Distance geometry and geometric algebra
NASA Astrophysics Data System (ADS)
Dress, Andreas W. M.; Havel, Timothy F.
1993-10-01
As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18)
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.
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…
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. PMID:26806075
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…
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…
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.
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.
Differential geometry on Hopf algebras and quantum groups
Watts, P.
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.
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.
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…
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.
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.
[Geometry and algebra of branches of the middle cerebral artery].
Blinkov, S M
1986-01-01
A classification of the cortical branches of the middle cerebral artery (MCA) is suggested by means of which each branch in any hemisphere can be qualified and identified in any variant of MCA branching. The principle of the classification consists in grouping the branches into arteries and trunks of the second, third, etc. order. Branches supplying blood to a certain sector of the lateral surface of the hemisphere are designated arteries. Their number and zone of branching are constant. Branches giving rise to 2 and more arteries are named trunks. Branching of the trunks, the number of trunks of the second, third, etc. order, and the site and type of origin of the arteries are extremely variable. Each trunk can be designated by a formula stating its order and the name of the artery supplied by this trunk. The arrangement of the MCA branches on the surface of the gyri and deep in the sulci, represented on the map of the lateral surface of the hemisphere, is designated conditionally as geometry of MCA branches. The order of branching of the trunks and the type of origin of the arteries, represented on abstract maps of the lateral surface of the hemisphere, are designated conditionally as algebra of the MCA branches. The variability of the geometry and algebra of the MCA branches must be taken into consideration in operations for extra-intracranial microanastomosis and in endovasal intervention on the MCA. PMID:3811741
Clifford Algebras in Symplectic Geometry and Quantum Mechanics
NASA Astrophysics Data System (ADS)
Binz, Ernst; de Gosson, Maurice A.; Hiley, Basil J.
2013-04-01
The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C 0,2. This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, {F}a of the Euclidean three-space. This enables us to construct a Poisson Clifford algebra, ℍ F , of a finite dimensional phase space which will carry the dynamics. The quantum dynamics appears as a realisation of ℍ F in terms of a Clifford algebra consisting of Hermitian operators.
Hermitian geometry of 6-dimensional submanifolds of the Cayley algebra
Banaru, M B
2002-06-30
Orientable 6-dimensional submanifolds (of general type) of the Cayley algebra are investigated on which the 3-fold vector cross products in the octave algebra induce a Hermitian structure. It is shown that such submanifolds of the Cayley algebra are minimal, non-compact, and para-Kaehler, their holomorphic bisectional curvature is positive and vanishes only at the geodesic points. It is also proved that cosymplectic hypersurfaces of 6-dimensional Hermitian submanifolds of the octave algebra are ruled. A simple test for the minimality of such surfaces is obtained. It is shown that 6-dimensional submanifolds of the Cayley algebra satisfying the axiom of g-cosymplectic hypersurfaces are Kaehler manifolds.
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.
NASA Astrophysics Data System (ADS)
Kao, M. J.; Yu, C. C.; Chang, H.; Tsung, T. T.; Lin, H. M.
2006-10-01
This paper describes the instrumentation and analysis of the Vehicle suspension's electrical signals. It will measure the Vehicle suspensions' Vertical Displacement, Track Change, Camber Angle, Caster Angle Steer Angle and convert physical quantity into electrical signals in a various vehicle load change. With using electrical signals for computer control, the electrical controlled vehicle has brought great convenience, great safety and thoughtful kindness vehicle system in our daily life. It will measure the Vehicle suspensions' Vertical Displacement, Track Change, Camber Angle, Caster Angle Steer Angle and convert physical quantity into electrical signals in a various vehicle load change. The function of a suspension system in an automobile is to improve ride comfort and stability. Advances in electronic control technology, applied to the automobile, can improve those functions. The results show that the photocell can convert the electrical signals of suspension for peripheral communications link between the vehicle driving and the electronic control unit (ECU) employed for processing.
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.
Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology
NASA Astrophysics Data System (ADS)
Manin, Yuri I.; Marcolli, Matilde
2014-07-01
We introduce some algebraic geometric models in cosmology related to the ''boundaries'' of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point x. This creates a boundary which consists of the projective space of tangent directions to x and possibly of the light cone of x. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from ''the end of previous aeon'' of the expanding and cooling Universe to the ''beginning of the next aeon'' is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary.
Quantum Phase Space from Schwinger's Measurement Algebra
NASA Astrophysics Data System (ADS)
Watson, P.; Bracken, A. J.
2014-07-01
Schwinger's algebra of microscopic measurement, with the associated complex field of transformation functions, is shown to provide the foundation for a discrete quantum phase space of known type, equipped with a Wigner function and a star product. Discrete position and momentum variables label points in the phase space, each taking distinct values, where is any chosen prime number. Because of the direct physical interpretation of the measurement symbols, the phase space structure is thereby related to definite experimental configurations.
Abstract Algebra, Projective Geometry and Time Encoding of Quantum Information
NASA Astrophysics Data System (ADS)
Planat, Michel; Saniga, Metod
2005-10-01
Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of the integers modulo a prime characteristic p. They can be used to generate efficient cyclic encoding, for transmitting secrete quantum keys, for quantum state recovery and for error correction in quantum computing. Finite projective planes and their generalization are the geometric counterpart to cyclotomic concepts, their coordinatization involves Galois fields, and they have been used repetitively for enciphering and coding. Finally, the characters over Galois fields are fundamental for generating complete sets of mutually unbiased bases, a generic concept of quantum information processing and quantum entanglement. Gauss sums over Galois fields ensure minimum uncertainty under such protocols. Some Galois rings which are cyclotomic extensions of the integers modulo 4 are also becoming fashionable for their role in time encoding and mutual unbiasedness.
Bridging Algebra & Geometry with "n"-Gram Proofs
ERIC Educational Resources Information Center
Craven, Joshua D.
2010-01-01
For many students, geometry is the first course in which mathematical proof takes center stage. To help ease students into writing proofs, the author tries to create lessons and activities throughout the year that challenge students to prove their own conjectures by using tools learned in previous mathematics courses. Teachers cannot get all…
Quantum error-correcting codes from algebraic geometry codes of Castle type
NASA Astrophysics Data System (ADS)
Munuera, Carlos; Tenório, Wanderson; Torres, Fernando
2016-07-01
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.
Algebras of Measurements: The Logical Structure of Quantum Mechanics
NASA Astrophysics Data System (ADS)
Lehmann, Daniel; Engesser, Kurt; Gabbay, Dov M.
2006-04-01
In quantum physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties of such operators are justified on epistemological grounds. Commutation of measurements is a central topic of interest. Classical logical systems may be viewed as measurement algebras in which all measurements commute.
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.
Sheaf-theoretic representation of quantum measure algebras
Zafiris, Elias
2006-09-15
We construct a sheaf-theoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measure algebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measure algebras and the category of quantum measure algebras. The sheaf-theoretic semantic transition of quantum structures shifts their physical significance from the orthoposet axiomatization at the level of events, to the sheaf-theoretic gluing conditions at the level of Boolean localization systems.
Algebraic geometry methods associated to the one-dimensional Hubbard model
NASA Astrophysics Data System (ADS)
Martins, M. J.
2016-06-01
In this paper we study the covering vertex model of the one-dimensional Hubbard Hamiltonian constructed by Shastry in the realm of algebraic geometry. We show that the Lax operator sits in a genus one curve which is not isomorphic but only isogenous to the curve suitable for the AdS/CFT context. We provide an uniformization of the Lax operator in terms of ratios of theta functions allowing us to establish relativistic like properties such as crossing and unitarity. We show that the respective R-matrix weights lie on an Abelian surface being birational to the product of two elliptic curves with distinct J-invariants. One of the curves is isomorphic to that of the Lax operator but the other is solely fourfold isogenous. These results clarify the reason the R-matrix can not be written using only difference of spectral parameters of the Lax operator.
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…
Gully geometry: what are we measuring?
NASA Astrophysics Data System (ADS)
Casalí, J.; Giménez, R.; Campo-Bescós, M. A.
2015-07-01
Much of the research on (ephemeral) gully erosion comprises the determination of the geometry of these eroded channels, especially their width and depth. This is not a simple task due to uncertainty generated by the wide range of variability in gully cross section shapes found in the field. However, in the literature, this uncertainty is not recognized so that no criteria for their measurement are indicated. The aim of this work is to make researchers aware of the ambiguity that arises when characterizing the geometry of an ephemeral gully and similar eroded channels. In addition, a measurement protocol is proposed with the ultimate goal of pooling criteria in future works. It is suggested that the geometry of a gully could be characterized through its mean equivalent width and mean equivalent depth, which, together with its length, define an "equivalent prismatic gully" (EPG). The latter would facilitate the comparison between different gullies.
Cumulants and the moment algebra: Tools for analyzing weak measurements
Aaberg, Johan; Mitchison, Graeme
2009-04-15
Recently it has been shown that cumulants significantly simplify the analysis of multipartite weak measurements. Here we consider the mathematical structure that underlies this and find that it can be formulated in terms of what we call the moment algebra. Apart from resulting in simpler proofs, the flexibility of this structure allows generalizations of the original results to a number of weak measurement scenarios, including one where the weakly interacting pointers reach thermal equilibrium with the probed system.
Measurement of quantum fluctuations in geometry
NASA Astrophysics Data System (ADS)
Hogan, Craig J.
2008-05-01
A particular form for the quantum indeterminacy of relative spacetime position of events is derived from the context of a holographic geometry with a minimum length at the Planck scale. The indeterminacy predicts fluctuations from a classically defined geometry in the form of “holographic noise” whose spatial character, absolute normalization, and spectrum are predicted with no parameters. The noise has a distinctive transverse spatial shear signature and a flat power spectral density given by the Planck time. An interferometer signal displays noise due to the uncertainty of relative positions of reflection events. The noise corresponds to an accumulation of phase offset with time that mimics a random walk of those optical elements that change the orientation of a wavefront. It only appears in measurements that compare transverse positions and does not appear at all in purely radial position measurements. A lower bound on holographic noise follows from a covariant upper bound on gravitational entropy. The predicted holographic noise spectrum is estimated to be comparable to measured noise in the currently operating interferometric gravitational-wave detector GEO600. Because of its transverse character, holographic noise is reduced relative to gravitational wave effects in other interferometer designs, such as the LIGO observatories, where beam power is much less in the beam splitter than in the arms.
External Variables as Predictors of Van Hiele Levels in Algebra and Geometry Students.
ERIC Educational Resources Information Center
Frykholm, Jeffrey A.
In the attempt to improve the quality of geometry instruction in schools, researchers and teachers alike have given considerable attention to the van Hiele theory of geometry learning and development, which proposes a series of cognitive levels through which every geometry student passes. This paper reports a study to determine the extent to which…
Super quantum measures on effect algebras with the Riesz decomposition properties
Xie, Yongjian Ren, Fang; Yang, Aili
2015-10-15
We give one basis of the space of super quantum measures on finite effect algebras with the Riesz decomposition properties (RDP for short). Then we prove that the super quantum measures and quantum interference functions on finite effect algebras with the RDP are determined each other. At last, we investigate the relationships between the super quantum measures and the diagonally positive signed measures on finite effect algebras with the RDP in detail.
Gully geometry: what are we measuring?
NASA Astrophysics Data System (ADS)
Casalí, Javier; Giménez, Rafael; Ángel Campo, Miguel
2014-05-01
Gully erosion has attracted the attention of many scientists during the last decades, and gullies are an important source of sediment within catchments. For succeeding in gully erosion research, gullies must be properly characterized. Characterization includes the determination of gully morphology and volume, being the definition of gully width (W) and depth (D) -and consequently related variables such as the well-known W/D ratio- key issues toward to this goal. However, and surprisingly, universally accepted criteria (rules or guidance) to define gully morphology are lacking. This because the protocol every researcher follows to measure the eroded channel geometry is generally taken for granted and most of the time even no explanation is given about it. For example, when analyzing a gully cross section we usually just identify gully depth with gully maximum depth. But, is this the right protocol? What does this length really represent? What is its meaning? All this uncertainties can lead to non-comparable results and then important inconsistencies. So, to define universal rules of procedure would allow gully scientists "speak the same language" and then deliver truly comparable gully geometry and volume. On the other hand, there are other misunderstandings. For example, very frequently we characterize or depict a whole gully only through some of its cross sections. Again, is this correct? The problem is even more complex when considering that gully geometry may (largely) change along the channel. The main aim of this presentation is to highlight some (unnoticed) common flaws when measuring and describing gully geometry, hoping ultimately to open a debate on that subject. For this last purpose, a conceptual approach to define gully cross section width and other derived variables is firstly proposed. It is based on the subtraction of a highly detailed digital elevation model of a landscape surface containing the studied gully (DEM1) from a detailed spatial
Algebraic Structures, Physics and Geometry from a Unified Field Theoretical Framework
NASA Astrophysics Data System (ADS)
Cirilo-Lombardo, Diego Julio
2015-10-01
Starting from a Unified Field Theory (UFT) proposed previously by the author, the possible fermionic representations arising from the same spacetime are considered from the algebraic and geometrical viewpoint. We specifically demonstrate in this UFT general context that the underlying basis of the single geometrical structure P( G, M) (the principal fiber bundle over the real spacetime manifold M with structural group G) reflecting the symmetries of the different fields carry naturally a biquaternionic structure instead of a complex one. This fact allows us to analyze algebraically and to interpret physically in a straighforward way the Majorana and Dirac representations and the relation of such structures with the spacetime signature and non-hermitian (CP) dynamic operators. Also, from the underlying structure of the tangent space, the existence of hidden (super) symmetries and the possibility of supersymmetric extensions of these UFT models are given showing that Rothstein's theorem is incomplete for that description. The importance of the Clifford algebras in the description of all symmetries, mainly the interaction of gravity with the other fields, is briefly discussed.
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.
Algebraic and algorithmic frameworks for optimized quantum measurements
NASA Astrophysics Data System (ADS)
Laghaout, Amine; Andersen, Ulrik L.
2015-10-01
von Neumann projections are the main operations by which information can be extracted from the quantum to the classical realm. They are, however, static processes that do not adapt to the states they measure. Advances in the field of adaptive measurement have shown that this limitation can be overcome by "wrapping" the von Neumann projectors in a higher-dimensional circuit which exploits the interplay between measurement outcomes and measurement settings. Unfortunately, the design of adaptive measurement has often been ad hoc and setup specific. We shall here develop a unified framework for designing optimized measurements. Our approach is twofold: The first is algebraic and formulates the problem of measurement as a simple matrix diagonalization problem. The second is algorithmic and models the optimal interaction between measurement outcomes and measurement settings as a cascaded network of conditional probabilities. Finally, we demonstrate that several figures of merit, such as Bell factors, can be improved by optimized measurements. This leads us to the promising observation that measurement detectors which—taken individually—have a low quantum efficiency can be arranged into circuits where, collectively, the limitations of inefficiency are compensated for.
The Lie algebraic significance of symmetric informationally complete measurements
NASA Astrophysics Data System (ADS)
Appleby, D. M.; Flammia, Steven T.; Fuchs, Christopher A.
2011-02-01
Examples of symmetric informationally complete positive operator-valued measures (SIC-POVMs) have been constructed in every dimension ⩽67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl(d,{C}). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl(d,{C}). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.
Schwinger's Measurement Algebra, Preons and the Lepton Masses
NASA Astrophysics Data System (ADS)
Brannen, Carl
2006-04-01
In the 1950s and 1960s, Julian Schwinger developed an elegant general scheme for quantum kinematics and dynamics appropriate to systems with a finite number of dynamical variables, now knowns as ``Schwinger's Measurement Algebra'' (SMA). The SMA has seen little use, largely because it is non relativistic in that it does not allow for particle creation. In this paper, we apply the SMA to the problem of modeling tightly bound subparticles (preons) of the leptons and quarks. We discuss the structure of the ideals of Clifford algebras and, applying this to the elementary fermions, derive a preon substructure for the quarks and leptons. We show that matrices of SMA type elements can be used to model the quarks and leptons under the assumption that the preons are of such high energy that they cannot be created in normal interactions. This gives a definition of the SMA for the composite particle in terms of the SMA of its constituents. We solve the resulting matrix equation for the quarks and leptons. We show that the mass operator for the charged leptons is related to the democratic mass matrix used in the Koide mass formula.
The Lie algebraic significance of symmetric informationally complete measurements
Appleby, D. M.; Flammia, Steven T.; Fuchs, Christopher A.
2011-02-15
Examples of symmetric informationally complete positive operator-valued measures (SIC-POVMs) have been constructed in every dimension {<=}67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl(d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl(d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.
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…
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…
An Algebraic Approach to Unital Quantities and their Measurement
NASA Astrophysics Data System (ADS)
Domotor, Zoltan; Batitsky, Vadim
2016-06-01
The goals of this paper fall into two closely related areas. First, we develop a formal framework for deterministic unital quantities in which measurement unitization is understood to be a built-in feature of quantities rather than a mere annotation of their numerical values with convenient units. We introduce this idea within the setting of certain ordered semigroups of physical-geometric states of classical physical systems. States are assumed to serve as truth makers of metrological statements about quantity values. A unital quantity is presented as an isomorphism from the target system's ordered semigroup of states to that of positive reals. This framework allows us to include various derived and variable quantities, encountered in engineering and the natural sciences. For illustration and ease of presentation, we use the classical notions of length, time, electric current and mean velocity as primordial examples. The most important application of the resulting unital quantity calculus is in dimensional analysis. Second, in evaluating measurement uncertainty due to the analog-to-digital conversion of the measured quantity's value into its measuring instrument's pointer quantity value, we employ an ordered semigroup framework of pointer states. Pointer states encode the measuring instrument's indiscernibility relation, manifested by not being able to distinguish the measured system's topologically proximal states. Once again, we focus mainly on the measurement of length and electric current quantities as our motivating examples. Our approach to quantities and their measurement is strictly state-based and algebraic in flavor, rather than that of a representationalist-style structure-preserving numerical assignment.
Development of an algorithm to measure defect geometry using a 3D laser scanner
NASA Astrophysics Data System (ADS)
Kilambi, S.; Tipton, S. M.
2012-08-01
Current fatigue life prediction models for coiled tubing (CT) require accurate measurements of the defect geometry. Three-dimensional (3D) laser imaging has shown promise toward becoming a nondestructive, non-contacting method of surface defect characterization. Laser imaging provides a detailed photographic image of a flaw, in addition to a detailed 3D surface map from which its critical dimensions can be measured. This paper describes algorithms to determine defect characteristics, specifically depth, width, length and projected cross-sectional area. Curve-fitting methods were compared and implicit algebraic fits have higher probability of convergence compared to explicit geometric fits. Among the algebraic fits, the Taubin circle fit has the least error. The algorithm was able to extract the dimensions of the flaw geometry from the scanned data of CT to within a tolerance of about 0.127 mm, close to the tolerance specified for the laser scanner itself, compared to measurements made using traveling microscopes. The algorithm computes the projected surface area of the flaw, which could previously only be estimated from the dimension measurements and the assumptions made about cutter shape. Although shadows compromised the accuracy of the shape characterization, especially for deep and narrow flaws, the results indicate that the algorithm with laser scanner can be used for non-destructive evaluation of CT in the oil field industry. Further work is needed to improve accuracy, to eliminate shadow effects and to reduce radial deviation.
Interpretation for scales of measurement linking with abstract algebra
2014-01-01
The Stevens classification of levels of measurement involves four types of scale: “Nominal”, “Ordinal”, “Interval” and “Ratio”. This classification has been used widely in medical fields and has accomplished an important role in composition and interpretation of scale. With this classification, levels of measurements appear organized and validated. However, a group theory-like systematization beckons as an alternative because of its logical consistency and unexceptional applicability in the natural sciences but which may offer great advantages in clinical medicine. According to this viewpoint, the Stevens classification is reformulated within an abstract algebra-like scheme; ‘Abelian modulo additive group’ for “Ordinal scale” accompanied with ‘zero’, ‘Abelian additive group’ for “Interval scale”, and ‘field’ for “Ratio scale”. Furthermore, a vector-like display arranges a mixture of schemes describing the assessment of patient states. With this vector-like notation, data-mining and data-set combination is possible on a higher abstract structure level based upon a hierarchical-cluster form. Using simple examples, we show that operations acting on the corresponding mixed schemes of this display allow for a sophisticated means of classifying, updating, monitoring, and prognosis, where better data mining/data usage and efficacy is expected. PMID:24987515
Application of Algebra Curriculum-Based Measurements for Decision Making in Middle and High School
ERIC Educational Resources Information Center
Johnson, Evelyn S.; Galow, Patricia A.; Allenger, Robert
2013-01-01
This article reports the results of a study examining the utility of curriculum-based measurement (CBM) in algebra for predicting performance on a state math assessment and informing instructional placement decisions for students in seventh, eighth, and tenth grades. Students completed six Basic Skills algebra probes across different time…
Chaves, Madalena; Sengupta, Anirvan; Sontag, Eduardo D.
2010-01-01
The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As a detailed case study, the segment polarity gene network is analyzed. PMID:18987858
Using Children's Literature to Teach Geometry and Measurement.
ERIC Educational Resources Information Center
Meconi, L. J.; Moss, Barbara
1991-01-01
Discusses selected children's literature dealing with geometry and measurement concepts. Suggests activities for grades three through six to be used as part of a learning center or to be completed in cooperative learning groups. (MG)
Measuring a van Hiele Geometry Sequence: A Reanalysis.
ERIC Educational Resources Information Center
Wilson, Mark
1990-01-01
Summarizes a reanalysis of the data from an investigation of a test designed to measure a learning sequence in geometry based on the work of van Hiele (1986). Discusses the test based on the Rasch model. (YP)
Work Measurements: Interdisciplinary Overlap in Manufacturing and Algebra I
ERIC Educational Resources Information Center
Rose, Mary Annette
2007-01-01
Manufacturing engineering provides a relevant context from which to envision interdisciplinary learning experiences because engineers integrate their knowledge and skills of manufacturing and algebra processes in order to plan the efficient manufacture of products. In this article, the author describes an interdisciplinary learning activity that…
Measuring Space-Time Geometry over the Ages
Stebbins, Albert; /Fermilab
2012-05-01
Theorists are often told to express things in the 'observational plane'. One can do this for space-time geometry, considering 'visual' observations of matter in our universe by a single observer over time, with no assumptions about isometries, initial conditions, nor any particular relation between matter and geometry, such as Einstein's equations. Using observables as coordinates naturally leads to a parametrization of space-time geometry in terms of other observables, which in turn prescribes an observational program to measure the geometry. Under the assumption of vorticity-free matter flow we describe this observational program, which includes measurements of gravitational lensing, proper motion, and redshift drift. Only 15% of the curvature information can be extracted without long time baseline observations, and this increases to 35% with observations that will take decades. The rest would likely require centuries of observations. The formalism developed is exact, non-perturbative, and more general than the usual cosmological analysis.
