of this model. Finally, I show that fine tuning the parameters ensure that these objects can be promoted to modular invariants and satisfy the holomorphic anomaly equation of the Kodaira-Spencer theory. This gives a new hint that the Dijkgraaf-Vafa conjecture is correct.
Geometry of moduli stacks of (k , l) -stable vector bundles over algebraic curves
NASA Astrophysics Data System (ADS)
Mata-Gutiérrez, O.; Neumann, Frank
2017-01-01
We study the geometry of the moduli stack of vector bundles of fixed rank and degree over an algebraic curve by introducing a filtration made of open substacks build from (k , l) -stable vector bundles. The concept of (k , l) -stability was introduced by Narasimhan and Ramanan to study the geometry of the coarse moduli space of stable bundles. We will exhibit the stacky picture and analyse the geometric and cohomological properties of the moduli stacks of (k , l) -stable vector bundles. For particular pairs (k , l) of integers we also show that these moduli stacks admit coarse moduli spaces and we discuss their interplay.
Finite Trigonometry: A Resource for Teachers.
ERIC Educational Resources Information Center
Malcom, Paul Scott
This investigation extends a 25-point geometric system for defining a 25-point trigonometry whose properties are analogous to those of the trigonometry of the Euclidean plane. These properties include definitions of trigonometric functions arising from ratios of sides of right triangles, the relations of elements of a given triangle through the…
The Trigonometry of Sensed Angles -- An Analogue to the Ciruclar Functions.
ERIC Educational Resources Information Center
Szabo, Steven
It is extremely difficult to extend in a natural way the concepts developed in high school geometry to motivate the concepts of trigonometry. To solve this difficultry, the author defines "sensed angles" as ordered pairs of rays and makes appropriate definitions which yield the familiar angle properties of plane geometry. He then shows how this…
Numerical algebraic geometry for model selection and its application to the life sciences
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
Axion experiments to algebraic geometry: Testing quantum gravity via the Weak Gravity Conjecture
NASA Astrophysics Data System (ADS)
Heidenreich, Ben; Reece, Matthew; Rudelius, Tom
2016-06-01
Common features of known quantum gravity theories may hint at the general nature of quantum gravity. The absence of continuous global symmetries is one such feature. This inspired the Weak Gravity Conjecture, which bounds masses of charged particles. We propose the Lattice Weak Gravity Conjecture, which further requires the existence of an infinite tower of particles of all possible charges under both abelian and nonabelian gauge groups and directly implies a cutoff for quantum field theory. It holds in a wide variety of string theory examples and has testable consequences for the real world and for pure mathematics. We sketch some implications of these ideas for models of inflation, for the QCD axion (and LIGO), for conformal field theory, and for algebraic geometry.
Numerical algebraic geometry for model selection and its application to the life sciences.
Gross, Elizabeth; Davis, Brent; Ho, Kenneth L; Bates, Daniel J; Harrington, Heather A
2016-10-01
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.
Trigonometry, A Tentative Guide Prepared for Use with the Text Plane Trigonometry with Tables.
ERIC Educational Resources Information Center
Brant, Vincent
This teacher's guide for a semester course in trigonometry is prepared for use with the text "Plane Trigonometry with Tables" by E. R. Heineman. Included is a daily schedule of topics for discussion and homework assignments. The scope of each lesson and teaching suggestions are provided. The content for the course includes trigonometric functions,…
Boolean Algebra. Geometry Module for Use in a Mathematics Laboratory Setting.
ERIC Educational Resources Information Center
Brotherton, Sheila; And Others
This module is recommended as an honors unit to follow a unit on logic. There are four basic parts: (1) What is a Boolean Algebra; (2) Using Boolean Algebra to Prove Theorems; (3) Using Boolean Algebra to Simplify Logical Statements; and (4) Circuit Problems with Logic and Boolean Algebra. Of these, sections 1, 2, and 3 are primarily written…
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…
Samoylovich, Mikhail; Talis, Alexander
2014-03-01
The chain of algebraic geometry and topology constructions is mapped on a structural level that allows one to single out a special class of discrete helicoidal structures. A structure that belongs to this class is locally periodic, topologically stable in three-dimensional Euclidean space and corresponds to the bifurcation domain. Singular points of its bounding minimal surface are related by transformations determined by symmetries of the second coordination sphere of the eight-dimensional crystallographic lattice E8. These points represent cluster vertices, whose helicoid joining determines the topology and structural parameters of linear biopolymers. In particular, structural parameters of the α-helix are determined by the seven-vertex face-to-face joining of tetrahedra with the E8 non-integer helical axis 40/11 having a rotation angle of 99°, and the development of its surface coincides with the cylindrical development of the α-helix. Also, packing models have been created which determine the topology of the A, B and Z forms of DNA.
Trigonometry with Year 8: Part 1
ERIC Educational Resources Information Center
Steer, Jessica; de Vila, Maria Antioneta; Eaton, James
2009-01-01
The authors explore the teaching of trigonometry using a method developed by Jeremy Burke of Kings College. A series of lessons was planned using an approach which looks at moving from a mathematical description of the topic, to a sequence plan, to a set of activities, which students can use to help them come to understand the topic. This is…
Trigonometry 2, Mathematics: 5219.12.
ERIC Educational Resources Information Center
Johnson, David J.
This booklet is the second of a two-part sequence of minimum content for trigonometry. It includes sum, difference, double-angle, and half-angle formulas; Law of Sines; Law of Cosines; inverse trigonometric functions; polar coordinates; and DeMoivre's Theorem. Goals, performance objectives for each unit, a course outline, references to…
A Module in Spherical Trigonometry.
ERIC Educational Resources Information Center
Congleton, C. A.; Broome, L. E.
1980-01-01
This module, designed for use at the high school level as a four- to eight-hour topic, includes: the geometry of a sphere, the coordinate system used to describe points on the earth's surface, parallel and meridian sailing, and the solution of right spherical triangles. (Author/MK)
The Trigonometry of Twistors and Elementary Particles
Gustafson, Karl
2009-03-10
A new trigonometry for twistors is presented. The operator-theoretic maximum twistor turning angle is shown to be related to the space-time geometric angle within the light cone. The corresponding maximally turned twistor antieigenvectors are calculated and interpretted. The two weak interaction CP eigenvectors of neutral kaons are shown to be exactly the two strong interaction strangeness antieigenvectors. Quark mixing is seen trigonometrically. 't Hooft's microcosmos model is connected to the theories of normal degree and complex dynamics.
ERIC Educational Resources Information Center
Schumann, Heinz; Green, David
2000-01-01
Discusses software for geometric construction, measurement, and calculation, and software for numerical calculation and symbolic analysis that allows for new approaches to the solution of geometric problems. Illustrates these computer-aided graphical, numerical, and algebraic methods of solution and discusses examples using the appropriate choice…
Math 1715/1613 (PIPI): Algebra and Trigonometry.
ERIC Educational Resources Information Center
Blose, Murray M.; And Others
This study guide, designed for use at Oklahoma State University, contains lists of activities for students to perform based on the "mastery of learning" concept. The activities include readings, problems, self evaluations, and assessment tasks. The units included are: Functions, Exponential and Logarithmic Functions, Trigonometric…
Math 3011--College Algebra and Trigonometry. Course Outline.
ERIC Educational Resources Information Center
New York Inst. of Tech., Old Westbury.
This document contains the course syllabus and 12 independent practice modules for a college level mathematics course designed to provide the necessary foundation for success in calculus, develop logical thinking skills, and enhance analytic skills through problem solving. Topics include relations and functions; inequalities; complex numbers;…
Investigating the Purpose of Trigonometry in the Modern Sciences
ERIC Educational Resources Information Center
Hertel, Joshua T.
2013-01-01
This dissertation reports the results of a qualitative research project that aimed to develop a research-based perspective on the purpose of trigonometry in the modern sciences. The investigation was guided by three objectives. First, the study sought to identify the purpose of trigonometry as described by educators and high school textbooks.…
Aslaksen, H.
1988-01-01
In this paper we will study triangles in SU(3). The orbit space of congruence classes of triangles in SU(3) has dimension 8. Each corner is made up of a pair of tangent vectors (X,Y), and we consider the 8 functions trX{sup 2}, i trX{sup 3}, trY{sup 2}, i trY{sup 3}, trXY, i trY{sup 2}Y, i trXY{sup 2}, trX{sup 2}Y{sup 2} which are invariant under the full isometry group of SU(3). We show that these 8 corner invariants determine the isometry class of the triangle. We give relations (laws of trigonometry) between the invariants at the different corners, enabling us to determine the invariants at the remaining corners, including the values of the remaining side and angles, if we know one set of corner invariants. The invariants that only depend on one tangent vector we will call side invariants, while those that depend on two tangent vectors will be called angular invariants. For each triangle we then have 6 side invariants and 12 angular invariants. Hence we need 18 {minus} 8 = 10 laws of trigonometry. The basic tool for deriving these laws is a formula expressing tr(exp X exp Y) in terms of the corner invariants.
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.
Methods of Power Geometry in Asymptotic Analysis of Solutions to Algebraic or Differential Equations
NASA Astrophysics Data System (ADS)
Goryuchkina, Irina
2010-06-01
Here we present some basic ideas of the plane Power Geometry to study asymptotic behavior of solutions to differential equations. We consider two examples for demonstration of these methods and two applications the methods.
NASA Astrophysics Data System (ADS)
Chekhov, L. O.; Eynard, B.; Marchal, O.
2011-02-01
We construct the solution of the loop equations of the β-ensemble model in a form analogous to the solution in the case of the Hermitian matrices β = 1. The solution for β = 1 is expressed in terms of the algebraic spectral curve given by y2 = U(x). The spectral curve for arbitrary β converts into the Schrödinger equation (ħ∂)2 - U(x) ψ(x) = 0, where ħ ∝ {{( {{{sqrt β - 1} {sqrt β }}} {{{sqrt β - 1} {sqrt β }}} {sqrt β }}} )} N}}. The basic ingredients of the method based on the algebraic solution retain their meaning, but we use an alternative approach to construct a solution of the loop equations in which the resolvents are given separately in each sector. Although this approach turns out to be more involved technically, it allows consistently defining the B-cycle structure for constructing the quantum algebraic curve (a D-module of the form y2 - U(x), where [y, x] = ħ) and explicitly writing the correlation functions and the corresponding symplectic invariants Fh or the terms of the free energy in an 1/N2-expansion at arbitrary ħ. The set of "flat" coordinates includes the potential times tk and the occupation numbers tilde \\varepsilon _α . We define and investigate the properties of the A- and B-cycles, forms of the first, second, and third kinds, and the Riemann bilinear identities. These identities allow finding the singular part of mathcal{F}_0 , which depends only on tilde \\varepsilon _α.
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…
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…
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…
ERIC Educational Resources Information Center
Ruthven, Kenneth
2008-01-01
This article examines three important facets of the incorporation of new technologies into educational practice, focusing on emergent usages of the mathematical tools of computer algebra and dynamic geometry. First, it illustrates the interpretative flexibility of these tools, highlighting important differences in ways of conceptualizing and…
Trigonometry--Mathematical Curricular Areas and Instructional Computing.
ERIC Educational Resources Information Center
Frandsen, Henry
1981-01-01
The computer has enormous potential for enhancing the secondary trigonometry curriculum. Areas that can be improved through use of the computer include: solutions of triangles, inductive investigations of functions, graphs of functions, and solution of trigonometric identities. (JN)
Trigonometry 1, Mathematics: 5219.11.
ERIC Educational Resources Information Center
Johnson, David J.
The first of a two-part sequence for the student who has had difficulty in second-year algebra, this booklet covers definitions and measurement of angles (in degrees and radians), the trigonometric functions, solving trigonometric equations and graphing functions, identities, and computation with base ten logarithms. Overall goals for the course…
Trigonometry and Advanced Math. De Soto Parish Curriculum Guide.
ERIC Educational Resources Information Center
DeSoto Parish School Board, Mansfield, LA.
The primary aim of this guide is to aid teachers in planning and preparing a senior high school mathematics course for students preparing for college work. It is divided into separate one-semester courses of seven chapters each. The first-semester course consists of a traditional approach to the introduction of trigonometry and trigonometric…
NASA Astrophysics Data System (ADS)
Dimitrov, B. G.
2010-02-01
On the base of the distinction between covariant and contravariant metric tensor components, a new (multivariable) cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been derived and parametrized with complicated non - elliptic functions, depending on the (elliptic) Weierstrass function and its derivative. This is different from standard algebraic geometry, where only two-dimensional cubic equations are parametrized with elliptic functions and not multivariable ones. Physical applications of the approach have been considered in reference to theories with extra dimensions. The s.c. "length function" l(x) has been introduced and found as a solution of quasilinear differential equations in partial derivatives for two different cases of "compactification + rescaling" and "rescaling + compactification". New physically important relations (inequalities) between the parameters in the action are established, which cannot be derived in the case $l=1$ of the standard gravitational theory, but should be fulfilled also for that case.
Learning Activity Package, Algebra 124, LAPs 46-55.
ERIC Educational Resources Information Center
Holland, Bill
A series of 10 teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, these units cover absolute value, inequalities, exponents, radicals, and complex numbers; functions; higher degree equations and the derivative; the trigonometric functions; graphs and applications of the trigonometric functions; sequences and…
Pseudo-Riemannian Novikov algebras
NASA Astrophysics Data System (ADS)
Chen, Zhiqi; Zhu, Fuhai
2008-08-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.
An algebraic approach to the study of weakly excited states for a condensate in a ring geometry
NASA Astrophysics Data System (ADS)
Buonsante, P.; Franco, R.; Penna, V.
2005-09-01
We determine the low-energy spectrum and the eigenstates for a two-bosonic mode nonlinear model by applying the Inönü-Wigner contraction method to the Hamiltonian algebra. This model is known to well represent a Bose-Einstein condensate rotating in a thin torus endowed with two angular-momentum modes as well as a condensate in a double-well potential characterized by two space modes. We consider such a model in the presence of both an attractive and a repulsive boson interaction and investigate regimes corresponding to different values of the inter-mode tunnelling parameter. We show that the results ensuing from our approach are in many cases extremely satisfactory. To this end, we compare our results with the ground state obtained both numerically and within a standard semiclassical approximation based on su(2) coherent states.
The Effect of the Math Emporium Instructional Method on Students' Performance in College Algebra
ERIC Educational Resources Information Center
Cousins-Cooper, Kathy; Staley, Katrina N.; Kim, Seongtae; Luke, Nicholas S.
2017-01-01
This study aims to investigate the effectiveness of the Emporium instructional method in a course of college algebra and trigonometry by comparing to the traditional lecture method. The math emporium method is a nontraditional instructional method of learning math that has been implemented at several universities with much success and has been…
NASA Astrophysics Data System (ADS)
Kalanov, Temur Z.
2015-04-01
Analysis of the foundations of standard trigonometry is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. It is shown that the foundations of trigonometry contradict to the principles of system approach and contain formal-logical errors. The principal logical error is that the definitions of trigonometric functions represent quantitative relationships between the different qualities: between qualitative determinacy of angle and qualitative determinacy of rectilinear segments (legs) in rectangular triangle. These relationships do not satisfy the standard definition of mathematical function because there are no mathematical operations that should be carry out on qualitative determinacy of angle to obtain qualitative determinacy of legs. Therefore, the left-hand and right-hand sides of the standard mathematical definitions have no the identical sense. The logical errors determine the essence of trigonometry: standard trigonometry is a false theory.
Teaching Algebra without Algebra
ERIC Educational Resources Information Center
Kalman, Richard S.
2008-01-01
Algebra is, among other things, a shorthand way to express quantitative reasoning. This article illustrates ways for the classroom teacher to convert algebraic solutions to verbal problems into conversational solutions that can be understood by students in the lower grades. Three reasonably typical verbal problems that either appeared as or…
Enumerative Algebraic Geometry: Counting Conics
2005-05-10
Problem. Appolonius of Perga posed the question, “Is it possible to construct all circles that are tangent to three given circles?” Although this... Perga was a Greek geometer who studied at Euclid’s School in Alexandria. His seminal work Conics is considered one of the greatest mathematical books...Mathematical Society, 1995. [EW] Weisstein, Eric W. “ Apollonius ’ Problem.” From MathWorld–A Wolfram Web Re- source. http://mathworld.wolfram.com
Enumerative Algebraic Geometry of Conics
2008-10-01
projective plane CP2 . When we first introduced the parameter space RP5 we noted that its points are in one-to-one correspondence with the equations of...example x2 + y2 + 1 = 0 and x2 + y2 + 3 = 0 both define the empty set. But in CP2 these equations become X 2 + Y 2 + Z2 = 0 and X 2 + Y 2 + 3Z2 = 0 and...they define different complex curves. Points in CP5 are in one-to-one correspondence with conic curves in CP2 . This fact follows from the observation
Emergent geometry from quantized spacetime
Yang, Hyun Seok; Sivakumar, M.
2010-08-15
We examine the picture of emergent geometry arising from a mass-deformed matrix model. Because of the mass deformation, a vacuum geometry turns out to be a constant curvature spacetime such as d-dimensional sphere and (anti-)de Sitter spaces. We show that the mass-deformed matrix model giving rise to the constant curvature spacetime can be derived from the d-dimensional Snyder algebra. The emergent geometry beautifully confirms all the rationale inferred from the algebraic point of view that the d-dimensional Snyder algebra is equivalent to the Lorentz algebra in (d+1)-dimensional flat spacetime. For example, a vacuum geometry of the mass-deformed matrix model is completely described by a G-invariant metric of coset manifolds G/H defined by the Snyder algebra. We also discuss a nonlinear deformation of the Snyder algebra.
ERIC Educational Resources Information Center
Exner, Robert; And Others
The sixteen chapters of this book provide the core material for the Elements of Mathematics Program, a secondary sequence developed for highly motivated students with strong verbal abilities. The sequence is based on a functional-relational approach to mathematics teaching, and emphasizes teaching by analysis of real-life situations. This text is…
Notes for Applied Mathematics in Trigonometry and Earth Geometry/Navigation
ERIC Educational Resources Information Center
Faulkner, Peter
2004-01-01
As time has progressed, the role of applied mathematics has become increasingly important. Indeed there are now more students enrolled in applied mathematics courses in senior high schools and colleges than in pure mathematics. Such courses become more relevant both to the student and to future employers, if the same constants and equations that…
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…
Cyclic homology for Hom-associative algebras
NASA Astrophysics Data System (ADS)
Hassanzadeh, Mohammad; Shapiro, Ilya; Sütlü, Serkan
2015-12-01
In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.
ERIC Educational Resources Information Center
James, Wendy Michelle
2013-01-01
Science and engineering instructors often observe that students have difficulty using or applying prerequisite mathematics knowledge in their courses. This qualitative project uses a case-study method to investigate the instruction in a trigonometry course and a physics course based on a different methodology and set of assumptions about student…
ERIC Educational Resources Information Center
Cetin, Omer Faruk
2015-01-01
This study aims to analyse university level mathematics education students' perceptions on conceptual understanding of trigonometry and trigonometric functions and their content development of these concepts. A case study was conducted with 90 freshman students of Elementary Mathematics Department. The data were gathered via a scale; they included…
ERIC Educational Resources Information Center
Sander, Elisabeth; Heiß, Andrea
2014-01-01
Three different versions of a learning program on trigonometry were compared, a program controlled, non-interactive version (CG), an interactive, conflict inducing version (EG 1), and an interactive one which was supposed to reduce the occurrence of a cognitive conflict regarding the central problem solution (EG 2). Pupils (N = 101) of a…
Teaching Harmonic Motion in Trigonometry: Inductive Inquiry Supported by Physics Simulations
ERIC Educational Resources Information Center
Sokolowski, Andrzej; Rackley, Robin
2011-01-01
In this article, the authors present a lesson whose goal is to utilise a scientific environment to immerse a trigonometry student in the process of mathematical modelling. The scientific environment utilised during this activity is a physics simulation called "Wave on a String" created by the PhET Interactive Simulations Project at…
ERIC Educational Resources Information Center
Dündar, Sefa
2015-01-01
Using multiple representations of a problem can reveal the relationship between complex concepts by expressing the same mathematical condition differently and can contribute to the meaningful learning of mathematical concepts. The purpose of this study is to assess the performances of mathematics teacher-candidates on trigonometry problems…
Clifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras
NASA Astrophysics Data System (ADS)
Catto, Sultan; Gürcan, Yasemin; Khalfan, Amish; Kurt, Levent; Kato La, V.
2016-10-01
We discuss a construction scheme for Clifford numbers of arbitrary dimension. The scheme is based upon performing direct products of the Pauli spin and identity matrices. Conjugate fermionic algebras can then be formed by considering linear combinations of the Clifford numbers and the Hermitian conjugates of such combinations. Fermionic algebras are important in investigating systems that follow Fermi-Dirac statistics. We will further comment on the applications of Clifford algebras to Fueter analyticity, twistors, color algebras, M-theory and Leech lattice as well as unification of ancient and modern geometries through them.
Cartan calculus on quantum Lie algebras
Schupp, P.; Watts, P.; Zumino, B.
1993-12-09
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions au into one big algebra, the ``Cartan Calculus.``
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
Graded geometry and Poisson reduction
Cattaneo, A. S.; Zambon, M.
2009-02-02
The main result extends the Marsden-Ratiu reduction theorem in Poisson geometry, and is proven by means of graded geometry. In this note we provide the background material about graded geometry necessary for the proof. Further, we provide an alternative algebraic proof for the main result.
Adaptive Algebraic Multigrid Methods
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
ERIC Educational Resources Information Center
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
The usage of image trigonometry in bone measurements.
Dymond, Ian W; Ashforth, James A; Dymond, Graeme F; Spirakis, Thanos; Learmonth, Ian D
2013-01-01
The entire musculo-skeletal system responds dynamically to stresses and strains applied to it. Restoring normal biomechanics contributes to the normal function that ensures that physiological stresses and strains are preserved. Appropriate preoperative planning is mandatory to restore normal biomechanics at reconstructive surgery. Effective preoperative planning depends on the ability to reproducibly make accurate measurements of lengths and angles from plain radiographs. Measurement has become an integral part of orthopaedics to define morphological abnormality, to plan for reconstruction and for comparative research. The most prevalent method of measurement is usually based on lines drawn on radiographs with no accurate reference to the actual geometry of the structures. This two-dimensional projection of an asymmetrical three-dimensional structure leads to inaccuracy and consequently to a compromise in the overall precision of many procedures. In addition it is also difficult to monitor the progression of disease as the exact relationship of the bones and joints to each other, and to prosthetics, cannot be accurately recorded. This paper presents a method of digitally measuring relevant bone parameters in a geometric manner in order to achieve accurate, repeatable measurements.
From operator algebras to superconformal field theory
Kawahigashi, Yasuyuki
2010-01-15
We survey operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory of Jones, and certain aspects of noncommutative geometry of Connes.
An algebraic approach to the scattering equations
NASA Astrophysics Data System (ADS)
Huang, Rijun; Rao, Junjie; Feng, Bo; He, Yang-Hui
2015-12-01
We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.
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…
Geometry of spinor regularization
NASA Technical Reports Server (NTRS)
Hestenes, D.; Lounesto, P.
1983-01-01
The Kustaanheimo theory of spinor regularization is given a new formulation in terms of geometric algebra. The Kustaanheimo-Stiefel matrix and its subsidiary condition are put in a spinor form directly related to the geometry of the orbit in physical space. A physically significant alternative to the KS subsidiary condition is discussed. Derivations are carried out without using coordinates.
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
NASA Astrophysics Data System (ADS)
Aiken, Brenda L.
The Commonwealth of Virginia requires high school students to receive a passing grade in core courses and a passing score on End-of-Course Standards of Learning (EOC SOL) tests to receive verified credits that lead to a Virginia high school diploma. These tests are believed to accurately reflect what students should know and be able to do in order to experience success in their endeavors beyond high school. For some students remediation is required to experience success on EOC SOL tests. This study sought to determine the effect of a County's public high school summer remediation program on student gains on EOC SOL tests in Algebra I, Biology, Chemistry, Geometry, and World History and Geography II. Specifically, the purpose of the study sought to determine the following: (a) If significant gains were made by students who attended the summer remediation program; (b) If significant gains were made by students who did not attend the summer remediation program; (c) If there were differences in gain scores of students who attended and those who did not attend the summer remediation program; and (d) If there were differences in gain scores among students who attended the summer remediation program related to school site, gender, ethnicity, learning ability group, socioeconomic status, and level of English proficiency. The results of the study indicate that students who attended and those who did not attend the summer remediation program made significant gains. However, the gains for students who attended the summer remediation program were significantly greater than the gains made by students who did not attend. The study also found that there were no significant differences in gain scores among students who attended the summer remediation program related to gender, ethnicity, learning ability group, socioeconomic status, and level of English proficiency. There were significant differences in Algebra I gain scores related to school site. Recommendations for
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.
Gustafson, K.
1994-12-31
By means of the author`s earlier theory of antieigenvalues and antieigenvectors, a new computational approach to iterative methods is presented. This enables an explicit trigonometric understanding of iterative convergence and provides new insights into the sharpness of error bounds. Direct applications to Gradient descent, Conjugate gradient, GCR(k), Orthomin, CGN, GMRES, CGS, and other matrix iterative schemes will be given.
Bilinear forms on fermionic Novikov algebras
NASA Astrophysics Data System (ADS)
Chen, Zhiqi; Zhu, Fuhai
2007-05-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian super-operator in a super-variable. In this paper, we show that there is a remarkable geometry on fermionic Novikov algebras with non-degenerate invariant symmetric bilinear forms, which we call pseudo-Riemannian fermionic Novikov algebras. They are related to pseudo-Riemannian Lie algebras. Furthermore, we obtain a procedure to classify pseudo-Riemannian fermionic Novikov algebras. As an application, we give the classification in dimension <=4. Motivated by the one in dimension 4, we construct some examples in high dimensions.
NASA Astrophysics Data System (ADS)
Mikhalev, A. V.; Pinchuk, I. A.
2005-06-01
The structure of Steinberg conformal algebras is studied; these are analogues of Steinberg groups (algebras, superalgebras).A Steinberg conformal algebra is defined as an abstract algebra by a system of generators and relations between the generators. It is proved that a Steinberg conformal algebra is the universal central extension of the corresponding conformal Lie algebra; the kernel of this extension is calculated.
ERIC Educational Resources Information Center
Capani, Antonio; De Dominicis, Gabriel
This paper proposes a model for a general interface between people and Computer Algebra Systems (CAS). The main features in the CAS interface are data navigation and the possibility of accessing powerful remote machines. This model is based on the idea of session management, in which the main engine of the tool enables interactions with the…
Multilinear Computing and Multilinear Algebraic Geometry
2016-08-10
the presentations. • “Hypermatrices,” Mathematics Colloquium, Department of Mathematics , University of Cal- ifornia, San Diego, CA, March 3, 2016...Blind multilinear identification and tensor nuclear norm,” Workshop on Low Complexity Models in Signal Processing, Hausdorff Center for Mathematics ...Institute for Mathematics and its Applications, University of Min- nesota, Minneapolis, MN, January 25–29, 2016. • “Grothendieck constant, quantum
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.
NASA Astrophysics Data System (ADS)
Beggs, Edwin J.; Majid, Shahn
2017-04-01
We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter λ, using a functorial approach. This leads us to field equations of 'Poisson-Riemannian geometry' between the classical metric, the Poisson bracket and a certain Poisson-compatible connection needed as initial data for the quantisation of the differential structure. We use such data to define a functor Q to O(λ2) from the monoidal category of all classical vector bundles equipped with connections to the monoidal category of bimodules equipped with bimodule connections over the quantised algebra. This is used to 'semiquantise' the wedge product of the exterior algebra and in the Riemannian case, the metric and the Levi-Civita connection in the sense of constructing a noncommutative geometry to O(λ2) . We solve our field equations for the Schwarzschild black-hole metric under the assumption of spherical symmetry and classical dimension, finding a unique solution and the necessity of nonassociativity at order λ2, which is similar to previous results for quantum groups. The paper also includes a nonassociative hyperboloid, nonassociative fuzzy sphere and our previously algebraic bicrossproduct model.
NASA Astrophysics Data System (ADS)
Deta, U. A.; Suparmi, Cari
2013-09-01
The approximate analytical solution of Schrodinger equation in D-Dimensions for Scarf trigonometry potential were investigated using Nikiforov-Uvarov method. The bound state energy are given in the close form and the corresponding wave function for arbitary l-state in D-dimensions are formulated in the form of generalized Jacobi Polynomials. The example of bound state energy and wave function in 3, 4, and 5 dimensions presented in condition of ground state to second excited state. The existence of arbitrary dimensions increase bound state energy and the amplitude of the wave function of this potential. The effect of the presence of Scarf trigonometry potential increase the energy spectrum of this potential.
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,…
Riemannian manifolds as Lie-Rinehart algebras
NASA Astrophysics Data System (ADS)
Pessers, Victor; van der Veken, Joeri
2016-07-01
In this paper, we show how Lie-Rinehart algebras can be applied to unify and generalize the elementary theory of Riemannian geometry. We will first review some necessary theory on a.o. modules, bilinear forms and derivations. We will then translate some classical theory on Riemannian geometry to the setting of Rinehart spaces, a special kind of Lie-Rinehart algebras. Some generalized versions of classical results will be obtained, such as the existence of a unique Levi-Civita connection, inducing a Levi-Civita connection on a submanifold, and the construction of spaces with constant sectional curvature.
Population Propensity Measurement Model
1993-12-01
school DQ702 Taken elementary algebra DQ703 Taken plane geometry DQ70 Taken computer science DQ706 Taken intermediate algebra DQ707 Taken trigonometry ...with separate models for distributing the arrival of applicants over FY’s, quarters, or months. The primary obstacle in these models is shifting the...to ŕ" = Otherwise DQ706 Binary: 1 = Taken intermediate Q706 is equal to ŕ" algebra, 0 = Otherwise DQ707 Binary: 1 = Taken trigonometry , 0 = Q707 is
Spectral Metric Spaces on Extensions of C*-Algebras
NASA Astrophysics Data System (ADS)
Hawkins, Andrew; Zacharias, Joachim
2017-03-01
We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones. Using our construction we find new spectral triples on the quantum 2- and 3-spheres giving a new perspective on these algebras in noncommutative geometry.
Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations
2014-07-01
non- linear hybrid systems by linear algebraic methods. In Radhia Cousot and Matthieu Martel, editors, SAS, volume 6337 of LNCS, pages 373–389. Springer...Tarski. A decision method for elementary algebra and geometry. Bulletin of the American Mathematical Society, 59, 1951. [36] Wolfgang Walter. Ordinary...Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 July 2014
ERIC Educational Resources Information Center
Schoen, Harold L.
1976-01-01
Seventeen studies of self-paced instruction in algebra, geometry, trigonometry, and college mathematics courses are reviewed. In general these studies did not find the individualized programs to be superior to theregular instruction with which they were compared. (SD)
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)
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
Analytic Methods in Investigative Geometry.
ERIC Educational Resources Information Center
Dobbs, David E.
2001-01-01
Suggests an alternative proof by analytic methods, which is more accessible than rigorous proof based on Euclid's Elements, in which students need only apply standard methods of trigonometry to the data without introducing new points or lines. (KHR)
The Geometry of Noncommutative Space-Time
NASA Astrophysics Data System (ADS)
Mendes, R. Vilela
2016-10-01
Stabilization, by deformation, of the Poincaré-Heisenberg algebra requires both the introduction of a fundamental lentgh and the noncommutativity of translations which is associated to the gravitational field. The noncommutative geometry structure that follows from the deformed algebra is studied both for the non-commutative tangent space and the full space with gravity. The contact points of this approach with the work of David Finkelstein are emphasized.
Profiles of Algebraic Competence
ERIC Educational Resources Information Center
Humberstone, J.; Reeve, R.A.
2008-01-01
The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…
ERIC Educational Resources Information Center
Miller, L. Diane; England, David A.
1989-01-01
Describes a study in a large metropolitan high school to ascertain what influence the use of regular writing in algebra classes would have on students' attitudes towards algebra and their skills in algebra. Reports the simpler and more direct the writing topics the better. (MVL)
NASA Technical Reports Server (NTRS)
Iachello, Franco
1995-01-01
An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.
Applied Algebra Curriculum Modules.
ERIC Educational Resources Information Center
Texas State Technical Coll., Marshall.
This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Ternary Virasoro - Witt algebra.
Zachos, C.; Curtright, T.; Fairlie, D.; High Energy Physics; Univ. of Miami; Univ. of Durham
2008-01-01
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
Problem Solving in Calculus with Symbolic Geometry and CAS
ERIC Educational Resources Information Center
Todd, Philip; Wiechmann, James
2008-01-01
Computer algebra systems (CAS) have been around for a number of years, as has dynamic geometry. Symbolic geometry software is new. It bears a superficial similarity to dynamic geometry software, but differs in that problems may be set up involving symbolic variables and constants, and measurements are given as symbolic expressions. Mathematical…
ERIC Educational Resources Information Center
Bot, Thomas D.; Eze, John E.
2016-01-01
This article presents the findings from an experimental study on the effectiveness of concept mapping and cooperative learning strategies on SSII students' achievement in trigonometry in mathematics. The research design used in conducting the study was quasi-experimental pre-test and post-test non-equivalent control group. The sample consisted of…
NASA Astrophysics Data System (ADS)
Sati, Hisham; Schreiber, Urs
2017-03-01
We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie ( p + 1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie ( p+1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.
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.
Weak Lie symmetry and extended Lie algebra
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).
Discrete Minimal Surface Algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Hoppe, Jens
2010-05-01
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
Gauging the Carroll algebra and ultra-relativistic gravity
NASA Astrophysics Data System (ADS)
Hartong, Jelle
2015-08-01
It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincaré algebra. Recently it has been shown that gauging the centrally extended Galilei algebra, known as the Bargmann algebra, leads to a geometrical framework that when made dynamical gives rise to Hořava-Lifshitz gravity. Here we consider the case where we contract the Poincaré algebra by sending the speed of light to zero leading to the Carroll algebra. We show how this algebra can be gauged and we construct the most general affine connection leading to the geometry of so-called Carrollian space-times. Carrollian space-times appear for example as the geometry on null hypersurfaces in a Lorentzian space-time of one dimension higher. We also construct theories of ultra-relativistic (Carrollian) gravity in 2+1 dimensions with dynamical exponent z < 1 including cases that have anisotropic Weyl invariance for z = 0.
Developing the concept of a parabola in Taxicab geometry
NASA Astrophysics Data System (ADS)
Ada, Tuba; Kurtuluş, Aytaç; Bahadır Yanik, H.
2015-02-01
The aim of this study was to observe the development process of the concept of a parabola in Taxicab geometry. The study was carried out in two stages. First, some activities related to Euclidean geometry and Taxicab geometry were designed based on concept development and real-life applications, and they were administered to a ninth-grade student. According to the findings, once the student learnt the definition of a parabola in Euclidean geometry, she was able to define a Taxicab parabola using the distance function in Taxicab geometry. Also, she came up with an algebraic definition of a Taxicab parabola based on this geometric definition of the concept of a parabola. Moving from algebraic definition to geometric representation, she configured the concept of a parabola in Taxicab geometry. By means of this application activity, the student had the opportunity to observe and practise the concept of a parabola in a real-life situation based on Euclidean geometry and Taxicab geometry.
Prediction of Algebraic Instabilities
NASA Astrophysics Data System (ADS)
Zaretzky, Paula; King, Kristina; Hill, Nicole; Keithley, Kimberlee; Barlow, Nathaniel; Weinstein, Steven; Cromer, Michael
2016-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A new methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to the governing differential equation (or dispersion relation) describing wave propagation in dispersive media. Several types of algebraic instabilities are explored in the context of both linear and nonlinear waves.
Connecting Algebra and Chemistry.
ERIC Educational Resources Information Center
O'Connor, Sean
2003-01-01
Correlates high school chemistry curriculum with high school algebra curriculum and makes the case for an integrated approach to mathematics and science instruction. Focuses on process integration. (DDR)
Noncommutative geometry and arithmetics
NASA Astrophysics Data System (ADS)
Almeida, P.
2009-09-01
We intend to illustrate how the methods of noncommutative geometry are currently used to tackle problems in class field theory. Noncommutative geometry enables one to think geometrically in situations in which the classical notion of space formed of points is no longer adequate, and thus a “noncommutative space” is needed; a full account of this approach is given in [3] by its main contributor, Alain Connes. The class field theory, i.e., number theory within the realm of Galois theory, is undoubtedly one of the main achievements in arithmetics, leading to an important algebraic machinery; for a modern overview, see [23]. The relationship between noncommutative geometry and number theory is one of the many themes treated in [22, 7-9, 11], a small part of which we will try to put in a more down-to-earth perspective, illustrating through an example what should be called an “application of physics to mathematics,” and our only purpose is to introduce nonspecialists to this beautiful area.
New symbolic tools for differential geometry, gravitation, and field theory
NASA Astrophysics Data System (ADS)
Anderson, I. M.; Torre, C. G.
2012-01-01
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, spinor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors, algebraic classification of solutions of the Einstein equations, and symmetry reduction of field equations.
On left Hopf algebras within the framework of inhomogeneous quantum groups for particle algebras
Rodriguez-Romo, Suemi
2012-10-15
We deal with some matters needed to construct concrete left Hopf algebras for inhomogeneous quantum groups produced as noncommutative symmetries of fermionic and bosonic creation/annihilation operators. We find a map for the bidimensional fermionic case, produced as in Manin's [Quantum Groups and Non-commutative Hopf Geometry (CRM Univ. de Montreal, 1988)] seminal work, named preantipode that fulfills all the necessary requirements to be left but not right on the generators of the algebra. Due to the complexity and importance of the full task, we consider our result as an important step that will be extended in the near future.
Bicovariant quantum algebras and quantum Lie algebras
NASA Astrophysics Data System (ADS)
Schupp, Peter; Watts, Paul; Zumino, Bruno
1993-10-01
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun(mathfrak{G}_q ) to U q g, given by elements of the pure braid group. These operators—the “reflection matrix” Y≡L + SL - being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for Y in SO q (N).
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…
Parastatistics Algebras and Combinatorics
NASA Astrophysics Data System (ADS)
Popov, T.
2005-03-01
We consider the algebras spanned by the creation parafermionic and parabosonic operators which give rise to generalized parastatistics Fock spaces. The basis of such a generalized Fock space can be labelled by Young tableaux which are combinatorial objects. By means of quantum deformations a nice combinatorial structure of the algebra of the plactic monoid that lies behind the parastatistics is revealed.
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
Learning Activity Package, Algebra.
ERIC Educational Resources Information Center
Evans, Diane
A set of ten teacher-prepared Learning Activity Packages (LAPs) in beginning algebra and nine in intermediate algebra, these units cover sets, properties of operations, number systems, open expressions, solution sets of equations and inequalities in one and two variables, exponents, factoring and polynomials, relations and functions, radicals,…
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.
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees
NASA Astrophysics Data System (ADS)
Agarwala, Susama; Delaney, Colleen
2015-04-01
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.
Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees
Agarwala, Susama; Delaney, Colleen
2015-04-15
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.
Hexagonal tessellations in image algebra
NASA Astrophysics Data System (ADS)
Eberly, David H.; Wenzel, Dennis J.; Longbotham, Harold G.
1990-11-01
In image algebra '' the concept of a coordinate set X is general in that such a set is simply a subset of ndimensional Euclidean space . The standard applications in 2-dimensional image processing use coordinate sets which are rectangular arrays X 72 x ZZm. However some applications may require other geometries for the coordinate set. We look at three such related applications in the context of image algebra. The first application is the modeling of photoreceptors in primate retinas. These receptors are inhomogeneously distributed on the retina. The largest receptor density occurs in the center of the fovea and decreases radially outwards. One can construct a hexagonal tessellation of the retina such that each hexagon contains approximately the same number of receptors. The resulting tessellation called a sunflower heart2 consists of concentric rings of hexagons whose sizes increase as the radius of the ring increases. The second application is the modeling of the primary visual . The neurons are assumed to be uniformly distributed as a regular hexagonal lattice. Cortical neural image coding is modeled by a recursive convolution of the retinal neural image using a special set of filters. The third application involves analysis of a hexagonally-tessellated image where the pixel resolution is variable .
Algebraic Nonlinear Collective Motion
NASA Astrophysics Data System (ADS)
Troupe, J.; Rosensteel, G.
1998-11-01
Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real numberΛ. TheΛ=0 solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear group action on Euclidean space transforms a certain family of deformed droplets among themselves. For positiveΛ, the droplets have a neck that becomes more pronounced asΛincreases; for negativeΛ, the droplets contain a spherical bubble of radius |Λ|1/3. The nonlinear vector field algebra is extended to the nonlinear general collective motion algebra gcm(3) which includes the inertia tensor. The quantum algebraic models of nonlinear nuclear collective motion are given by irreducible unitary representations of the nonlinear gcm(3) Lie algebra. These representations model fissioning isotopes (Λ>0) and bubble and two-fluid nuclei (Λ<0).
Algebraic invariants for homotopy types
NASA Astrophysics Data System (ADS)
Blanc, David
1999-11-01
We define a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the [Pi]-algebra [pi][low asterisk]X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract [Pi]-algebra can be realized as the homotopy [Pi]-algebra of a space.
A Richer Understanding of Algebra
ERIC Educational Resources Information Center
Foy, Michelle
2008-01-01
Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number,…
Authorized Course of Instruction for the Quinmester Program, Mathematics: Survey of Algebra 1.
ERIC Educational Resources Information Center
Moore, Mary N.; Rose, Patricia
Outlined are the minimum requirements for a quinmester course intended to strengthen a student's experience in a first algebra course, prior to entry to high school geometry and the second algebra course. After a brief description of overall goals and strategies, further details are presented in eight sections. Each section gives performance…
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.…
ERIC Educational Resources Information Center
Desseyn, H. O.; And Others
1985-01-01
Compares linear-nonlinear and planar-nonplanar geometry through the valence-shell electron pairs repulsion (V.S.E.P.R.), Mulliken-Walsh, and electrostatic force theories. Indicates that although the V.S.E.P.R. theory has more advantages for elementary courses, an explanation of the best features of the different theories offers students a better…
Trigonometry for Non-Trigonometry Students.
ERIC Educational Resources Information Center
Boyes, G. R.
1994-01-01
Presents a team-oriented student activity designed to introduce trigonometric ratios and applications of selected trigonometric functions by constructing tables of values. Can be used at the middle school level. (MKR)
NASA Astrophysics Data System (ADS)
Markarian, Nikita
2017-03-01
We introduce Weyl n-algebras and show how their factorization complex may be used to define invariants of manifolds. In the appendix, we heuristically explain why these invariants must be perturbative Chern-Simons invariants.
Developing Algebraic Thinking.