Measurement of proton momentum distributions using a direct geometry instrument
NASA Astrophysics Data System (ADS)
Senesi, R.; Kolesnikov, A. I.; Andreani, C.
2014-12-01
We report the results of inelastic neutron scattering measurements on bulk water and ice using the direct geometry SEQUOIA chopper spectrometer at the Spallation Neutron Source (USA), with incident energy Ei= 6 eV. In this set up the measurements allow to access the Deep Inelastic Neutron Scattering regime. The scattering is centred at the proton recoil energy given by the impulse approximation, and the shape of the recoil peak conveys information on the proton momentum distribution in the system. The comparison with the performance of inverse geometry instruments, such as VESUVIO at the ISIS source (UK), shows that complementary information can be accessed by the use of direct and inverse geometry instruments. Analysis of the neutron Compton profiles shows that the proton kinetic energy in ice at 271 K is larger than in room temperature liquid water, in agreement with previous measurements on VESUVIO.
NASA Astrophysics Data System (ADS)
Cat Ho, Nguyen; Van Long, Nguyen
2006-06-01
In the paper, we shall examine fuzziness measure of terms in linear and complete hedge algebras of a linguistic variable. A notion of the so-called semantically quantifying mappings will be redefined more generally and it will be established a closed relationship between fuzziness measure and a class of semantically quantifying mappings defined by a recursive expression with parameters to be fuzziness measure of primary terms and linguistic hedges. An application of fuzziness measure and semantically quantifying mappings in solving fuzzy the multiple conditional reasoning problem will be presented to show an applicability of hedge algebras.
Measuring finite quantum geometries via quasi-coherent states
NASA Astrophysics Data System (ADS)
Schneiderbauer, Lukas; Steinacker, Harold C.
2016-07-01
We develop a systematic approach to determine and measure numerically the geometry of generic quantum or ‘fuzzy’ geometries realized by a set of finite-dimensional Hermitian matrices. The method is designed to recover the semi-classical limit of quantized symplectic spaces embedded in {{{R}}}d including the well-known examples of fuzzy spaces, but it applies much more generally. The central tool is provided by quasi-coherent states, which are defined as ground states of Laplace- or Dirac operators corresponding to localized point branes in target space. The displacement energy of these quasi-coherent states is used to extract the local dimension and tangent space of the semi-classical geometry, and provides a measure for the quality and self-consistency of the semi-classical approximation. The method is discussed and tested with various examples, and implemented in an open-source Mathematica package.
Acoustic Liner Drag: Measurements on Novel Facesheet Perforate Geometries
NASA Technical Reports Server (NTRS)
Howerton, Brian M.; Jones, Michael G.
2016-01-01
Interest in characterization of the aerodynamic drag of acoustic liners has increased in the past several years. This paper details experiments in the NASA Langley Grazing Flow Impedance Tube to quantify the relative drag of several perforate-over-honeycomb liner configurations at flow speeds of centerline flow Mach number equals 0.3 and 0.5. Various perforate geometries and orientations are investigated to determine their resistance factors using a static pressure drop approach. Comparison of these resistance factors gives a relative measurement of liner drag. For these same flow conditions, acoustic measurements are performed with tonal excitation from 400 to 3000 hertz at source sound pressure levels of 140 and 150 decibels. Educed impedance and attenuation spectra are used to determine the impact of variations in perforate geometry on acoustic performance.
NASA Astrophysics Data System (ADS)
Zhu, Huangjun
2014-09-01
Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank 1. They are interesting originally because of their connection with rank-1 SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal informationally complete (IC) measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs are optimal among minimal IC measurements with given average purity of measurement outcomes. Besides its significance to the current study, this result may help us to understand tomographic efficiencies of minimal IC measurements under the influence of noise. When minimal IC measurements are taken as bases for the Lie algebra of the unitary group, generalized SICs are uniquely characterized by the antisymmetry of the associated structure constants.
Interpretation of the prominence differential emissions measure for 3 geometries
NASA Technical Reports Server (NTRS)
Schmahl, E. J.; Orrall, F. Q.
1986-01-01
Researchers have used prominence extreme ultraviolet line intensities observed from Skylab to derive the differential emission measure Q(T) in the prominence-corona (PC) interface from 3 x 10,000 to 3 times 1 million K, including the effects of Lyman Continuum absorption. Using lines both shortward and longward of the Lyman limit, researchers have estimated the importance of absorption as function of temperature. The magnitude of the absorption, as well as its rate of increase as a function of temperature, place limits on the thread scales and the character of the interfilar medium. Researchers have calculated models based on three assumed geometries: (1) threads with hot sheaths and cool cores; (2) isothermal threads; and (3) threads with longitudinal temperature gradients along the magnetic field. Comparison of the absorption computed from these models with the observed absorption in prominences shows that none of the geometries is totally satisfactory.
Coincidence-Summing Corrections for Close Geometry Measurements
Gueray, R. Taygun
2008-11-11
For a given stellar temperature, nuclear reactions take place in the energy range of the Gamow window with the relatively low energies of the astrophysical interest for charged particle induced reactions. In order to measure the nuclear reaction cross sections with the activation method at projectile energies as low as possible, a gamma counting system that consists of Ge detectors and the irradiated target in close geometry is required. The presence of cascade transitions requires coincidence summing corrections that can not be ignored because of the very large solid angle. In this study, the determination of the summing correction factor and photopeak efficiency for a gamma spectrometer, as an example, composed of two Ge clover detectors in close geometry is briefly described.
Retrieving Stratospheric Aerosol Extinction from SCIAMACHY Measurements in Limb Geometry
NASA Astrophysics Data System (ADS)
Dörner, Steffen; Penning de Vries, Marloes; Pukite, Janis; Beirle, Steffen; Wagner, Thomas
2015-04-01
Techniques for retrieving height resolved information on stratospheric aerosol improved significantly in the past decade with the availability of satellite measurements in limb geometry. Instruments like OMPS, OSIRIS and SCIAMACHY provide height resolved radiance spectra with global coverage. Long term data sets of stratospheric aerosol extinction profiles are important for a detailed investigation of spatial and temporal variation and formation processes (e.g. after volcanic eruptions or in polar stratospheric clouds). Resulting data sets contain vital information for climate models (radiative effect) or chemistry models (reaction surface for heterogeneous chemistry). This study focuses on the SCIAMACHY instrument which measured scattered sunlight in the ultra-violet, visible and near infra-red spectral range since the launch on EnviSat in 2002 until an instrumental error occurred in April 2012. SCIAMACHY's unique method of alternating measurements in limb and nadir geometry provides co-located profile and column information respectively that can be used to characterize plumes with small horizontal extents. The covered wavelength range potentially provides information on effective micro-physical properties of the aerosol particles. However, scattering on background aerosol constitutes only a small fraction of detected radiance and assumptions on particle characteristics (e.g. size distribution) have to be made which results in large uncertainties especially for wavelengths below 700nm and for measurements in backscatter geometry. Methods to reduce these uncertainties are investigated and applied to our newly developed retrieval algorithm. In addition, so called spatial straylight contamination of the measured signal was identified as a significant error source and an empirical correction scheme was developed. A large scale comparison study with SAGE II for the temporal overlap of both instruments (2002 to 2005) shows promising results.
Measurement of Fracture Geometry for Accurate Computation of Hydraulic Conductivity
NASA Astrophysics Data System (ADS)
Chae, B.; Ichikawa, Y.; Kim, Y.
2003-12-01
Fluid flow in rock mass is controlled by geometry of fractures which is mainly characterized by roughness, aperture and orientation. Fracture roughness and aperture was observed by a new confocal laser scanning microscope (CLSM; Olympus OLS1100). The wavelength of laser is 488nm, and the laser scanning is managed by a light polarization method using two galvano-meter scanner mirrors. The system improves resolution in the light axis (namely z) direction because of the confocal optics. The sampling is managed in a spacing 2.5 μ m along x and y directions. The highest measurement resolution of z direction is 0.05 μ m, which is the more accurate than other methods. For the roughness measurements, core specimens of coarse and fine grained granites were provided. Measurements were performed along three scan lines on each fracture surface. The measured data were represented as 2-D and 3-D digital images showing detailed features of roughness. Spectral analyses by the fast Fourier transform (FFT) were performed to characterize on the roughness data quantitatively and to identify influential frequency of roughness. The FFT results showed that components of low frequencies were dominant in the fracture roughness. This study also verifies that spectral analysis is a good approach to understand complicate characteristics of fracture roughness. For the aperture measurements, digital images of the aperture were acquired under applying five stages of uniaxial normal stresses. This method can characterize the response of aperture directly using the same specimen. Results of measurements show that reduction values of aperture are different at each part due to rough geometry of fracture walls. Laboratory permeability tests were also conducted to evaluate changes of hydraulic conductivities related to aperture variation due to different stress levels. The results showed non-uniform reduction of hydraulic conductivity under increase of the normal stress and different values of
Current Density Measurements of an Annular-Geometry Ion Engine
NASA Technical Reports Server (NTRS)
Shastry, Rohit; Patterson, Michael J.; Herman, Daniel A.; Foster, John E.
2012-01-01
The concept of the annular-geometry ion engine, or AGI-Engine, has been shown to have many potential benefits when scaling electric propulsion technologies to higher power. However, the necessary asymmetric location of the discharge cathode away from thruster centerline could potentially lead to non-uniformities in the discharge not present in conventional geometry ion thrusters. In an effort to characterize the degree of this potential nonuniformity, a number of current density measurements were taken on a breadboard AGI-Engine. Fourteen button probes were used to measure the ion current density of the discharge along a perforated electrode that replaced the ion optics during conditions of simulated beam extraction. Three Faraday probes spaced apart in the vertical direction were also used in a separate test to interrogate the plume of the AGI-Engine during true beam extraction. It was determined that both the discharge and the plume of the AGI-Engine are highly uniform, with variations under most conditions limited to 10% of the average current density in the discharge and 5% of the average current density in the plume. Beam flatness parameter measured 30 mm from the ion optics ranged from 0.85 0.95, and overall uniformity was shown to generally increase with increasing discharge and beam currents. These measurements indicate that the plasma is highly uniform despite the asymmetric location of the discharge cathode.
Current Density Measurements of an Annular-Geometry Ion Engine
NASA Technical Reports Server (NTRS)
Shastry, Rohit; Patterson, Michael J.; Herman, Daniel A.; Foster, John E.
2012-01-01
The concept of the annular-geometry ion engine, or AGI-Engine, has been shown to have many potential benefits when scaling electric propulsion technologies to higher power. However, the necessary asymmetric location of the discharge cathode away from thruster centerline could potentially lead to non-uniformities in the discharge not present in conventional geometry ion thrusters. In an effort to characterize the degree of this potential non-uniformity, a number of current density measurements were taken on a breadboard AGI-Engine. Fourteen button probes were used to measure the ion current density of the discharge along a perforated electrode that replaced the ion optics during conditions of simulated beam extraction. Three Faraday probes spaced apart in the vertical direction were also used in a separate test to interrogate the plume of the AGI-Engine during true beam extraction. It was determined that both the discharge and the plume of the AGI-Engine are highly uniform, with variations under most conditions limited to +/-10% of the average current density in the discharge and +/-5% of the average current density in the plume. Beam flatness parameter measured 30 mm from the ion optics ranged from 0.85 - 0.95, and overall uniformity was shown to generally increase with increasing discharge and beam currents. These measurements indicate that the plasma is highly uniform despite the asymmetric location of the discharge cathode.
NASA Astrophysics Data System (ADS)
Terzis, Petros A.; Christodoulakis, T.
2012-12-01
Lie-group symmetry analysis for systems of coupled, nonlinear ordinary differential equations is performed in order to obtain the entire solution space to Einstein’s field equations for vacuum Bianchi spacetime geometries. The symmetries used are the automorphisms of the Lie algebra of the corresponding three-dimensional isometry group acting on the hyper-surfaces of simultaneity for each Bianchi type, as well as the scaling and the time reparametrization symmetry. A detailed application of the method is presented for Bianchi type IV. The result is the acquisition of the general solution of type IV in terms of sixth Painlevé transcendent PVI, along with the known pp-wave solution. For Bianchi types I, II, V the known entire solution space is attained and very briefly listed, along with two new type V solutions of Euclidean and neutral signature and a type I pp-wave metric.
Algebraic Singularity Method for Mass Measurements with Missing Energy
Kim, Ian-Woo
2010-02-26
We propose a novel generalized method for mass measurements based on phase space singularity structures that can be applied to any event topology with missing energy. Our method subsumes the well-known end point and transverse mass methods and yields new techniques for studying 'missing particle' events, such as the double chain production of stable neutral particles at the LHC.
NASA Astrophysics Data System (ADS)
McLenaghan, Raymond G.; Smirnov, Roman G.; The, Dennis
2004-03-01
We develop a new approach to the study of Killing tensors defined in pseudo-Riemannian spaces of constant curvature that is ideologically close to the classical theory of invariants. The main idea, which provides the foundation of the new approach, is to treat a Killing tensor as an algebraic object determined by a set of parameters of the corresponding vector space of Killing tensors under the action of the isometry group. The spaces of group invariants and conformal group invariants of valence two Killing tensors defined in the Minkowski plane are described. The group invariants, which are the generators of the space of invariants, are applied to the problem of classification of orthogonally separable Hamiltonian systems defined in the Minkowski plane. Transformation formulas to separable coordinates expressed in terms of the parameters of the corresponding space of Killing tensors are presented. The results are applied to the problem of orthogonal separability of the Drach superintegrable potentials.
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…
TPV efficiency predictions and measurements for a closed cavity geometry
Gethers, C.K.; Ballinger, C.T.; Postlethwait, M.A.; DePoy, D.M.; Baldasaro, P.F.
1997-05-01
A thermophotovoltaic (TPV) efficiency measurement, within a closed cavity, is an integrated test which incorporates four fundamental parameters of TPV direct energy conversion. These are: (1) the TPV devices, (2) spectral control, (3) a radiation/photon source, and (4) closed cavity geometry affects. The overall efficiency of the TPV device is controlled by the TPV cell performance, the spectral control characteristics, the radiator temperature and the geometric arrangement. Controlled efficiency measurements and predictions provide valuable feedback on all four. This paper describes and compares two computer codes developed to model 16, 1 cm{sup 2} TPV cells (in a 4x4 configuration) in a cavity geometry. The first code subdivides the infrared spectrum into several bands and then numerically integrates over the spectrum to provide absorbed heat flux and cell performance predictions (assuming infinite parallel plates). The second utilizes a Monte Carlo Ray-Tracing code that tracks photons, from birth at the radiation source, until they either escape or are absorbed. Absorption depends upon energy dependent reflection probabilities assigned to every geometrical surface within the cavity. The model also has the capability of tallying above and below bandgap absorptions (as a function of location) and can support various radiator temperature profiles. The arrays are fabricated using 0.55 eV InGaAs cells with Si/SiO interference filters for spectral control and at steady state conditions, array efficiency was calculated as the ratio of the load matched power to its absorbed heat flux. Preliminary experimental results are also compared with predictions.
TPV efficiency measurements and predictions for a closed cavity geometry
Gethers, C.K.; Ballinger, C.T.; Postlethwait, M.A.; DePoy, D.M.; Baldasaro, P.F.
1997-05-01
A thermophotovoltaic (TPV) efficiency measurement, within a closed cavity, is an integrated test which incorporates four fundamental parameters of TPV direct energy conversion. These are: (1) the TPV devices, (2) spectral control, (3) a radiation/photon source, and (4) closed cavity geometry effects. The overall efficiency of the TPV device is controlled by the TP cell performance, the spectral control characteristics, the radiator temperature and the geometric arrangement. Controlled efficiency measurements and predictions provide valuable feedback on all four. This paper describes and compares two computer codes developed to model 16, 1 cm{sup 2} TPV cells (in a 4 x 4 configuration) in a cavity geometry. The first code, subdivides the infrared spectrum into several bands and then numerically integrates over the spectrum to provide absorbed heat flux and cell electrical output performance predictions (assuming infinite parallel plates). The second code, utilizes a Monte Carlo Photon Transport code that tracks photons, from birth at the radiation source, until they either escape or are absorbed. Absorption depends upon energy dependent reflection probabilities assigned to every geometrical surface within the cavity. The model also has the capability of tallying above and below bandgap absorptions (as a function of location) and can support various radiator temperature profiles. The arrays were fabricated using 0.55 eV InGaAs cells with Si/SiO interference filters for spectral control and at steady state conditions, array efficiency was calculated as the ratio of the load matched power to its absorbed heat flux. Preliminary experimental results are also compared with predictions.
ERIC Educational Resources Information Center
Ketterlin-Geller, Leanne R.; Gifford, Diane B.; Perry, Lindsey
2015-01-01
Students' understanding and proficiency with rational number concepts and operations is considered a key foundational skill for future success in algebra. As middle school students work with these concepts, teachers need timely data to determine whether students are making adequate progress. The purpose of this article is to document the content…
DG Poisson algebra and its universal enveloping algebra
NASA Astrophysics Data System (ADS)
Lü, JiaFeng; Wang, XingTing; Zhuang, GuangBin
2016-05-01
In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.
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…
Geometric Algebra for Physicists
NASA Astrophysics Data System (ADS)
Doran, Chris; Lasenby, Anthony
2007-11-01
Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.
Geometry and molecular architecture effects in nanobubble inflation measurements
NASA Astrophysics Data System (ADS)
Xu, Shanhong; Castagnet, Sylvie; McKenna, Gregory
2011-03-01
Confinement effects on the mechanical properties of ultrathin polymer films were investigated by a bubble inflation technique developed in our lab. Prior studies of ultrathin films of poly(vinyl acetate) (PVAc) and linear polystyrene (PS) were performed on circular bubbles of different diameters. Here the creep behaviors of ultrathin films of linear PS were investigated on rectangular bubbles. The modulus of the thin film rectangular bubbles was analyzed by approximation methods. The inflation of rectangular bubbles was simulated by finite element analysis (FEA). The mechanical properties of the thin films with the same thickness for circular and rectangular bubbles are compared and we find that the rubbery plateau compliance is geometry independent. We also investigated the creep behaviors of ultrathin films of 3-arm star PS on circular bubbles. We find the rubbery plateau compliance is molecular architecture independent.
Uncertainties in aspheric profile measurements with the geometry measuring machine at NIST.
Griesmann, U.; Machkour-Deshayes, N.; Soons, J.; Kim, B. C.; Wang, Q.; Stoup, J. R.; Assoufid, L.; Experimental Facilities Division; NIST
2005-01-01
The Geometry Measuring Machine (GEMM) of the National Institute of Standards and Technology (NIST) is a profilometer for free-form surfaces. A profile is reconstructed from the local curvature of a test part surface, measured at several locations along a line. For profile measurements of free-form surfaces, methods based on local part curvature sensing have strong appeal. Unlike full-aperture interferometry they do not require customized null optics. The uncertainty of a reconstructed profile is critically dependent upon the uncertainty of the curvature measurement and, to a lesser extent, on curvature sensor positioning accuracy. For an instrument of the GEMM type, we evaluate the measurement uncertainties for a curvature sensor based on a small aperture interferometer and then estimate the uncertainty that can be achieved in the reconstructed profile. In addition, profile measurements of a free-form mirror using GEMM are compared with measurements using a long-trace profiler, a coordinate measuring machine, and subaperture-stitching interferometry.
Koc, Ramazan . E-mail: koc@gantep.edu.tr; Tuetuencueler, Hayriye; Koca, Mehmet; Olgar, Eser
2005-10-01
We consider solutions of the 2 x 2 matrix Hamiltonians of the physical systems within the context of the su (2) and su (1, 1) Lie algebras. Our technique is relatively simple when compared with those of others and treats those Hamiltonians which can be treated in a unified framework of the Sp (4, R) algebra. The systematic study presented here reproduces a number of earlier results in a natural way as well as leads to a novel finding. Possible generalizations of the method are also suggested.
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…
ERIC Educational Resources Information Center
Solomon, Frederick
This module applies linear algebraic methods to solve the following problem: If an object in a three-dimensional coordinate system is first rotated about a given axis through the origin by a given angle, and then rotated about another axis through the origin by another angle, there is a straightforward way to calculate the combined result of the…
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…
Intraglottal geometry and velocity measurements in canine larynges
Oren, Liran; Khosla, Sid; Gutmark, Ephraim
2014-01-01
Previous flow velocity measurements during phonation in canine larynges were done above the glottal exit. These studies found that vortical structures are present in the flow above the glottis at different phases of the glottal cycle. Some vortices were observed to leave the glottis during the closing phase and assumptions were proposed regarding their formation mechanism. In the current study, intraglottal velocity measurements are performed using PIV, and the intraglottal flow characteristics are determined. Results from five canine larynges show that at low subglottal pressure the glottis assumes a minimal divergence angle during closing and the flow separates at the glottal exit. Vortical structures are observed above the glottis but not inside. As the subglottal pressure is increased, the divergence angle between the folds during closing increases and the location of the flow separation moves upstream into the glottis. Entrainment flow enters the glottis to fill the void that is formed between the glottal jet and the fold. Vortical structures develop near the superior edge at medium and high subglottal pressures from the flow separation. The magnitude of their swirling strength changes as a function of the wall dynamics. PMID:24437778
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.
Geometry of Logarithmic Strain Measures in Solid Mechanics
NASA Astrophysics Data System (ADS)
Neff, Patrizio; Eidel, Bernhard; Martin, Robert J.