ERIC Educational Resources Information Center
Alejandre, Suzanne
2002-01-01
Presents a teaching experience that resulted in students getting to a point of full understanding of the kinesthetic activity and the algebra behind it. Includes a lesson plan for a traffic jam activity. (KHR)
Jordan Algebraic Quantum Categories
NASA Astrophysics Data System (ADS)
Graydon, Matthew; Barnum, Howard; Ududec, Cozmin; Wilce, Alexander
2015-03-01
State cones in orthodox quantum theory over finite dimensional complex Hilbert spaces enjoy two particularly essential features: homogeneity and self-duality. Orthodox quantum theory is not, however, unique in that regard. Indeed, all finite dimensional formally real Jordan algebras -- arenas for generalized quantum theories with close algebraic kinship to the orthodox theory -- admit homogeneous self-dual positive cones. We construct categories wherein these theories are unified. The structure of composite systems is cast from universal tensor products of the universal C*-algebras enveloping ambient spaces for the constituent state cones. We develop, in particular, a notion of composition that preserves the local distinction of constituent systems in quaternionic quantum theory. More generally, we explicitly derive the structure of hybrid quantum composites with subsystems of arbitrary Jordan algebraic type.
Accounting Equals Applied Algebra.
ERIC Educational Resources Information Center
Roberts, Sondra
1997-01-01
Argues that students should be given mathematics credits for completing accounting classes. Demonstrates that, although the terminology is different, the mathematical concepts are the same as those used in an introductory algebra class. (JOW)
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
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.
Double conformal space-time algebra
NASA Astrophysics Data System (ADS)
Easter, Robert Benjamin; Hitzer, Eckhard
2017-01-01
The Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra G 4,8that extends the concepts introduced with the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA) G 8,2 with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the Space-Time Algebra (STA) G 1,3. Two Conformal Space-Time subalgebras (CSTA) G 2,4 provide spacetime entities for points, flats (incl. worldlines), and hyperbolics, and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reflection, and (pseudo)spherical inversion in rounds or hyperbolics. The DCSTA G 4,8 is a doubling product of two G 2,4 CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new bivector entities for (pseudo)quadrics and Darboux (pseudo)cyclides in spacetime that are also transformed by the doubled versors. The "pseudo" surface entities are spacetime hyperbolics or other surface entities using the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in rounds or hyperbolics. An operation for the directed non-uniform scaling (anisotropic dilation) of the bivector general quadric entities is defined using the boost operator and a spatial projection. DCSTA allows general quadric surfaces to be transformed in spacetime by the same complete set of doubled CSTA versor (i.e., DCSTA versor) operations that are also valid on the doubled CSTA point entity (i.e., DCSTA point) and the other doubled CSTA entities. The new DCSTA bivector entities are formed by extracting values from the DCSTA point entity using specifically defined inner product extraction operators. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the length
Teaching and Learning a New Algebra with Understanding.
ERIC Educational Resources Information Center
Kaput, James J.
This paper suggests a route to deep, long-term algebra reform that begins not with more new approaches but with elementary school teachers and the reform efforts that currently exist. This route involves generalization and expression of that generality using increasingly formal languages, beginning with arithmetic, modeling situations, geometry,…
FINAL REPORT: GEOMETRY AND ELEMENTARY PARTICLE PHYSICS
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 Energy 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.
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,…
Sequential Mathematics Course III. Part 1. Pilot Edition.
ERIC Educational Resources Information Center
New York City Board of Education, Brooklyn, NY. Div. of Curriculum and Instruction.
This document outlines part one of the third year course of the newly implemented New York State Mathematics Curriculum. The three-year Sequential Mathematics sequence designed by New York State is a move in the direction of fusing the formerly separate topics found in algebra, geometry, and intermediate algebra/trigonometry, and introducing…
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)
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)
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…
Algebraic mesh quality metrics
KNUPP,PATRICK
2000-04-24
Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally-invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. the singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. Condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Combined metrics for shape and volume, shape-volume-orientation are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to non-simplical elements. A series of numerical tests verify the theoretical properties of the metrics defined.
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
ERIC Educational Resources Information Center
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
NASA Astrophysics Data System (ADS)
Durka, R.
2017-04-01
The S-expansion framework is analyzed in the context of a freedom in closing the multiplication tables for the abelian semigroups. Including the possibility of the zero element in the resonant decomposition, and associating the Lorentz generator with the semigroup identity element, leads to a wide class of the expanded Lie algebras introducing interesting modifications to the gauge gravity theories. Among the results, we find all the Maxwell algebras of type {{B}m} , {{C}m} , and the recently introduced {{D}m} . The additional new examples complete the resulting generalization of the bosonic enlargements for an arbitrary number of the Lorentz-like and translational-like generators. Some further prospects concerning enlarging the algebras are discussed, along with providing all the necessary constituents for constructing the gravity actions based on the obtained results.
NASA Astrophysics Data System (ADS)
Roytenberg, Dmitry
2007-11-01
A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and certain "up to homotopy" structures on the complex; for strictly skew-symmetric Lie 2-algebras these are L∞-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer-Cartan equation in some differential graded Lie algebras and L∞-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples.
Algebra for Gifted Third Graders.
ERIC Educational Resources Information Center
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
A Holistic Approach to Algebra.
ERIC Educational Resources Information Center
Barbeau, Edward J.
1991-01-01
Described are two examples involving recursive mathematical sequences designed to integrate a holistic approach to learning algebra. These examples promote pattern recognition with algebraic justification, full class participation, and mathematical values that can be transferred to other situations. (MDH)
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.
An algebra of reversible computation.
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.
Handy elementary algebraic properties of the geometry of entanglement
NASA Astrophysics Data System (ADS)
Blair, Howard A.; Alsing, Paul M.
2013-05-01
The space of separable states of a quantum system is a hyperbolic surface in a high dimensional linear space, which we call the separation surface, within the exponentially high dimensional linear space containing the quantum states of an n component multipartite quantum system. A vector in the linear space is representable as an n-dimensional hypermatrix with respect to bases of the component linear spaces. A vector will be on the separation surface iff every determinant of every 2-dimensional, 2-by-2 submatrix of the hypermatrix vanishes. This highly rigid constraint can be tested merely in time asymptotically proportional to d, where d is the dimension of the state space of the system due to the extreme interdependence of the 2-by-2 submatrices. The constraint on 2-by-2 determinants entails an elementary closed formformula for a parametric characterization of the entire separation surface with d-1 parameters in the char- acterization. The state of a factor of a partially separable state can be calculated in time asymptotically proportional to the dimension of the state space of the component. If all components of the system have approximately the same dimension, the time complexity of calculating a component state as a function of the parameters is asymptotically pro- portional to the time required to sort the basis. Metric-based entanglement measures of pure states are characterized in terms of the separation hypersurface.
Teaching Algebra and Geometry Concepts by Modeling Telescope Optics
ERIC Educational Resources Information Center
Siegel, Lauren M.; Dickinson, Gail; Hooper, Eric J.; Daniels, Mark
2008-01-01
This article describes preparation and delivery of high school mathematics lessons that integrate mathematics and astronomy through The Geometer's Sketchpad models, traditional proof, and inquiry-based activities. The lessons were created by a University of Texas UTeach preservice teacher as part of a project-based field experience in which high…
ERIC Educational Resources Information Center
Ketterlin-Geller, Leanne R.; Jungjohann, Kathleen; Chard, David J.; Baker, Scott
2007-01-01
Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…
ERIC Educational Resources Information Center
Nwabueze, Kenneth K.
2004-01-01
The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…
Algebraic Thinking through Origami.
ERIC Educational Resources Information Center
Higginson, William; Colgan, Lynda
2001-01-01
Describes the use of paper folding to create a rich environment for discussing algebraic concepts. Explores the effect that changing the dimensions of two-dimensional objects has on the volume of related three-dimensional objects. (Contains 13 references.) (YDS)
Computer Algebra versus Manipulation
ERIC Educational Resources Information Center
Zand, Hossein; Crowe, David
2004-01-01
In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…
Geometry and Symmetric Coherent States of Three Qubits Systems
NASA Astrophysics Data System (ADS)
Guo, Xiao-Kan
2016-06-01
In this paper, we first generalize the previous results that relate 1- and 2-qubit geometries to complex and quaternionic Möbius transformations respectively, to the case of 3-qubit states under octonionic Möbius transformations. This completes the correspondence between the qubit geometries and the four normed division algebras. Thereby, new systems of symmetric coherent states with 2 and 3 qubits can be constructed by mapping the spin coherent states to their antipodal symmetric ponits on the generalized Bloch spheres via Möbius transformations in corresponding dimensions. Finally, potential applications of the normed division algebras in physics are discussed.
Large N Duality, Lagrangian Cycles, and Algebraic Knots
NASA Astrophysics Data System (ADS)
Diaconescu, D.-E.; Shende, V.; Vafa, C.
2013-05-01
We consider knot invariants in the context of large N transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicitly constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case.
Enrichment Activities for Geometry.
ERIC Educational Resources Information Center
Usiskin, Zalman
1983-01-01
Enrichment activities that teach about geometry as they instruct in geometry are given for some significant topics. The facets of geometry included are tessellations, round robin tournaments, geometric theorems on triangles, and connections between geometry and complex numbers. (MNS)
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
On Dunkl angular momenta algebra
NASA Astrophysics Data System (ADS)
Feigin, Misha; Hakobyan, Tigran
2015-11-01
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
ERIC Educational Resources Information Center
Carson, Cristi L.; Day, Judith
This paper argues that operations with negative numbers should be taught using a curriculum that is grounded in algebraic geometry. This position is supported by the results from a study that compared the conceptual understanding of grade 9 students who received the Algebra Project transition curriculum to a control group of grade 6 gifted…
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…
Marquette, Ian
2013-07-15
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.
Universal Algebraic Varieties and Ideals in Physics:. Field Theory on Algebraic Varieties
NASA Astrophysics Data System (ADS)
Iguchi, Kazumoto
A class of universal algebraic varieties in physics is discussed herein using the concepts of determinant ideals in algebraic geometry. It is shown that these algebraic varieties arise with very different physical contexts in many branches of physics and mathematics from high energy physics theory to chaos theory. In these physical systems the models are constructed by using the fields on usual manifolds such as vector fields in a Euclidean space and a Minkowskian space. But there is a universal mathematical aspect of linear algebra for linear vector spaces, where the linear independency and dependency are described using the Gramians of the vectors. These Gramians form a class of hypersurfaces in a higher-dimensional mathematical space: If there exist g vectors vi in an n-dimensional Euclidean space, the Gramian Gg is given as a g × g determinant Gg=Det[xij] with the inner products xij=(vi,vj), and exists in a g(g-1)/2-[g(g+1)/2-] dimensional space if the vectors are (not) normalized, xii=1 (xii ≠ 1). It is also shown that the Gramians are invariant under automorphisms of the vectors. The mathematical structure of the Gramians is revealed to be equivalent to the concepts of determinant ideals Ig(v), each element of which is a g × g determinant constructed from components of an arbitrary N×N matrix with N>n and which have inclusion relation: R=I0(v)⊃ I1(v) ⊃···⊃ Ig(v) ⊃···, and Ig(v)=0 if g>n. In the various physical systems the ideals naturally emerge to give us dynamical flows on the hypersurfaces, and therefore, it is called the field theory on algebraic varieties. This viewpoint provides us a grand viewpoint in physics and mathematics.
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
ERIC Educational Resources Information Center
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
NASA Technical Reports Server (NTRS)
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
The Z_2 -Orbifold of the W_3-Algebra
NASA Astrophysics Data System (ADS)
Al-Ali, Masoumah; Linshaw, Andrew R.
2016-12-01
The Zamolodchikov W_3-algebra W^c_3 with central charge c has full automorphism group Z_2. It was conjectured in the physics literature over 20 years ago that the orbifold (W^c_3)^{Z_2} is of type W(2,6,8,10,12) for generic values of c. We prove this conjecture for all c ≠ 559 ± 7 √{76657}/95, and we show that for these two values, the orbifold is of type W(2,6,8,10,12,14). This paper is part of a larger program of studying orbifolds and cosets of vertex algebras that depend continuously on a parameter. Minimal strong generating sets for orbifolds and cosets are often easy to find for generic values of the parameter, but determining which values are generic is a difficult problem. In the example of (W^c_3)^{Z_2} , we solve this problem using tools from algebraic geometry.
Quantum computation using geometric algebra
NASA Astrophysics Data System (ADS)
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
ERIC Educational Resources Information Center
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
Noncommutative Riemannian geometry on graphs
NASA Astrophysics Data System (ADS)
Majid, Shahn
2013-07-01
We show that arising out of noncommutative geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative Laplace-Beltrami operator on differential 1-forms, where we use the language of differential algebras to functorially interpret a graph as providing a 'finite manifold structure' on the set of vertices. We equip any graph with a canonical 'Euclidean metric' and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations.
Applications of algebraic grid generation
NASA Technical Reports Server (NTRS)
Eiseman, Peter R.; Smith, Robert E.
1990-01-01
Techniques and applications of algebraic grid generation are described. The techniques are univariate interpolations and transfinite assemblies of univariate interpolations. Because algebraic grid generation is computationally efficient, the use of interactive graphics in conjunction with the techniques is advocated. A flexible approach, which works extremely well in an interactive environment, called the control point form of algebraic grid generation is described. The applications discussed are three-dimensional grids constructed about airplane and submarine configurations.
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.
Coincidence technique to reduce geometry and matrix effects in assay
Zucker, M.S.; Gozani, T.; Bernatowicz, H.
1983-01-01
Algebraic combinations of coincidence multiplicities can be formed which are relatively independent of detection efficiency, yet proportional to the amount of nuclear material being assayed. Considering these combinations, rather than the coincidence alone as signatures, has the demonstrable advantage that the assay results are comparatively independent of sample geometry or even matrix.
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Algebraic nonlinear growth of the resistive kink instability
NASA Astrophysics Data System (ADS)
Biskamp, Dieter
1991-12-01
It is derived from a simple model that the resistive kink mode grows algebraically W∝t2 for island size W exceeding the resistive layer width. The model only uses the properties of the linear eigenfunction and of current-sheet reconnection. Because of the geometry of the inflow velocity, the usual quasisingular behavior in the current sheet edge region vanishes. The theory is in quantitative agreement with high-S number numerical simulations.
Implementation of Geometric Algebra in MATLAB (registered trademark) with Applications
2014-09-01
indefinite scalar product with signature (+,−,−,−) models Minkowski spacetime , so the multiplication tables and the dual operator are computed...differently. Alternatively, the Minkowski spacetime can be identified with the subspace C0 ( R3 ) ⊕ C1 ( R3 ) whose elements are called paravectors. 3...the algebra multiplication rules. The extension of the code to an indefinite scalar product which arises in the geometry of spacetime is straightforward
Geometrical description of algebraic structures: Applications to Quantum Mechanics
Carinena, J. F.; Ibort, A.; Marmo, G.; Morandi, G.
2009-05-06
Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to quantum mechanics. We will concentrate our attention into quantum theories and we will show how to use in a systematic way the transition from algebraic to geometrical structures to explore their geometry, mainly its Jordan-Lie structure.
Structured adaptive grid generation using algebraic methods
NASA Technical Reports Server (NTRS)
Yang, Jiann-Cherng; Soni, Bharat K.; Roger, R. P.; Chan, Stephen C.
1993-01-01
of points to the grid redistribution scheme. The evaluation of the weighting mesh is accomplished by utilizing the weight function representing the solution variation and the equidistribution law. The selection of the weight function plays a key role in grid adaptation. A new weight function utilizing a properly weighted boolean sum of various flowfield characteristics is defined. The redistribution scheme is developed utilizing Non-Uniform Rational B-Splines (NURBS) representation. The application of NURBS representation results in a well distributed smooth grid by maintaining the fidelity of the geometry associated with boundary curves. Several algebraic methods are applied to smooth and/or nearly orthogonalize the grid lines. An elliptic solver is utilized to smooth the grid lines if there are grid crossings. Various computational examples of practical interest are presented to demonstrate the success of these methods.
Statecharts Via Process Algebra
NASA Technical Reports Server (NTRS)
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
Chin, P W; Spezi, E; Lewis, D G
2003-08-21
A software solution has been developed to carry out Monte Carlo simulations of portal dosimetry using the BEAMnrc/DOSXYZnrc code at oblique gantry angles. The solution is based on an integrated phantom, whereby the effect of incident beam obliquity was included using geometric transformations. Geometric transformations are accurate within +/- 1 mm and +/- 1 degrees with respect to exact values calculated using trigonometry. An application in portal image prediction of an inhomogeneous phantom demonstrated good agreement with measured data, where the root-mean-square of the difference was under 2% within the field. Thus, we achieved a dose model framework capable of handling arbitrary gantry angles, voxel-by-voxel phantom description and realistic particle transport throughout the geometry.
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
Viterbi/algebraic hybrid decoder
NASA Technical Reports Server (NTRS)
Boyd, R. W.; Ingels, F. M.; Mo, C.
1980-01-01
Decoder computer program is hybrid between optimal Viterbi and optimal algebraic decoders. Tests have shown that hybrid decoder outperforms any strictly Viterbi or strictly algebraic decoder and effectively handles compound channels. Algorithm developed uses syndrome-detecting logic to direct two decoders to assume decoding load alternately, depending on real-time channel characteristics.
Online Algebraic Tools for Teaching
ERIC Educational Resources Information Center
Kurz, Terri L.
2011-01-01
Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)
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…
Elementary maps on nest algebras
NASA Astrophysics Data System (ADS)
Li, Pengtong
2006-08-01
Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.
Linear algebra and image processing
NASA Astrophysics Data System (ADS)
Allali, Mohamed
2010-09-01
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.
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.)
Learning Algebra from Worked Examples
ERIC Educational Resources Information Center
Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.
2014-01-01
For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…
ERIC Educational Resources Information Center
Buerman, Margaret
2007-01-01
Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…
The Algebra of Complex Numbers.
ERIC Educational Resources Information Center
LePage, Wilbur R.
This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…
Thermodynamics. [algebraic structure
NASA Technical Reports Server (NTRS)
Zeleznik, F. J.
1976-01-01
The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.
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…
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.
Symplectic Clifford Algebraic Field Theory.
NASA Astrophysics Data System (ADS)
Dixon, Geoffrey Moore
We develop a mathematical framework on which is built a theory of fermion, scalar, and gauge vector fields. This field theory is shown to be equivalent to the original Weinberg-Salam model of weak and electromagnetic interactions, but since the new framework is more rigid than that on which the original Weinberg-Salam model was built, a concomitant reduction in the number of assumptions lying outside of the framework has resulted. In particular, parity violation is actually hiding within our framework, and with little difficulty we are able to manifest it. The mathematical framework upon which we build our field theory is arrived at along two separate paths. The first is by the marriage of a Clifford algebra and a Lie superalgebra, the result being called a super Clifford algebra. The second is by providing a new characterization for a Clifford algebra employing its generators and a symmetric array of metric coefficients. Subsequently we generalize this characterization to the case of an antisymmetric array of metric coefficients, and we call the algebra which results a symplectic Clifford algebra. It is upon one of these that we build our field theory, and it is shown that this symplectic Clifford algebra is a particular subalgebra of a super Clifford algebra. The final ingredient is the operation of bracketing which involves treating the elements of our algebra as endomorphisms of a particular inner product space, and employing this space and its inner product to provide us with maps from our algebra to the reals. It is this operation which enables us to manifest the parity violation hiding in our algebra.
Quanta of geometry: noncommutative aspects.
Chamseddine, Ali H; Connes, Alain; Mukhanov, Viatcheslav
2015-03-06
In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres, which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin manifolds with large quantized volume are then obtained as solutions. The two algebras M_{2}(H) and M_{4}(C) are obtained, which are the exact constituents of the standard model. Using the two maps from M_{4} to S^{4} the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter, and area quantization of black holes.
Tits Satake projections of homogeneous special geometries
NASA Astrophysics Data System (ADS)
Fré, Pietro; Gargiulo, Floriana; Rosseel, Jan; Rulik, Ksenya; Trigiante, Mario; Van Proeyen, Antoine
2007-01-01
We organize the homogeneous special geometries, describing as well the couplings of D = 6, 5, 4 and 3 supergravities with eight supercharges, in a small number of universality classes. This relates manifolds on which similar types of dynamical solutions can exist. The mathematical ingredient is the Tits Satake projection of real simple Lie algebras, which we extend to all solvable Lie algebras occurring in these homogeneous special geometries. Apart from some exotic cases all the other, 'very special', homogeneous manifolds can be grouped into seven universality classes. The organization of these classes, which capture the essential features of their basic dynamics, commutes with the r- and c-map. Different members are distinguished by different choices of the paint group, a notion discovered in the context of cosmic billiard dynamics of non-maximally supersymmetric supergravities. We comment on the usefulness of this organization in universality class both in relation with cosmic billiard dynamics and with configurations of branes and orbifolds defining special geometry backgrounds.
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)
ERIC Educational Resources Information Center
Cukier, Mimi; Asdourian, Tony; Thakker, Anand
2012-01-01
Geometry provides a natural window into what it is like to do mathematics. In the world of geometry, playful experimentation is often more fruitful than following a procedure, and logic plus a few axioms can open new worlds. Nonetheless, teaching a geometry course in a way that combines both rigor and play can be difficult. Many geometry courses…
A New Reynolds Stress Algebraic Equation Model
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Zhu, Jiang; Lumley, John L.