2016-07-01
We consider the two logarithmic strain measures {ω_{iso}} = ||{dev_n log U} || = ||{dev_n log √{F^TF}}|| quad and quad {ω_{vol}} = |{tr(log U)} = |{tr(log√{F^TF})}| = |log(det U)|, which are isotropic invariants of the Hencky strain tensor U, and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group {GL(n)} . Here, {F} is the deformation gradient, {U=√{F^TF}} is the right Biot-stretch tensor, log denotes the principal matrix logarithm, {| \\cdot |} is the Frobenius matrix norm, tr is the trace operator and {{dev}_n X = X- 1/n {tr}(X)\\cdot {{1}}} is the {n} -dimensional deviator of {Xin{{R}}^{n × n}} . This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor {ɛ={sym}nabla u} , which is the symmetric part of the displacement gradient {nabla u} , and reveals a close geometric relation between the classical quadratic isotropic energy potential μ {| {dev}_n {sym} nabla u |}^2 + κ/2{[{tr}({sym} nabla u)]}^2 = μ {| {dev}_n ɛ |}^2 + κ/2 {[{tr} (ɛ)]}^2 in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy μ {| {dev}_n log U |}^2 + κ/2{[{tr}(log U)]}^2 = μ {ω_{{iso}}^2} + κ/2{ω_{{vol}}^2}, where {μ} is the shear modulus and {κ} denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor {R} , where {F=RU} is the polar decomposition of {F} . We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity.
Geometry and Texture Measures for Interactive Virtualized Reality Indoor Modeler
NASA Astrophysics Data System (ADS)
Thangamania, K.; Ichikari, R.; Okuma, T.; Ishikawa, T.; Kurata, T.
2015-05-01
This paper discusses the algorithm to detect the distorted textures in the virtualized reality indoor models and automatically generate the necessary 3D planes to hold the undistorted textures. Virtualized reality (VR) interactive indoor modeler, our previous contribution enables the user to interactively create their desired indoor VR model from a single 2D image. The interactive modeler uses the projective texture mapping for mapping the textures over the manually created 3D planes. If the user has not created the necessary 3D planes, then the texture that belong to various objects are projected to the available 3D planes, which leads to the presence of distorted textures. In this paper, those distorted textures are detected automatically by the suitable principles from the shape from texture research. The texture distortion features such as the slant, tilt and the curvature parameters are calculated from the 2D image by means of affine transformation measured between the neighboring texture patches within the single image. This kind of affine transform calculation from a single image is useful in the case of deficient multiple view images. The usage of superpixels in clustering the textures corresponding to different objects, reduces the modeling labor cost. A standby database also stores the repeated basic textures that are found in the indoor model, and provides texture choices for the distorted floor, wall and other regions. Finally, this paper documents the prototype implementation and experiments with the automatic 3D plane creation and distortion detection with the above mentioned principles in the virtualized reality indoor environment.
KEELE BD
2005-02-01
A collimated portable gamma-ray detector will be used to quantify the plutonium content of items that can be approximated as a point, line, or area geometry with respect to the detector. These items can include ducts, piping, glove boxes, isolated equipment inside of gloveboxes, and HEPA filters. The Generalized Geometry Holdup (GGH) model is used for the reduction of counting data. This document specifies the calculations to reduce counting data into contained plutonium and the associated total measurement uncertainty.
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.``
NASA Technical Reports Server (NTRS)
Grauer, Jared A.; Morelli, Eugene A.
2013-01-01
A nonlinear simulation of the NASA Generic Transport Model was used to investigate the effects of errors in sensor measurements, mass properties, and aircraft geometry on the accuracy of dynamic models identified from flight data. Measurements from a typical system identification maneuver were systematically and progressively deteriorated and then used to estimate stability and control derivatives within a Monte Carlo analysis. Based on the results, recommendations were provided for maximum allowable errors in sensor measurements, mass properties, and aircraft geometry to achieve desired levels of dynamic modeling accuracy. Results using other flight conditions, parameter estimation methods, and a full-scale F-16 nonlinear aircraft simulation were compared with these recommendations.
Locally finite dimensional Lie algebras
NASA Astrophysics Data System (ADS)
Hennig, Johanna
We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L. Then we prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic p which give sufficient conditions for the algebras to be of the form [K(R, *), K( R, *)] / (Z(R) ∩ [ K(R, *), K(R, *)]) for a simple, locally finite dimensional associative algebra R with involution *. Lastly, we explore the noncommutative geometry of locally simple representations of the diagonal locally finite Lie algebras sl(ninfinity), o( ninfinity), and sp(n infinity).
ERIC Educational Resources Information Center
Norton, Anderson; Rutledge, Zachary
2006-01-01
In a secondary school mathematics teaching methods course, a research team engaged 22 preservice secondary teachers (PSTs) in designing and posing tasks to algebra students through weekly letter writing. The goal of the tasks was for PSTs to elicit responses that would indicate student engagement in the mathematical processes described by NCTM…
Effects of detector geometry on measured lineshapes and intensities in surface scattering
NASA Astrophysics Data System (ADS)
Hinch, B. J.; Frankl, D. R.; Allison, W.
1987-02-01
A general expression for the detector response to a given beam flux distribution is given. Illustrative examples are worked out for some simple idealized cases and it is shown that both the measured lineshape and the measured intensity depend on the details of incident beam and detector geometry.
NASA Astrophysics Data System (ADS)
Hiley, B. J.
In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.
Mathematical Art-O-Facts: Activities to Introduce, Reinforce, or Assess Geometry & Measurement
ERIC Educational Resources Information Center
Kuhns, Catherine Jones
2008-01-01
This book is loaded with hands-on measurement and geometry activities that will get students rolling up their sleeves, excited to use their math skills. The author has created these activities to supplement a teacher's existing math curriculum. They are flexibly designed for use as either introductory, reinforcement, or assessment activities. This…
Measurement and reconstruction of the leaflet geometry for a pericardial artificial heart valve.
Jiang, Hongjun; Campbell, Gord; Xi, Fengfeng
2005-03-01
This paper describes the measurement and reconstruction of the leaflet geometry for a pericardial heart valve. Tasks involved include mapping the leaflet geometries by laser digitizing and reconstructing the 3D freeform leaflet surface based on a laser scanned profile. The challenge is to design a prosthetic valve that maximizes the benefits offered to the recipient as compared to the normally operating naturally-occurring valve. This research was prompted by the fact that artificial heart valve bioprostheses do not provide long life durability comparable to the natural heart valve, together with the anticipated benefits associated with defining the valve geometries, especially the leaflet geometries for the bioprosthetic and human valves, in order to create a replicate valve fabricated from synthetic materials. Our method applies the concept of reverse engineering in order to reconstruct the freeform surface geometry. A Brown & Shape coordinate measuring machine (CMM) equipped with a HyMARC laser-digitizing system was used to measure the leaflet profiles of a Baxter Carpentier-Edwards pericardial heart valve. The computer software, Polyworks was used to pre-process the raw data obtained from the scanning, which included merging images, eliminating duplicate points, and adding interpolated points. Three methods, creating a mesh model from cloud points, creating a freeform surface from cloud points, and generating a freeform surface by B-splines are presented in this paper to reconstruct the freeform leaflet surface. The mesh model created using Polyworks can be used for rapid prototyping and visualization. To fit a freeform surface to cloud points is straightforward but the rendering of a smooth surface is usually unpredictable. A surface fitted by a group of B-splines fitted to cloud points was found to be much smoother. This method offers the possibility of manually adjusting the surface curvature, locally. However, the process is complex and requires additional
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.
Tomographic PIV measurements of flow patterns in a nasal cavity with geometry acquisition
NASA Astrophysics Data System (ADS)
Im, Sunghyuk; Heo, Go Eun; Jeon, Young Jin; Sung, Hyung Jin; Kim, Sung Kyun
2014-01-01
The flow patterns inside a scaled transparent model of a nasal cavity were measured by tomographic particle image velocimetry (PIV) with three-dimensional (3D) geometry acquisition. The model was constructed using transparent silicone. The refractive index of the working fluid was matched to the index of silicone by mixing water and glycerol. Four cameras and a double-pulse laser system were used for tomographic PIV. Red fluorescent particles and long-pass filters were used to obtain a high signal-to-noise ratio. The complex geometry of the 3D nasal model was acquired by accumulating triangulated 3D particle positions, obtained through a least square-based triangulation method. Certain morphological operations, such as the opening and closing of the nasal cavity, were used to improve the quality of acquired nasal geometry data. The geometry information was used to distinguish the fluid from the solid regions during the tomographic reconstruction procedure. The quality of the model geometry acquisition and tomographic reconstruction algorithms was evaluated using a synthetic image test. Synthetic images were generated by fitting a computational model (stereolithography file) to the virtual 3D coordinates and by randomly seeding particles inside the nasal region. A perspective transformation matrix of each camera was used to generate the synthetic images based on the experimental configuration of the camera. The synthetic image test showed that the voxel reconstruction quality could be improved by applying acquired model geometry in the tomographic reconstruction step. The nasal geometry was acquired and a flow velocity field was determined by cross-correlating the reconstructed 3D voxel intensities.
InGaAs spin light emitting diodes measured in the Faraday and oblique Hanle geometries
NASA Astrophysics Data System (ADS)
Mansell, R.; Laloë, J.-B.; Holmes, S. N.; Petrou, A.; Farrer, I.; Jones, G. A. C.; Ritchie, D. A.; Barnes, C. H. W.
2016-04-01
InGaAs quantum well light emitting diodes (LED) with spin-injecting, epitaxial Fe contacts were fabricated using an in situ wafer transfer process where the semiconductor wafer was transferred under ultrahigh vacuum (UHV) conditions to a metals growth chamber to achieve a high quality interface between the two materials. The spin LED devices were measured optically with applied magnetic fields in either the Faraday or the oblique Hanle geometries in two experimental set-ups. Optical polarizations efficiencies of 4.5% in the Faraday geometry and 1.5% in the Hanle geometry are shown to be equivalent. The polarization efficiency of the electroluminescence is seen to decay as the temperature increases although the spin lifetime remains constant due to the influence of the D’yakonov–Perel’ spin scattering mechanism in the quantum well.
Wake Geometry Measurements and Analytical Calculations on a Small-Scale Rotor Model
NASA Technical Reports Server (NTRS)
Ghee, Terence A.; Berry, John D.; Zori, Laith A. J.; Elliott, Joe W.
1996-01-01
An experimental investigation was conducted in the Langley 14- by 22-Foot Subsonic Tunnel to quantify the rotor wake behind a scale model helicopter rotor in forward level flight at one thrust level. The rotor system in this test consisted of a four-bladed fully articulated hub with blades of rectangular planform and an NACA 0012 airfoil section. A laser light sheet, seeded with propylene glycol smoke, was used to visualize the vortex geometry in the flow in planes parallel and perpendicular to the free-stream flow. Quantitative measurements of wake geometric proper- ties, such as vortex location, vertical skew angle, and vortex particle void radius, were obtained as well as convective velocities for blade tip vortices. Comparisons were made between experimental data and four computational method predictions of experimental tip vortex locations, vortex vertical skew angles, and wake geometries. The results of these comparisons highlight difficulties of accurate wake geometry predictions.
Wei, Q.; Dalvit, D. A. R.; Lombardo, F. C.; Mazzitelli, F. D.; Onofrio, R.
2010-05-15
We report on measurements performed on an apparatus aimed to study the Casimir force in the cylinder-plane configuration. The electrostatic calibrations evidence anomalous behaviors in the dependence of the electrostatic force and the minimizing potential upon distance. We discuss analogies and differences of these anomalies with respect to those already observed in the sphere-plane configuration. At the smallest explored distances we observe frequency shifts of non-Coulombian nature preventing the measurement of the Casimir force in the same range. We also report on measurements performed in the parallel-plane configuration, showing that the dependence on distance of the minimizing potential, if present at all, is milder than in the sphere-plane or cylinder-plane geometries. General considerations on the interplay between the distance-dependent minimizing potential and the precision of Casimir force measurements in the range relevant to detect the thermal corrections for all geometries are finally reported.
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.
ERIC Educational Resources Information Center
Schaufele, Christopher; Zumoff, Nancy
Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through…
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.…
NASA Technical Reports Server (NTRS)
Grauer, Jared A.; Morelli, Eugene A.
2013-01-01
The NASA Generic Transport Model (GTM) nonlinear simulation was used to investigate the effects of errors in sensor measurements, mass properties, and aircraft geometry on the accuracy of identified parameters in mathematical models describing the flight dynamics and determined from flight data. Measurements from a typical flight condition and system identification maneuver were systematically and progressively deteriorated by introducing noise, resolution errors, and bias errors. The data were then used to estimate nondimensional stability and control derivatives within a Monte Carlo simulation. Based on these results, recommendations are provided for maximum allowable errors in sensor measurements, mass properties, and aircraft geometry to achieve desired levels of dynamic modeling accuracy. Results using additional flight conditions and parameter estimation methods, as well as a nonlinear flight simulation of the General Dynamics F-16 aircraft, were compared with these recommendations
Evaluation of the effect of geometry for measuring section thickness in tomosynthesis.
Fukui, Ryohei; Ishii, Rie; Kishimoto, Junichi; Yamato, Shinichiro; Takahata, Akira; Kohama, Chiyuki
2014-01-01
Our aim in this study was to evaluate the effect of geometry for measuring section thickness in tomosynthesis by using a metal bead device (bead method). Tomosynthesis images were obtained from two types of tomosynthesis equipment, Safire17 (ST, Shimadzu, Kyoto, Japan) and XR650 (GT, GE Healthcare, Milwaukee, WI). After tomosynthesis radiography with each device, the bead tomosynthesis images were obtained by image reconstruction. The digital profile was obtained from the digital value of the bead central coordinate in the perpendicular direction, and we acquired the slice sensitivity profile (SSP). The section thickness was defined with the full width at half maximum obtained from the SSP. We investigated the change in section thickness under different evaluation conditions: the angular range, the height of the bead position, the source-image receptor distance (SID), and image processing. The section thickness decreased when the angular range and height of the bead position increased. Also, the section thickness varied with a change in the SID. The section thickness differed according to the geometry for measuring the section thickness. Thus, the effect of the geometry used for measurement should be considered when the section thickness in tomosynthesis is measured by the bead method. PMID:24254729
High-resolution, real-time simultaneous 3D surface geometry and temperature measurement.
An, Yatong; Zhang, Song
2016-06-27
This paper presents a method to simultaneously measure three-dimensional (3D) surface geometry and temperature in real time. Specifically, we developed 1) a holistic approach to calibrate both a structured light system and a thermal camera under exactly the same world coordinate system even though these two sensors do not share the same wavelength; and 2) a computational framework to determine the sub-pixel corresponding temperature for each 3D point as well as discard those occluded points. Since the thermal 2D imaging and 3D visible imaging systems do not share the same spectrum of light, they can perform sensing simultaneously in real time: we developed a hardware system that can achieve real-time 3D geometry and temperature measurement at 26 Hz with 768 × 960 points per frame. PMID:27410608
Online measurement of bead geometry in GMAW-based additive manufacturing using passive vision
NASA Astrophysics Data System (ADS)
Xiong, Jun; Zhang, Guangjun
2013-11-01
Additive manufacturing based on gas metal arc welding is an advanced technique for depositing fully dense components with low cost. Despite this fact, techniques to achieve accurate control and automation of the process have not yet been perfectly developed. The online measurement of the deposited bead geometry is a key problem for reliable control. In this work a passive vision-sensing system, comprising two cameras and composite filtering techniques, was proposed for real-time detection of the bead height and width through deposition of thin walls. The nozzle to the top surface distance was monitored for eliminating accumulated height errors during the multi-layer deposition process. Various image processing algorithms were applied and discussed for extracting feature parameters. A calibration procedure was presented for the monitoring system. Validation experiments confirmed the effectiveness of the online measurement system for bead geometry in layered additive manufacturing.
Gas Flow Measurements of a Novel Geometry for Neutral Beam Neutralizers.
NASA Astrophysics Data System (ADS)
Pirkle, David Ross
The gas flow characteristics of a novel geometry (pumped neutralizer) for decreasing the flow of gas from neutral beam neutralizers were measured and compared with a conventional (passive) neutralizer. A passive neutralizer is typically a duct attached to the ion source. For the pumped neutralizer the top and bottom surfaces of the duct are replaced by a Venetian blind geometry which opens into ballast vacuum pumping volumes. With guidance from a Monte Carlo program which models gas flow at low pressure, a one-half scale model with pumped neutralizer geometry was built and compared to a passive neutralizer with comparable dimensions. With the vanes on the pumped neutralizer opened to 55 degrees, the line density of the pumped neutralizer was 1.6 times less than the passive neutralizer. The amount of gas flowing from the exit of the pumped neutralizer was from 2 to 5 times less than the amount flowing from the pumped neutralizer. Hence, the pumped neutralizer geometry appears to be a promising method of limiting the flow of gas from neutral beam gas cell neutralizers.
Yamamoto, Ken; Togawa, Ryotaro; Fujimura, Ryushi; Kajikawa, Kotaro
2016-08-22
Strong temperature dependence of anti-Stokes luminescence intensity from Rhodamine 101 is used to probe local temperature variation at a surface region in the attenuated total reflection geometry (ATR), when heating with laser light. In this method, the measured region can be limited by observing evanescent luminescence. The near-field depth (penetration depth) was changed by the observation angle θ_{out} of the evanescent luminescence and the spatial temperature variation was observed. PMID:27557182
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.
Computer-aided evaluation of the railway track geometry on the basis of satellite measurements
NASA Astrophysics Data System (ADS)
Specht, Cezary; Koc, Władysław; Chrostowski, Piotr
2016-05-01
In recent years, all over the world there has been a period of intensive development of GNSS (Global Navigation Satellite Systems) measurement techniques and their extension for the purpose of their applications in the field of surveying and navigation. Moreover, in many countries a rising trend in the development of rail transportation systems has been noticed. In this paper, a method of railway track geometry assessment based on mobile satellite measurements is presented. The paper shows the implementation effects of satellite surveying railway geometry. The investigation process described in the paper is divided on two phases. The first phase is the GNSS mobile surveying and the analysis obtained data. The second phase is the analysis of the track geometry using the flat coordinates from the surveying. The visualization of the measured route, separation and quality assessment of the uniform geometric elements (straight sections, arcs), identification of the track polygon (main directions and intersection angles) are discussed and illustrated by the calculation example within the article.
NASA Astrophysics Data System (ADS)
Nyssen, F.; Golinval, J.-C.
2016-02-01
In this work, an experimental modal analysis is performed on an academic bladed disk using a base excitation to identify the mistuning of each blade. Optical measurement is used to obtain the exact geometry of the structure and to be able to associate geometric mistuning to each blade. Differences are observed between the experimentally identified mistuning and the geometric mistuning. Since the bladed disk is a one-piece structure, there are no welded connections between the blades and the disk and the material properties can be assumed to be uniform. It can be shown that these differences come from non uniform clamping conditions, and that this mistuning is of the same order of magnitude than the variations in the geometry of the structure. It follows that the precise characterization of mistuning for industrial structures is in practice illusory because of the numerous factors introducing mistuning, such as the clamping conditions, aerodynamic damping, wear in service, etc.
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.
Black Saturday bushfire smoke plumes as seen from SCIAMACHY measurements in limb geometry
NASA Astrophysics Data System (ADS)
Dörner, Steffen; Pukite, Janis; Penning de Vries, Marloes; Fromm, Mike; Wagner, Thomas
2016-04-01
The so called Black Saturday bushfires started on the 7th of February 2009 in southeastern Victoria, Australia. Resulting smoke plumes contaminated the lower stratosphere in the following weeks as measured by a variety satellite instruments. Particle extinction profiles retrieved from SCIAMACHY measurements in limb geometry provide a complementary view on the development of the smoke plume, especially on the first days of the event when measurements of other instruments were sparse. Earlier studies showed that commonly used 1D retrieval algorithms for limb observations of particle extinction potentially underestimate optical thickness and altitude of such injections into the stratosphere. In this study, a 2D particle extinction retrieval algorithm for SCIAMACHY limb measurements is used to track optical thickness and plume altitude of the Black Saturday bushfires over the month of February. The required information about the horizontal distribution of the plume is determined by the absorbing aerosol index (AAI) derived from SCIAMACHY measurements in nadir geometry. First results indicate enhanced particle scattering above 18 km on the 9th of February while the smoke plume is drifting to the north east above the Pacific ocean.
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
Steele, Michael D.
2013-01-01
While recent national and international assessments have shown mathematical progress being made by US students, little to no gains are evident in the areas of geometry and measurement. These reports also suggest that practicing teachers have traditionally had few opportunities to engage in content learning around topics in geometry and…
Salmingo, Remel Alingalan; Tadano, Shigeru; Abe, Yuichiro; Ito, Manabu
2016-05-12
Treatment for severe scoliosis is usually attained when the scoliotic spine is deformed and fixed by implant rods. Investigation of the intraoperative changes of implant rod shape in three-dimensions is necessary to understand the biomechanics of scoliosis correction, establish consensus of the treatment, and achieve the optimal outcome. The objective of this study was to measure the intraoperative three-dimensional geometry and deformation of implant rod during scoliosis corrective surgery.A pair of images was obtained intraoperatively by the dual camera system before rotation and after rotation of rods during scoliosis surgery. The three-dimensional implant rod geometry before implantation was measured directly by the surgeon and after surgery using a CT scanner. The images of rods were reconstructed in three-dimensions using quintic polynomial functions. The implant rod deformation was evaluated using the angle between the two three-dimensional tangent vectors measured at the ends of the implant rod.The implant rods at the concave side were significantly deformed during surgery. The highest rod deformation was found after the rotation of rods. The implant curvature regained after the surgical treatment.Careful intraoperative rod maneuver is important to achieve a safe clinical outcome because the intraoperative forces could be higher than the postoperative forces. Continuous scoliosis correction was observed as indicated by the regain of the implant rod curvature after surgery. PMID:27175467
NASA Technical Reports Server (NTRS)
Rhodes, D. L.; Lilley, D. G.
1985-01-01
Numerical predictions, flow visualization experiments and time-mean velocity measurements were obtained for six basic nonreacting flowfields (with inlet swirl vane angles of 0 (swirler removed), 45 and 70 degrees and sidewall expansion angles of 90 and 45 degrees) in an idealized axisymmetric combustor geometry. A flowfield prediction computer program was developed which solves appropriate finite difference equations including a conventional two equation k-epsilon eddy viscosity turbulence model. The wall functions employed were derived from previous swirling flow measurements, and the stairstep approximation was employed to represent the sloping wall at the inlet to the test chamber. Recirculation region boundaries have been sketched from the entire flow visualization photograph collection. Tufts, smoke, and neutrally buoyant helium filled soap bubbles were employed as flow tracers. A five hole pitot probe was utilized to measure the axial, radial, and swirl time mean velocity components.