1994-01-01
A general turbulent constitutive relation is directly applied to propose a new Reynolds stress algebraic equation model. In the development of this model, the constraints based on rapid distortion theory and realizability (i.e. the positivity of the normal Reynolds stresses and the Schwarz' inequality between turbulent velocity correlations) are imposed. Model coefficients are calibrated using well-studied basic flows such as homogeneous shear flow and the surface flow in the inertial sublayer. The performance of this model is then tested in complex turbulent flows including the separated flow over a backward-facing step and the flow in a confined jet. The calculation results are encouraging and point to the success of the present model in modeling turbulent flows with complex geometries.
Quantum algebra of N superspace
Hatcher, Nicolas; Restuccia, A.; Stephany, J.
2007-08-15
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.
Chen, J.; Safro, I.
2011-01-01
Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
Investigating Teacher Noticing of Student Algebraic Thinking
ERIC Educational Resources Information Center
Walkoe, Janet Dawn Kim
2013-01-01
Learning algebra is critical for students in the U.S. today. Algebra concepts provide the foundation for much advanced mathematical content. In addition, algebra serves as a gatekeeper to opportunities such as admission to college. Yet many students in the U.S. struggle in algebra classes. Researchers claim that one reason for these difficulties…
Twisted spectral geometry for the standard model
NASA Astrophysics Data System (ADS)
Martinetti, Pierre
2015-07-01
In noncommutative geometry, the spectral triple of a manifold does not generate bosonic fields, for fluctuations of the Dirac operator vanish. A Connes-Moscovici twist forces the commutative algebra to be multiplied by matrices. Keeping the space of spinors untouched, twisted-fluctuations then yield perturbations of the spin connection. Applied to the spectral triple of the Standard Model, a similar twist yields the scalar field needed to stabilize the vacuum and to make the computation of the Higgs mass compatible with its experimental value.
Central extensions of Lax operator algebras
NASA Astrophysics Data System (ADS)
Schlichenmaier, M.; Sheinman, O. K.
2008-08-01
Lax operator algebras were introduced by Krichever and Sheinman as a further development of Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this paper local cocycles and associated almost-graded central extensions of Lax operator algebras are classified. It is shown that in the case when the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives.
Learning Geometry through Dynamic Geometry Software
ERIC Educational Resources Information Center
Forsythe, Sue
2007-01-01
In this article, the author investigates effective teaching and learning of geometrical concepts using dynamic geometry software (DGS). Based from her students' reactions to her project, the author found that her students' understanding of the concepts was better than if they had learned geometry through paper-based tasks. However, mixing computer…
2015-06-23
AFRL-OSR-VA-TR-2015-0143 DYNAMIC INFORMATION NETWORKS: GEOMETRY, TOPOLOGY, AND STATISTICAL LEARNING FOR THE ARTICULATION OF STRUCTURE Daniel Rockmore...and machine learning . 15. SUBJECT TERMS Information networks, machine learning , link prediction, hyperbolic geometry, multiscale networks, complex...ideas from linear algebra, markov processes, diffusion networks, differential geometry, and machine learning
Asymptotic aspect of derivations in Banach algebras.
Roh, Jaiok; Chang, Ick-Soon
2017-01-01
We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.
Computing Matrix Representations of Filiform Lie Algebras
NASA Astrophysics Data System (ADS)
Ceballos, Manuel; Núñez, Juan; Tenorio, Ángel F.
In this paper, we compute minimal faithful unitriangular matrix representations of filiform Lie algebras. To do it, we use the nilpotent Lie algebra, g_n, formed of n ×n strictly upper-triangular matrices. More concretely, we search the lowest natural number n such that the Lie algebra g_n contains a given filiform Lie algebra, also computing a representative of this algebra. All the computations in this paper have been done using MAPLE 9.5.
Quantum groups: Geometry and applications
Chu, Chong -Sun
1996-05-13
The main theme of this thesis is a study of the geometry of quantum groups and quantum spaces, with the hope that they will be useful for the construction of quantum field theory with quantum group symmetry. The main tool used is the Faddeev-Reshetikhin-Takhtajan description of quantum groups. A few content-rich examples of quantum complex spaces with quantum group symmetry are treated in details. In chapter 1, the author reviews some of the basic concepts and notions for Hopf algebras and other background materials. In chapter 2, he studies the vector fields of quantum groups. A compact realization of these vector fields as pseudodifferential operators acting on the linear quantum spaces is given. In chapter 3, he describes the quantum sphere as a complex quantum manifold by means of a quantum stereographic projection. A covariant calculus is introduced. An interesting property of this calculus is the existence of a one-form realization of the exterior differential operator. The concept of a braided comodule is introduced and a braided algebra of quantum spheres is constructed. In chapter 4, the author considers the more general higher dimensional quantum complex projective spaces and the quantum Grassman manifolds. Differential calculus, integration and braiding can be introduced as in the one dimensional case. Finally, in chapter 5, he studies the framework of quantum principal bundle and construct the q-deformed Dirac monopole as a quantum principal bundle with a quantum sphere as the base and a U(1) with non-commutative calculus as the fiber. The first Chern class can be introduced and integrated to give the monopole charge.
Cartooning in Algebra and Calculus
ERIC Educational Resources Information Center
Moseley, L. Jeneva
2014-01-01
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
GCD, LCM, and Boolean Algebra?
ERIC Educational Resources Information Center
Cohen, Martin P.; Juraschek, William A.
1976-01-01
This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)
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.
Coherent States for Hopf Algebras
NASA Astrophysics Data System (ADS)
Škoda, Zoran
2007-07-01
Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. If, in addition, the Hopf algebra has a left Haar integral, then a formula for noncommutative resolution of identity in terms of the family of coherent states holds. Examples come from quantum groups.
Multiplier operator algebras and applications
Blecher, David P.; Zarikian, Vrej
2004-01-01
The one-sided multipliers of an operator space X are a key to “latent operator algebraic structure” in X. We begin with a survey of these multipliers, together with several of the applications that they have had to operator algebras. We then describe several new results on one-sided multipliers, and new applications, mostly to one-sided M-ideals. PMID:14711990
Hopf algebras and topological recursion
NASA Astrophysics Data System (ADS)
Esteves, João N.
2015-11-01
We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).
Triangles with Integer Side Lengths and Rational Internal Radius P and External Radius R
ERIC Educational Resources Information Center
Zelator, Konstantine
2005-01-01
This paper is written on a level accessible to college/university students of mathematics who are taking second-year, algebra based, mathematics courses beyond calculus I. This article combines material from geometry, trigonometry, and number theory. This integration of various techniques is an excellent experience for the serious student. The…
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…
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.…
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…
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)
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…
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,…
46 CFR 310.55 - Scholastic requirements.
Code of Federal Regulations, 2012 CFR
2012-10-01
... Mathematics (from algebra, geometry and trigonometry); (B) 3 units of English; and (C) 1 unit of Physics or Chemistry. (ii) 8 other units, preferably chosen from the following fields: (A) Additional mathematics and... candidate electing to use the College Board shall take both the mathematics and the verbal section of...
46 CFR 310.55 - Scholastic requirements.
Code of Federal Regulations, 2014 CFR
2014-10-01
... Mathematics (from algebra, geometry and trigonometry); (B) 3 units of English; and (C) 1 unit of Physics or Chemistry. (ii) 8 other units, preferably chosen from the following fields: (A) Additional mathematics and... candidate electing to use the College Board shall take both the mathematics and the verbal section of...
46 CFR 310.55 - Scholastic requirements.
Code of Federal Regulations, 2013 CFR
2013-10-01
... Mathematics (from algebra, geometry and trigonometry); (B) 3 units of English; and (C) 1 unit of Physics or Chemistry. (ii) 8 other units, preferably chosen from the following fields: (A) Additional mathematics and... candidate electing to use the College Board shall take both the mathematics and the verbal section of...
46 CFR 310.55 - Scholastic requirements.
Code of Federal Regulations, 2011 CFR
2011-10-01
... Mathematics (from algebra, geometry and trigonometry); (B) 3 units of English; and (C) 1 unit of Physics or Chemistry. (ii) 8 other units, preferably chosen from the following fields: (A) Additional mathematics and... candidate electing to use the College Board shall take both the mathematics and the verbal section of...
46 CFR 310.55 - Scholastic requirements.
Code of Federal Regulations, 2010 CFR
2010-10-01
... Mathematics (from algebra, geometry and trigonometry); (B) 3 units of English; and (C) 1 unit of Physics or Chemistry. (ii) 8 other units, preferably chosen from the following fields: (A) Additional mathematics and... candidate electing to use the College Board shall take both the mathematics and the verbal section of...
The local geometry of compact homogeneous Lorentz spaces
NASA Astrophysics Data System (ADS)
Günther, Felix
2015-03-01
In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.
Novikov algebras with associative bilinear forms
NASA Astrophysics Data System (ADS)
Zhu, Fuhai; Chen, Zhiqi
2007-11-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. The goal of this paper is to study Novikov algebras with non-degenerate associative symmetric bilinear forms, which we call quadratic Novikov algebras. Based on the classification of solvable quadratic Lie algebras of dimension not greater than 4 and Novikov algebras in dimension 3, we show that quadratic Novikov algebras up to dimension 4 are commutative. Furthermore, we obtain the classification of transitive quadratic Novikov algebras in dimension 4. But we find that not every quadratic Novikov algebra is commutative and give a non-commutative quadratic Novikov algebra in dimension 6.
ERIC Educational Resources Information Center
McDonald, Nathaniel J.
2001-01-01
Chronicles a teacher's first year teaching geometry at the Hershey Montessori Farm School in Huntsburg, Ohio. Instructional methods relied on Euclid primary readings and combined pure abstract logic with practical applications of geometry on the land. The course included geometry background imparted by Montessori elementary materials as well as…
Noncommutative algebras, nano-structures, and quantum dynamics generated by resonances. III
NASA Astrophysics Data System (ADS)
Karasev, M.
2006-04-01
Quantum geometry, algebras with nonlinear commutation relations, representation theory, the coherent transform, and operator averaging are used to solve the perturbation problem for wave (quantum) systems near a resonance equilibrium point or a resonance equilibrium ray in two and three-dimensional spaces.
Learning Activity Package, Algebra 103-104, LAPs 23-33.
ERIC Educational Resources Information Center
Evans, Diane
This set of 11 teacher-prepared Learning Activity Packages (LAPs) in intermediate algebra covers number systems; exponents and radicals; polynomials and factoring; rational expressions; coordinate geometry; relations, functions, and inequalities; quadratic equations and inequalities; Quadratic functions; systems of equations and inequalities;…
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…
Quantum Q systems: from cluster algebras to quantum current algebras
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Kedem, Rinat
2017-02-01
This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the A_r quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593-2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra U_{√{q}}({n}[u,u^{-1}])subset U_{√{q}}(widehat{{sl}}_2), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97-152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.
Algebraically special solutions in AdS/CFT
NASA Astrophysics Data System (ADS)
de Freitas, Gabriel Bernardi; Reall, Harvey S.
2014-06-01
We investigate the AdS/CFT interpretation of the class of algebraically special solutions of Einstein gravity with a negative cosmological constant. Such solutions describe a CFT living in a 2 + 1 dimensional time-dependent geometry that, generically, has no isometries. The algebraically special condition implies that the expectation value of the CFT energy-momentum tensor is a local function of the boundary metric. When such a spacetime is slowly varying, the fluid/gravity approximation is valid and one can read off the values of certain higher order transport coefficients. To do this, we introduce a formalism for studying conformal, relativistic fluids in 2 + 1 dimensions that reduces everything to the manipulation of scalar quantities.
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.
Lambda modes of the neutron diffusion equation in hexagonal geometry
Barrachina, T.; Ginestar, D.; Verdu, G.
2006-07-01
A nodal collocation method is proposed to compute the dominant Lambda modes of nuclear reactor core with a hexagonal geometry. This method is based on a triangular mesh and assumes that the neutronic flux can be approximated as a finite expansion in terms of Dubiner's polynomials. The method transforms the initial differential eigenvalue problem into a generalized algebraic one, from which the dominant modes of the reactor can be computed. The performance of the method is tested with two benchmark problems. (authors)
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-07-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
ERIC Educational Resources Information Center
Portnoy, Neil; Grundmeier, Todd A.; Graham, Karen J.
2006-01-01
This research explored students' views of geometric objects through the implementation of a curriculum module that allowed them to explore the relationships between transformational geometry and linear algebra. The majority of the students were middle and secondary mathematics education majors enrolled in a one-semester geometry course that is…
Colored Quantum Algebra and Its Bethe State
NASA Astrophysics Data System (ADS)
Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun
2014-12-01
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.
Using Number Theory to Reinforce Elementary Algebra.
ERIC Educational Resources Information Center
Covillion, Jane D.
1995-01-01
Demonstrates that using the elementary number theory in algebra classes helps students to use acquired algebraic skills as well as helping them to more clearly understand concepts that are presented. Discusses factoring, divisibility rules, and number patterns. (AIM)
Algebraic orbifold conformal field theories
Xu, Feng
2000-01-01
The unitary rational orbifold conformal field theories in the algebraic quantum field theory and subfactor theory framework are formulated. Under general conditions, it is shown that the orbifold of a given unitary rational conformal field theory generates a unitary modular category. Many new unitary modular categories are obtained. It is also shown that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time. PMID:11106383
Scalable Parallel Algebraic Multigrid Solvers
Bank, R; Lu, S; Tong, C; Vassilevski, P
2005-03-23
The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.
Symmetry algebras of linear differential equations
NASA Astrophysics Data System (ADS)
Shapovalov, A. V.; Shirokov, I. V.
1992-07-01
The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.
Spatial-Operator Algebra For Robotic Manipulators
NASA Technical Reports Server (NTRS)
Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.
1991-01-01
Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.
Applications of Algebraic Logic and Universal Algebra to Computer Science
1989-06-21
conference, with roughly equal representation from Mathematics and Computer Science . The conference consisted of eight invited lectures (60 minutes...each) and 26 contributed talks (20-40 minutes each). There was also a round-table discussion on the role of algebra and logic in computer science . Keywords
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.
A Balancing Act: Making Sense of Algebra
ERIC Educational Resources Information Center
Gavin, M. Katherine; Sheffield, Linda Jensen
2015-01-01
For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…
Algebra? A Gate! A Barrier! A Mystery!
ERIC Educational Resources Information Center
Mathematics Educatio Dialogues, 2000
2000-01-01
This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…
Unifying the Algebra for All Movement
ERIC Educational Resources Information Center
Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.
2015-01-01
There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…
Constraint-Referenced Analytics of Algebra Learning
ERIC Educational Resources Information Center
Sutherland, Scot M.; White, Tobin F.
2016-01-01
The development of the constraint-referenced analytics tool for monitoring algebra learning activities presented here came from the desire to firstly, take a more quantitative look at student responses in collaborative algebra activities, and secondly, to situate those activities in a more traditional introductory algebra setting focusing on…
Embedding Algebraic Thinking throughout the Mathematics Curriculum
ERIC Educational Resources Information Center
Vennebush, G. Patrick; Marquez, Elizabeth; Larsen, Joseph
2005-01-01
This article explores the algebra that can be uncovered in many middle-grades mathematics tasks that, on first inspection, do not appear to be algebraic. It shows connections to the other four Standards that occur in traditional algebra problems, and it offers strategies for modifying activities so that they can be used to foster algebraic…
Teaching Strategies to Improve Algebra Learning
ERIC Educational Resources Information Center
Zbiek, Rose Mary; Larson, Matthew R.
2015-01-01
Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…
Build an Early Foundation for Algebra Success
ERIC Educational Resources Information Center
Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela
2016-01-01
Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…
Teacher Actions to Facilitate Early Algebraic Reasoning
ERIC Educational Resources Information Center
Hunter, Jodie
2015-01-01
In recent years there has been an increased emphasis on integrating the teaching of arithmetic and algebra in primary school classrooms. This requires teachers to develop links between arithmetic and algebra and use pedagogical actions that facilitate algebraic reasoning. Drawing on findings from a classroom-based study, this paper provides an…
Difficulties in Initial Algebra Learning in Indonesia
ERIC Educational Resources Information Center
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-01-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…
Developments in special geometry
NASA Astrophysics Data System (ADS)
Mohaupt, Thomas; Vaughan, Owen
2012-02-01
We review the special geometry of Script N = 2 supersymmetric vector and hypermultiplets with emphasis on recent developments and applications. A new formulation of the local c-map based on the Hesse potential and special real coordinates is presented. Other recent developments include the Euclidean version of special geometry, and generalizations of special geometry to non-supersymmetric theories. As applications we disucss the proof that the local r-map and c-map preserve geodesic completeness, and the construction of four- and five-dimensional static solutions through dimensional reduction over time. The shared features of the real, complex and quaternionic version of special geometry are stressed throughout.
Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*
Bank, R.; Falgout, R. D.; Jones, T.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. W.
2015-10-29
In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.
Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*
Bank, R.; Falgout, R. D.; Jones, T.; ...
2015-10-29
In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods inmore » Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.« less
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.
Carry Groups: Abstract Algebra Projects
ERIC Educational Resources Information Center
Miller, Cheryl Chute; Madore, Blair F.
2004-01-01
Carry Groups are a wonderful collection of groups to introduce in an undergraduate Abstract Algebra course. These groups are straightforward to define but have interesting structures for students to discover. We describe these groups and give examples of in-class group projects that were developed and used by Miller.
Algebra, Home Mortgages, and Recessions
ERIC Educational Resources Information Center
Mariner, Jean A. Miller; Miller, Richard A.
2009-01-01
The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…
Exploring Algebraic Misconceptions with Technology
ERIC Educational Resources Information Center
Sakow, Matthew; Karaman, Ruveyda
2015-01-01
Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…
Easing Students' Transition to Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2006-01-01
Traditionally, students learn arithmetic throughout their primary schooling, and this is seen as the ideal preparation for the learning of algebra in the junior secondary school. The four operations are taught and rehearsed in the early years and from this, it is assumed, "children will induce the fundamental structure of arithmetic" (Warren &…
Algebra for All. Research Brief
ERIC Educational Resources Information Center
Bleyaert, Barbara
2009-01-01
The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless…
Algebraic methods in system theory
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Willems, J. C.; Willsky, A. S.
1975-01-01
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
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…
Adventures in Flipping College Algebra
ERIC Educational Resources Information Center
Van Sickle, Jenna
2015-01-01
This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…
Elementary Algebra Connections to Precalculus
ERIC Educational Resources Information Center
Lopez-Boada, Roberto; Daire, Sandra Arguelles
2013-01-01
This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…
Celestial mechanics with geometric algebra
NASA Technical Reports Server (NTRS)
Hestenes, D.
1983-01-01
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
Algebraic Activities Aid Discovery Lessons
ERIC Educational Resources Information Center
Wallace-Gomez, Patricia
2013-01-01
After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…
Teachers' Understanding of Algebraic Generalization
NASA Astrophysics Data System (ADS)
Hawthorne, Casey Wayne
Generalization has been identified as a cornerstone of algebraic thinking (e.g., Lee, 1996; Sfard, 1995) and is at the center of a rich conceptualization of K-8 algebra (Kaput, 2008; Smith, 2003). Moreover, mathematics teachers are being encouraged to use figural-pattern generalizing tasks as a basis of student-centered instruction, whereby teachers respond to and build upon the ideas that arise from students' explorations of these activities. Although more and more teachers are engaging their students in such generalizing tasks, little is known about teachers' understanding of generalization and their understanding of students' mathematical thinking in this domain. In this work, I addressed this gap, exploring the understanding of algebraic generalization of 4 exemplary 8th-grade teachers from multiple perspectives. A significant feature of this investigation is an examination of teachers' understanding of the generalization process, including the use of algebraic symbols. The research consisted of two phases. Phase I was an examination of the teachers' understandings of the underlying quantities and quantitative relationships represented by algebraic notation. In Phase II, I observed the instruction of 2 of these teachers. Using the lens of professional noticing of students' mathematical thinking, I explored the teachers' enacted knowledge of algebraic generalization, characterizing how it supported them to effectively respond to the needs and queries of their students. Results indicated that teachers predominantly see these figural patterns as enrichment activities, disconnected from course content. Furthermore, in my analysis, I identified conceptual difficulties teachers experienced when solving generalization tasks, in particular, connecting multiple symbolic representations with the quantities in the figures. Moreover, while the teachers strived to overcome the challenges of connecting different representations, they invoked both productive and unproductive
Explicit field realizations of W algebras
NASA Astrophysics Data System (ADS)
Wei, Shao-Wen; Liu, Yu-Xiao; Zhang, Li-Jie; Ren, Ji-Rong
2009-06-01
The fact that certain nonlinear W2,s algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize W2,s algebras from linear W1,2,s algebras. In this paper, we first construct the explicit field realizations of linear W1,2,s algebras with double scalar and double spinor, respectively. Then, after a change of basis, the realizations of W2,s algebras are presented. The results show that all these realizations are Romans-type realizations.
Array algebra estimation in signal processing
NASA Astrophysics Data System (ADS)
Rauhala, U. A.
A general theory of linear estimators called array algebra estimation is interpreted in some terms of multidimensional digital signal processing, mathematical statistics, and numerical analysis. The theory has emerged during the past decade from the new field of a unified vector, matrix and tensor algebra called array algebra. The broad concepts of array algebra and its estimation theory cover several modern computerized sciences and technologies converting their established notations and terminology into one common language. Some concepts of digital signal processing are adopted into this language after a review of the principles of array algebra estimation and its predecessors in mathematical surveying sciences.
On special classes of n-algebras
NASA Astrophysics Data System (ADS)
Vainerman, L.; Kerner, R.
1996-05-01
We define n-algebras as linear spaces on which the internal composition law involves n elements: m:V⊗n■V. It is known that such algebraic structures are interesting for their applications to problems of modern mathematical physics. Using the notion of a commutant of two subalgebras of an n-algebra, we distinguish certain classes of n-algebras with reasonable properties: semisimple, Abelian, nilpotent, solvable. We also consider a few examples of n-algebras of different types, and show their properties.
Geometry of multihadron production
Bjorken, J.D.
1994-10-01
This summary talk only reviews a small sample of topics featured at this symposium: Introduction; The Geometry and Geography of Phase space; Space-Time Geometry and HBT; Multiplicities, Intermittency, Correlations; Disoriented Chiral Condensate; Deep Inelastic Scattering at HERA; and Other Contributions.
ERIC Educational Resources Information Center
Lyublinskaya, Irina; Funsch, Dan
2012-01-01
Several interactive geometry software packages are available today to secondary school teachers. An example is The Geometer's Sketchpad[R] (GSP), also known as Dynamic Geometry[R] software, developed by Key Curriculum Press. This numeric based technology has been widely adopted in the last twenty years, and a vast amount of creativity has been…
Euclidean Geometry via Programming.
ERIC Educational Resources Information Center
Filimonov, Rossen; Kreith, Kurt
1992-01-01
Describes the Plane Geometry System computer software developed at the Educational Computer Systems laboratory in Sofia, Bulgaria. The system enables students to use the concept of "algorithm" to correspond to the process of "deductive proof" in the development of plane geometry. Provides an example of the software's capability…
ERIC Educational Resources Information Center
Morris, Barbara H.
2004-01-01
This article describes a geometry project that used the beauty of stained-glass-window designs to teach middle school students about geometric figures and concepts. Three honors prealgebra teachers and a middle school mathematics gifted intervention specialist created a geometry project that covered the curriculum and also assessed students'…
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.
HOMAR: A computer code for generating homotopic grids using algebraic relations: User's manual
NASA Technical Reports Server (NTRS)
Moitra, Anutosh
1989-01-01
A computer code for fast automatic generation of quasi-three-dimensional grid systems for aerospace configurations is described. The code employs a homotopic method to algebraically generate two-dimensional grids in cross-sectional planes, which are stacked to produce a three-dimensional grid system. Implementation of the algebraic equivalents of the homotopic relations for generating body geometries and grids are explained. Procedures for controlling grid orthogonality and distortion are described. Test cases with description and specification of inputs are presented in detail. The FORTRAN computer program and notes on implementation and use are included.
HOMAR: A computer code for generating homotopic grids using algebraic relations: User's manual
NASA Astrophysics Data System (ADS)
Moitra, Anutosh
1989-07-01
A computer code for fast automatic generation of quasi-three-dimensional grid systems for aerospace configurations is described. The code employs a homotopic method to algebraically generate two-dimensional grids in cross-sectional planes, which are stacked to produce a three-dimensional grid system. Implementation of the algebraic equivalents of the homotopic relations for generating body geometries and grids are explained. Procedures for controlling grid orthogonality and distortion are described. Test cases with description and specification of inputs are presented in detail. The FORTRAN computer program and notes on implementation and use are included.
On the applications of algebraic grid generation methods based on transfinite interpolation
NASA Technical Reports Server (NTRS)
Nguyen, Hung Lee
1989-01-01
Algebraic grid generation methods based on transfinite interpolation called the two-boundary and four-boundary methods are applied for generating grids with highly complex boundaries. These methods yield grid point distributions that allow for accurate application to regions of sharp gradients in the physical domain or time-dependent problems with small length scale phenomena. Algebraic grids are derived using the two-boundary and four-boundary methods for applications in both two- and three-dimensional domains. Grids are developed for distinctly different geometrical problems and the two-boundary and four-boundary methods are demonstrated to be applicable to a wide class of geometries.
Frolov, Vadim A; Escalada, Artur; Akimov, Sergey A; Shnyrova, Anna V
2015-01-01
Cellular membranes define the functional geometry of intracellular space. Formation of new membrane compartments and maintenance of complex organelles require division and disconnection of cellular membranes, a process termed membrane fission. Peripheral membrane proteins generally control membrane remodeling during fission. Local membrane stresses, reflecting molecular geometry of membrane-interacting parts of these proteins, sum up to produce the key membrane geometries of fission: the saddle-shaped neck and hour-glass hemifission intermediate. Here, we review the fundamental principles behind the translation of molecular geometry into membrane shape and topology during fission. We emphasize the central role the membrane insertion of specialized protein domains plays in orchestrating fission in vitro and in cells. We further compare individual to synergistic action of the membrane insertion during fission mediated by individual protein species, proteins complexes or membrane domains. Finally, we describe how local geometry of fission intermediates defines the functional design of the protein complexes catalyzing fission of cellular membranes.
Recursion and feedback in image algebra
NASA Astrophysics Data System (ADS)
Ritter, Gerhard X.; Davidson, Jennifer L.
1991-04-01
Recursion and feedback are two important processes in image processing. Image algebra, a unified algebraic structure developed for use in image processing and image analysis, provides a common mathematical environment for expressing image processing transforms. It is only recently that image algebra has been extended to include recursive operations [1]. Recently image algebra was shown to incorporate neural nets [2], including a new type of neural net, the morphological neural net [3]. This paper presents the relationship of the recursive image algebra to the field of fractions of the ring of matrices, and gives the two dimensional moving average filter as an example. Also, the popular multilayer perceptron with back propagation and a morphology neural network with learning rule are presented in image algebra notation. These examples show that image algebra can express these important feedback concepts in a succinct way.
Deformed Kac Moody and Virasoro algebras
NASA Astrophysics Data System (ADS)
Balachandran, A. P.; Queiroz, A. R.; Marques, A. M.; Teotonio-Sobrinho, P.
2007-07-01
Whenever the group {\\bb R}^n acts on an algebra {\\cal A} , there is a method to twist \\cal A to a new algebra {\\cal A}_\\theta which depends on an antisymmetric matrix θ (θμν = -θνμ = constant). The Groenewold-Moyal plane {\\cal A}_\\theta({\\bb R}^{d+1}) is an example of such a twisted algebra. We give a general construction to realize this twist in terms of {\\cal A} itself and certain 'charge' operators Qμ. For {\\cal A}_\\theta({\\bb R}^{d+1}), Q_\\mu are translation generators. This construction is then applied to twist the oscillators realizing the Kac-Moody (KM) algebra as well as the KM currents. They give different deformations of the KM algebra. From one of the deformations of the KM algebra, we construct, via the Sugawara construction, the Virasoro algebra. These deformations have an implication for statistics as well.
Algebraic complexities and algebraic curves over finite fields
Chudnovsky, D. V.; Chudnovsky, G. V.
1987-01-01
We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields. PMID:16593816
On the Hermitian Projective Line as a Home for the Geometry of Quantum Theory
Bertram, Wolfgang
2008-11-18
In the paper [1], generalized projective geometries have been proposed as a framework for a geometric formulation of Quantum Theory. In the present note, we refine this proposition by discussing further structural features of Quantum Theory: the link with associative involutive algebras A and with Jordan-Lie and Lie-Jordan algebas. The associated geometries are (Hermitian) projective lines over A; their axiomatic definition and theory will be given in subsequent work with M. Kinyon [2].
Differential geometry of groups in string theory
Schmidke, W.B. Jr.
1990-09-01
Techniques from differential geometry and group theory are applied to two topics from string theory. The first topic studied is quantum groups, with the example of GL (1{vert bar}1). The quantum group GL{sub q}(1{vert bar}1) is introduced, and an exponential description is derived. The algebra and coproduct are determined using the invariant differential calculus method introduced by Woronowicz and generalized by Wess and Zumino. An invariant calculus is also introduced on the quantum superplane, and a representation of the algebra of GL{sub q}(1{vert bar}1) in terms of the super-plane coordinates is constructed. The second topic follows the approach to string theory introduced by Bowick and Rajeev. Here the ghost contribution to the anomaly of the energy-momentum tensor is calculated as the Ricci curvature of the Kaehler quotient space Diff(S{sup 1})/S{sup 1}. We discuss general Kaehler quotient spaces and derive an expression for their Ricci curvatures. Application is made to the string and superstring diffeomorphism groups, considering all possible choices of subgroup. The formalism is extended to associated holomorphic vector bundles, where the Ricci curvature corresponds to the anomaly for different ghost sea levels. 26 refs.
Alternative algebraic approaches in quantum chemistry
Mezey, Paul G.
2015-01-22
Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.
Flyby Geometry Optimization Tool
NASA Technical Reports Server (NTRS)
Karlgaard, Christopher D.
2007-01-01
The Flyby Geometry Optimization Tool is a computer program for computing trajectories and trajectory-altering impulsive maneuvers for spacecraft used in radio relay of scientific data to Earth from an exploratory airplane flying in the atmosphere of Mars.
ERIC Educational Resources Information Center
Chern, Shiing-Shen
1990-01-01
Discussed are the major historical developments of geometry. Euclid, Descartes, Klein's Erlanger Program, Gaus and Riemann, globalization, topology, Elie Cartan, and an application to molecular biology are included as topics. (KR)
ERIC Educational Resources Information Center
Emenaker, Charles E.
1999-01-01
Describes a sixth-grade interdisciplinary geometry unit based on Charles Dickens's "A Christmas Carol". Focuses on finding area, volume, and perimeter, and working with estimation, decimals, and fractions in the context of making gingerbread houses. (ASK)
Facilitating Understandings of Geometry.
ERIC Educational Resources Information Center
Pappas, Christine C.; Bush, Sara
1989-01-01
Illustrates some learning encounters for facilitating first graders' understanding of geometry. Describes some of children's approaches using Cuisenaire rods and teacher's intervening. Presents six problems involving various combinations of Cuisenaire rods and cubes. (YP)
Proof in Transformation Geometry
ERIC Educational Resources Information Center
Bell, A. W.
1971-01-01
The first of three articles showing how inductively-obtained results in transformation geometry may be organized into a deductive system. This article discusses two approaches to enlargement (dilatation), one using coordinates and the other using synthetic methods. (MM)
Algebraic Methods to Design Signals
2015-08-27
group theory are employed to investigate the theory of their construction methods leading to new families of these arrays and some generalizations...sequences and arrays with desirable correlation properties. The methods used are very algebraic and number theoretic. Many new families of sequences...context of optical quantum computing, we prove that infinite families of anticirculant block weighing matrices can be obtained from generic weighing
A Sourcebook of Problems for Geometry Based upon Industrial Design and Architectural Ornament.
ERIC Educational Resources Information Center
Sykes, Mabel
This updated reprint of a classic work presents design analysis of geometric patterns and information helpful to constructing mathematical drawings of industrial and achitectural features. Both simple and complex designs are given. Problems combine both algebra and geometry. The work is divided into six chapters which are further divided into…
Computer algebra and transport theory.
Warsa, J. S.
2004-01-01
Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.
BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1994-01-01
The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.
Introduction to Image Algebra Ada
NASA Astrophysics Data System (ADS)
Wilson, Joseph N.
1991-07-01
Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.
Buchele, S.F.; Ellingson, W.A.
1997-06-01
Recent advances in reverse engineering have focused on recovering a boundary representation (b-rep) of an object, often for integration with rapid prototyping. This boundary representation may be a 3-D point cloud, a triangulation of points, or piecewise algebraic or parametric surfaces. This paper presents work in progress to develop an algorithm to extend the current state of the art in reverse engineering of mechanical parts. This algorithm will take algebraic surface representations as input and will produce a constructive solid geometry (CSG) description that uses solid primitives such as rectangular block, pyramid, sphere, cylinder, and cone. The proposed algorithm will automatically generate a CSG solid model of a part given its algebraic b-rep, thus allowing direct input into a CAD system and subsequent CSG model generation.
Tautges, Timothy J.
2005-01-01
The Common Geometry Module (CGM) is a code library which provides geometry functionality used for mesh generation and other applications. This functionality includes that commonly found in solid modeling engines, like geometry creation, query and modification; CGM also includes capabilities not commonly found in solid modeling engines, like geometry decomposition tools and support for shared material interfaces. CGM is built upon the ACIS solid modeling engine, but also includes geometry capability developed beside and on top of ACIS. CGM can be used as-is to provide geometry functionality for codes needing this capability. However, CGM can also be extended using derived classes in C++, allowing the geometric model to serve as the basis for other applications, for example mesh generation. CGM is supported on Sun Solaris, SGI, HP, IBM, DEC, Linux and Windows NT platforms. CGM also indudes support for loading ACIS models on parallel computers, using MPI-based communication. Future plans for CGM are to port it to different solid modeling engines, including Pro/Engineer or SolidWorks. CGM is being released into the public domain under an LGPL license; the ACIS-based engine is available to ACIS licensees on request.
Software Geometry in Simulations
NASA Astrophysics Data System (ADS)
Alion, Tyler; Viren, Brett; Junk, Tom
2015-04-01
The Long Baseline Neutrino Experiment (LBNE) involves many detectors. The experiment's near detector (ND) facility, may ultimately involve several detectors. The far detector (FD) will be significantly larger than any other Liquid Argon (LAr) detector yet constructed; many prototype detectors are being constructed and studied to motivate a plethora of proposed FD designs. Whether it be a constructed prototype or a proposed ND/FD design, every design must be simulated and analyzed. This presents a considerable challenge to LBNE software experts; each detector geometry must be described to the simulation software in an efficient way which allows for multiple authors to easily collaborate. Furthermore, different geometry versions must be tracked throughout their use. We present a framework called General Geometry Description (GGD), written and developed by LBNE software collaborators for managing software to generate geometries. Though GGD is flexible enough to be used by any experiment working with detectors, we present it's first use in generating Geometry Description Markup Language (GDML) files to interface with LArSoft, a framework of detector simulations, event reconstruction, and data analyses written for all LAr technology users at Fermilab. Brett is the other of the framework discussed here, the General Geometry Description (GGD).
Algebra: A Challenge at the Crossroads of Policy and Practice
ERIC Educational Resources Information Center
Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.
2011-01-01
The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
NASA Astrophysics Data System (ADS)
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
Numerical linear algebra algorithms and software
NASA Astrophysics Data System (ADS)
Dongarra, Jack J.; Eijkhout, Victor
2000-11-01
The increasing availability of advanced-architecture computers has a significant effect on all spheres of scientific computation, including algorithm research and software development in numerical linear algebra. Linear algebra - in particular, the solution of linear systems of equations - lies at the heart of most calculations in scientific computing. This paper discusses some of the recent developments in linear algebra designed to exploit these advanced-architecture computers. We discuss two broad classes of algorithms: those for dense, and those for sparse matrices.
Symbolic Lie algebras manipulations using COMMON LISP
NASA Astrophysics Data System (ADS)
Cecchini, R.; Tarlini, M.
1989-01-01
We present a description and an implementation of a program in COMMON LISP to perform symbolic computations in a given Lie algebra. Using the general definitions of vector space Lie algebra and enveloping algebra, the program is able to compute commutators, to evaluate similarity transformations and the general Baker-Campbell-Hausdorff formula. All the computations are exact, including numerical coefficients. For the interactive user an optional menu facility and online help are available. LISP knowledge is unnecessary.
Lie algebras of classical and stochastic electrodynamics
NASA Astrophysics Data System (ADS)
Neto, J. J. Soares; Vianna, J. D. M.
1994-03-01
The Lie algebras associated with infinitesimal symmetry transformations of third-order differential equations of interest to classical electrodynamics and stochastic electrodynamics have been obtained. The structure constants for a general case are presented and the Lie algebra for each particular application is easily achieved. By the method used here it is not necessary to know the explicit expressions of the infinitesimal generators in order to determine the structure constants of the Lie algebra.
NASA Astrophysics Data System (ADS)
Manerowska, Anna; Nieznański, Edward; Mulawka, Jan
2013-10-01
Our aim is to present the algebra of concepts in two formal languages. First, after introducing a primary relation between concepts, which is subsumption, we shall specify in a language that uses quantifiers, the Boolean algebra of general concepts. Next, we shall note down the same algebra in simplified non-quantifying language, in order to use it as basis for two specific implementations, i.e. to create the Boolean algebras of deontic concepts and axiological concepts.
Dispersion Operators Algebra and Linear Canonical Transformations
NASA Astrophysics Data System (ADS)
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-04-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
Dispersion Operators Algebra and Linear Canonical Transformations
NASA Astrophysics Data System (ADS)
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-02-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
Banach Algebras Associated to Lax Pairs
NASA Astrophysics Data System (ADS)
Glazebrook, James F.
2015-04-01
Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall- Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose-Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand-Naimark-Segal construction. Instrumental is the Plücker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan-Lie-Banach algebras structures can be compared.
Difficulties in initial algebra learning in Indonesia
NASA Astrophysics Data System (ADS)
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-12-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.
Multicloning and Multibroadcasting in Operator Algebras
NASA Astrophysics Data System (ADS)
Kaniowski, Krzysztof; Lubnauer, Katarzyna; Łuczak, Andrzej
2015-12-01
We investigate multicloning and multibroadcasting in the general operator algebra framework in arbitrary dimension, generalizing thus results obtained in this framework for simple cloning and broadcasting.
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…
ERIC Educational Resources Information Center
Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos
2016-01-01
This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…
Integrable Background Geometries
NASA Astrophysics Data System (ADS)
Calderbank, David M. J.
2014-03-01
This work has its origins in an attempt to describe systematically the integrable geometries and gauge theories in dimensions one to four related to twistor theory. In each such dimension, there is a nondegenerate integrable geometric structure, governed by a nonlinear integrable differential equation, and each solution of this equation determines a background geometry on which, for any Lie group G, an integrable gauge theory is defined. In four dimensions, the geometry is selfdual conformal geometry and the gauge theory is selfdual Yang-Mills theory, while the lower-dimensional structures are nondegenerate (i.e., non-null) reductions of this. Any solution of the gauge theory on a k-dimensional geometry, such that the gauge group H acts transitively on an ℓ-manifold, determines a (k+ℓ)-dimensional geometry (k+ℓ≤4) fibering over the k-dimensional geometry with H as a structure group. In the case of an ℓ-dimensional group H acting on itself by the regular representation, all (k+ℓ)-dimensional geometries with symmetry group H are locally obtained in this way. This framework unifies and extends known results about dimensional reductions of selfdual conformal geometry and the selfdual Yang-Mills equation, and provides a rich supply of constructive methods. In one dimension, generalized Nahm equations provide a uniform description of four pole isomonodromic deformation problems, and may be related to the {SU}(∞) Toda and dKP equations via a hodograph transformation. In two dimensions, the {Diff}(S^1) Hitchin equation is shown to be equivalent to the hyperCR Einstein-Weyl equation, while the {SDiff}(Σ^2) Hitchin equation leads to a Euclidean analogue of Plebanski's heavenly equations. In three and four dimensions, the constructions of this paper help to organize the huge range of examples of Einstein-Weyl and selfdual spaces in the literature, as well as providing some new ! ones. The nondegenerate reductions have a long ancestry. More ! recently
2011-01-01
Cells are highly complex and orderly machines, with defined shapes and a startling variety of internal organizations. Complex geometry is a feature of both free-living unicellular organisms and cells inside multicellular animals. Where does the geometry of a cell come from? Many of the same questions that arise in developmental biology can also be asked of cells, but in most cases we do not know the answers. How much of cellular organization is dictated by global cell polarity cues as opposed to local interactions between cellular components? Does cellular structure persist across cell generations? What is the relationship between cell geometry and tissue organization? What ensures that intracellular structures are scaled to the overall size of the cell? Cell biology is only now beginning to come to grips with these questions. PMID:21880160
NASA Astrophysics Data System (ADS)
Ochiai, T.; Nacher, J. C.
2011-09-01
Recently, the application of geometry and conformal mappings to artificial materials (metamaterials) has attracted the attention in various research communities. These materials, characterized by a unique man-made structure, have unusual optical properties, which materials found in nature do not exhibit. By applying the geometry and conformal mappings theory to metamaterial science, it may be possible to realize so-called "Harry Potter cloaking device". Although such a device is still in the science fiction realm, several works have shown that by using such metamaterials it may be possible to control the direction of the electromagnetic field at will. We could then make an object hidden inside of a cloaking device. Here, we will explain how to design invisibility device using differential geometry and conformal mappings.
Students Discovering Spherical Geometry Using Dynamic Geometry Software
ERIC Educational Resources Information Center
Guven, Bulent; Karatas, Ilhan
2009-01-01
Dynamic geometry software (DGS) such as Cabri and Geometers' Sketchpad has been regularly used worldwide for teaching and learning Euclidean geometry for a long time. The DGS with its inductive nature allows students to learn Euclidean geometry via explorations. However, with respect to non-Euclidean geometries, do we need to introduce them to…
Trigonometry from a Different Angle
ERIC Educational Resources Information Center
Cavanagh, Michael
2008-01-01
The mathematics methodology subjects the author undertook in the early 1980s encouraged him to adopt a very expository style of teaching in which each new concept is introduced by its formal definition. The teacher should then explain a few carefully chosen examples for students to copy into their books, and then provide plenty of graded practice…
Classification of filiform Lie algebras of order 3
NASA Astrophysics Data System (ADS)
Navarro, Rosa María
2016-12-01
Lie algebras of order 3 constitute a generalization of Lie algebras and superalgebras. Throughout this paper the classification problem of filiform Lie algebras of order 3 is considered and therefore this work is a continuation papers seen in the literature. We approach this classification by extending Vergne's result for filiform Lie algebras and by considering algebras of order 3 of high nilindex. We find the expression of the law to which any elementary filiform Lie algebra of order 3 is isomorphic.