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
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.
NASA Astrophysics Data System (ADS)
Zhang, Shixue
2015-10-01
The method to get high-orbit satellite basic information such as geometry and material characteristic, is an important goal in the field of space posture apperception. In this paper, we calculate the satellite magnitude by comparing the output value of camera's CCD between the known fixed star and the satellite. We select certain reference stars to calculate the luminance value of a certain object on the acquired image using a background-removing method. We make time-domain analysis of the measurement data, and get the statistic result. With the knowledge of the theory brightness of the target, we estimate the geometric characteristics of the target. We have got a serious of the images of a certain satellite on large telescope. The experimental results demonstrate that, the accuracy of the measured magnitude is better than 0.12Mv, and the estimation error of the target reflection surface size is less than 15%.
SU-E-I-79: Source Geometry Dependence of Gamma Well-Counter Measurements
Park, M; Belanger, A; Kijewski, M
2015-06-15
Purpose: To determine the effect of liquid sample volume and geometry on counting efficiency in a gamma well-counter, and to assess the relative contributions of sample geometry and self-attenuation. Gamma wellcounters are standard equipment in clinical and preclinical studies, for measuring patient blood radioactivity and quantifying animal tissue uptake for tracer development and other purposes. Accurate measurements are crucial. Methods: Count rates were measured for aqueous solutions of 99m- Tc at four liquid volume values in a 1-cm-diam tube and at six volume values in a 2.2-cm-diam vial. Total activity was constant for all volumes, and data were corrected for decay. Count rates from a point source in air, supported by a filter paper, were measured at seven heights between 1.3 and 5.7 cm from the bottom of a tube. Results: Sample volume effects were larger for the tube than for the vial. For the tube, count efficiency relative to a 1-cc volume ranged from 1.05 at 0.05 cc to 0.84 at 3 cc. For the vial, relative count efficiency ranged from 1.02 at 0.05 cc to 0.87 at 15 cc. For the point source, count efficiency relative to 1.3 cm from the tube bottom ranged from 0.98 at 1.8 cm to 0.34 at 5.7 cm. The relative efficiency of a 3-cc liquid sample in a tube compared to a 1-cc sample is 0.84; the average relative efficiency for the solid sample in air between heights in the tube corresponding to the surfaces of those volumes (1.3 and 4.8 cm) is 0.81, implying that the major contribution to efficiency loss is geometry, rather than attenuation. Conclusion: Volume-dependent correction factors should be used for accurate quantitation radioactive of liquid samples. Solid samples should be positioned at the bottom of the tube for maximum count efficiency.
Cooper, Robert F.; Sulai, Yusufu N.; Dubis, Adam M.; Chui, Toco Y.; Rosen, Richard B.; Michaelides, Michel; Dubra, Alfredo; Carroll, Joseph
2016-01-01
Purpose To characterize the effects of intraframe distortion due to involuntary eye motion on measures of cone mosaic geometry derived from adaptive optics scanning light ophthalmoscope (AOSLO) images. Methods We acquired AOSLO image sequences from 20 subjects at 1.0, 2.0, and 5.0° temporal from fixation. An expert grader manually selected 10 minimally distorted reference frames from each 150-frame sequence for subsequent registration. Cone mosaic geometry was measured in all registered images (n = 600) using multiple metrics, and the repeatability of these metrics was used to assess the impact of the distortions from each reference frame. In nine additional subjects, we compared AOSLO-derived measurements to those from adaptive optics (AO)-fundus images, which do not contain system-imposed intraframe distortions. Results We observed substantial variation across subjects in the repeatability of density (1.2%–8.7%), inter-cell distance (0.8%–4.6%), percentage of six-sided Voronoi cells (0.8%–10.6%), and Voronoi cell area regularity (VCAR) (1.2%–13.2%). The average of all metrics extracted from AOSLO images (with the exception of VCAR) was not significantly different than those derived from AO-fundus images, though there was variability between individual images. Conclusions Our data demonstrate that the intraframe distortion found in AOSLO images can affect the accuracy and repeatability of cone mosaic metrics. It may be possible to use multiple images from the same retinal area to approximate a “distortionless” image, though more work is needed to evaluate the feasibility of this approach. Translational Relevance Even in subjects with good fixation, images from AOSLOs contain intraframe distortions due to eye motion during scanning. The existence of these artifacts emphasizes the need for caution when interpreting results derived from scanning instruments. PMID:26933523
ERIC Educational Resources Information Center
Durkin, Kelley; Pollack, Courtney; Star, Jon R.; Rittle-Johnson, Bethany
2012-01-01
The current paper investigated the following research questions regarding measures of fidelity: (1) Is there a significant relationship between two different measures of fidelity of implementation: a survey of instructional practices and coded videos of classroom lessons? Does the strength of this relationship differ between treatment and control…
Validating Measures of Algebra Teacher Subject Matter Knowledge and Pedagogical Content Knowledge
ERIC Educational Resources Information Center
Buschang, Rebecca E.; Chung, Gregory K. W. K.; Delacruz, Girlie C.; Baker, Eva L.
2012-01-01
The purpose of this study was to validate inferences about scores of one task designed to measure subject matter knowledge and three tasks designed to measure aspects of pedagogical content knowledge. Evidence for the validity of inferences was based on two expectations. First, if tasks were sensitive to expertise, we would find group differences.…
Differential geometry measures of nonlinearity for the bearing-only tracking problem
NASA Astrophysics Data System (ADS)
Mallick, Mahendra; La Scala, Barbara F.; Arulampalam, M. S.
2005-05-01
The bearing-only tracking problem arises in many radar and sonar tracking applications. Since the bearing measurement model is a nonlinear function of the target state, the filtering problem is nonlinear in nature. A great deal of attention has been focused on this problem due to the difficulty posed by the so-called high degree of nonlinearity (DoN) in the problem. However, a quantitative measure of the DoN is not calculated in previous works. It has been observed that the extended Kalman filter (EKF) in which the state vector consists of the Cartesian components of position and velocity is unstable and diverges in some cases. The range parametrized EKF (RPEKF) and particle filter (PF) have been shown to produce improved estimates for the bearing-only tracking problem. In this paper, we calculate two measures of nonlinearity, (1) the parameter-effects curvature and (2) intrinsic curvature for the bearing-only tracking problem using the differential geometry measures of nonlinearity. We present numerical results using simulated data for the constant velocity motion of a target in 2D with bearing-only measurements where the sensor platform uses a higher order motion than the target to achieve observability. We analyze the DoN by varying the distance between the target and sensor.
Measurement of the PPN parameter γ by testing the geometry of near-Earth space
NASA Astrophysics Data System (ADS)
Luo, Jie; Tian, Yuan; Wang, Dian-Hong; Qin, Cheng-Gang; Shao, Cheng-Gang
2016-06-01
The Beyond Einstein Advanced Coherent Optical Network (BEACON) mission was designed to achieve an accuracy of 10^{-9} in measuring the Eddington parameter γ , which is perhaps the most fundamental Parameterized Post-Newtonian parameter. However, this ideal accuracy was just estimated as a ratio of the measurement accuracy of the inter-spacecraft distances to the magnitude of the departure from Euclidean geometry. Based on the BEACON concept, we construct a measurement model to estimate the parameter γ with the least squares method. Influences of the measurement noise and the out-of-plane error on the estimation accuracy are evaluated based on the white noise model. Though the BEACON mission does not require expensive drag-free systems and avoids physical dynamical models of spacecraft, the relatively low accuracy of initial inter-spacecraft distances poses a great challenge, which reduces the estimation accuracy in about two orders of magnitude. Thus the noise requirements may need to be more stringent in the design in order to achieve the target accuracy, which is demonstrated in the work. Considering that, we have given the limits on the power spectral density of both noise sources for the accuracy of 10^{-9}.
Nozzle geometry and injection duration effects on diesel sprays measured by x-ray radiography.
Kastengren, A. L.; Powell, C. F.; Riedel, T.; Cheong, S.-K.; Im, K.-S.; Liu, X.; Wang, Y. J.; Wang, J.; Robert Bosch GmbH
2008-04-01
X-ray radiography was used to measure the behavior of four fuel sprays from a light-duty common-rail diesel injector. The sprays were at 250 bar injection pressure and 1 bar ambient pressure. Injection durations of 400 {micro}s and 1000 {micro}s were tested, as were axial single-hole nozzles with hydroground and nonhydroground geometries. The X-ray data provide quantitative measurements of the internal mass distribution of the spray, including near the injector orifice. Such measurements are not possible with optical diagnostics. The 400 {micro}s sprays from the hydroground and nonhydroground nozzles appear qualitatively similar. The 1000 {micro}s spray from the nonhydroground nozzle has a relatively consistent moderate width, while that from the hydroground nozzle is quite wide before transitioning into a narrow jet. The positions of the leading- and trailing-edges of the spray have also been determined, as has the amount of fuel residing in a concentrated structure near the leading edge of the spray.
Application of Computer Axial Tomography (CAT) to measuring crop canopy geometry. [corn and soybeans
NASA Technical Reports Server (NTRS)
Bauer, M. E.; Vanderbilt, V. C. (Principal Investigator); Kilgore, R. W.
1981-01-01
The feasibility of using the principles of computer axial topography (CAT) to quantify the structure of crop canopies was investigated because six variables are needed to describe the position-orientation with time of a small piece of canopy foliage. Several cross sections were cut through the foliage of healthy, green corn and soybean canopies in the dent and full pod development stages, respectively. A photograph of each cross section representing the intersection of a plane with the foliage was enlarged and the air-foliage boundaries delineated by the plane were digitized. A computer program was written and used to reconstruct the cross section of the canopy. The approach used in applying optical computer axial tomography to measuring crop canopy geometry shows promise of being able to provide needed geometric information for input data to canopy reflectance models. The difficulty of using the CAT scanner to measure large canopies of crops like corn is discussed and a solution is proposed involving the measurement of plants one at a time.
Noël, Mario; Zwinkels, Joanne; Liu, Jian
2006-06-01
A reference instrument has been developed at the National Research Council of Canada for rapid, reproducible specular gloss measurements. The design and validation of this instrument for specular gloss measurements in accordance with standard methods for paints and plastics at 20 degree, 60 degree, and 85 degree geometries [American Society for Testing and Materials (ASTM) D523 and the International Organization for Standards (ISO) 2813] have been recently reported. These standard methods require a collimated beam geometry. Here we present the optical design considerations and characterization of this instrument to extend its gloss measurement capabilities to specular gloss measurements of paper samples at 75 degree geometry in accordance with standard test methods requiring a converging beam geometry (ASTM D1223 and TAPPI T480). This is, to the best of our knowledge, the first reported reference instrument that provides direct traceability for both types of standard gloss method and applications. The design challenge was to convert from a collimated beam to converging beam geometry while meeting the rigorous requirements of beam uniformity at the sample and receptor apertures specified in the 75 degree geometry test methods. We describe the innovative design to achieve this degree of functionality and reference instrument performance. The instrument's optical performance has been characterized theoretically and by comparison with measurement results. The light collection and detection systems have been analyzed via Monte Carlo simulation and ray tracing. The instrument validation includes comparison of the measurement results with theoretical gloss values for quartz, black glass, Vitrolite, and mirror gloss working standards, giving agreement of better than 0.32%. Measurement validation also involved participation in the Collaborative Testing Services program interlaboratory comparison measurements of 75 degree gloss for white papers. PMID:16724127
NASA Astrophysics Data System (ADS)
Noël, Mario; Zwinkels, Joanne; Liu, Jian
2006-06-01
A reference instrument has been developed at the National Research Council of Canada for rapid, reproducible specular gloss measurements. The design and validation of this instrument for specular gloss measurements in accordance with standard methods for paints and plastics at 20°, 60°, and 85° geometries [American Society for Testing and Materials (ASTM) D523 and the International Organization for Standards (ISO) 2813] have been recently reported. These standard methods require a collimated beam geometry. Here we present the optical design considerations and characterization of this instrument to extend its gloss measurement capabilities to specular gloss measurements of paper samples at 75° geometry in accordance with standard test methods requiring a converging beam geometry (ASTM D1223 and TAPPI T480). This is, to the best of our knowledge, the first reported reference instrument that provides direct traceability for both types of standard gloss method and applications. The design challenge was to convert from a collimated beam to converging beam geometry while meeting the rigorous requirements of beam uniformity at the sample and receptor apertures specified in the 75° geometry test methods. We describe the innovative design to achieve this degree of functionality and reference instrument performance. The instrument's optical performance has been characterized theoretically and by comparison with measurement results. The light collection and detection systems have been analyzed via Monte Carlo simulation and ray tracing. The instrument validation includes comparison of the measurement results with theoretical gloss values for quartz, black glass, Vitrolite, and mirror gloss working standards, giving agreement of better than 0.32%. Measurement validation also involved participation in the Collaborative Testing Services program interlaboratory comparison measurements of 75° gloss for white papers.
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.
Measurements of laser-driven magnetic fields in quasi-hohlraum geometries
NASA Astrophysics Data System (ADS)
Pollock, Bradley; Turnbull, D.; Goyon, C.; Ross, S.; Farmer, W.; Hazi, A.; Tubman, E.; Woolsey, N.; Law, K.; Fujioka, S.; Moody, J.
2015-11-01
Magnetic fields of 10-100 T have been produced with a laser-driven scheme using a parallel-plate target geometry, where a laser is directed through a hole in the front plate and irradiates the plate behind it. Hot electrons generated from the rear plate collect on the front plate, creating a voltage difference (~ 10-100 keV) between them. When the plates are connected via a quasi-loop conductor, this voltage sources current in the range of ~ 0.1-1 MA which produces a magnetic field along the axis of the loop. The field is generated on fast (~ ns) timescales, and can be scaled by changing the drive laser parameters. Recent experiments at the Jupiter Laser Facility have allowed temporally-resolved measurements of the voltage between the plates with ~ 1 J laser drive. Separate experiments at the Omega EP laser system have allowed direct Faraday rotation (in fused SiO2) measurements of the field strength inside the current loop by employing the 4w polarimetry capability of EP. We have also measured the extent and structure of the field with proton deflectometry at EP. The maximum field recorded along the axis of the quasi-loop is ~ 5 T at moderate (100 J) laser drive, and measurements of fringing fields outside the loop at 1 kJ indicate that the field increases to ~ 40 T. These results are compared with modeling to determine the current driven in the target, and infer information about the plasma conditions which sourced the current. This work was performed under the auspices of the United States Department of Energy by the Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.
An Instrument for Measuring Performance in Geometry Based on the Van Hiele Model
ERIC Educational Resources Information Center
Sánchez-García, Ana B.; Cabello, Ana Belén
2016-01-01
In this paper we present the process of constructing a test for assessing student performance in geometry corresponding to the first year of Secondary Education. The main goal was to detect student errors in the understanding of geometry in order to develop a proposal according to the Van Hiele teaching model, explained in this paper. Our research…
Simulation of large x-ray fields using independently measured source and geometry details
Sawkey, D.; Faddegon, B. A.
2009-01-01
Purpose: Obtain an accurate simulation of the dose from the 6 and 18 MV x-ray beams from a Siemens Oncor linear accelerator by comparing simulation to measurement. Constrain the simulation by independently determining parameters of the treatment head and incident beam, in particular, the energy and spot size. Methods: Measurements were done with the treatment head in three different configurations: (1) The clinical configuration, (2) the flattening filter removed, and (3) the target and flattening filter removed. Parameters of the incident beam and treatment head were measured directly. Incident beam energy and spectral width were determined from the percent-depth ionization of the raw beam (as described previously), spot size was determined using a spot camera, and the densities of the flattening filters were determined by weighing them. Simulations were done with EGSnrc∕BEAMnrc code. An asymmetric simulation was used, including offsets of the spot, primary collimator, and flattening filter from the collimator rotation axis. Results: Agreement between measurement and simulation was obtained to the least restrictive of 1% or 1 mm at 6 MV, both with and without the flattening filter in place, except for the buildup region. At 18 MV, the agreement was 1.5%∕1.5 mm with the flattening filter in place and 1%∕1 mm with it removed, except for in the buildup region. In the buildup region, the discrepancy was 2%∕2 mm at 18 MV and 1.5%∕1.5 mm at 6 MV with the flattening filter either removed or in place. The methodology for measuring the source and geometry parameters for the treatment head simulation is described. Except to determine the density of the flattening filter, no physical modification of the treatment head is necessary to obtain those parameters. In particular, the flattening filter does not need to be removed as was done in this work. Conclusions: Good agreement between measured and simulated dose distributions was obtained, even in the buildup region
Twisted Quantum Toroidal Algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Liu, Rongjia
2014-09-01
We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.
NASA Technical Reports Server (NTRS)
Willett, J. C.; LeVine, D. M.
2002-01-01
Direct current measurements are available near the attachment point from both natural cloud-to-ground lightning and rocket-triggered lightning, but little is known about the rise time and peak amplitude of return-stroke currents aloft. We present, as functions of height, current amplitudes, rise times, and effective propagation velocities that have been estimated with a novel remote-sensing technique from data on 24 subsequent return strokes in six different lightning flashes that were triggering at the NASA Kennedy Space Center, FL, during 1987. The unique feature of this data set is the stereo pairs of still photographs, from which three-dimensional channel geometries were determined previously. This has permitted us to calculate the fine structure of the electric-field-change (E) waveforms produced by these strokes, using the current waveforms measured at the channel base together with physically reasonable assumptions about the current distributions aloft. The computed waveforms have been compared with observed E waveforms from the same strokes, and our assumptions have been adjusted to maximize agreement. In spite of the non-uniqueness of solutions derived by this technique, several conclusions seem inescapable: 1) The effective propagation speed of the current up the channel is usually significantly (but not unreasonably) faster than the two-dimensional velocity measured by a streak camera for 14 of these strokes. 2) Given the deduced propagation speed, the peak amplitude of the current waveform often must decrease dramatically with height to prevent the electric field from being over-predicted. 3) The rise time of the current wave front must always increase rapidly with height in order to keep the fine structure of the calculated field consistent with the observations.
NASA Astrophysics Data System (ADS)
Moura, M.; Fiorentino, E.-A.; Mâløy, K. J.; Schäfer, G.; Toussaint, R.
2015-11-01
In this paper, we study the influence of sample geometry on the measurement of pressure-saturation relationships, by analyzing the drainage of a two-phase flow from a quasi-2-D random porous medium. The medium is transparent, which allows for the direct visualization of the invasion pattern during flow, and is initially saturated with a viscous liquid (a dyed glycerol-water mix). As the pressure in the liquid is gradually reduced, air penetrates from an open inlet, displacing the liquid which leaves the system from an outlet on the opposite side. Pressure measurements and images of the flow are recorded and the pressure-saturation relationship is computed. We show that this relationship depends on the system size and aspect ratio. The effects of the system's boundaries on this relationship are measured experimentally and compared with simulations produced using an invasion percolation algorithm. The pressure build up at the beginning and end of the invasion process are particularly affected by the boundaries of the system whereas at the central part of the model (when the air front progresses far from these boundaries), the invasion happens at a statistically constant capillary pressure. These observations have led us to propose a much simplified pressure-saturation relationship, valid for systems that are large enough such that the invasion is not influenced by boundary effects. The properties of this relationship depend on the capillary pressure thresholds distribution, sample dimensions, and average pore connectivity and its applications may be of particular interest for simulations of two-phase flow in large porous media.
ERIC Educational Resources Information Center
Bierschenk, Bernhard
Topological and algebraic scales were compared in the representation of the concept of human worth in behavioral-semantic terms. In a first experiment, seven doctoral students of Business Administration in Sweden explored the notion of worth using definitions from at least 10 dictionaries as the intentional-semantic content. Each subject served as…
Algebraic vs physical N = 6 3-algebras
Cantarini, Nicoletta; Kac, Victor G.
2014-01-15
In our previous paper, we classified linearly compact algebraic simple N = 6 3-algebras. In the present paper, we classify their “physical” counterparts, which actually appear in the N = 6 supersymmetric 3-dimensional Chern-Simons theories.
Measuring the geometry of the universe in the presence of isocurvature modes.
Dunkley, J; Bucher, M; Ferreira, P G; Moodley, K; Skordis, C
2005-12-31
The cosmic microwave background (CMB) anisotropy constrains the geometry of the Universe because the positions of the acoustic peaks of the angular power spectrum depend strongly on the curvature of three-dimensional space. In this Letter we exploit current observations to determine the geometry in the presence of isocurvature modes. Most previous analyses assumed that the primordial perturbations were adiabatic. A priori one might expect that allowing isocurvature modes would substantially degrade constraints on the curvature. We find, however, that with additional data sets, the geometry remains well constrained. When the most general isocurvature perturbation is allowed, the CMB alone can only poorly constrain the geometry to . Including large-scale structure data, one obtains Ohm(0) = 1.07 +/- 0.03, and 1.06 +/- 0.02 when supplemented by supernova data and the determination of H(0). PMID:16486336
NASA Technical Reports Server (NTRS)
Willett, J. C.; LeVine, D. M.; Idone, V. P.
2006-01-01
Three-dimensional reconstructions of six rocket-triggered lightning channels are derived from stereo photographs. These reconstructed channels are used to infer the behavior of the current in return strokes above the ground from current waveforms measured at the channel base and electric-field-change waveforms measured at a range of 5.2 kilometers for 24 return strokes in these channels. Streak photographs of 14 of the same strokes are analyzed to determine the rise times, propagation speeds, and amplitudes of relative light intensity for comparison with the electrical inferences. Results include the following: 1) The fine structure of the field-change waveforms that were radiated by these subsequent return strokes can be explained, in large part, by channel geometry. 2) The average 10 - 90% rise time of the stroke current increased by about a factor of seven in our sample, from an observed 0.31 plus or minus 0.17 microseconds at the surface to an inferred 2.2 plus or minus 0.5 microcseconds at 1 kilometer path length above the surface. 3) The three-dimensional propagation speed of the current front averaged 1.80 plus or minus 0.24 X 10(exp 8) meters per second over channel lengths typically greater than 1 kilometer. 4) Assuming that the measured current was entirely due to the return stroke forced an unreasonably large and abrupt reduction in inferred current amplitude over the first few tens of meters above the surface, especially in cases when the leader was bright relative to its stroke. Therefore, a significant fraction of the current at the surface was probably due to the leader, at least in such cases. 5) Peak return-stroke currents decreased by approximately 37 plus or minus 12% from 100 meters to 1 kilometer of path length above the surface. Because of uncertainty about how to partition the measured current between leader and return stroke, we are unable to infer the variation of current amplitude near the ground.