The Algebra of Lexical Semantics
NASA Astrophysics Data System (ADS)
Kornai, András
The current generative theory of the lexicon relies primarily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the generative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of describing discrepancies between syntactic type (lexical category) and semantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR.
Strengthening Effect Algebras in a Logical Perspective: Heyting-Wajsberg Algebras
NASA Astrophysics Data System (ADS)
Konig, Martinvaldo
2014-10-01
Heyting effect algebras are lattice-ordered pseudoboolean effect algebras endowed with a pseudocomplementation that maps on the center (i.e. Boolean elements). They are the algebraic counterpart of an extension of both Łukasiewicz many-valued logic and intuitionistic logic. We show that Heyting effect algebras are termwise equivalent to Heyting-Wajsberg algebras where the two different logical implications are defined as primitive operators. We prove this logic to be decidable, to be strongly complete and to have the deduction-detachment theorem.
Emergent Hyperbolic Network Geometry.
Bianconi, Ginestra; Rahmede, Christoph
2017-02-07
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry.
Sliding vane geometry turbines
Sun, Harold Huimin; Zhang, Jizhong; Hu, Liangjun; Hanna, Dave R
2014-12-30
Various systems and methods are described for a variable geometry turbine. In one example, a turbine nozzle comprises a central axis and a nozzle vane. The nozzle vane includes a stationary vane and a sliding vane. The sliding vane is positioned to slide in a direction substantially tangent to an inner circumference of the turbine nozzle and in contact with the stationary vane.
Hsü, K J; Hsü, A J
1990-01-01
Music critics have compared Bach's music to the precision of mathematics. What "mathematics" and what "precision" are the questions for a curious scientist. The purpose of this short note is to suggest that the mathematics is, at least in part, Mandelbrot's fractal geometry and the precision is the deviation from a log-log linear plot. PMID:11607061
ERIC Educational Resources Information Center
Martin, John
2010-01-01
The cycloid has been called the Helen of Geometry, not only because of its beautiful properties but also because of the quarrels it provoked between famous mathematicians of the 17th century. This article surveys the history of the cycloid and its importance in the development of the calculus.
Emergent Hyperbolic Network Geometry
NASA Astrophysics Data System (ADS)
Bianconi, Ginestra; Rahmede, Christoph
2017-02-01
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry.
Emergent Hyperbolic Network Geometry
Bianconi, Ginestra; Rahmede, Christoph
2017-01-01
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry. PMID:28167818
ERIC Educational Resources Information Center
MacKeown, P. K.
1984-01-01
Clarifies two concepts of gravity--those of a fictitious force and those of how space and time may have geometry. Reviews the position of Newton's theory of gravity in the context of special relativity and considers why gravity (as distinct from electromagnetics) lends itself to Einstein's revolutionary interpretation. (JN)
ERIC Educational Resources Information Center
Fielker, David
2007-01-01
Geoff Giles died suddenly in 2005. He was a highly original thinker in the field of geometry teaching. As early as 1964, when teaching at Strathallen School in Perth, he was writing in "MT27" about constructing tessellations by modifying the sides of triangles and (irregular) quadrilaterals to produce what he called "trisides" and "quadrisides".…
ERIC Educational Resources Information Center
Hartz, Viggo
1981-01-01
Allowing students to use a polystyrene cutter to fashion their own three-dimensional models is suggested as a means of allowing individuals to experience problems and develop ideas related to solid geometry. A list of ideas that can lead to mathematical discovery is provided. (MP)
ERIC Educational Resources Information Center
Cooper, Brett D.; Barger, Rita
2009-01-01
The many connections between music and mathematics are well known. The length of a plucked string determines its tone, the time signature of a piece of music is a ratio, and note durations are measured in fractions. One connection commonly overlooked is that between music and geometry--specifically, geometric transformations, including…
ERIC Educational Resources Information Center
KLIER, KATHERINE M.
PRESENTED IS A FUSED COURSE IN PLANE, SOLID, AND COORDINATE GEOMETRY. ELEMENTARY SET THEORY, LOGIC, AND THE PRINCIPLE OF SEPARATION PROVIDE UNIFYING THREADS THROUGHOUT THE TEXT. THE TWO CURRICULUM GUIDES HAVE BEEN PREPARED FOR USE WITH TWO DIFFERENT TEXTS. EITHER CURRICULUM GUIDE MAY BE USED DEPENDING UPON THE CHOICE OF THE TEACHER AND THE NEEDS…
ERIC Educational Resources Information Center
Hirata, Li Ann
Core Geometry is a course offered in the Option Y sequence of the high school mathematics program described by the Hawaii State Department of Education's guidelines. The emphasis of this course is on the general awareness and use of the relationships among points, lines, and figures in planes and space. This sample course is based on the…
ERIC Educational Resources Information Center
Case, Christine L.
1991-01-01
Presented is an activity in which students make models of viruses, which allows them to visualize the shape of these microorganisms. Included are some background on viruses, the biology and geometry of viruses, directions for building viruses, a comparison of cells and viruses, and questions for students. (KR)
Advanced geometries and regimes
Bulanov, S. S.; Bulanov, S. V.; Turchetti, G.; Limpouch, J.; Klimo, O.; Psikal, J.; Margarone, D.; Korn, G.
2013-07-26
We review and discuss different schemes of laser ion acceleration as well as advanced target geometries in connection with the development of the laser-driven proton source for hadron therapy of oncological diseases, which is a part of the ELIMED project.
Algebraic Thinking: A Problem Solving Approach
ERIC Educational Resources Information Center
Windsor, Will
2010-01-01
Algebraic thinking is a crucial and fundamental element of mathematical thinking and reasoning. It initially involves recognising patterns and general mathematical relationships among numbers, objects and geometric shapes. This paper will highlight how the ability to think algebraically might support a deeper and more useful knowledge, not only of…
Learning from Student Approaches to Algebraic Proofs
ERIC Educational Resources Information Center
D'Ambrosio, Beatriz S.; Kastberg, Signe E.; Viola dos Santos, Joao Ricardo
2010-01-01
Many mathematics teachers struggle to support their students' developing understanding of proof as an essential element in investigations of mathematics. The area of mathematics where the development of an understanding of proof is most challenging is algebra. In the case of algebraic proof, analysis of student written work on tasks that demand…
A Technology-Intensive Approach to Algebra.
ERIC Educational Resources Information Center
Heid, M. Kathleen; Zbiek, Rose Mary
1995-01-01
Computer-Intensive Algebra (CIA) focuses on the use of technology to help develop a rich understanding of fundamental algebraic concepts in real-world settings using computing tools for easy access to numerical, graphical, and symbolic representations of mathematical ideas. (MKR)
Focus on Fractions to Scaffold Algebra
ERIC Educational Resources Information Center
Ooten, Cheryl Thomas
2013-01-01
Beginning algebra is a gatekeeper course into the pipeline to higher mathematics courses required for respected professions in engineering, science, statistics, mathematics, education, and technology. Beginning algebra can also be a perfect storm if the necessary foundational skills are not within a student's grasp. What skills ensure beginning…
Post-Lie Algebras and Isospectral Flows
NASA Astrophysics Data System (ADS)
Ebrahimi-Fard, Kurusch; Lundervold, Alexander; Mencattini, Igor; Munthe-Kaas, Hans Z.
2015-11-01
In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical R-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation.
Teaching Modeling and Axiomatization with Boolean Algebra.
ERIC Educational Resources Information Center
De Villiers, Michael D.
1987-01-01
Presented is an alternative approach to the traditional teaching of Boolean algebra for secondary school mathematics. The main aim of the approach is to use Boolean algebra to teach pupils such mathematical processes as modeling and axiomatization. A course using the approach is described. (RH)
Arithmetic and Cognitive Contributions to Algebra
ERIC Educational Resources Information Center
Cirino, Paul T.; Tolar, Tammy D.; Fuchs, Lynn S.
2013-01-01
Algebra is a prerequisite for access to STEM careers and occupational success (NMAP, 2008a), yet algebra is difficult for students through high school (US DOE, 2008). Growth in children's conceptual and procedural arithmetical knowledge is reciprocal, although conceptual knowledge has more impact on procedural knowledge than the reverse…
Algebraic Thinking through Koch Snowflake Constructions
ERIC Educational Resources Information Center
Ghosh, Jonaki B.
2016-01-01
Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…
Calif. Laws Shift Gears on Algebra, Textbooks
ERIC Educational Resources Information Center
Robelen, Erik W.
2012-01-01
New laws in California have set the state on a course for some potentially significant changes to the curriculum, including a measure that revisits the matter of teaching Algebra 1 in 8th grade and another that revamps the state's textbook-adoption process and hands districts greater leeway in choosing instructional materials. The algebra-related…
Using Students' Interests as Algebraic Models
ERIC Educational Resources Information Center
Whaley, Kenneth A.
2012-01-01
Fostering algebraic thinking is an important goal for middle-grades mathematics teachers. Developing mathematical reasoning requires that teachers cultivate students' habits of mind. Teachers develop students' understanding of algebra by engaging them in tasks that involve modeling and representation. This study was designed to investigate how…
THE RADICAL OF A JORDAN ALGEBRA
McCrimmon, Kevin
1969-01-01
In this paper we define a Jacobson radical for Jordan algebras analogous to that for associative algebras and show that it enjoys many of the properties of the associative radical. We then relate the corresponding notion of “semisimplicity” to the previously defined notion of “nondegeneracy” (Jacobson, N., these Proceedings, 55, 243-251 (1966)). PMID:16591736
The operator algebra approach to quantum groups
Kustermans, Johan; Vaes, Stefaan
2000-01-01
A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory. PMID:10639116
Using the Internet To Investigate Algebra.
ERIC Educational Resources Information Center
Sherwood, Walter
The lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approach algebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…
Teaching Algebra to Students with Learning Disabilities
ERIC Educational Resources Information Center
Impecoven-Lind, Linda S.; Foegen, Anne
2010-01-01
Algebra is a gateway to expanded opportunities, but it often poses difficulty for students with learning disabilities. Consequently, it is essential to identify evidence-based instructional strategies for these students. The authors begin by identifying three areas of algebra difficulty experienced by students with disabilities: cognitive…
Gary M. Klingler Algebra Teacher Assistance Packages
ERIC Educational Resources Information Center
Klingler, Gary
2005-01-01
Several packages designed by Elizabeth Marquez for mathematics teachers of grades 6-12, officially entitled the Teacher Assistance Package in Support of Better Algebra Assessment, is a series of resources developed to accompany ET's End-of-Course Algebra Assessment. It is designed to enhance teachers classroom assessment by providing examples of…
Just Say Yes to Early Algebra!
ERIC Educational Resources Information Center
Stephens, Ana; Blanton, Maria; Knuth, Eric; Isler, Isil; Gardiner, Angela Murphy
2015-01-01
Mathematics educators have argued for some time that elementary school students are capable of engaging in algebraic thinking and should be provided with rich opportunities to do so. Recent initiatives like the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) have taken up this call by reiterating the place of early algebra in…
Symbolic Notations and Students' Achievements in Algebra
ERIC Educational Resources Information Center
Peter, Ebiendele E.; Olaoye, Adetunji A.
2013-01-01
This study focuses on symbolic notations and its impact on students' achievement in Algebra. The main reason for this study rests on the observation from personal and professional experiences on students' increasing hatred for Algebra. One hundred and fifty (150) Senior Secondary School Students (SSS) from Ojo Local Education District, Ojo, Lagos,…
SAYD Modules over Lie-Hopf Algebras
NASA Astrophysics Data System (ADS)
Rangipour, Bahram; Sütlü, Serkan
2012-11-01
In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.
Algebraic Formulas for Areas between Curves.
ERIC Educational Resources Information Center
Gabai, Hyman
1982-01-01
Korean secondary school students preparing for college learn about a simple algebraic formula for area bounded by a parabola and line. The approach does not seem well-known among American students. It is noted that, while the formula derivations rely on integration, algebra students could use the formulas without proofs. (MP)
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…
An Inquiry-Based Linear Algebra Class
ERIC Educational Resources Information Center
Wang, Haohao; Posey, Lisa
2011-01-01
Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…
Practicing Algebraic Skills: A Conceptual Approach
ERIC Educational Resources Information Center
Friedlander, Alex; Arcavi, Abraham
2012-01-01
Traditionally, a considerable part of teaching and learning algebra has focused on routine practice and the application of rules, procedures, and techniques. Although today's computerized environments may have decreased the need to master algebraic skills, procedural competence is still a central component in any mathematical activity. However,…
Success in Algebra among Community College Students
ERIC Educational Resources Information Center
Reyes, Czarina
2010-01-01
College algebra is a required course for most majors, but is viewed by many as a gatekeeper course for degree completion by students. With almost half a million students taking college algebra each year, faculty are experimenting with new course lengths of time that might result in higher success, completion, and retention rates for college…
Is Algebra Really Difficult for All Students?
ERIC Educational Resources Information Center
Egodawatte, Gunawardena
2009-01-01
Research studies have shown that students encounter difficulties in transitioning from arithmetic to algebra. Errors made by high school students were analyzed for patterns and their causes. The origins of errors were: intuitive assumptions, failure to understand the syntax of algebra, analogies with other familiar symbol systems such as the…
NASA Astrophysics Data System (ADS)
Prástaro, Agostino
2008-02-01
Following our previous results on this subject [R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(I): Webs on PDE's and integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17 (2007) 239-266; R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(II): Webs on PDE's and integral bordism groups. Applications to Riemannian geometry PDE's, Adv. Math. Sci. Appl. 17 (2007) 267-285; A. Prástaro, Geometry of PDE's and Mechanics, World Scientific, Singapore, 1996; A. Prástaro, Quantum and integral (co)bordism in partial differential equations, Acta Appl. Math. (5) (3) (1998) 243-302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59 (2) (1999) 111-201; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co, Singapore, 2004, 500 pp.; A. Prástaro, Geometry of PDE's. I: Integral bordism groups in PDE's, J. Math. Anal. Appl. 319 (2006) 547-566; A. Prástaro, Geometry of PDE's. II: Variational PDE's and integral bordism groups, J. Math. Anal. Appl. 321 (2006) 930-948; A. Prástaro, Th.M. Rassias, Ulam stability in geometry of PDE's, Nonlinear Funct. Anal. Appl. 8 (2) (2003) 259-278; I. Stakgold, Boundary Value Problems of Mathematical Physics, I, The MacMillan Company, New York, 1967; I. Stakgold, Boundary Value Problems of Mathematical Physics, II, Collier-MacMillan, Canada, Ltd, Toronto, Ontario, 1968], integral bordism groups of the Navier-Stokes equation are calculated for smooth, singular and weak solutions, respectively. Then a characterization of global solutions is made on this ground. Enough conditions to assure existence of global smooth solutions are given and related to nullity of integral characteristic numbers of the boundaries. Stability of global solutions are related to some characteristic numbers of the space-like Cauchy dataE Global solutions of variational problems constrained by (NS) are classified by means of suitable integral bordism groups too.
MODEL IDENTIFICATION AND COMPUTER ALGEBRA.
Bollen, Kenneth A; Bauldry, Shawn
2010-10-07
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods.
LINPACK. Simultaneous Linear Algebraic Equations
Miller, M.A.
1990-05-01
LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic equations. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).
LINPACK. Simultaneous Linear Algebraic Equations
Dongarra, J.J.
1982-05-02
LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic equations. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).
Generalization of n-ary Nambu algebras and beyond
Ataguema, H.; Makhlouf, A.; Silvestrov, S.
2009-08-15
The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.
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.
Ordered Incidence Geometry and the Geometric Foundations of Convexity Theory.
1984-03-01
needed in the sequel. 5. Lineal hulls. Characterization of affine sets in terms of lineal hulls, (T2). 6. Convex sets. Definitions (D5)-(D7) and basic...wo) 5. LINEAL HULLS In the Euclidean geometry R" a set A is affine if and only if A = {Xiz,: ;EA, N"Xi=1J51 i.e. A coincides with the set of affine...combination, of its elements. The analogous geometrical representation in an QIG (where algebraic constructions such as (5.1) are not available) is
Noncommutative geometry, Grand Symmetry and twisted spectral triple
NASA Astrophysics Data System (ADS)
Devastato, Agostino
2015-08-01
In the noncommutative geometry approach to the standard model we discuss the possibility to derive the extra scalar field sv - initially suggested by particle physicist to stabilize the electroweak vacuum - from a “grand algebra” that contains the usual standard model algebra. We introduce the Connes-Moscovici twisted spectral triples for the Grand Symmetry model, to cure a technical problem, that is the appearance, together with the field sv, of unbounded vectorial terms. The twist makes these terms bounded, and also permits to understand the breaking making the computation of the Higgs mass compatible with the 126 GeV experimental value.
Cylindrical geometry hall thruster
Raitses, Yevgeny; Fisch, Nathaniel J.
2002-01-01
An apparatus and method for thrusting plasma, utilizing a Hall thruster with a cylindrical geometry, wherein ions are accelerated in substantially the axial direction. The apparatus is suitable for operation at low power. It employs small size thruster components, including a ceramic channel, with the center pole piece of the conventional annular design thruster eliminated or greatly reduced. Efficient operation is accomplished through magnetic fields with a substantial radial component. The propellant gas is ionized at an optimal location in the thruster. A further improvement is accomplished by segmented electrodes, which produce localized voltage drops within the thruster at optimally prescribed locations. The apparatus differs from a conventional Hall thruster, which has an annular geometry, not well suited to scaling to small size, because the small size for an annular design has a great deal of surface area relative to the volume.
Geometry of thermodynamic control.
Zulkowski, Patrick R; Sivak, David A; Crooks, Gavin E; DeWeese, Michael R
2012-10-01
A deeper understanding of nonequilibrium phenomena is needed to reveal the principles governing natural and synthetic molecular machines. Recent work has shown that when a thermodynamic system is driven from equilibrium then, in the linear response regime, the space of controllable parameters has a Riemannian geometry induced by a generalized friction tensor. We exploit this geometric insight to construct closed-form expressions for minimal-dissipation protocols for a particle diffusing in a one-dimensional harmonic potential, where the spring constant, inverse temperature, and trap location are adjusted simultaneously. These optimal protocols are geodesics on the Riemannian manifold and reveal that this simple model has a surprisingly rich geometry. We test these optimal protocols via a numerical implementation of the Fokker-Planck equation and demonstrate that the friction tensor arises naturally from a first-order expansion in temporal derivatives of the control parameters, without appealing directly to linear response theory.
Freezing in confined geometries
NASA Technical Reports Server (NTRS)
Sokol, P. E.; Ma, W. J.; Herwig, K. W.; Snow, W. M.; Wang, Y.; Koplik, Joel; Banavar, Jayanth R.
1992-01-01
Results of detailed structural studies, using elastic neutron scattering, of the freezing of liquid O2 and D2 in porous vycor glass, are presented. The experimental studies have been complemented by computer simulations of the dynamics of freezing of a Lennard-Jones liquid in narrow channels bounded by molecular walls. Results point to a new simple physical interpretation of freezing in confined geometries.
NASA Astrophysics Data System (ADS)
Cederwall, Martin; Rosabal, J. A.
2015-07-01
We investigate exceptional generalised diffeomorphisms based on E 8(8) in a geometric setting. The transformations include gauge transformations for the dual gravity field. The surprising key result, which allows for a development of a tensor formalism, is that it is possible to define field-dependent transformations containing connection, which are covariant. We solve for the spin connection and construct a curvature tensor. A geometry for the Ehlers symmetry SL( n + 1) is sketched. Some related issues are discussed.
Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis
NASA Astrophysics Data System (ADS)
Mehta, Dhagash; Daleo, Noah S.; Dörfler, Florian; Hauenstein, Jonathan D.
2015-05-01
Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system of equations, which is an important yet challenging problem. We translate this into an algebraic geometry problem and use numerical methods to find all of the equilibria for various choices of coupling constants K, natural frequencies, and on different graphs. We note that for even modest sizes (N ˜ 10-20), the number of equilibria is already more than 100 000. We analyze the stability of each computed equilibrium as well as the configuration of angles. Our exploration of the equilibrium landscape leads to unexpected and possibly surprising results including non-monotonicity in the number of equilibria, a predictable pattern in the indices of equilibria, counter-examples to conjectures, multi-stable equilibrium landscapes, scenarios with only unstable equilibria, and multiple distinct extrema in the stable equilibrium distribution as a function of the number of cycles in the graph.
Degeneracy measures for the algebraic classification of numerical spacetimes
Owen, Robert
2010-06-15
We study the issue of algebraic classification of the Weyl curvature tensor, with a particular focus on numerical relativity simulations. The spacetimes of interest in this context, binary black hole mergers, and the ringdowns that follow them, present subtleties in that they are generically, strictly speaking, type I, but in many regions approximately, in some sense, type D. To provide meaning to any claims of ''approximate'' Petrov class, one must define a measure of degeneracy on the space of null rays at a point. We will investigate such a measure, used recently to argue that certain binary black hole merger simulations ring down to the Kerr geometry, after hanging up for some time in Petrov type II. In particular, we argue that this hangup in Petrov type II is an artefact of the particular measure being used, and that a geometrically better-motivated measure shows a black hole merger produced by our group settling directly to Petrov type D.
Integral geometry and holography
Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel; ...