ERIC Educational Resources Information Center
Cawley, John F.; Foley, Teresa E.; Hayes, Anne Marie
2009-01-01
The purpose of this paper is to present a summary of selected facets of geometry and measurement in elementary school programs and to describe curricula content options designed to demonstrate the feasibility of seeking high level outcomes and meanings for students with learning disabilities. While there are a multitude of published papers…
Sequential products on effect algebras
NASA Astrophysics Data System (ADS)
Gudder, Stan; Greechie, Richard
2002-02-01
A sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The properties of sequential products on Hilbert space effect algebras are discussed. For a general SEA, relationships between sequential independence, coexistence and compatibility are given. It is shown that the sharp elements of a SEA form an orthomodular poset. The sequential center of a SEA is discussed and a characterization of when the sequential center is isomorphic to a fuzzy set system is presented. It is shown that the existence, of a sequential product is a strong restriction that eliminates many effect algebras from being SEA's. For example, there are no finite nonboolean SEA's, A measure of sharpness called the sharpness index is studied. The existence of horizontal sums of SEA's is characterized and examples of horizontal sums and tensor products are presented.
Liang, J.; Edelsbrunner, H.; Woodward, C.
1998-01-01
Identification and size characterization of surface pockets and occluded cavities are initial steps in protein structure-based ligand design. A new program, CAST, for automatically locating and measuring protein pockets and cavities, is based on precise computational geometry methods, including alpha shape and discrete flow theory. CAST identifies and measures pockets and pocket mouth openings, as well as cavities. The program specifies the atoms lining pockets, pocket openings, and buried cavities; the volume and area of pockets and cavities; and the area and circumference of mouth openings. CAST analysis of over 100 proteins has been carried out; proteins examined include a set of 51 monomeric enzyme-ligand structures, several elastase-inhibitor complexes, the FK506 binding protein, 30 HIV-1 protease-inhibitor complexes, and a number of small and large protein inhibitors. Medium-sized globular proteins typically have 10-20 pockets/cavities. Most often, binding sites are pockets with 1-2 mouth openings; much less frequently they are cavities. Ligand binding pockets vary widely in size, most within the range 10(2)-10(3)A3. Statistical analysis reveals that the number of pockets and cavities is correlated with protein size, but there is no correlation between the size of the protein and the size of binding sites. Most frequently, the largest pocket/cavity is the active site, but there are a number of instructive exceptions. Ligand volume and binding site volume are somewhat correlated when binding site volume is < or =700 A3, but the ligand seldom occupies the entire site. Auxiliary pockets near the active site have been suggested as additional binding surface for designed ligands (Mattos C et al., 1994, Nat Struct Biol 1:55-58). Analysis of elastase-inhibitor complexes suggests that CAST can identify ancillary pockets suitable for recruitment in ligand design strategies. Analysis of the FK506 binding protein, and of compounds developed in SAR by NMR (Shuker SB et
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, Inc., Reston, VA.
This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…
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
The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration
Figueroa-O'Farrill, Jose Miguel
2009-11-15
We phrase deformations of n-Leibniz algebras in terms of the cohomology theory of the associated Leibniz algebra. We do the same for n-Lie algebras and for the metric versions of n-Leibniz and n-Lie algebras. We place particular emphasis on the case of n=3 and explore the deformations of 3-algebras of relevance to three-dimensional superconformal Chern-Simons theories with matter.
New directions in algebraic dynamical systems
NASA Astrophysics Data System (ADS)
Schmidt, Klaus; Verbitskiy, Evgeny
2011-02-01
The logarithmic Mahler measure of certain multivariate polynomials occurs frequently as the entropy or the free energy of solvable lattice models (especially dimer models). It is also known that the entropy of an algebraic dynamical system is the logarithmic Mahler measure of the defining polynomial. The connection between the lattice models and the algebraic dynamical systems is still rather mysterious.
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
Study of the Effects of Photometric Geometry on Spectral Reflectance Measurements
NASA Technical Reports Server (NTRS)
Helfenstein, Paul
1998-01-01
The objective of this research is to investigate how the spectrophotometric properties of planetary surface materials depend on photometric geometry by refining and applying radiative transfer theory to data obtained from spacecraft and telescope observations of planetary surfaces, studies of laboratory analogs, and computer simulations. The goal is to perfect the physical interpretation of photometric parameters in the context of planetary surface geological properties and processes. The purpose of this report is to document the research achievements associated with this study.
NASA Astrophysics Data System (ADS)
Huang, C.; Chan, Y.; Hu, J.; Lee, J.
2006-12-01
In the past decade, improvements in geodesy allow geologists to measure the surface displacement of the hanging wall of a fault more precisely. The improved geodetic observations provide opportunities to better characterize the deformational behaviors of the hanging wall block due to earthquake fault slip. In the case of the 1999 Chi-Chi earthquake, the hanging wall block of the earthquake fault showed complex deformation pattern at the kilometer scale. Because previous studies mainly characterize on the fault at the regional scale, it is of interest and also challenge to characterize the fault at a smaller scale with a higher resolution. In this study we reconstructed the geometry of a kilometer-scale patch of the fault plane using the displacement data collected from the densely distributed city planning benchmarks. The study area is approximately 4 km by 8 km in size, and contains as many as 924 benchmarks. Among the benchmarks, 62 have both horizontal and vertical displacement data, and the rest of the benchmarks have the horizontal displacement data. Based on the assumption of rigid block motion, we established the earthquake fault geometry using the 62 slip vectors. We then use fault parallel flow method to test the derived fault geometry model with satisfied results. The derived fault geometry model is rather consistent with the borehole data from the nearby 450 m well.
Nonlinear Elastic J-Integral Measurements in Mode I Using a Tapered Double Cantilever Beam Geometry
NASA Technical Reports Server (NTRS)
Macon, David J.
2006-01-01
An expression for the J-integral of a nonlinear elastic material is derived for an advancing crack in a tapered double cantilever beam fracture specimen. The elastic and plastic fracture energies related to the test geometry and how these energies correlates to the crack position are discussed. The dimensionless shape factors eta(sub el and eta(sub p) are shown to be equivalent and the deformation J-integral is analyzed in terms of the eta(sub el) function. The fracture results from a structural epoxy are interpreted using the discussed approach. The magnitude of the plastic dissipation is found to strongly depend upon the initial crack shape.
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…
NASA Astrophysics Data System (ADS)
Hanson, D. L.; Vesey, R. A.; Slutz, S. A.; Cuneo, M. E.; Porter, J. L.; Adams, R. G.; Chandler, G. A.; Dropinski, S. C.; Johnson, D. W.; Keller, K. L.; McGurn, J. S.; Rambo, P. K.; Ruggles, L. E.; Simpson, W. W.; Speas, C. S.; Torres, J. A.; Smith, I. C.; Bennett, G. R.; Green, R.; Seamen, H.; Smelser, R. M.; Gilliland, T. L.; Cowan, T. E.; Schroen, D. G.; Tanner, D. L.
2002-11-01
In the fast ignitor approach to inertial fusion [Tabak et al., Phys. Plasmas 1, 1626 (1994)], ignition is produced by heating highly-compressed fuel with a fast, ultra-high power laser pulse. By separating the fuel compression and fast heating processes, symmetry and energy requirements for ignition are significantly relaxed. Laser propagation issues can be avoided by maintaining a plasma-free path for the short-pulse laser [Kodama et al., Nature 412, 798 (2001)]. In experiments on the Z accelerator at Sandia, we are exploring a fast ignitor hohlraum geometry uniquely adapted to fuel compression with a single-sided z-pinch radiation drive [Hanson et al., Phys. Plasmas 9, 2173 (2002)]. In this geometry, a hemispherical capsule mounted on a pedestal (short-pulse laser channel) is symmetrically imploded in a cylindrical secondary hohlraum heated by a single-wire-array z-pinch. Z-Beamlet point projection backlighter images of initial hemispherical capsule implosions on Z will be presented.
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.
NASA Astrophysics Data System (ADS)
Kumar, Jagadeesha; Attridge, Alex; Wood, P. K. C.; Williams, Mark A.
2011-03-01
Industrial x-ray computed tomography (CT) scanners are used for non-contact dimensional measurement of small, fragile components and difficult-to-access internal features of castings and mouldings. However, the accuracy and repeatability of measurements are influenced by factors such as cone-beam system geometry, test object configuration, x-ray power, material and size of test object, detector characteristics and data analysis methods. An attempt is made in this work to understand the measurement errors of a CT scanner over the complete scan volume, taking into account only the errors in system geometry and the object configuration within the scanner. A cone-beam simulation model is developed with the radiographic image projection and reconstruction steps. A known amount of errors in geometrical parameters were introduced in the model to understand the effect of geometry of the cone-beam CT system on measurement accuracy for different positions, orientations and sizes of the test object. Simulation analysis shows that the geometrical parameters have a significant influence on the dimensional measurement at specific configurations of the test object. Finally, the importance of system alignment and estimation of correct parameters for accurate CT measurements is outlined based on the analysis.
Tomkowiak, Michael T.; Raval, Amish N.; Van Lysel, Michael S.; Funk, Tobias; Speidel, Michael A.
2014-01-01
Proper sizing of interventional devices to match coronary vessel dimensions improves procedural efficiency and therapeutic outcomes. We have developed a novel method using inverse geometry x-ray fluoroscopy to automatically determine vessel dimensions without the need for magnification calibration or optimal views. To validate this method in vivo, we compared results to intravascular ultrasound (IVUS) and coronary computed tomography angiography (CCTA) in a healthy porcine model. Coronary angiography was performed using Scanning-Beam Digital X-ray (SBDX), an inverse geometry fluoroscopy system that performs multiplane digital x-ray tomosynthesis in real time. From a single frame, 3D reconstruction of the arteries was performed by localizing the depth of vessel lumen edges. The 3D model was used to directly calculate length and to determine the best imaging plane to use for diameter measurements, where out-of-plane blur was minimized and the known pixel spacing was used to obtain absolute vessel diameter. End-diastolic length and diameter measurements were compared to measurements from CCTA and IVUS, respectively. For vessel segment lengths measuring 6 mm to 73 mm by CCTA, the SBDX length error was −0.49 ± 1.76 mm (SBDX − CCTA, mean ± 1 SD). For vessel diameters measuring 2.1 mm to 3.6 mm by IVUS, the SBDX diameter error was 0.07 ± 0.27 mm (SBDX − minimum IVUS diameter, mean ± 1 SD). The in vivo agreement between SBDX-based vessel sizing and gold standard techniques supports the feasibility of calibration-free coronary vessel sizing using inverse geometry x-ray fluoroscopy. PMID:24999298
NASA Astrophysics Data System (ADS)
Tomkowiak, Michael T.; Raval, Amish N.; Van Lysel, Michael S.; Funk, Tobias; Speidel, Michael A.
2014-03-01
Proper sizing of interventional devices to match coronary vessel dimensions improves procedural efficiency and therapeutic outcomes. We have developed a novel method using inverse geometry x-ray fluoroscopy to automatically determine vessel dimensions without the need for magnification calibration or optimal views. To validate this method in vivo, we compared results to intravascular ultrasound (IVUS) and coronary computed tomography angiography (CCTA) in a healthy porcine model. Coronary angiography was performed using Scanning-Beam Digital X-ray (SBDX), an inverse geometry fluoroscopy system that performs multiplane digital x-ray tomosynthesis in real time. From a single frame, 3D reconstruction of the arteries was performed by localizing the depth of vessel lumen edges. The 3D model was used to directly calculate length and to determine the best imaging plane to use for diameter measurements, where outof- plane blur was minimized and the known pixel spacing was used to obtain absolute vessel diameter. End-diastolic length and diameter measurements were compared to measurements from CCTA and IVUS, respectively. For vessel segment lengths measuring 6 mm to 73 mm by CCTA, the SBDX length error was -0.49 +/- 1.76 mm (SBDX - CCTA, mean +/- 1 SD). For vessel diameters measuring 2.1 mm to 3.6 mm by IVUS, the SBDX diameter error was 0.07 +/- 0.27 mm (SBDX - minimum IVUS diameter, mean +/- 1 SD). The in vivo agreement between SBDX-based vessel sizing and gold standard techniques supports the feasibility of calibration-free coronary vessel sizing using inverse geometry x-ray fluoroscopy.
Learning Algebra in a Computer Algebra Environment
ERIC Educational Resources Information Center
Drijvers, Paul
2004-01-01
This article summarises a doctoral thesis entitled "Learning algebra in a computer algebra environment, design research on the understanding of the concept of parameter" (Drijvers, 2003). It describes the research questions, the theoretical framework, the methodology and the results of the study. The focus of the study is on the understanding of…
Realizations of Galilei algebras
NASA Astrophysics Data System (ADS)
Nesterenko, Maryna; Pošta, Severin; Vaneeva, Olena
2016-03-01
All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations.
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.
Orientation in operator algebras
Alfsen, Erik M.; Shultz, Frederic W.
1998-01-01
A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. PMID:9618457
Developing Thinking in Algebra
ERIC Educational Resources Information Center
Mason, John; Graham, Alan; Johnson-Wilder, Sue
2005-01-01
This book is for people with an interest in algebra whether as a learner, or as a teacher, or perhaps as both. It is concerned with the "big ideas" of algebra and what it is to understand the process of thinking algebraically. The book has been structured according to a number of pedagogic principles that are exposed and discussed along the way,…
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…
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…
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…
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.
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.
Entropy algebras and Birkhoff factorization
NASA Astrophysics Data System (ADS)
Marcolli, Matilde; Tedeschi, Nicolas
2015-11-01
We develop notions of Rota-Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.
Boerger, D.M.; Kramer, J.J.; Partain, L.D.
1981-01-01
A rigorous derivation is given to generalize the allowed, Hall effect, sample shapes from the restrictive, rectangular parallelepiped configurations to a much more general class of geometries characterized by mirror symmetry for materials whose mobile carriers have surfaces of constant energy in k-bar space that are well described by ellipsoids. However, this mirror symmetry condition is more restrictive than the almost arbitrary sample shapes proposed with the van der Pauw technique for thin films. Experimental data taken on n-type CdS at liquid-nitrogen temperatures in magnetic field strengths of 8 and 145 kG show that errors ranging from 1 to 600% can result from van der Pauw-type geometries depending on how much the sample shape and/or contact arrangement differs from the mirror symmetry. An empirically derived averaging technique is described that reduces the observed errors to less than 13% even with van der Pauw-type shapes that do not meet the mirror symmetry conditions.
Poswal, A. K.; Agrawal, A.; Bhattachryya, D.; Jha, S. N.; Sahoo, N. K.
2013-02-05
We have designed and fabricated parallel plate ionization chamber detectors and voltage vs. current characteristics (V-I curve) of the detectors were recorded with synchrotron radiation to qualify for use in X-ray Absorption Spectroscopy (XAS) measurements. After qualifying the ionization chambers, the detectors were used in the dispersive EXAFS beamline (BL-08) at INDUS-2 SRS in Turbo-XAS geometry. Using the same setup and under the same setting, XAS spectra were also recorded with a CCD detector and the observation on relative performance of the ionization chamber vis-a-vis the CCD detector is presented in this paper.
NASA Astrophysics Data System (ADS)
Poswal, A. K.; Agrawal, A.; Bhattachryya, D.; Jha, S. N.; Sahoo, N. K.
2013-02-01
We have designed and fabricated parallel plate ionization chamber detectors and voltage vs. current characteristics (V-I curve) of the detectors were recorded with synchrotron radiation to qualify for use in X-ray Absorption Spectroscopy (XAS) measurements. After qualifying the ionization chambers, the detectors were used in the dispersive EXAFS beamline (BL-08) at INDUS-2 SRS in Turbo-XAS geometry. Using the same setup and under the same setting, XAS spectra were also recorded with a CCD detector and the observation on relative performance of the ionization chamber vis-a-vis the CCD detector is presented in this paper.
Barclay, R.K.
1984-06-01
A device for measuring the exposure rate from neutron-activated indium foil, under constant geometry, has been designed, constructed, and tested. The device is intended for use with the Juno ionization chambers, although it adapts to Victoreen CDV-700 and Victoreen 193 G-M instruments. Juno dose-response data for low (53 rad) and high (226 rad) doses were compiled and modeled. This model was compared to that assumed from the indium foil dose-response model in current use; plots of fitted and assumed models are congruent. An analysis of data from both Juno and CDV-700 instruments indicates that the constant geometry device may be used effectively to monitor the decay of In-116m. Tolerance limits for the Juno dose-response curve increase with time after activation, which results in diminished precision of dose estimates made by indium foil measurement. From the data collected in these experiments, the system appears to be most useful if activation is measured within 250 min after exposure. 5 references, 7 figures, 1 table.
Optimal Energy Measurement in Nonlinear Systems: An Application of Differential Geometry
NASA Technical Reports Server (NTRS)
Fixsen, Dale J.; Moseley, S. H.; Gerrits, T.; Lita, A.; Nam, S. W.
2014-01-01
Design of TES microcalorimeters requires a tradeoff between resolution and dynamic range. Often, experimenters will require linearity for the highest energy signals, which requires additional heat capacity be added to the detector. This results in a reduction of low energy resolution in the detector. We derive and demonstrate an algorithm that allows operation far into the nonlinear regime with little loss in spectral resolution. We use a least squares optimal filter that varies with photon energy to accommodate the nonlinearity of the detector and the non-stationarity of the noise. The fitting process we use can be seen as an application of differential geometry. This recognition provides a set of well-developed tools to extend our work to more complex situations. The proper calibration of a nonlinear microcalorimeter requires a source with densely spaced narrow lines. A pulsed laser multi-photon source is used here, and is seen to be a powerful tool for allowing us to develop practical systems with significant detector nonlinearity. The combination of our analysis techniques and the multi-photon laser source create a powerful tool for increasing the performance of future TES microcalorimeters.
Influence of measurement geometry on the human skin reflectance spectra detection
NASA Astrophysics Data System (ADS)
Borisova, E.; Troyanova, P.; Avramov, L.
2007-06-01
One of the optical techniques applied in skin lesion investigations is reflectance spectroscopy of cutaneous tissues. Diffuse reflectance signals could be applied for absolute determination of absorption and scattering coefficients of biological tissue and give reliable results for wide range of wavelengths. However, light distribution anisotropy leads to significant influence over the reflectance spectra appearance depending on the used geometry. To obtain good reproducibility and repeatability of the results, one needs to fix carefully the geometrical conditions. We present skin reflectance spectra obtained at different angles and distances from tissue surface and discuss possible reasons for differences observed. In the case of small angles of incidence, we have significant losses of the backscattered light related to the anisotropy of the tissue and the intensity observed of the reflected signal is much lower than in the case of diffuse reflectance signal detected at higher angles. In the case of direct contact between skin surface and end tip of the fibers great reduction of the short wavelength component is observed. The appearance of higher intensity blue signal as the distance between end tip of the optical fibers and skin surface is increased could be related to the specular component of the reflectance spectra. An optimum distance exists where the registered signal has the greatest value, which is related to the specific parameters of the optical fibers used, such as diameter and optical aperture.
Velocity and Scalar Measurements of Strut-Based Hypermixing Geometries in a Mach 3 Flow
NASA Astrophysics Data System (ADS)
Burns, Ross; Clemens, Noel
2011-11-01
Strut-based fuel injection with hypermixing exhibits great potential as a fuel-injection strategy for future scramjet engine design. Hypermixing entails the introduction of strong streamwise vorticity by means of geometrically-induced pressure gradients at the trailing edge of the strut; however, these complex flowfields are not well understood. An experimental investigation is being conducted on the flowfield characteristics of several strut-based hypermixers in a Mach 3 freestream. The hypermixing flowfields are generated from an injection pylon with interchangeable trailing-edge geometries including compressive and expansive wedges. Particle image velocimetry (PIV) in conjunction with two scalar visualization techniques are used to obtain velocity and scalar field data. The scalar imaging techniques include two-photon absorption planar laser-induced fluorescence (PLIF) of krypton gas, which simulates fuel injection into the wake, and planar laser scattering (PLS) from condensed carbon dioxide fog, which marks the outer flow structures. The velocity and scalar data reveal details of the underlying flow physics as well as the turbulent mixing characteristics. This work was supported by NASA under cooperative agreement NNX08AB41A.
Fast laser systems for measuring the geometry of complex-shaped objects
NASA Astrophysics Data System (ADS)
Galiulin, Ravil M.; Galiulin, Rishat M.; Bakirov, J. M.; Vorontsov, A. V.; Ponomarenko, I. V.
1999-01-01
The technical characteristics, advantages and applications of an automated optoelectronic measuring system designed by 'Optel' company, State Aviation University of Ufa, are presented in this paper. The measuring apparatus can be applied for industrial development and research, for example, in rapid prototyping, and for obtaining geometrical parameters in medicine and criminalistics. It essentially is a non-contact and rapid scanning system, allowing measurements of complex shaped objects like metal and plastic workpieces or parts of human body.
Algebraic models of flexible manufacturing systems
NASA Astrophysics Data System (ADS)
Leskin, Aleksei Alekseevich
Various aspects of the use of mathematical methods in the development of flexible manufacturing systems are examined. Attention is given to dynamical and structural models of flexible manufacturing systems developed by using methods of algebraic and differential geometry, topology, polynomial algebra, and extreme value problem theory. The principles of model integration are discussed, and approaches are proposed for solving problems related to the selection of flexible manufacturing equipment, real-time modeling of the manufacturing process, and optimization of local automation systems. The discussion is illustrated by examples.
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).
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.
Gillespie, D.R.H.; Wang, Z.; Ireland, P.T.; Kohler, S.T.
1998-01-01
Cast impingement cooling geometries offer the gas turbine designer higher structural integrity and improved convective cooling when compared to traditional impingement cooling systems, which rely on plate inserts. In this paper, it is shown that the surface that forms the jets contributes significantly to the total cooling. Local heat transfer coefficient distributions have been measured in a model of an engine wall cooling geometry using the transient heat transfer technique. The method employs temperature-sensitive liquid crystals to measure the surface temperature of large-scale perspex models during transient experiments. Full distributions of local Nusselt number on both surfaces of the impingement plate, and on the impingement target plate, are presented at engine representative Reynolds numbers. The relative effects of the impingement plate thermal boundary condition and the coolant supply temperature on the target plate heat transfer have been determined by maintaining an isothermal boundary condition at the impingement plate during the transient tests. The results are discussed in terms of the interpreted flow field.
NASA Astrophysics Data System (ADS)
Richter, Wolf-Dieter
2016-06-01
The local approach to the notion of a star generalized surface measure, consisting of taking derivatives of sector volumes, is proved to be equivalent to a suitable generalization of the well known integral (or diffential geometric) approach to the common notion of surface content. For star-shaped probability laws having a density contour defining star body K, a known geometric measure representation which is based upon the local approach to the star-generalized surface measure, in consequence appears in the new light of being a representation in the space (Rn, ĥK*) where ĥK* is a slight modification of the Minkowski functional of a certain generalized ball K* which is constructed in dependence of K.
Measurement of noise and its correlation to performance and geometry of small aircraft propellers
NASA Astrophysics Data System (ADS)
Štorch, Vít; Nožička, Jiří; Brada, Martin; Gemperle, Jiří; Suchý, Jakub
2016-03-01
A set of small model and UAV propellers is measured both in terms of aerodynamic performance and acoustic noise under static conditions. Apart from obvious correlation of noise to tip speed and propeller diameter the influence of blade pitch, blade pitch distribution, efficiency and shape of the blade is sought. Using the measured performance data a computational model for calculation of aerodynamic noise of propellers will be validated. The range of selected propellers include both propellers designed for nearly static conditions and propellers that are running at highly offdesign conditions, which allows to investigate i.e. the effect of blade stall on both noise level and performance results.
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, pre-algebra, and algebra…
Tomkowiak, Michael T.; Raval, Amish N.; Van Lysel, Michael S.; Funk, Tobias; Speidel, Michael A.
2014-01-01
Abstract. Proper sizing of interventional devices to match coronary vessel dimensions improves procedural efficiency and therapeutic outcomes. We have developed a method that uses an inverse geometry x-ray fluoroscopy system [scanning beam digital x-ray (SBDX)] to automatically determine vessel dimensions from angiograms without the need for magnification calibration or optimal views. For each frame period (1/15th of a second), SBDX acquires a sequence of narrow beam projections and performs digital tomosynthesis at multiple plane positions. A three-dimensional model of the vessel is reconstructed by localizing the depth of the vessel edges from the tomosynthesis images, and the model is used to calculate the length and diameter in units of millimeters. The in vivo algorithm performance was evaluated in a healthy porcine model by comparing end-diastolic length and diameter measurements from SBDX to coronary computed tomography angiography (CCTA) and intravascular ultrasound (IVUS), respectively. The length error was −0.49±1.76 mm (SBDX – CCTA, mean±1 SD). The diameter error was 0.07±0.27 mm (SBDX − minimum IVUS diameter, mean±1 SD). The in vivo agreement between SBDX-based vessel sizing and gold standard techniques supports the feasibility of calibration-free coronary vessel sizing using inverse geometry x-ray fluoroscopy. PMID:25544948
Renga, Alfredo; Moccia, Antonio
2009-01-01
During the last decade a methodology for the reconstruction of surface relief by Synthetic Aperture Radar (SAR) measurements – SAR interferometry – has become a standard. Different techniques developed before, such as stereo-radargrammetry, have been experienced from space only in very limiting geometries and time series, and, hence, branded as less accurate. However, novel formation flying configurations achievable by modern spacecraft allow fulfillment of SAR missions able to produce pairs of monostatic-bistatic images gathered simultaneously, with programmed looking angles. Hence it is possible to achieve large antenna separations, adequate for exploiting to the utmost the stereoscopic effect, and to make negligible time decorrelation, a strong liming factor for repeat-pass stereo-radargrammetric techniques. This paper reports on design of a monostatic-bistatic mission, in terms of orbit and pointing geometry, and taking into account present generation SAR and technology for accurate relative navigation. Performances of different methods for monostatic-bistatic stereo-radargrammetry are then evaluated, showing the possibility to determine the local surface relief with a metric accuracy over a wide range of Earth latitudes. PMID:22389594
The geometry of three-dimensional measurement from paired coplanar x-ray images.
Baumrind, S; Moffitt, F H; Curry, S
1983-10-01
This article outlines the geometric principles which underlie the process of making craniofacial measurements in three dimensions by combining information from pairs of coplanar x-ray images. The main focus is upon the rationale of the method rather than upon the computational details. We stress particularly the importance of having available accurate measurements as to the relative positions of the x-ray tubes and the film plane. The use of control arrays of radiopaque "points" whose projected images upon the film plane allow the retrospective calculation of the spatial relationship between the x-ray tubes and the film plane is explained. Finally, the question of correcting for movement of the subject between two films of an image pair is considered briefly. PMID:6578682
NASA Astrophysics Data System (ADS)
Facsko, Gabor; Reshetnyk, Volodymyr; Agapitov, Oleksiy; Opitz, Andrea; Szabo, Adam; McComas, David
Tangential discontinuties (TDs) are usually considered as thin planar current sheets frozen in the solar wind flow. Previous studies based on the magnetic field measurements onboard of ACE, Wind, and STEREO A, and B proved that this hypotesis is not valid. The curvature of the TDs were determined in several cases. After applying minimum variance and the cross product methods for Ulysses, ACE and STEREO A and B magnetometer measurements, numerous TDs are identified in 2008 and 2009. The time shift of the TD observations is determinated by correlation analysis of the solar wind speed and the magnetic field variations. The 3D topology of the TD is then determinated in some special cases when the four spacecraft are on the same side of the Sun. After fitting a simple model, the location of the TD formation region can be outlined.
Flow and heat transfer measurements in a swirl chamber with different outlet geometries
NASA Astrophysics Data System (ADS)
Biegger, Christoph; Weigand, Bernhard
2015-04-01
In technical applications, an efficient cooling is necessary for high thermal load components such as turbine blades. One potential and promising technique is a swirling tube flow in comparison with an axial flow. The additional circumferential velocity and enhanced turbulent mixing increase the heat transfer. But the complex flow field and heat transfer mechanisms are still under research. Furthermore, the reliability of a swirl chamber regarding different outlet conditions is of great interest for a robust cooling design. Therefore, we investigated the influence of a straight, a tangential and a bend outlet. To gain understanding of the flow phenomena, we measured the velocity field by means of stereo-PIV (particle image velocimetry). We experimentally studied the cooling capability measuring the heat transfer coefficients using thermochromic liquid crystals. For an accurate cooling design, we used the local adiabatic wall temperature as the correct reference temperature for calculating the heat transfer coefficients. We will show the velocity field, the pressure loss and the heat transfer results for realistic Reynolds numbers from 10,000 to 40,000 and for swirl numbers between and . The obtained heat transfer is more than four times higher compared to an axial tube flow. Our measurements indicate that the here investigated outlet redirection has no significant influence on the flow field and the heat transfer coefficients.
Xu, Chen; Kumavor, Patrick D; Aguirre, Andres; Zhu, Quing
2012-06-01
Photoacoustic tomography provides the distribution of absorbed optical energy density, which is the product of optical absorption coefficient and optical fluence distribution. We report the experimental investigation of a novel fitting procedure that quantitatively determines the optical absorption coefficient of chromophores. The experimental setup consisted of a hybrid system of a 64-channel photoacoustic imaging system with a frequency-domain diffused optical measurement system. The fitting procedure included a complete photoacoustic forward model and an analytical solution of a target chromophore using the diffusion approximation. The fitting procedure combines the information from the photoacoustic image and the background information from the diffuse optical measurements to minimize the photoacoustic measurements and forward model data and recover the target absorption coefficient quantitatively. 1-cm-cube phantom absorbers of high and low contrasts were imaged at depths of up to 3.0 cm. The fitted absorption coefficient results were at least 80% of their true values. The sensitivities of this fitting procedure to target location, target radius, and background optical properties were also investigated. We found that this fitting procedure was most sensitive to the accurate determination of the target radius and depth. Blood sample in a thin tube of radius 0.58 mm, simulating a blood vessel, was also studied. The photoacoustic images and fitted absorption coefficients are presented. These results demonstrate the clinical potential of this fitting procedure to quantitatively characterize small lesions in breast imaging. PMID:22734743
NASA Astrophysics Data System (ADS)
Xu, Chen; Kumavor, Patrick D.; Aguirre, Andres; Zhu, Quing
2012-06-01
Photoacoustic tomography provides the distribution of absorbed optical energy density, which is the product of optical absorption coefficient and optical fluence distribution. We report the experimental investigation of a novel fitting procedure that quantitatively determines the optical absorption coefficient of chromophores. The experimental setup consisted of a hybrid system of a 64-channel photoacoustic imaging system with a frequency-domain diffused optical measurement system. The fitting procedure included a complete photoacoustic forward model and an analytical solution of a target chromophore using the diffusion approximation. The fitting procedure combines the information from the photoacoustic image and the background information from the diffuse optical measurements to minimize the photoacoustic measurements and forward model data and recover the target absorption coefficient quantitatively. 1-cm-cube phantom absorbers of high and low contrasts were imaged at depths of up to 3.0 cm. The fitted absorption coefficient results were at least 80% of their true values. The sensitivities of this fitting procedure to target location, target radius, and background optical properties were also investigated. We found that this fitting procedure was most sensitive to the accurate determination of the target radius and depth. Blood sample in a thin tube of radius 0.58 mm, simulating a blood vessel, was also studied. The photoacoustic images and fitted absorption coefficients are presented. These results demonstrate the clinical potential of this fitting procedure to quantitatively characterize small lesions in breast imaging.
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)
Fuzzy-algebra uncertainty assessment
Cooper, J.A.; Cooper, D.K.
1994-12-01
A significant number of analytical problems (for example, abnormal-environment safety analysis) depend on data that are partly or mostly subjective. Since fuzzy algebra depends on subjective operands, we have been investigating its applicability to these forms of assessment, particularly for portraying uncertainty in the results of PRA (probabilistic risk analysis) and in risk-analysis-aided decision-making. Since analysis results can be a major contributor to a safety-measure decision process, risk management depends on relating uncertainty to only known (not assumed) information. The uncertainties due to abnormal environments are even more challenging than those in normal-environment safety assessments; and therefore require an even more judicious approach. Fuzzy algebra matches these requirements well. One of the most useful aspects of this work is that we have shown the potential for significant differences (especially in perceived margin relative to a decision threshold) between fuzzy assessment and probabilistic assessment based on subtle factors inherent in the choice of probability distribution models. We have also shown the relation of fuzzy algebra assessment to ``bounds`` analysis, as well as a description of how analyses can migrate from bounds analysis to fuzzy-algebra analysis, and to probabilistic analysis as information about the process to be analyzed is obtained. Instructive examples are used to illustrate the points.
NASA Astrophysics Data System (ADS)
Bedington, Robert; Kataria, Dhiren; Smith, Alan
2015-09-01
The CATS (Cylindrical And Tiny Spectrometer) electrostatic optics geometry features multiple nested cylindrical analysers to simultaneously measure multiple energies of electron and multiple energies of ion in a configuration that is targeted at miniaturisation and MEMS fabrication. In the prototyped model, two configurations of cylindrical analyser were used, featuring terminating side-plates that caused particle trajectories to either converge (C type) or diverge (D type) in the axial direction. Simulations show how these different electrode configurations affect the particle focussing and instrument parameters; C-type providing greater throughputs but D-type providing higher resolving powers. The simulations were additionally used to investigate unexpected plate spacing variations in the as-built model, revealing that the k-factors are most sensitive to the width of the inter-electrode spacing at its narrowest point.
Measurement of Bedding Geometry of Upper Aeolis Mons, Gale Crater, Mars
NASA Astrophysics Data System (ADS)
Anderson, R. B.; Lewis, K. W.; Rubin, D. M.
2014-12-01
Aeolis Mons, informally called "Mount Sharp," is a >5 km tall mound of layered sedimentary rock in Gale crater. The mound can be divided into lower and upper formations, with a clear unconformity between the two formations identified by Malin and Edgett (2000). Multiple authors (e.g. Anderson and Bell, 2010; Thomson et al., 2011; Wray, 2012) have noted that the upper formation may have a distinct origin from the lower mound. Although the Curiosity rover is expected to explore the base of the lower formation, the upper portion of Aeolis Mons is likely unreachable. HiRISE observations of the upper formation reveal sinuous bedding patterns on a scale of 100s of meters with apparent truncations. These patterns have been interpreted to be cross-beds (Anderson and Bell, 2010). However, identifying cross-bedding in orbital images is not always straightforward. Planar beds intersecting eroded topography can produce complex patterns of exposed bedding that may look superficially like cross-bedding. To confirm the presence of cross-bedding, the exposure must be studied in three dimensions. We present initial results of an investigation using a HiRISE Digital Terrain Model (DTM) based on the HiRISE stereo pair PSP_001620_1750 and PSP_001422_1750 test the hypothesis that the upper formation of Aeolis Mons represents aeolian cross-bedding. By tracing the intersection of the beds with a plane, we will determine whether the observed patterns might be explained by the interaction of planar beds and complex erosion or if the observed structures require cross-bedded deposits. These measurements are ongoing. If the complex bedding patterns observed in the upper formation are confirmed to be cross-beds, we will present measurements of the bedding orientation and use computer models to interpret the depositional conditions for the upper formation of Aeolis Mons. Anderson, R., Bell, J.F., 2010. Mars J. 5, 76-128. doi:10.1555/mars.2010.0004 Malin, M.C., Edgett, K.S., 2000. Science 290
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…
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…
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
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…
C*-algebras of holonomy-diffeomorphisms and quantum gravity: I
NASA Astrophysics Data System (ADS)
Aastrup, Johannes; Møller Grimstrup, Jesper
2013-04-01
A new approach to a unified theory of quantum gravity based on noncommutative geometry and canonical quantum gravity is presented. The approach is built around a *-algebra generated by local holonomy-diffeomorphisms on a 3-manifold and a quantized Dirac-type operator, the two capturing the kinematics of quantum gravity formulated in terms of Ashtekar variables. We prove that the separable part of the spectrum of the algebra is contained in the space of measurable connections modulo gauge transformations and we give limitations to the non-separable part. The construction of the Dirac-type operator—and thus the application of noncommutative geometry—is motivated by the requirement of diffeomorphism invariance. We conjecture that a semi-finite spectral triple, which is invariant under volume-preserving diffeomorphisms, arises from a GNS construction of a semi-classical state. Key elements of quantum field theory emerge from the construction in a semi-classical limit, as does an almost commutative algebra. Finally, we note that the spectrum of loop quantum gravity emerges from a discretization of our construction. Certain convergence issues are left unresolved. This paper is the first of two where the second paper [1] is concerned with mathematical details and proofs concerning the spectrum of the holonomy-diffeomorphism algebra.
Davidson, R. L.; Earle, G. D.; Heelis, R. A.; Klenzing, J. H.
2010-08-15
Planar retarding potential analyzers (RPAs) have been utilized numerous times on high profile missions such as the Communications/Navigation Outage Forecast System and the Defense Meteorological Satellite Program to measure plasma composition, temperature, density, and the velocity component perpendicular to the plane of the instrument aperture. These instruments use biased grids to approximate ideal biased planes. These grids introduce perturbations in the electric potential distribution inside the instrument and when unaccounted for cause errors in the measured plasma parameters. Traditionally, the grids utilized in RPAs have been made of fine wires woven into a mesh. Previous studies on the errors caused by grids in RPAs have approximated woven grids with a truly flat grid. Using a commercial ion optics software package, errors in inferred parameters caused by both woven and flat grids are examined. A flat grid geometry shows the smallest temperature and density errors, while the double thick flat grid displays minimal errors for velocities over the temperature and velocity range used. Wire thickness along the dominant flow direction is found to be a critical design parameter in regard to errors in all three inferred plasma parameters. The results shown for each case provide valuable design guidelines for future RPA development.
Semigroups and computer algebra in algebraic structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
Lie algebra extensions of current algebras on S3
NASA Astrophysics Data System (ADS)
Kori, Tosiaki; Imai, Yuto
2015-06-01
An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory.
An introduction to Minkowski geometries
NASA Astrophysics Data System (ADS)
Farnsworth, David L.
2016-07-01
The fundamental ideas of Minkowski geometries are presented. Learning about Minkowski geometries can sharpen our students' understanding of concepts such as distance measurement. Many of its ideas are important and accessible to undergraduate students. Following a brief overview, distance and orthogonality in Minkowski geometries are thoroughly discussed and many illustrative examples and applications are supplied. Suggestions for further study of these geometries are given. Indeed, Minkowski geometries are an excellent source of topics for undergraduate research and independent study.
Massive neutrinos in almost-commutative geometry
NASA Astrophysics Data System (ADS)
Stephan, Christoph A.
2007-02-01
In the noncommutative formulation of the standard model of particle physics by Chamseddine and Connes [Commun. Math. Phys. 182, 155 (1996), e-print hep-th/9606001], one of the three generations of fermions has to possess a massless neutrino. [C. P. Martin et al., Phys. Rep. 29, 363 (1998), e-print hep-th-9605001]. This formulation is consistent with neutrino oscillation experiments and the known bounds of the Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS matrix). But future experiments which may be able to detect neutrino masses directly and high-precision measurements of the PMNS matrix might need massive neutrinos in all three generations. In this paper we present an almost-commutative geometry which allows for a standard model with massive neutrinos in all three generations. This model does not follow in a straightforward way from the version of Chamseddine and Connes since it requires an internal algebra with four summands of matrix algebras, instead of three summands for the model with one massless neutrino.
Massive neutrinos in almost-commutative geometry
Stephan, Christoph A.
2007-02-15
In the noncommutative formulation of the standard model of particle physics by Chamseddine and Connes [Commun. Math. Phys. 182, 155 (1996), e-print hep-th/9606001], one of the three generations of fermions has to possess a massless neutrino. [C. P. Martin et al., Phys. Rep. 29, 363 (1998), e-print hep-th-9605001]. This formulation is consistent with neutrino oscillation experiments and the known bounds of the Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS matrix). But future experiments which may be able to detect neutrino masses directly and high-precision measurements of the PMNS matrix might need massive neutrinos in all three generations. In this paper we present an almost-commutative geometry which allows for a standard model with massive neutrinos in all three generations. This model does not follow in a straightforward way from the version of Chamseddine and Connes since it requires an internal algebra with four summands of matrix algebras, instead of three summands for the model with one massless neutrino.
Leibniz algebras associated with representations of filiform Lie algebras
NASA Astrophysics Data System (ADS)
Ayupov, Sh. A.; Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A.
2015-12-01
In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1. We introduce a Fock module for the algebra nn,1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra nn,1 with condition that the ideal I is a Fock nn,1-module, where I is the ideal generated by squares of elements from L. We also consider Leibniz algebras with corresponding Lie algebra nn,1 and such that the action I ×nn,1 → I gives rise to a minimal faithful representation of nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4.
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.
Coreflections in Algebraic Quantum Logic
NASA Astrophysics Data System (ADS)
Jacobs, Bart; Mandemaker, Jorik
2012-07-01
Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.
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.
Algebraic Systems Biology: A Case Study for the Wnt Pathway.
Gross, Elizabeth; Harrington, Heather A; Rosen, Zvi; Sturmfels, Bernd
2016-01-01
Steady-state analysis of dynamical systems for biological networks gives rise to algebraic varieties in high-dimensional spaces whose study is of interest in their own right. We demonstrate this for the shuttle model of the Wnt signaling pathway. Here, the variety is described by a polynomial system in 19 unknowns and 36 parameters. It has degree 9 over the parameter space. This case study explores multistationarity, model comparison, dynamics within regions of the state space, identifiability, and parameter estimation, from a geometric point of view. We employ current methods from computational algebraic geometry, polyhedral geometry, and combinatorics. PMID:26645985
NASA Astrophysics Data System (ADS)
Nakonieczna, Anna; Yeom, Dong-han
2016-05-01
Investigating the dynamics of gravitational systems, especially in the regime of quantum gravity, poses a problem of measuring time during the evolution. One of the approaches to this issue is using one of the internal degrees of freedom as a time variable. The objective of our research was to check whether a scalar field or any other dynamical quantity being a part of a coupled multi-component matter-geometry system can be treated as a `clock' during its evolution. We investigated a collapse of a self-gravitating electrically charged scalar field in the Einstein and Brans-Dicke theories using the 2+2 formalism. Our findings concentrated on the spacetime region of high curvature existing in the vicinity of the emerging singularity, which is essential for the quantum gravity applications. We investigated several values of the Brans-Dicke coupling constant and the coupling between the Brans-Dicke and the electrically charged scalar fields. It turned out that both evolving scalar fields and a function which measures the amount of electric charge within a sphere of a given radius can be used to quantify time nearby the singularity in the dynamical spacetime part, in which the apparent horizon surrounding the singularity is spacelike. Using them in this respect in the asymptotic spacetime region is possible only when both fields are present in the system and, moreover, they are coupled to each other. The only nonzero component of the Maxwell field four-potential cannot be used to quantify time during the considered process in the neighborhood of the whole central singularity. None of the investigated dynamical quantities is a good candidate for measuring time nearby the Cauchy horizon, which is also singular due to the mass inflation phenomenon.
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.
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.
Differential Geometry and Lie Groups for Physicists
NASA Astrophysics Data System (ADS)
Fecko, Marián.
2011-03-01
Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.
Differential Geometry and Lie Groups for Physicists
NASA Astrophysics Data System (ADS)
Fecko, Marián.
2006-10-01
Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.
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)
Algebraic integrability: a survey.
Vanhaecke, Pol
2008-03-28
We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems. PMID:17588863
Algebraic Semantics for Narrative
ERIC Educational Resources Information Center
Kahn, E.
1974-01-01
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
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.
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.