2015-10-27
We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS3/CFT2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulkmore » curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS3 whose kinematic space is two-dimensional de Sitter space.« less
Emergent Complex Network Geometry
Wu, Zhihao; Menichetti, Giulia; Rahmede, Christoph; Bianconi, Ginestra
2015-01-01
Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant for routing problems, inference and data mining. In real growing networks, topological, structural and geometrical properties emerge spontaneously from their dynamical rules. Nevertheless we still miss a model in which networks develop an emergent complex geometry. Here we show that a single two parameter network model, the growing geometrical network, can generate complex network geometries with non-trivial distribution of curvatures, combining exponential growth and small-world properties with finite spectral dimensionality. In one limit, the non-equilibrium dynamical rules of these networks can generate scale-free networks with clustering and communities, in another limit planar random geometries with non-trivial modularity. Finally we find that these properties of the geometrical growing networks are present in a large set of real networks describing biological, social and technological systems. PMID:25985280
Integral geometry and holography
Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel; Sully, James
2015-10-27
We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS_{3}/CFT_{2} correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS_{3} whose kinematic space is two-dimensional de Sitter space.
Light polarization: A geometric-algebra approach
NASA Astrophysics Data System (ADS)
Baylis, W. E.; Bonenfant, J.; Derbyshire, J.; Huschilt, J.
1993-06-01
The geometric algebra of three-dimensional space (the ``Pauli algebra'') is known to provide an efficient geometric description of electromagnetic phenomena. Here, it is applied to the three-dimensional Stokes subspace to describe the polarization of an approximately monochromatic collimated beam of electromagnetic radiation. The coherency density ρ is a real element of the algebra whose components are the four Stokes parameters: a scalar representing the total photon flux density plus a three-dimensional vector whose direction and length in the Poincaré sphere give the type and degree of polarization. The detection of the radiation and the incoherent and coherent modification of the polarization by various optical elements are calculated by algebraic multiplication which has faithful representations in 2×2 matrices. One matrix representation of ρ is the coherency matrix with which Jones and Mueller matrices are related whereas another representation is the spin density matrix. However, the calculations are simplest to perform and interpret in the algebraic form independent of any particular matrix representation. It is shown that any possible change in the Stokes parameters can be treated algebraically by a combination of attenuation, depolarization, polarization, and rotation transformations of ρ. The geometric algebra thus unifies Stokes parameters, the Poincaré sphere, Jones and Mueller matrices, and the coherency and density matrices in a single, simple formalism.
Conformal current algebra in two dimensions
NASA Astrophysics Data System (ADS)
Ashok, Sujay K.; Benichou, Raphael; Troost, Jan
2009-06-01
We construct a non-chiral current algebra in two dimensions consistent with conformal invariance. We show that the conformal current algebra is realized in non-linear sigma-models on supergroup manifolds with vanishing Killing form, with or without a Wess-Zumino term. The current algebra is computed using two distinct methods. First we exploit special algebraic properties of supergroups to compute the exact two- and three-point functions of the currents and from them we infer the current algebra. The algebra is also calculated by using conformal perturbation theory about the Wess-Zumino-Witten point and resumming the perturbation series. We also prove that these models realize a non-chiral Kac-Moody algebra and construct an infinite set of commuting operators that is closed under the action of the Kac-Moody generators. The supergroup models that we consider include models with applications to statistical mechanics, condensed matter and string theory. In particular, our results may help to systematically solve and clarify the quantum integrability of PSU(n|n) models and their cosets, which appear prominently in string worldsheet models on anti-deSitter spaces.
Working memory, worry, and algebraic ability.
Trezise, Kelly; Reeve, Robert A
2014-05-01
Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.
Three-dimensional polarization algebra.
R Sheppard, Colin J; Castello, Marco; Diaspro, Alberto
2016-10-01
If light is focused or collected with a high numerical aperture lens, as may occur in imaging and optical encryption applications, polarization should be considered in three dimensions (3D). The matrix algebra of polarization behavior in 3D is discussed. It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system, which are in the literature. Explicit transformation matrices for converting the column vector form of these different matrices are extended to the 3D case, where they are large (81×81) but can be generated using simple rules. It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell-Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily. Explicit expressions are given for 3D complex field components in terms of Chandrasekhar-Stokes parameters.
Jucys-Murphy elements for Birman-Murakami-Wenzl algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Ogievetsky, O. V.
2011-05-01
The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.
Kinematical superalgebras and Lie algebras of order 3
Campoamor-Stursberg, R.; Rausch de Traubenberg, M.
2008-06-15
We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonue-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order 3.
Becchi-Rouet-Stora-Tyutin operators for W algebras
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2008-07-15
The study of quantum Lie algebras motivates a use of noncanonical ghosts and antighosts for nonlinear algebras, such as W-algebras. This leads, for the W{sub 3} and W{sub 3}{sup (2)} algebras, to the Becchi-Rouet-Stora-Tyutin operator having the conventional cubic form.
On computational complexity of Clifford algebra
NASA Astrophysics Data System (ADS)
Budinich, Marco
2009-05-01
After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced by Budinich and Budinich ["A spinorial formulation of the maximum clique problem of a graph," J. Math. Phys. 47, 043502 (2006)].
Gauged Ads-Maxwell Algebra and Gravity
NASA Astrophysics Data System (ADS)
Durka, R.; Kowalski-Glikman, J.; Szczachor, M.
We deform the anti-de Sitter algebra by adding additional generators {Z}ab, forming in this way the negative cosmological constant counterpart of the Maxwell algebra. We gauge this algebra and construct a dynamical model with the help of a constrained BF theory. It turns out that the resulting theory is described by the Einstein-Cartan action with Holst term, and the gauge fields associated with the Maxwell generators {Z}ab appear only in topological terms that do not influence dynamical field equations. We briefly comment on the extension of this construction, which would lead to a nontrivial Maxwell fields dynamics.
Homomorphisms between C*-algebras and linear derivations on C*-algebras
NASA Astrophysics Data System (ADS)
Park, Choonkil; Boo, Deok-Hoon; An, Jong Su
2008-01-01
It is shown that every almost unital almost linear mapping of a unital C*-algebra to a unital C*-algebra is a homomorphism when h(3nuy)=h(3nu)h(y) holds for all unitaries , all , and all , and that every almost unital almost linear continuous mapping of a unital C*-algebra of real rank zero to a unital C*-algebra is a homomorphism when h(3nuy)=h(3nu)h(y) holds for all , and v is invertible}, all , and all . Furthermore, we prove the Hyers-Ulam-Rassias stability of *-homomorphisms between unital C*-algebras, and -linear *-derivations on unital C*-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300.
Lie algebra type noncommutative phase spaces are Hopf algebroids
NASA Astrophysics Data System (ADS)
Meljanac, Stjepan; Škoda, Zoran; Stojić, Martina
2016-11-01
For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.
A note on derivations of Murray–von Neumann algebras
Kadison, Richard V.; Liu, Zhe
2014-01-01
A Murray–von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray–von Neumann algebras. We show that the “extended derivations” of a Murray–von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray–von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer’s seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements. PMID:24469831
A double commutant theorem for Murray–von Neumann algebras
Liu, Zhe
2012-01-01
Murray–von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra of the Murray–von Neumann algebra associated with a finite von Neumann algebra is the Murray–von Neumann algebra , where is a maximal abelian self-adjoint subalgebra of and, in addition, is . We also prove that the Murray–von Neumann algebra with the center of is the center of the Murray–von Neumann algebra . Von Neumann’s celebrated double commutant theorem characterizes von Neumann algebras as those for which , where , the commutant of , is the set of bounded operators on the Hilbert space that commute with all operators in . At the end of this article, we present a double commutant theorem for Murray–von Neumann algebras. PMID:22543165
NASA Technical Reports Server (NTRS)
Moitra, Anutosh
1989-01-01
A fast and versatile procedure for algebraically generating boundary conforming computational grids for use with finite-volume Euler flow solvers is presented. A semi-analytic homotopic procedure is used to generate the grids. Grids generated in two-dimensional planes are stacked to produce quasi-three-dimensional grid systems. The body surface and outer boundary are described in terms of surface parameters. An interpolation scheme is used to blend between the body surface and the outer boundary in order to determine the field points. The method, albeit developed for analytically generated body geometries is equally applicable to other classes of geometries. The method can be used for both internal and external flow configurations, the only constraint being that the body geometries be specified in two-dimensional cross-sections stationed along the longitudinal axis of the configuration. Techniques for controlling various grid parameters, e.g., clustering and orthogonality are described. Techniques for treating problems arising in algebraic grid generation for geometries with sharp corners are addressed. A set of representative grid systems generated by this method is included. Results of flow computations using these grids are presented for validation of the effectiveness of the method.
Highest-weight representations of Brocherd`s algebras
Slansky, R.
1997-01-01
General features of highest-weight representations of Borcherd`s algebras are described. to show their typical features, several representations of Borcherd`s extensions of finite-dimensional algebras are analyzed. Then the example of the extension of affine- su(2) to a Borcherd`s algebra is examined. These algebras provide a natural way to extend a Kac-Moody algebra to include the hamiltonian and number-changing operators in a generalized symmetry structure.
BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras
NASA Astrophysics Data System (ADS)
Graziani, Giacomo; Makhlouf, Abdenacer; Menini, Claudia; Panaite, Florin
2015-10-01
A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α,β\\colon A→ A such that α (a)(bc)=(ab)β (c), for all a, b, cin A. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).
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.
NASA Astrophysics Data System (ADS)
Calcagni, Gianluca
2012-08-01
The change of the effective dimension of spacetime with the probed scale is a universal phenomenon shared by independent models of quantum gravity. Using tools of probability theory and multifractal geometry, we show how dimensional flow is controlled by a multiscale fractional diffusion equation, and physically interpreted as a composite stochastic process. The simplest example is a fractional telegraph process, describing quantum spacetimes with a spectral dimension equal to 2 in the ultraviolet and monotonically rising to 4 towards the infrared. The general profile of the spectral dimension of the recently introduced multifractional spaces is constructed for the first time.
Geometrie verstehen: statisch - kinematisch
NASA Astrophysics Data System (ADS)
Kroll, Ekkehard
Dem Allgemeinen steht begrifflich das Besondere gegenüber. In diesem Sinne sind allgemeine Überlegungen zum Verstehen von Mathematik zu ergänzen durch Untersuchungen hinsichtlich des Verstehens der einzelnen mathematischen Disziplinen, insbesondere der Geometrie. Hier haben viele Schülerinnen und Schüler Probleme. Diese rühren hauptsächlich daher, dass eine fertige geometrische Konstruktion in ihrer statischen Präsentation auf Papier nicht mehr die einzelnen Konstruktionsschritte erkennen lässt; zum Nachvollzug müssen sie daher ergänzend in einer Konstruktionsbeschreibung festgehalten werden.
Computer-Aided Geometry Modeling
NASA Technical Reports Server (NTRS)
Shoosmith, J. N. (Compiler); Fulton, R. E. (Compiler)
1984-01-01
Techniques in computer-aided geometry modeling and their application are addressed. Mathematical modeling, solid geometry models, management of geometric data, development of geometry standards, and interactive and graphic procedures are discussed. The applications include aeronautical and aerospace structures design, fluid flow modeling, and gas turbine design.
Teaching of Geometry in Bulgaria
ERIC Educational Resources Information Center
Bankov, Kiril
2013-01-01
Geometry plays an important role in the school mathematics curriculum all around the world. Teaching of geometry varies a lot (Hoyls, Foxman, & Kuchemann, 2001). Many countries revise the objectives, the content, and the approaches to the geometry in school. Studies of the processes show that there are not common trends of these changes…
Using Schemas to Develop Algebraic Thinking
ERIC Educational Resources Information Center
Steele, Diana F.
2005-01-01
This article describes ways in which students develop schemas as they generalize and formalize patterns when solving related algebraic problems that involve size, shape, growth, and change. (Contains 7 figures and 3 tables.)
Cohomological invariants of central simple algebras
NASA Astrophysics Data System (ADS)
Merkurjev, A. S.
2016-10-01
We determine the indecomposable degree 3 cohomological invariants of tuples of central simple algebras with linear relations. Equivalently, we determine the degree 3 reductive cohomological invariants of all split semisimple groups of type A.
Cyclic Cocycles on Twisted Convolution Algebras
NASA Astrophysics Data System (ADS)
Angel, Eitan
2013-01-01
We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper étale groupoids, Tu and Xu (Adv Math 207(2):455-483, 2006) provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to the construction of Mathai and Stevenson (Adv Math 200(2):303-335, 2006). When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.
ALGEBRAIC DEPENDENCE THEOREMS ON COMPLEX PSEUDOCONCAVE SPACES
The notion of pseudoconcave space is introduced and classical theorems on algebraic dependence of meromorphic functions are extended for this new class of spaces and for sections in a coherent sheaf. (Author)
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)
Lisa's Lemonade Stand: Exploring Algebraic Ideas.
ERIC Educational Resources Information Center
Billings, Esther M. H.; Lakatos, Tracy
2003-01-01
Presents an activity, "Lisa's Lemonade Stand," that actively engages students in algebraic thinking as they analyze change by investigating relationships between variables and gain experience describing and representing these relationships graphically. (YDS)
Algebraic operator approach to gas kinetic models
NASA Astrophysics Data System (ADS)
Il'ichov, L. V.
1997-02-01
Some general properties of the linear Boltzmann kinetic equation are used to present it in the form ∂ tϕ = - Â†Âϕ with the operators ÂandÂ† possessing some nontrivial algebraic properties. When applied to the Keilson-Storer kinetic model, this method gives an example of quantum ( q-deformed) Lie algebra. This approach provides also a natural generalization of the “kangaroo model”.
Dynamical systems and quantum bicrossproduct algebras
NASA Astrophysics Data System (ADS)
Arratia, Oscar; del Olmo, Mariano A.
2002-06-01
We present a unified study of some aspects of quantum bicrossproduct algebras of inhomogeneous Lie algebras, such as Poincaré, Galilei and Euclidean in N dimensions. The action associated with the bicrossproduct structure allows us to obtain a nonlinear action over a new group linked to the translations. This new nonlinear action associates a dynamical system with each generator which is the object of our study.
Algebra and topology for applications to physics
NASA Technical Reports Server (NTRS)
Rozhkov, S. S.
1987-01-01
The principal concepts of algebra and topology are examined with emphasis on applications to physics. In particular, attention is given to sets and mapping; topological spaces and continuous mapping; manifolds; and topological groups and Lie groups. The discussion also covers the tangential spaces of the differential manifolds, including Lie algebras, vector fields, and differential forms, properties of differential forms, mapping of tangential spaces, and integration of differential forms.
Vague Congruences and Quotient Lattice Implication Algebras
Qin, Xiaoyan; Xu, Yang
2014-01-01
The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given. PMID:25133207
Edge covers and independence: Algebraic approach
NASA Astrophysics Data System (ADS)
Kalinina, E. A.; Khitrov, G. M.; Pogozhev, S. V.
2016-06-01
In this paper, linear algebra methods are applied to solve some problems of graph theory. For ordinary connected graphs, edge coverings and independent sets are considered. Some results concerning minimum edge covers and maximum matchings are proved with the help of linear algebraic approach. The problem of finding a maximum matching of a graph is fundamental both practically and theoretically, and has numerous applications, e.g., in computational chemistry and mathematical chemistry.
Algebraic Sub-Structuring for Electromagnetic Applications
Yang, C.; Gao, W.G.; Bai, Z.J.; Li, X.Y.S.; Lee, L.Q.; Husbands, P.; Ng, E.G.; /LBL, Berkeley /UC, Davis /SLAC
2006-06-30
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, they show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Algebraic sub-structuring for electromagnetic applications
Yang, Chao; Gao, Weiguo; Bai, Zhaojun; Li, Xiaoye; Lee, Lie-Quan; Husbands, Parry; Ng, Esmond G.
2004-09-14
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, we show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Dillon, Moira R.; Spelke, Elizabeth S.
2015-01-01
Research on animals, infants, children, and adults provides evidence that distinct cognitive systems underlie navigation and object recognition. Here we examine whether and how these systems interact when children interpret 2D edge-based perspectival line drawings of scenes and objects. Such drawings serve as symbols early in development, and they preserve scene and object geometry from canonical points of view. Young children show limits when using geometry both in non-symbolic tasks and in symbolic map tasks that present 3D contexts from unusual, unfamiliar points of view. When presented with the familiar viewpoints in perspectival line drawings, however, do children engage more integrated geometric representations? In three experiments, children successfully interpreted line drawings with respect to their depicted scene or object. Nevertheless, children recruited distinct processes when navigating based on the information in these drawings, and these processes depended on the context in which the drawings were presented. These results suggest that children are flexible but limited in using geometric information to form integrated representations of scenes and objects, even when interpreting spatial symbols that are highly familiar and faithful renditions of the visual world. PMID:25441089
Noncommutative geometry of Zitterbewegung
NASA Astrophysics Data System (ADS)
Eckstein, Michał; Franco, Nicolas; Miller, Tomasz
2017-03-01
Drawing from the advanced mathematics of noncommutative geometry, we model a "classical" Dirac fermion propagating in a curved spacetime. We demonstrate that the inherent causal structure of the model encodes the possibility of Zitterbewegung—the "trembling motion" of the fermion. We recover the well-known frequency of Zitterbewegung as the highest possible speed of change in the fermion's "internal space." Furthermore, we show that the bound does not change in the presence of an external electromagnetic field and derive its explicit analogue when the mass parameter is promoted to a Yukawa field. We explain the universal character of the model and discuss a table-top experiment in the domain of quantum simulation to test its predictions.
Critique of information geometry
Skilling, John
2014-12-05
As applied to probability, information geometry fails because probability distributions do not form a metric space. Probability theory rests on a compelling foundation of elementary symmetries, which also support information (aka minus entropy, Kullback-Leibler) H(p;q) as the unique measure of divergence from source probability distribution q to destination p. Because the only compatible connective H is from≠to asymmetric, H(p;q)≠H(q;p), there can be no compatible geometrical distance (which would necessarily be from=to symmetric). Hence there is no distance relationship compatible with the structure of probability theory. Metrics g and densities sqrt(det(g)) interpreted as prior probabilities follow from the definition of distance, and must fail likewise. Various metrics and corresponding priors have been proposed, Fisher's being the most popular, but all must behave unacceptably. This is illustrated with simple counter-examples.
Topological string theory and enumerative geometry
NASA Astrophysics Data System (ADS)
Song, Yun S.
2001-10-01
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 Hurwitz numbers and Hodge integrals, we derive a generating function for the latter on the moduli space overline Mg,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 P1 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.
Magnetism in curved geometries
NASA Astrophysics Data System (ADS)
Streubel, Robert; Fischer, Peter; Kronast, Florian; Kravchuk, Volodymyr P.; Sheka, Denis D.; Gaididei, Yuri; Schmidt, Oliver G.; Makarov, Denys
2016-09-01
Extending planar two-dimensional structures into the three-dimensional space has become a general trend in multiple disciplines, including electronics, photonics, plasmonics and magnetics. This approach provides means to modify conventional or to launch novel functionalities by tailoring the geometry of an object, e.g. its local curvature. In a generic electronic system, curvature results in the appearance of scalar and vector geometric potentials inducing anisotropic and chiral effects. In the specific case of magnetism, even in the simplest case of a curved anisotropic Heisenberg magnet, the curvilinear geometry manifests two exchange-driven interactions, namely effective anisotropy and antisymmetric exchange, i.e. Dzyaloshinskii-Moriya-like interaction. As a consequence, a family of novel curvature-driven effects emerges, which includes magnetochiral effects and topologically induced magnetization patterning, resulting in theoretically predicted unlimited domain wall velocities, chirality symmetry breaking and Cherenkov-like effects for magnons. The broad range of altered physical properties makes these curved architectures appealing in view of fundamental research on e.g. skyrmionic systems, magnonic crystals or exotic spin configurations. In addition to these rich physics, the application potential of three-dimensionally shaped objects is currently being explored as magnetic field sensorics for magnetofluidic applications, spin-wave filters, advanced magneto-encephalography devices for diagnosis of epilepsy or for energy-efficient racetrack memory devices. These recent developments ranging from theoretical predictions over fabrication of three-dimensionally curved magnetic thin films, hollow cylinders or wires, to their characterization using integral means as well as the development of advanced tomography approaches are in the focus of this review.
Magnetism in curved geometries
Streubel, Robert; Fischer, Peter; Kronast, Florian; Kravchuk, Volodymyr P.; Sheka, Denis D.; Gaididei, Yuri; Schmidt, Oliver G.; Makarov, Denys
2016-08-17
Extending planar two-dimensional structures into the three-dimensional space has become a general trend in multiple disciplines, including electronics, photonics, plasmonics and magnetics. This approach provides means to modify conventional or to launch novel functionalities by tailoring the geometry of an object, e.g. its local curvature. In a generic electronic system, curvature results in the appearance of scalar and vector geometric potentials inducing anisotropic and chiral effects. In the specific case of magnetism, even in the simplest case of a curved anisotropic Heisenberg magnet, the curvilinear geometry manifests two exchange-driven interactions, namely effective anisotropy and antisymmetric exchange, i.e. Dzyaloshinskii–Moriya-like interaction. As a consequence, a family of novel curvature-driven effects emerges, which includes magnetochiral effects and topologically induced magnetization patterning, resulting in theoretically predicted unlimited domain wall velocities, chirality symmetry breaking and Cherenkov-like effects for magnons. The broad range of altered physical properties makes these curved architectures appealing in view of fundamental research on e.g. skyrmionic systems, magnonic crystals or exotic spin configurations. In addition to these rich physics, the application potential of three-dimensionally shaped objects is currently being explored as magnetic field sensorics for magnetofluidic applications, spin-wave filters, advanced magneto-encephalography devices for diagnosis of epilepsy or for energy-efficient racetrack memory devices. Finally, these recent developments ranging from theoretical predictions over fabrication of three-dimensionally curved magnetic thin films, hollow cylinders or wires, to their characterization using integral means as well as the development of advanced tomography approaches are in the focus of this review.