NASA Astrophysics Data System (ADS)
Xu, Chen; Kumavor, Patrick D.; Zhu, Quing
2012-02-01
Traditional Photoacoustic tomography provides the distribution of absorbed optical energy densities which are the products of the optical absorption coefficients and fluences. However, the absorption coefficient is the only functional parameter that is related to disease diagnosis, such as cancer. In this paper, we report the experimental investigation of an improved fitting procedure which can quantitatively characterize optical absorption coefficients of multiple targets. The original fitting procedure was proposed by us and used for a single target. The fitting procedure included a complete photoacoustic forward model, which incorporated an analytical model of light transport and a model of acoustic propagation. Using the target information from the PAT images and the background information from diffuse optical measurements (DOM), the fitting method minimizes the photoacoustic measurements and forward model data and recovers the target absorption coefficient quantitatively. The fitting errors in the absorption coefficients can reach 20% to 100% if the original fitting procedure is directly used on multiple targets. In our improved fitting method, the ratio between the photoacoustic intensities is introduced and served as extra input to the fitting procedure. As a result, the total number of unknown parameters is reduced and fitting accuracy is improved. The hybrid system used in the experiment combines a 64-channel photoacoustic system with a frequency-domain diffused optical system. The experiment was performed in the reflection geometry suitable for breast imaging. Phantom experiments include the combination of high contrast and low contrast targets with absorption coefficients ranging from 0.07 to 0.28 cm-1 and with different spatial separations. The phantoms were inserted into a chicken breast tissue. The fitting errors of multiple targets were reduced to less than 20% for both high and low contrast targets. These results illustrate the potential application
Geometry and the quantum: basics
NASA Astrophysics Data System (ADS)
Chamseddine, Ali H.; Connes, Alain; Mukhanov, Viatcheslav
2014-12-01
Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M 2(ℍ) and M 4(ℂ) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume > 4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.
Woodhead, H J; Kemp, A F; Blimkie CJR; Briody, J N; Duncan, C S; Thompson, M; Lam, A; Howman-Giles, R; Cowell, C T
2001-12-01
Although macroscopic geometric architecture is an important determinant of bone strength, there is limited published information relating to the validation of the techniques used in its measurement. This study describes new techniques for assessing geometry at the midfemur using magnetic resonance imaging (MRI) and dual-energy X-ray absorptiometry (DXA) and examines both the repeatability and the accuracy of these and previously described DXA methods. Contiguous transverse MRI (Philips 1.5T) scans of the middle one-third femur were made in 13 subjects, 3 subjects with osteoporosis. Midpoint values for total width (TW), cortical width (CW), total cross-sectional area (TCSA), cortical cross-sectional area (CCSA), and volumes from reconstructed three-dimensional (3D) images (total volume [TV] and cortical volume [CVol]) were derived. Midpoint TW and CW also were determined using DXA (Lunar V3.6, lumbar software) by visual and automated edge detection analysis. Repeatability was assessed on scans made on two occasions and then analyzed twice by two independent observers (blinded), with intra- and interobserver repeatability expressed as the CV (CV +/- SD). Accuracy was examined by comparing MRI and DXA measurements of venison bone (and Perspex phantom for MRI), against "gold standard" measures made by vernier caliper (width), photographic image digitization (area) and water displacement (volume). Agreement between methods was analyzed using mean differences (MD +/- SD%). MRI CVs ranged from 0.5 +/- 0.5% (TV) to 3.1 +/- 3.1% (CW) for intraobserver and 0.55 +/- 0.5% (TV) to 3.6 +/- 3.6% (CW) for interobserver repeatability. DXA results ranged from 1.6 +/- 1.5% (TW) to 4.4 +/- 4.5% (CW) for intraobserver and 3.8 +/- 3.8% (TW) to 8.3 +/- 8.1% (CW) for interobserver variation. MRI accuracy was excellent for TV (3.3 +/- 6.4%), CVol (3.5 +/- 4.0%), TCSA (1.8 +/- 2.6%), and CCSA (1.6 +/- 4.2%) but not TW (4.1 +/- 1.4%) or CW (16.4 +/14.9%). DXA results were TW (6.8 +/- 2
Polynomial Extensions of the Weyl C*-Algebra
NASA Astrophysics Data System (ADS)
Accardi, Luigi; Dhahri, Ameur
2015-09-01
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl C*-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.
Hou, Chang-Yu; Chamon, Claudio
2006-10-01
We study a tunneling geometry defined by a single point-contact constriction that brings to close vicinity two points sitting at the same edge of a quantum Hall liquid, shortening the trip between the otherwise spatially separated points along the normal chiral edge path. This wormhole-like geometry allows for entrapping bulk quasiparticles between the edge path and the tunnel junction, possibly realizing a topologically protected qubit if the quasiparticles have non-Abelian statistics. We show how either noise or simpler voltage measurements along the edge can probe the non-Abelian nature of the trapped quasiparticles. PMID:17155280
Spinors in Physics and Geometry
NASA Astrophysics Data System (ADS)
Trautman, A.; Furlan, G.
1988-11-01
The Table of Contents for the full book PDF is as follows: * Preface * Killing Spinors According to O. Hijazi and Applications * Self-Duality Conditions Satisfied by the Spin Connections on Spheres * Maslov Index and Half - Forms * Spin - 3/2 Fields on Black Hole Spacetimes * Indecomposable Conformal Spinors and Operator Product Expansions in a Massless QED Model * Nonlinear Spinor Representations * Nonlinear Wave Equations for Intrinsic Spinor Coordinates * Twistors - "Spinors" of SU(2,2), Their Generalizations and Achievements * Spinors, Reflections and Clifford Algebras: A Review * overline {SL}(n, R) Spinors for Particles, Gravity and Superstrings * Spinors on Compact Riemann Surfaces * Simple Spinors as Urfelder * Applications of Cartan Spinors to Differential Geometry in Higher Dimensions * Killing Spinors on Spheres and Projective Spaces * Spinor Structures on Homogeneous Riemannian Spaces * Classical Strings and Minimal Surfaces * Representing Spinors with Differential Forms * Inequalities for Spinors Norms in Clifford Algebras * The Importance of Spin * The Theory of World Spinors * Final List of Participants
The three-dimensional origin of the classifying algebra
NASA Astrophysics Data System (ADS)
Fuchs, Jürgen; Schweigert, Christoph; Stigner, Carl
2010-01-01
It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization of correlators of bulk fields on the disk. This allows us to derive explicit expressions for the structure constants of the classifying algebra as invariants of ribbon graphs in the three-manifold S×S. Our result unravels a precise relation between intertwiners of the action of the mapping class group on spaces of conformal blocks and boundary conditions in rational conformal field theories.
Quantized Nambu-Poisson manifolds and n-Lie algebras
DeBellis, Joshua; Saemann, Christian; Szabo, Richard J.
2010-12-15
We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms of foliations of R{sup n} by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.
Quantized Nambu-Poisson manifolds and n-Lie algebras
NASA Astrophysics Data System (ADS)
DeBellis, Joshua; Sämann, Christian; Szabo, Richard J.
2010-12-01
We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms of foliations of {{R}}^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.
High School Algebra Readiness Program: A Quasi-Experimental Study
ERIC Educational Resources Information Center
Birnbohm, Carol L.
2010-01-01
This quasi-experimental study measured the effectiveness of a locally created summer Algebra readiness program in a large suburban high school district in New Jersey. Incoming ninth grade students who were not ready for high school algebra were invited to participate in the summer program. The program was designed to provide access to more…
Putting Algebra Progress Monitoring into Practice: Insights from the Field
ERIC Educational Resources Information Center
Foegen, Anne; Morrison, Candee
2010-01-01
Algebra progress monitoring is a research-based practice that extends a long history of research in curriculum-based measurement (CBM). This article describes the theoretical foundations and research evidence for algebra progress monitoring, along with critical features of the practice. A detailed description of one practitioner's implementation…
NASA Astrophysics Data System (ADS)
Roitman, Michael
2008-08-01
In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
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…
Ulanovsky, A.V.; Minenko, V.F.; Korneev, S.V.
1997-01-01
An approach for evaluating the influence of measurement geometry on estimates of {sup 131}I in the thyroid from measurements with survey meters was developed using Monte Carlo simulation of radiation transport in the human body and the radiation detector. The modified Monte Carlo code, EGS4, including a newly developed mathematical model of detector, thyroid gland, and neck, was used for the computations. The approach was tested by comparing calculated and measured differential and integral detector characteristics. This procedure was applied to estimate uncertainties in direct thyroid-measurement results due to geometrical errors. 14 refs., 11 figs., 4 tabs.
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…
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.
NASA Technical Reports Server (NTRS)
Shahshahani, M.
1991-01-01
The performance characteristics are discussed of certain algebraic geometric codes. Algebraic geometric codes have good minimum distance properties. On many channels they outperform other comparable block codes; therefore, one would expect them eventually to replace some of the block codes used in communications systems. It is suggested that it is unlikely that they will become useful substitutes for the Reed-Solomon codes used by the Deep Space Network in the near future. However, they may be applicable to systems where the signal to noise ratio is sufficiently high so that block codes would be more suitable than convolutional or concatenated codes.
NASA Astrophysics Data System (ADS)
Bouwknegt, Peter
1988-06-01
We investigate extensions of the Virasoro algebra by a single primary field of integer or halfinteger conformal dimension Δ. We argue that for vanishing structure constant CΔΔΔ, the extended conformal algebra can only be associative for a generic c-value if Δ=1/2, 1, 3/2, 2 or 3. For the other Δ<=5 we compute the finite set of allowed c-values and identify the rational solutions. The case CΔΔΔ≠0 is also briefly discussed. I would like to thank Kareljan Schoutens for discussions and Sander Bais for a careful reading of the manuscript.
Teaching Arithmetic and Algebraic Expressions
ERIC Educational Resources Information Center
Subramaniam, K.; Banerjee, Rakhi
2004-01-01
A teaching intervention study was conducted with sixth grade students to explore the interconnections between students' growing understanding of arithmetic expressions and beginning algebra. Three groups of students were chosen, with two groups receiving instruction in arithmetic and algebra, and one group in algebra without arithmetic. Students…
Assessing Elementary Algebra with STACK
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2007-01-01
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…
Spinors in the hyperbolic algebra
NASA Astrophysics Data System (ADS)
Ulrych, S.
2006-01-01
The three-dimensional universal complex Clifford algebra Cbar3,0 is used to represent relativistic vectors in terms of paravectors. In analogy to the Hestenes spacetime approach spinors are introduced in an algebraic form. This removes the dependance on an explicit matrix representation of the algebra.
NASA Astrophysics Data System (ADS)
Chaudhuri, A.; Sekhar, M.; Descloitres, M.; Godderis, Y.; Ruiz, L.; Braun, J. J.
2013-11-01
Stochastic modelling is a useful way of simulating complex hard-rock aquifers as hydrological properties (permeability, porosity etc.) can be described using random variables with known statistics. However, very few studies have assessed the influence of topological uncertainty (i.e. the variability of thickness of conductive zones in the aquifer), probably because it is not easy to retrieve accurate statistics of the aquifer geometry, especially in hard rock context. In this paper, we assessed the potential of using geophysical surveys to describe the geometry of a hard rock-aquifer in a stochastic modelling framework. The study site was a small experimental watershed in South India, where the aquifer consisted of a clayey to loamy-sandy zone (regolith) underlain by a conductive fissured rock layer (protolith) and the unweathered gneiss (bedrock) at the bottom. The spatial variability of the thickness of the regolith and fissured layers was estimated by electrical resistivity tomography (ERT) profiles, which were performed along a few cross sections in the watershed. For stochastic analysis using Monte Carlo simulation, the generated random layer thickness was made conditional to the available data from the geophysics. In order to simulate steady state flow in the irregular domain with variable geometry, we used an isoparametric finite element method to discretize the flow equation over an unstructured grid with irregular hexahedral elements. The results indicated that the spatial variability of the layer thickness had a significant effect on reducing the simulated effective steady seepage flux and that using the conditional simulations reduced the uncertainty of the simulated seepage flux. As a conclusion, combining information on the aquifer geometry obtained from geophysical surveys with stochastic modelling is a promising methodology to improve the simulation of groundwater flow in complex hard-rock aquifers.
ERIC Educational Resources Information Center
Glick, David
1995-01-01
Presents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)
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.…
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…
ERIC Educational Resources Information Center
Boiteau, Denise; Stansfield, David
This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative Numbers"; and (5) "Using Positive…
Thinking Visually about Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2015-01-01
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
ERIC Educational Resources Information Center
Kennedy, John
This text provides information and exercises on arithmetic topics which should be mastered before a student enrolls in an Elementary Algebra course. Section I describes the fundamental properties and relationships of whole numbers, focusing on basic operations, divisibility tests, exponents, order of operations, prime numbers, greatest common…
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…
NASA Astrophysics Data System (ADS)
Cembranos, J. A. R.; Dobado, A.; Maroto, A. L.
Extra-dimensional theories contain additional degrees of freedom related to the geometry of the extra space which can be interpreted as new particles. Such theories allow to reformulate most of the fundamental problems of physics from a completely different point of view. In this essay, we concentrate on the brane fluctuations which are present in brane-worlds, and how such oscillations of the own space-time geometry along curved extra dimensions can help to resolve the Universe missing mass problem. The energy scales involved in these models are low compared to the Planck scale, and this means that some of the brane fluctuations distinctive signals could be detected in future colliders and in direct or indirect dark matter searches.
C∗-algebras of Penrose hyperbolic tilings
NASA Astrophysics Data System (ADS)
Oyono-Oyono, Hervé; Petite, Samuel
2011-02-01
Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the C∗-algebras and of the K-theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these C∗-algebras are traceless. Nevertheless, harmonic currents give rise to 3-cyclic cocycles and we discuss in this setting a higher-order version of the gap-labeling.
NASA Astrophysics Data System (ADS)
Ratkowski, Anthony J.; Anderson, Gail P.; Chetwynd, James H.; Nadile, Richard M.; Devir, Adam D.; Conley, T. D.
1998-01-01
Radiance measurements conducted during tropospheric operations to detect objects on the Earth's surface from a manned aircraft or from an unmanned airborne vehicle (UAV) will involve long, near-horizontal viewing geometries. The computer code MODTRAN is widely used for the prediction of the propagation of infrared radiation through the lower atmosphere. Consequently, we have undertaken to test the predictions of MODTRAN for the 3 - 5 and 8 - 12 micron spectral regions under mid-Eastern desert conditions.
NASA Astrophysics Data System (ADS)
Zhang, Ming; Yao, JingTao
2004-04-01
The XML is a new standard for data representation and exchange on the Internet. There are studies on XML query languages as well as XML algebras in literature. However, attention has not been paid to research on XML algebras for data mining due to partially the fact that there is no widely accepted definition of XML mining tasks. This paper tries to examine the XML mining tasks and provide guidelines to design XML algebras for data mining. Some summarization and comparison have been done to existing XML algebras. We argue that by adding additional operators for mining tasks, XML algebras may work well for data mining with XML documents.
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.
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.
Promoting Problem Solving across Geometry and Algebra by Using Technology
ERIC Educational Resources Information Center
Erbas, A. Kursat; Ledford, Sara D.; Orrill, Chandra Hawley; Polly, Drew
2005-01-01
Technology is a powerful tool in assisting students in problem solving by allowing for multiple representations. The vignette offered in this article provides insight into ways to solve open-ended problems using multiple technologies.
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…
Tafra, E; Culo, M; Basletić, M; Korin-Hamzić, B; Hamzić, A; Jacobsen, C S
2012-02-01
We have measured the Hall effect on recently synthesized single crystals of the quasi-one-dimensional organic conductor TTF-TCNQ (tetrathiafulvalene-tetracyanoquinodimethane), a well known charge transfer complex that has two kinds of conductive stacks: the donor (TTF) and the acceptor (TCNQ) chains. The measurements were performed in the temperature interval 30 K < T < 300 K and for several different magnetic field and current directions through the crystal. By applying the equivalent isotropic sample approach, we have demonstrated the importance of the choice of optimal geometry for accurate Hall effect measurements. Our results show, contrary to past belief, that the Hall coefficient does not depend on the geometry of measurements and that the Hall coefficient value is approximately zero in the high temperature region (T > 150 K), implying that there is no dominance of either the TTF or the TCNQ chain. At lower temperatures our measurements clearly prove that all three phase transitions of TTF-TCNQ could be identified from Hall effect measurements. PMID:22214728
NASA Astrophysics Data System (ADS)
Tafra, E.; Čulo, M.; Basletić, M.; Korin-Hamzić, B.; Hamzić, A.; Jacobsen, C. S.
2012-02-01
We have measured the Hall effect on recently synthesized single crystals of the quasi-one-dimensional organic conductor TTF-TCNQ (tetrathiafulvalene-tetracyanoquinodimethane), a well known charge transfer complex that has two kinds of conductive stacks: the donor (TTF) and the acceptor (TCNQ) chains. The measurements were performed in the temperature interval 30 K < T < 300 K and for several different magnetic field and current directions through the crystal. By applying the equivalent isotropic sample approach, we have demonstrated the importance of the choice of optimal geometry for accurate Hall effect measurements. Our results show, contrary to past belief, that the Hall coefficient does not depend on the geometry of measurements and that the Hall coefficient value is approximately zero in the high temperature region (T > 150 K), implying that there is no dominance of either the TTF or the TCNQ chain. At lower temperatures our measurements clearly prove that all three phase transitions of TTF-TCNQ could be identified from Hall effect measurements.
Pilot study on algebra learning among junior secondary students
NASA Astrophysics Data System (ADS)
Poon, Kin-Keung; Leung, Chi-Keung
2010-01-01
The purpose of the study reported herein was to identify the common mistakes made by junior secondary students in Hong Kong when learning algebra and to compare teachers' perceptions of students' ability with the results of an algebra test. An algebra test was developed and administered to a sample of students (aged between 13 and 14 years). From the responses of the participating students (N = 815), it was found that students in schools with a higher level of academic achievement had better algebra test results than did those in schools with a lower level of such achievement. Moreover, it was found that a teacher's perception of a student's ability has a correlation with that student's level of achievement. Based on this finding, an instrument that measures teaching effectiveness is discussed. Last but not least, typical errors in algebra are identified, and some ideas for an instructional design based on these findings are discussed.
Zhao, Qile; Wang, Guangxing; Liu, Zhizhao; Hu, Zhigang; Dai, Zhiqiang; Liu, Jingnan
2016-01-01
Using GNSS observable from some stations in the Asia-Pacific area, the carrier-to-noise ratio (CNR) and multipath combinations of BeiDou Navigation Satellite System (BDS), as well as their variations with time and/or elevation were investigated and compared with those of GPS and Galileo. Provided the same elevation, the CNR of B1 observables is the lowest among the three BDS frequencies, while B3 is the highest. The code multipath combinations of BDS inclined geosynchronous orbit (IGSO) and medium Earth orbit (MEO) satellites are remarkably correlated with elevation, and the systematic “V” shape trends could be eliminated through between-station-differencing or modeling correction. Daily periodicity was found in the geometry-free ionosphere-free (GFIF) combinations of both BDS geostationary Earth orbit (GEO) and IGSO satellites. The variation range of carrier phase GFIF combinations of GEO satellites is −2.0 to 2.0 cm. The periodicity of carrier phase GFIF combination could be significantly mitigated through between-station differencing. Carrier phase GFIF combinations of BDS GEO and IGSO satellites might also contain delays related to satellites. Cross-correlation suggests that the GFIF combinations’ time series of some GEO satellites might vary according to their relative geometries with the sun. PMID:26805831
Zhao, Qile; Wang, Guangxing; Liu, Zhizhao; Hu, Zhigang; Dai, Zhiqiang; Liu, Jingnan
2016-01-01
Using GNSS observable from some stations in the Asia-Pacific area, the carrier-to-noise ratio (CNR) and multipath combinations of BeiDou Navigation Satellite System (BDS), as well as their variations with time and/or elevation were investigated and compared with those of GPS and Galileo. Provided the same elevation, the CNR of B1 observables is the lowest among the three BDS frequencies, while B3 is the highest. The code multipath combinations of BDS inclined geosynchronous orbit (IGSO) and medium Earth orbit (MEO) satellites are remarkably correlated with elevation, and the systematic "V" shape trends could be eliminated through between-station-differencing or modeling correction. Daily periodicity was found in the geometry-free ionosphere-free (GFIF) combinations of both BDS geostationary Earth orbit (GEO) and IGSO satellites. The variation range of carrier phase GFIF combinations of GEO satellites is -2.0 to 2.0 cm. The periodicity of carrier phase GFIF combination could be significantly mitigated through between-station differencing. Carrier phase GFIF combinations of BDS GEO and IGSO satellites might also contain delays related to satellites. Cross-correlation suggests that the GFIF combinations' time series of some GEO satellites might vary according to their relative geometries with the sun. PMID:26805831
Queisser, Gillian; Wittmann, Malte; Bading, Hilmar; Wittum, Gabriel
2008-01-01
The cell nucleus is often considered a spherical structure. However, the visualization of proteins associated with the nuclear envelope in rat hippocampal neurons indicates that the geometry of nuclei is far more complex. The shape of cell nuclei is likely to influence the nucleo-cytoplasmic exchange of macromolecules and ions, in particular calcium, a key regulator of neuronal gene expression. We developed a tool to retrieve the 3-D view of cell nuclei from laser scanning confocal microscopy data. By applying an inertia-based filter, based on a special structure detection mechanism, the signal-to-noise ratio of the image is enhanced, the signal is smoothed, gaps in the membrane are closed, while at the same time the geometric properties, such as diameters of the membrane, are preserved. After segmentation of the image data, the microscopy data are sufficiently processed to extract surface information of the membrane by creating an isosurface with a marching tetrahedra algorithm combined with a modified Dijkstra graph-search algorithm. All methods are tested on artificial data, as well as on real data, which are recorded with a laser scanning confocal microscope. Significant advantages of the inertia-based filter can be observed when comparing it to other state of the art nonlinear diffusion filters. An additional program is written to calculate surface and volume of cell nuclei. These results represent the first step toward establishing a geometry-based model of the-dynamics of cytoplasmic and nuclear calcium. PMID:18315367
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.
NASA Astrophysics Data System (ADS)
Dankova, T. S.; Rosensteel, G.
1998-10-01
Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for an arbitrary Lie algebra L of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space L^* of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The mean field theories for the Elliott SU(3) and symplectic Sp(3,R) algebras are constructed explicitly in the coadjoint orbit framework.
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®,…
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.
Vertex Algebras, Kac-Moody Algebras, and the Monster
NASA Astrophysics Data System (ADS)
Borcherds, Richard E.
1986-05-01
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
NASA Astrophysics Data System (ADS)
Palmkvist, Jakob
2014-01-01
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D - 2 - p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
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.