Magnetism in curved geometries
Streubel, Robert; Fischer, Peter; Kronast, Florian; ...
2016-08-17
Extending planar two-dimensional structures into the three-dimensional space has become a general trend in multiple disciplines, including electronics, photonics, plasmonics and magnetics. This approach provides means to modify conventional or to launch novel functionalities by tailoring the geometry of an object, e.g. its local curvature. In a generic electronic system, curvature results in the appearance of scalar and vector geometric potentials inducing anisotropic and chiral effects. In the specific case of magnetism, even in the simplest case of a curved anisotropic Heisenberg magnet, the curvilinear geometry manifests two exchange-driven interactions, namely effective anisotropy and antisymmetric exchange, i.e. Dzyaloshinskii–Moriya-like interaction. Asmore » a consequence, a family of novel curvature-driven effects emerges, which includes magnetochiral effects and topologically induced magnetization patterning, resulting in theoretically predicted unlimited domain wall velocities, chirality symmetry breaking and Cherenkov-like effects for magnons. The broad range of altered physical properties makes these curved architectures appealing in view of fundamental research on e.g. skyrmionic systems, magnonic crystals or exotic spin configurations. In addition to these rich physics, the application potential of three-dimensionally shaped objects is currently being explored as magnetic field sensorics for magnetofluidic applications, spin-wave filters, advanced magneto-encephalography devices for diagnosis of epilepsy or for energy-efficient racetrack memory devices. Finally, these recent developments ranging from theoretical predictions over fabrication of three-dimensionally curved magnetic thin films, hollow cylinders or wires, to their characterization using integral means as well as the development of advanced tomography approaches are in the focus of this review.« less
I CAN Learn[R] Pre-Algebra and Algebra. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2007
2007-01-01
The "I CAN Learn[R] Pre-Algebra" and "Algebra" computerized curricula are designed to cover mathematics and problem-solving skills for ethnically diverse, inner-city students in grades 6-12. The curricula are designed to equip students with the skills they need to meet district, state, and national math objectives through an…
ERIC Educational Resources Information Center
Okpube, Nnaemeka Michael; Anugwo, M. N.
2016-01-01
This study investigated the Card Games and Algebra tic-Tacmatics on Junior Secondary II Students' Achievement in Algebraic Expressions. Three research questions and three null hypotheses guided the study. The study adopted the pre-test, post-test control group design. A total of two hundred and forty (240) Junior Secondary School II students were…
ERIC Educational Resources Information Center
Ormond, Christine
2012-01-01
Primary teachers play a key role in their students' future mathematical success in the early secondary years. While the word "algebra" may make some primary teachers feel uncomfortable or worried, the basic arithmetic ideas underlying algebra are vitally important for older primary students as they are increasingly required to use "algebraic…
The Application of a Computer Algebra System as a Tool in College Algebra.
ERIC Educational Resources Information Center
Mayes, Robert L.
1995-01-01
Students (n=61) in an experimental course stressing active student involvement and the use of a computer algebra system scored higher than students (n=76) in a traditional college algebra course on final measures of inductive reasoning, visualization, and problem solving while maintaining equivalent manipulation and computation skills. (Author/MLB)
ERIC Educational Resources Information Center
Davies Gomez, Lisa
2012-01-01
Algebra is the gatekeeper of access to higher-level math and science courses, higher education and future earning opportunities. Unequal numbers of African-American males drop out of Algebra and mathematics courses and underperform on tests of mathematical competency and are thus denied both essential skills and a particularly important pathway to…
Matrix Methods to Analytic Geometry.
ERIC Educational Resources Information Center
Bandy, C.
1982-01-01
The use of basis matrix methods to rotate axes is detailed. It is felt that persons who have need to rotate axes often will find that the matrix method saves considerable work. One drawback is that most students first learning to rotate axes will not yet have studied linear algebra. (MP)
Algebraic dual-energy magnetic analysis with application to variable reluctance motor design
Tolikas, M.; Lang, J.H.; Kirtley, J.L. Jr.
1995-12-31
The dual-energy method has been successfully employed in the calculation of static resistances, capacitances and inductances, yielding fast and accurate solutions. Two different directions can be followed in applying the method, one is the more widely known method of tubes and slices, the other is the algebraic approach and is explored further in this paper. The present literature on the algebraic dual-energy method involves trivial examples with boundaries, boundary conditions and source distributions that are easily handled. However, the geometry of the variable reluctance motor at the unaligned position, the dominating curvatures characterizing its electromagnetic field distribution and the presence of a large number of boundary conditions arising at the steel boundaries, provide a challenging exercise in the application, behavior and effectiveness of the dual-energy method in a realistic framework.
Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Makhlouf, Abdenacer; Silvestrov, Sergei
2010-04-01
The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom-Nambu-Lie algebras obtained using this construction.
Classification of central extensions of Lax operator algebras
Schlichenmaier, Martin
2008-11-18
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
Linearizing W2,4 and WB2 algebras
NASA Astrophysics Data System (ADS)
Bellucci, S.; Krivonos, S.; Sorin, A.
1995-02-01
It has recently been shown that the W3 and W3(2) algebras can be considered as subalgebras in some linear conformal algebras. In this paper we show that the nonlinear algebras W2,4 and WB2 as well as Zamolodchikov's spin {5}/{2} superalgebra also can be embedded as subalgebras into some linear conformal algebras with a finite set of currents. These linear algebras give rise to new realizations of the nonlinear algebras which could be suitable in the construction of W-string theories.
Classification of central extensions of Lax operator algebras
NASA Astrophysics Data System (ADS)
Schlichenmaier, Martin
2008-11-01
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION
NASA Technical Reports Server (NTRS)
Smith, R. E.
1994-01-01
TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.
NASA Astrophysics Data System (ADS)
Gualtieri, Marco
2014-10-01
Generalized Kähler geometry is the natural analogue of Kähler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We prove that generalized Kähler geometry is equivalent to the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry. We also prove the existence of natural holomorphic Courant algebroids for each of the underlying complex structures, and that these split into a sum of transverse holomorphic Dirac structures. Finally, we explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.
Planetary Image Geometry Library
NASA Technical Reports Server (NTRS)
Deen, Robert C.; Pariser, Oleg
2010-01-01
The Planetary Image Geometry (PIG) library is a multi-mission library used for projecting images (EDRs, or Experiment Data Records) and managing their geometry for in-situ missions. A collection of models describes cameras and their articulation, allowing application programs such as mosaickers, terrain generators, and pointing correction tools to be written in a multi-mission manner, without any knowledge of parameters specific to the supported missions. Camera model objects allow transformation of image coordinates to and from view vectors in XYZ space. Pointing models, specific to each mission, describe how to orient the camera models based on telemetry or other information. Surface models describe the surface in general terms. Coordinate system objects manage the various coordinate systems involved in most missions. File objects manage access to metadata (labels, including telemetry information) in the input EDRs and RDRs (Reduced Data Records). Label models manage metadata information in output files. Site objects keep track of different locations where the spacecraft might be at a given time. Radiometry models allow correction of radiometry for an image. Mission objects contain basic mission parameters. Pointing adjustment ("nav") files allow pointing to be corrected. The object-oriented structure (C++) makes it easy to subclass just the pieces of the library that are truly mission-specific. Typically, this involves just the pointing model and coordinate systems, and parts of the file model. Once the library was developed (initially for Mars Polar Lander, MPL), adding new missions ranged from two days to a few months, resulting in significant cost savings as compared to rewriting all the application programs for each mission. Currently supported missions include Mars Pathfinder (MPF), MPL, Mars Exploration Rover (MER), Phoenix, and Mars Science Lab (MSL). Applications based on this library create the majority of operational image RDRs for those missions. A
Spinor representations of affine Lie algebras
Frenkel, I. B.
1980-01-01
Let [unk] be an infinite-dimensional Kac-Moody Lie algebra of one of the types Dl+1(2), Bl(1), or Dl(1). These algebras are characterized by the property that an elimination of any endpoint of their Dynkin diagrams gives diagrams of types Bl or Dl of classical orthogonal Lie algebras. We construct two representations of a Lie algebra [unk], which we call spinor representations, following the analogy with the classical case. We obtain that every spinor representation is either irreducible or has two irreducible components. This provides us with an explicit construction of fundamental representations of [unk], two for the type Dl+1(2), three for Bl(1), and four for Dl(1). We note the profound connection of our construction with quantum field theory—in particular, with fermion fields. Comparing the character formulas of our representations with another construction of the fundamental representations of Kac-Moody Lie algebras of types Al(1), Dl(1), El(1), we obtain classical Jacobi identities and addition formulas for elliptic θ-functions. PMID:16592912
A process algebra model of QED
NASA Astrophysics Data System (ADS)
Sulis, William
2016-03-01
The process algebra approach to quantum mechanics posits a finite, discrete, determinate ontology of primitive events which are generated by processes (in the sense of Whitehead). In this ontology, primitive events serve as elements of an emergent space-time and of emergent fundamental particles and fields. Each process generates a set of primitive elements, using only local information, causally propagated as a discrete wave, forming a causal space termed a causal tapestry. Each causal tapestry forms a discrete and finite sampling of an emergent causal manifold (space-time) M and emergent wave function. Interactions between processes are described by a process algebra which possesses 8 commutative operations (sums and products) together with a non-commutative concatenation operator (transitions). The process algebra possesses a representation via nondeterministic combinatorial games. The process algebra connects to quantum mechanics through the set valued process and configuration space covering maps, which associate each causal tapestry with sets of wave functions over M. Probabilities emerge from interactions between processes. The process algebra model has been shown to reproduce many features of the theory of non-relativistic scalar particles to a high degree of accuracy, without paradox or divergences. This paper extends the approach to a semi-classical form of quantum electrodynamics.
ADA interpretative system for image algebra
NASA Astrophysics Data System (ADS)
Murillo, Juan J.; Wilson, Joseph N.
1992-06-01
An important research problem in image processing is to find appropriate tools to support algorithm development. There have been efforts to build algorithm development support systems for image algebra in several languages, but these systems still have the disadvantage of the time consuming algorithm development style associated with compilation-oriented programming. This paper starts with a description of the Run-Time Support Library (RTSL), which serves as the base for executing programs on both the Image Algebra Ada Translator (IAAT) and Image Algebra Ada Interpreter (IAAI). A presentation on the current status of IAAT and its capabilities is followed by a brief introduction to the utilization of the Image Display Manager (IDM) for image manipulation and analysis. We then discuss in detail the current development stage of IAAI and its relation with RTSL and IDM. The last section describes the design of a syntax-directed graphical user interface for IAAI. We close with an analysis of the current performance of IAAI, and future trends are discussed. Appendix A gives a brief introduction to Image Algebra (IA), and in Appendix B the reader is presented to the Image Algebra Ada (IAA) grammar.
Permutation centralizer algebras and multimatrix invariants
NASA Astrophysics Data System (ADS)
Mattioli, Paolo; Ramgoolam, Sanjaye
2016-03-01
We introduce a class of permutation centralizer algebras which underly the combinatorics of multimatrix gauge-invariant observables. One family of such noncommutative algebras is parametrized by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of two-matrix models. The structure of the algebra, notably its dimension, its center and its maximally commuting subalgebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The center of the algebra allows efficient computation of a sector of multimatrix correlators. These generate the counting of a certain class of bicoloured ribbon graphs with arbitrary genus.
Thermodynamics of Asymptotically Conical Geometries.
Cvetič, Mirjam; Gibbons, Gary W; Saleem, Zain H
2015-06-12
We study the thermodynamical properties of a class of asymptotically conical geometries known as "subtracted geometries." We derive the mass and angular momentum from the regulated Komar integral and the Hawking-Horowitz prescription and show that they are equivalent. By deriving the asymptotic charges, we show that the Smarr formula and the first law of thermodynamics hold. We also propose an analog of Christodulou-Ruffini inequality. The analysis can be generalized to other asymptotically conical geometries.
Investigating Fractal Geometry Using LOGO.
ERIC Educational Resources Information Center
Thomas, David A.
1989-01-01
Discusses dimensionality in Euclidean geometry. Presents methods to produce fractals using LOGO. Uses the idea of self-similarity. Included are program listings and suggested extension activities. (MVL)
Kligfield, R.; Geiser, P.; Geiser, J.
1985-01-01
Blind thrusts are structures which at no time in their history broke the erosion surface and along which displacement progressively changes upwards. Faults of the stiff layer along which displacement progressively decreases to zero (tip) are one prominent type of blind thrust structure. Shortening above such tips is accommodated entirely by folding whereas shortening below the tip is partitioned between folding and faulting. For these types of faults it is possible to determine the original length of the stiff layer for balancing purposes. A systematic methodology for line length and area restoration is outlined for determining blind thrust geometry. Application of the methodology is particularly suitable for use with microcomputers. If the folded form of the cover is known along with the position of the fault and its tip, then it is possible to locate hanging and footwall cutoffs. If the fault trajectory, tip, and a single hanging wall footwall cutoff pair are known, then the folded form of the cover layer can be determined. In these constructions it is necessary to specify pin lines for balancing purposes. These pin lines may or may not have a zero displacement gradient, depending upon the amount of simple shear deformation. Examples are given from both Laramide structures of the western USA and the Appalachians.
Higher spin approaches to quantum field theory and (psuedo)-Riemannian geometries
NASA Astrophysics Data System (ADS)
Hallowell, Karl Evan
In this thesis, we study a number of higher spin quantum field theories and some of their algebraic and geometric consequences. These theories apply mostly either over constant curvature or more generally symmetric pseudo-Riemannian manifolds. The first part of this dissertation covers a superalgebra coming from a family of particle models over symmetric spaces. These theories are novel in that the symmetries of the (super)algebra osp( Q|2p) are larger and more elaborate than traditional symmetries. We construct useful (super)algebras related to and generalizing old work by Lichnerowicz and describe their role in developing the geometry of massless models with osp(Q|2 p) symmetry. The result is two practical applications of these (super)algebras: (1) a lunch more concise description of a family of higher spin quantum field theories; and (2) an interesting algebraic probe of underlying background geometries. We also consider massive models over constant curvature spaces. We use a radial dimensional reduction process which converts massless models into massive ones over a lower dimensional space. In our case, we take from the family of theories above the particular free, massless model over flat space associated with sp(2, R ) and derive a massive model. In the process, we develop a novel associative algebra, which is a deformation of the original differential operator algebra associated with the sp(2, R ) model. This algebra is interesting in its own right since its operators realize the representation structure of the sp(2, R ) group. The massive model also has implications for a sequence of unusual, "partially massless" theories. The derivation illuminates how reduced degrees of freedom become manifest in these particular models. Finally, we study a Yang-Mills model using an on-shell Poincare Yang-Mills twist of the Maxwell complex along with a non-minimal coupling. This is a special, higher spin case of a quantum field theory called a Yang-Mills detour complex
Computational algebraic geometry for statistical modeling FY09Q2 progress.
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 in more detail; the next section provides an overview of the project and how the current progress fits into it.
ERIC Educational Resources Information Center
Dyer, Richard R.; Reed, Philip A.; Berry, Robert Q.
2006-01-01
The standards-based reform movement in education that began in the 1980s has evolved. In the 1990s, the focus was on producing subject-area content standards and modifying instruction. Today, the focus has shifted to assessment, and for technology education, demonstrating the impact on children and the efficacy of the discipline within general…
Real Numerical Algebraic Geometry: Finding All Real Solutions of a Polynomial System
2014-02-01
self -conjugate, and performing isosingular deflation (if necessary). This ensures that the remainder of the algorithm will function properly...of the repository is available for checkout to the public without credentials, at svn://paramotopy.com/bertini_real. Documentation, produced
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…
Can You Fathom This? Connecting Data Analysis, Algebra, and Geometry with Probability Simulation
ERIC Educational Resources Information Center
Edwards, Michael Todd; Phelps, Steve
2008-01-01
Data analysis plays a prominent role in various facets of modern life: Schools evaluate and revise programs on the basis of test scores; policymakers make decisions on the basis of information gleaned from polling data; supermarkets stock shelves on the basis of data collected at checkout lanes. Data analysis provides teachers with new tools and…
An algebra of discrete event processes
NASA Technical Reports Server (NTRS)
Heymann, Michael; Meyer, George
1991-01-01
This report deals with an algebraic framework for modeling and control of discrete event processes. The report consists of two parts. The first part is introductory, and consists of a tutorial survey of the theory of concurrency in the spirit of Hoare's CSP, and an examination of the suitability of such an algebraic framework for dealing with various aspects of discrete event control. To this end a new concurrency operator is introduced and it is shown how the resulting framework can be applied. It is further shown that a suitable theory that deals with the new concurrency operator must be developed. In the second part of the report the formal algebra of discrete event control is developed. At the present time the second part of the report is still an incomplete and occasionally tentative working paper.
Operator algebra in logarithmic conformal field theory
Nagi, Jasbir
2005-10-15
For some time now, conformal field theories in two dimensions have been studied as integrable systems. Much of the success of these studies is related to the existence of an operator algebra of the theory. In this paper, some of the extensions of this machinery to the logarithmic case are studied and used. More precisely, from Moebius symmetry constraints, the generic three- and four-point functions of logarithmic quasiprimary fields are calculated in closed form for arbitrary Jordan rank. As an example, c=0 disordered systems with nondegenerate vacua are studied. With the aid of two-, three-, and four-point functions, the operator algebra is obtained and associativity of the algebra studied.
Algebraic quantum gravity (AQG): II. Semiclassical analysis
NASA Astrophysics Data System (ADS)
Giesel, K.; Thiemann, T.
2007-05-01
In the previous paper (Giesel and Thiemann 2006 Conceptual setup Preprint gr-qc/0607099) a new combinatorial and thus purely algebraical approach to quantum gravity, called algebraic quantum gravity (AQG), was introduced. In the framework of AQG, existing semiclassical tools can be applied to operators that encode the dynamics of AQG such as the master constraint operator. In this paper, we will analyse the semiclassical limit of the (extended) algebraic master constraint operator and show that it reproduces the correct infinitesimal generators of general relativity. Therefore, the question of whether general relativity is included in the semiclassical sector of the theory, which is still an open problem in LQG, can be significantly improved in the framework of AQG. For the calculations, we will substitute SU(2) with U(1)3. That this substitution is justified will be demonstrated in the third paper (Giesel and Thiemann 2006 Semiclassical perturbation theory Preprint gr-qc/0607101) of this series.
Sound Off! A Dialogue between Calculator and Algebra
ERIC Educational Resources Information Center
Wade, William R.
2006-01-01
This article illustrates the fact that unless tempered by algebraic reasoning, a graphing calculator can lead one to erroneous conclusions. It also demonstrates that some problems can be solved by combining technology with algebra.
A new algebra core for the minimal form' problem
Purtill, M.R. . Center for Communications Research); Oliveira, J.S.; Cook, G.O. Jr. )
1991-12-20
The demands of large-scale algebraic computation have led to the development of many new algorithms for manipulating algebraic objects in computer algebra systems. For instance, parallel versions of many important algorithms have been discovered. Simultaneously, more effective symbolic representations of algebraic objects have been sought. Also, while some clever techniques have been found for improving the speed of the algebraic simplification process, little attention has been given to the issue of restructuring expressions, or transforming them into minimal forms.'' By minimal form,'' we mean that form of an expression that involves a minimum number of operations. In a companion paper, we introduce some new algorithms that are very effective at finding minimal forms of expressions. These algorithms require algebraic and combinatorial machinery that is not readily available in most algebra systems. In this paper we describe a new algebra core that begins to provide the necessary capabilities.
Infinitesimal deformations of naturally graded filiform Leibniz algebras
NASA Astrophysics Data System (ADS)
Khudoyberdiyev, A. Kh.; Omirov, B. A.
2014-12-01
In the present paper we describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any n-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra Fn3(0) . We establish that in the same way any n-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras Fn1,Fn2and Fn3(α) . Moreover, we describe the linear integrable deformations of the above-mentioned algebras with a fixed basis of HL2 in the set of all n-dimensional Leibniz algebras. Among these deformations one new rigid algebra has been found.
Current algebra and the nonlinear σ-model
NASA Astrophysics Data System (ADS)
Ghosh, S.
2007-06-01
We present the current algebra of a particular form in the nonlinear σ-model. The algebra has a non-Abelian form with field-dependent structure functions. We comment on the connection of the model with noncommutative space.
Kac-Moody algebra and nonlinear sigma model
NASA Astrophysics Data System (ADS)
Ogura, Waichi; Hosoya, Akio
1985-12-01
We investigate the nonlinear sigma model over an arbitrary homogeneous space. Then it is shown that the sigma model realizes the Kac-Moody algebra as current algebra only if the homogeneous space is restricted to the group manifold.
Upper bound for the length of commutative algebras
NASA Astrophysics Data System (ADS)
Markova, Ol'ga V.
2009-12-01
By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field one means the least positive integer k such that the words of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, an upper bound for the length of a commutative algebra in terms of a function of two invariants of the algebra, the dimension and the maximal degree of the minimal polynomial for the elements of the algebra, is obtained. As a corollary, a formula for the length of the algebra of diagonal matrices over an arbitrary field is obtained. Bibliography: 8 titles.
Rees algebras, Monomial Subrings and Linear Optimization Problems
NASA Astrophysics Data System (ADS)
Dupont, Luis A.
2010-06-01
In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.