Palmkvist, Jakob
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
NASA Technical Reports Server (NTRS)
Heffernan, Ruth M.; Gaubert, Michel
1986-01-01
A flight test program was conducted to obtain data from an upgraded Gazelle helicopter with an advanced geometry, three bladed rotor. Data were acquired on upper and lower surface chordwise blade pressure, blade bending and torsion moments, and fuselage structural loads. Results are presented from 16 individual flight conditions, including level flights ranging from 10 to 77 m/sec at 50 to 3000 m altitude, turning flights up to 2.0 g, and autorotation. Rotor aerodynamic data include information from 51 pressure transducers distributed chordwise at 75, 88, and 97% radial stations. Individual tranducer pressure coefficients and airfoil section lift and pitching moment coefficients are presented, as are steady state flight condition parameters and time dependence rotor loads. All dynamic data are presented as harmonic analysis coefficients.
Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras
NASA Astrophysics Data System (ADS)
Paseka, Jan; Riečanová, Zdenka
2010-12-01
We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.
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…
On the cohomology of Leibniz conformal algebras
NASA Astrophysics Data System (ADS)
Zhang, Jiao
2015-04-01
We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.
Assessing Algebraic Solving Ability: A Theoretical Framework
ERIC Educational Resources Information Center
Lian, Lim Hooi; Yew, Wun Thiam
2012-01-01
Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…
The Idea of Order at Geometry Class.
ERIC Educational Resources Information Center
Rishel, Thomas
The idea of order in geometry is explored using the experience of assignments given to undergraduates in a college geometry course "From Space to Geometry." Discussed are the definition of geometry, and earth measurement using architecture, art, and common experience. This discussion concludes with a consideration of the question of whether…
Teaching Activity-Based Taxicab Geometry
ERIC Educational Resources Information Center
Ada, Tuba
2013-01-01
This study aimed on the process of teaching taxicab geometry, a non-Euclidean geometry that is easy to understand and similar to Euclidean geometry with its axiomatic structure. In this regard, several teaching activities were designed such as measuring taxicab distance, defining a taxicab circle, finding a geometric locus in taxicab geometry, and…
ERIC Educational Resources Information Center
Actuarial Foundation, 2013
2013-01-01
"Setting the Stage with Geometry" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards that is designed to help students in grades 6-8 build and reinforce basic geometry skills for measuring 2D and 3D shapes. Developed by The Actuarial Foundation, this program seeks to provide skill-building math…
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.
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…
Higher level twisted Zhu algebras
Ekeren, Jethro van
2011-05-15
The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra V, graded by {Gamma}/Z for some subgroup {Gamma} of R containing Z, and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras Zhu{sub p,{Gamma}}(V), and we prove the following theorems: For each p, there is a bijection between the irreducible Zhu{sub p,{Gamma}}(V)-modules and the irreducible {Gamma}-twisted positive energy V-modules, and V is ({Gamma}, H)-rational if and only if all its Zhu algebras Zhu{sub p,{Gamma}}(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra Vir{sup c} and the universal affine Kac-Moody vertex algebra V{sup k}(g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.
ERIC Educational Resources Information Center
Polignano, Joy C.; Hojnoski, Robin L.
2012-01-01
There has been increased attention to the development of assessment measures for evaluating mathematical skills in young children in order to inform instruction and intervention. However, existing tools have focused primarily on number sense with little attention to other areas of mathematical thinking such as geometry and algebra. The purpose of…
NASA Astrophysics Data System (ADS)
Ning, T.; Elgered, G.; Johansson, J. M.
2011-01-01
We have used microwave absorbing material in different geometries around ground-based Global Navigation Satellite System (GNSS) antennas in order to mitigate multipath effects on the estimates of station coordinates and atmospheric water vapour. The influence of a hemispheric radome - of the same type as in the Swedish GPS network SWEPOS - was also investigated. Two GNSS stations at the Onsala Space Observatory were used forming a 12 m baseline. GPS data from October 2008 to November 2009 were analyzed by the GIPSY/OASIS II software using the Precise Point Positioning (PPP) processing strategy for five different elevation cutoff angles from 5° to 25°. We found that the use of the absorbing material decreases the offset in the estimated vertical component of the baseline from ˜27 mm to ˜4 mm when the elevation cutoff angle varies from 5° to 20°. The horizontal components are much less affected. The corresponding offset in the estimates of the atmospheric Integrated Water Vapour (IWV) decreases from ˜1.6 kg/m2 to ˜0.3 kg/m2. Changes less than 5 mm in the offsets in the vertical component of the baseline are seen for all five elevation cutoff angle solutions when the antenna was covered by a hemispheric radome. Using the radome affects the IWV estimates less than 0.4 kg/m2 for all different solutions. IWV comparisons between a Water Vapour Radiometer (WVR) and the GPS data give consistent results.
Balduzzi, Mathilde A.F.; Van der Zande, Dimitry; Stuckens, Jan; Verstraeten, Willem W.; Coppin, Pol
2011-01-01
Light Detection and Ranging (LiDAR) technology can be a valuable tool for describing and quantifying vegetation structure. However, because of their size, extraction of leaf geometries remains complicated. In this study, the intensity data produced by the Terrestrial Laser System (TLS) FARO LS880 is corrected for the distance effect and its relationship with the angle of incidence between the laser beam and the surface of the leaf of a Conference Pear tree (Pyrus Commmunis) is established. The results demonstrate that with only intensity, this relationship has a potential for determining the angle of incidence with the leaves surface with a precision of ±5° for an angle of incidence smaller than 60°, whereas it is more variable for an angle of incidence larger than 60°. It appears that TLS beam footprint, leaf curvatures and leaf wrinkles have an impact on the relationship between intensity and angle of incidence, though, this analysis shows that the intensity of scanned leaves has a potential to eliminate ghost points and to improve their meshing. PMID:22319374
Handheld Computer Algebra Systems in the Pre-Algebra Classroom
ERIC Educational Resources Information Center
Gantz, Linda Ann Galofaro
2010-01-01
This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…
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…
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…
Philip, Bobby; Chartier, Dr Timothy
2012-01-01
methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coecient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem.
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
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 jumpsmore » and an anisotropy in one part.« less
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.
NASA Astrophysics Data System (ADS)
Nakonieczna, Anna; Yeom, Dong-han
2016-02-01
There does not exist a notion of time which could be transferred straightforwardly from classical to quantum gravity. For this reason, a method of time quantification which would be appropriate for gravity quantization is being sought. One of the existing proposals is using the evolving matter as an intrinsic `clock' while investigating the dynamics of gravitational systems. The objective of our research was to check whether scalar fields can serve as time variables during a dynamical evolution of a coupled multicomponent matter-geometry system. We concentrated on a neutral case, which means that the elaborated system was not charged electrically nor magnetically. For this purpose, we investigated a gravitational collapse of a self-interacting complex and real scalar fields in the Brans-Dicke theory using the 2+2 spacetime foliation. We focused mainly on the region of high curvature appearing nearby the emerging singularity, which is essential from the perspective of quantum gravity. We investigated several formulations of the theory for various values of the Brans-Dicke coupling constant and the coupling between the Brans-Dicke field and the matter sector of the theory. The obtained results indicated that the evolving scalar fields can be treated as time variables in close proximity of the singularity due to the following reasons. The constancy hypersurfaces of the Brans-Dicke field are spacelike in the vicinity of the singularity apart from the case, in which the equation of motion of the field reduces to the wave equation due to a specific choice of free evolution parameters. The hypersurfaces of constant complex and real scalar fields are spacelike in the regions nearby the singularities formed during the examined process. The values of the field functions change monotonically in the areas, in which the constancy hypersurfaces are spacelike.
Piecewise Principal Coactions of Co-Commutative Hopf Algebras
NASA Astrophysics Data System (ADS)
Zieliński, Bartosz
2014-08-01
Principal comodule algebras can be thought of as objects representing principal bundles in non-commutative geometry. A crucial component of a principal comodule algebra is a strong connection map. For some applications it suffices to prove that such a map exists, but for others, such as computing the associated bundle projectors or Chern-Galois characters, an explicit formula for a strong connection is necessary. It has been known for some time how to construct a strong connection map on a multi-pullback comodule algebra from strong connections on multi-pullback components, but the known explicit general formula is unwieldy. In this paper we derive a much easier to use strong connection formula, which is not, however, completely general, but is applicable only in the case when a Hopf algebra is co-commutative. Because certain linear splittings of projections in multi-pullback comodule algebras play a crucial role in our construction, we also devote a significant part of the paper to the problem of existence and explicit formulas for such splittings. Finally, we show example application of our work.
Matrix De Rham Complex and Quantum A-infinity algebras
NASA Astrophysics Data System (ADS)
Barannikov, S.
2014-04-01
I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A ∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin-Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006-48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein-Leites simple associative algebras with odd trace q( N), and gl( N| N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin-Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.
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…
Algebraic Squares: Complete and Incomplete.
ERIC Educational Resources Information Center
Gardella, Francis J.
2000-01-01
Illustrates ways of using algebra tiles to give students a visual model of competing squares that appear in algebra as well as in higher mathematics. Such visual representations give substance to the symbolic manipulation and give students who do not learn symbolically a way of understanding the underlying concepts of completing the square. (KHR)
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…
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.)
Condensing Algebra for Technical Mathematics.
ERIC Educational Resources Information Center
Greenfield, Donald R.
Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-week…
Algebraic Thinking in Adult Education
ERIC Educational Resources Information Center
Manly, Myrna; Ginsburg, Lynda
2010-01-01
In adult education, algebraic thinking can be a sense-making tool that introduces coherence among mathematical concepts for those who previously have had trouble learning math. Further, a modeling approach to algebra connects mathematics and the real world, demonstrating the usefulness of math to those who have seen it as just an academic…
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.)
ERIC Educational Resources Information Center
Instructional Objectives Exchange, Los Angeles, CA.
A complete set of behavioral objectives for first-year algebra taught in any of grades 8 through 12 is presented. Three to six sample test items and answers are provided for each objective. Objectives were determined by surveying the most used secondary school algebra textbooks. Fourteen major categories are included: (1) whole numbers--operations…
Exploring Algebraic Patterns through Literature.
ERIC Educational Resources Information Center
Austin, Richard A.; Thompson, Denisse R.
1997-01-01
Presents methods for using literature to develop algebraic thinking in an environment that connects algebra to various situations. Activities are based on the book "Anno's Magic Seeds" with additional resources listed. Students express a constant function, exponential function, and a recursive function in their own words as well as writing about…
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…
Invariants of triangular Lie algebras
NASA Astrophysics Data System (ADS)
Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman
2007-07-01
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated.
NASA Astrophysics Data System (ADS)
Milathianaki, D.; Hawreliak, J.; McNaney, J. M.; El-Dasher, B. S.; Saculla, M. D.; Swift, D. C.; Lorenzana, H. E.; Ditmire, T.
2009-09-01
We report on a focusing x-ray diffraction geometry capable of high-resolution in situ lattice probing from dynamically loaded polycrystalline and amorphous materials. The Seeman-Bohlin-type camera presented here is ideally suited for time-resolved x-ray diffraction measurements performed on high energy multibeam laser platforms. Diffraction from several lattice planes of ablatively shock-loaded 25 μm thick Cu foils was recorded on a focusing circle of diameter D =100 mm with exceptional angular resolution limited only by the spectral broadening of the x-ray source. Excellent agreement was found between the density measured using x-ray diffraction and that inferred from Doppler velocimetry and the known shock Hugoniot of Cu. In addition, x-ray diffraction signal was captured from an amorphous material under static conditions.
Milathianaki, D; Hawreliak, J; McNaney, J M; El-Dasher, B S; Saculla, M D; Swift, D C; Lorenzana, H E; Ditmire, T
2009-09-01
We report on a focusing x-ray diffraction geometry capable of high-resolution in situ lattice probing from dynamically loaded polycrystalline and amorphous materials. The Seeman-Bohlin-type camera presented here is ideally suited for time-resolved x-ray diffraction measurements performed on high energy multibeam laser platforms. Diffraction from several lattice planes of ablatively shock-loaded 25 mum thick Cu foils was recorded on a focusing circle of diameter D=100 mm with exceptional angular resolution limited only by the spectral broadening of the x-ray source. Excellent agreement was found between the density measured using x-ray diffraction and that inferred from Doppler velocimetry and the known shock Hugoniot of Cu. In addition, x-ray diffraction signal was captured from an amorphous material under static conditions. PMID:19791950
Milathianaki, D.; Hawreliak, J.; McNaney, J. M.; El-Dasher, B. S.; Saculla, M. D.; Swift, D. C.; Lorenzana, H. E.; Ditmire, T.
2009-09-15
We report on a focusing x-ray diffraction geometry capable of high-resolution in situ lattice probing from dynamically loaded polycrystalline and amorphous materials. The Seeman-Bohlin-type camera presented here is ideally suited for time-resolved x-ray diffraction measurements performed on high energy multibeam laser platforms. Diffraction from several lattice planes of ablatively shock-loaded 25 {mu}m thick Cu foils was recorded on a focusing circle of diameter D=100 mm with exceptional angular resolution limited only by the spectral broadening of the x-ray source. Excellent agreement was found between the density measured using x-ray diffraction and that inferred from Doppler velocimetry and the known shock Hugoniot of Cu. In addition, x-ray diffraction signal was captured from an amorphous material under static conditions.
NASA Astrophysics Data System (ADS)
Oswald, Patrick; Poy, Guilhem
2015-12-01
Shape measurements after the coalescence of isotropic droplets embedded in a thin sample of a homeotropic nematic phase provides a tool to measure the nematic-isotropic surface tension. In addition, this experiment allows us to check the scaling laws recently given by Brun et al. [P.-T. Brun, M. Nagel, and F. Gallaire, Phys. Rev. E 88, 043009 (2013), 10.1103/PhysRevE.88.043009] to explain the relaxation of ellipsoidal droplets in a Hele-Shaw cell.
On Clifford-algebraic dimensional extension and SUSY holography
NASA Astrophysics Data System (ADS)
Gates, S. J.; Hübsch, T.; Stiffler, K.
2015-03-01
We analyze the group of maximal automorphisms of the N-extended worldline supersymmetry algebra, and its action on off-shell supermultiplets. This defines a concept of "holoraumy" that extends the notions of holonomy and curvature in a novel way and provides information about the geometry of the supermultiplet field-space. In turn, the "holoraumy" transformations of 0-brane dimensionally reduced supermultiplets provide information about Lorentz transformations in the higher-dimensional space-time from which the 0-brane supermultiplets are descended. Specifically, Spin(3) generators are encoded within 0-brane "holoraumy" tensors. Worldline supermultiplets are thus able to holographically encrypt information about higher-dimensional space-time geometry.
Near-horizon geometry and warped conformal symmetry
NASA Astrophysics Data System (ADS)
Afshar, Hamid; Detournay, Stéphane; Grumiller, Daniel; Oblak, Blagoje
2016-03-01
We provide boundary conditions for three-dimensional gravity including boosted Rindler spacetimes, representing the near-horizon geometry of non-extremal black holes or flat space cosmologies. These boundary conditions force us to make some unusual choices, like integrating the canonical boundary currents over retarded time and periodically identifying the latter. The asymptotic symmetry algebra turns out to be a Witt algebra plus a twisted u(1) current algebra with vanishing level, corresponding to a twisted warped CFT that is qualitatively different from the ones studied so far in the literature. We show that this symmetry algebra is related to BMS by a twisted Sugawara construction and exhibit relevant features of our theory, including matching micro- and macroscopic calculations of the entropy of zero-mode solutions. We confirm this match in a generalization to boosted Rindler-AdS. Finally, we show how Rindler entropy emerges in a suitable limit.
NASA Astrophysics Data System (ADS)
Gomes, Jay
A power measurement system has been designed for an ultra-high temperature inductively heated molten oxide electrolysis (MOE) reactor. The work presented in this research contributes to three different aspects of the induction heated MOE reactor facility: mathematical modeling of coil-to-workpiece power transfer, numerical modeling of heat transfer within the reactor, and experiments to measure the total hemispherical emittance of potential crucible materials. Facility-specific coupling coefficients for various samples have been experimentally determined for the MOE reactor facility. An analytical model coupling the predicted power input with heat transfer software was developed using COMSOL Multiphysics, and validated with experimental measurements of the steady state temperature gradient inside the reactor. These models were used to support the design of an experiment to measure the total hemispherical emissivity (epsilon) of conductive samples using a transient calorimetric technique. Results of epsilon are presented over a wide range of temperatures for copper, nickel, graphite and molybdenum. Furthermore, an investigation into optimizing the reactor system for heating will be discussed.
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)
Constraint algebra in bigravity
Soloviev, V. O.
2015-07-15
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
Geometric calculus: a new computational tool for Riemannian geometry
Moussiaux, A.; Tombal, P.
1988-05-01
We compare geometric calculus applied to Riemannian geometry with Cartan's exterior calculus method. The correspondence between the two methods is clearly established. The results obtained by a package written in an algebraic language and doing general manipulations on multivectors are compared. We see that the geometric calculus is as powerful as exterior calculus.
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.
Assessing Readiness for Geometry in Mathematically Talented Middle School Students.
ERIC Educational Resources Information Center
Mason, Marguerite M.; Moore, Sara Delano
1997-01-01
Based on research on the geometric understanding of regular and gifted middle school students, this article describes a procedure for assessing geometry readiness in mathematically able middle school students. The procedure utilizes the vanHiele model of stages of geometric understanding and distinguishes between readiness for algebra and…
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.
Cha, Y.S.; Hull, J.R.; Mulcahy, T.M.; Rossing, T.D. )
1991-11-15
A series of experiments measuring the levitation force between a permanent magnet (PM) and a high-temperature superconductor (HTS) and between pairs of PMs, coupled with finite-element calculations of the forces and fields, has identified factors that influence the levitation force. The self-demagnetizing factor within the HTS and, to some extent, within the PM has a profound effect on magnetic pressure. For large HTSs with strong flux-pinning, the demagnetizing effect of the diamagnetic image of the PM is substantial. For short distances between the HTS and PM, compression of magnetic flux produces a dependence on PM diameter.
Cha, Y.S.; Hull, J.R.; Mulcahy, T.M.; Rossing, T.D. Northern Illinois Univ., De Kalb, IL . Dept. of Physics)
1991-01-01
A series of experiments measuring the levitation force between a permanent magnet (PM) and a high temperature superconductor (HTS) and between pairs of PMs, coupled with finite-element analysis of the experiments, has identified factors that influence the levitation force. The self demagnetizing factor within the HTS and, to some extent, within the PM has a profound effect on magnetic pressure. For large HTSs with strong flux-pinning, the demagnetizing effect of the diamagnetic image of the PM is substantial. For short distances between the HTS and PM, compression of magnetic flux produces a dependence on PM diameter. 8 refs.
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.
Using Homemade Algebra Tiles To Develop Algebra and Prealgebra Concepts.
ERIC Educational Resources Information Center
Leitze, Annette Ricks; Kitt, Nancy A.
2000-01-01
Describes how to use homemade tiles, sketches, and the box method to reach a broader group of students for successful algebra learning. Provides a list of concepts appropriate for such an approach. (KHR)
Loop Virasoro Lie conformal algebra
Wu, Henan Chen, Qiufan; Yue, Xiaoqing
2014-01-15
The Lie conformal algebra of loop Virasoro algebra, denoted by CW, is introduced in this paper. Explicitly, CW is a Lie conformal algebra with C[∂]-basis (L{sub i} | i∈Z) and λ-brackets [L{sub i} {sub λ} L{sub j}] = (−∂−2λ)L{sub i+j}. Then conformal derivations of CW are determined. Finally, rank one conformal modules and Z-graded free intermediate series modules over CW are classified.
NASA Astrophysics Data System (ADS)
Guittienne, Ph; Jacquier, R.; Howling, A. A.; Furno, I.
2015-12-01
Measurements and analysis of a radio-frequency planar antenna are presented for applications in inductively-coupled plasma processing. The network of inductive and capacitive elements exhibits high currents under resonance which are efficient for plasma generation. Mode frequencies and impedances are accurately calculated by accounting for the mutual partial inductances using the impedance matrix. The effect of plasma inductive coupling on mode frequency shift and mode impedance is estimated using the complex image method, giving good agreement with experiment. It is proposed that the complex image method combined with the partial inductance concept (see the accompanying paper, Part I (Howling et al 2015 Plasma Sources Sci. Technol. 24 065014)) offers a general way to calculate the impedance characteristics of inductively-coupled plasma sources in planar geometry.
Fewster, P.F.; Andrew, N.L.; Holy, V.; Barmak, K.
2005-11-01
A method to measure the crystallite size and its distribution as a function of depth in multilayer thin films is described. The principle relies on the idea that when x-rays are scattered at an interface the incident and scattered waves create a standing wave whose periodicity can be varied and thereby enhance the scattering at certain depths. Practical examples of this method are given for Nb/Al periodic multilayers, one of which indicates considerable macrostrain for the surface layer and a variation in microstrain as a function of depth. The theoretical modeling of the scattering process is presented, which includes the influence of the general density modulation and interfacial roughness. Both these contributions are shown to be necessary to account for the experimental scattering profiles.
Kanick, S. C.; Krishnaswamy, V.; Gamm, U. A.; Sterenborg, H. J. C. M.; Robinson, D. J.; Amelink, A.; Pogue, B. W.
2012-01-01
Reflectance spectra measured in Intralipid (IL) close to the source are sensitive to wavelength-dependent changes in reduced scattering coefficient (μ′s) and scattering phase function (PF). Experiments and simulations were performed using device designs with either single or separate optical fibers for delivery and collection of light in varying concentrations of IL. Spectral reflectance is not consistently linear with varying IL concentration, with PF-dependent effects observed for single fiber devices with diameters smaller than ten transport lengths and for separate source-detector devices that collected light at less than half of a transport length from the source. Similar effects are thought to be seen in tissue, limiting the ability to quantitatively compare spectra from different devices without compensation. PMID:22567598
Taylor, James E.; Massey, Richard J.; Leauthaud, Alexie; Tanaka, Masayuki; George, Matthew R.; Rhodes, Jason; Ellis, Richard; Scoville, Nick; Kitching, Thomas D.; Capak, Peter; Finoguenov, Alexis; Ilbert, Olivier; Kneib, Jean-Paul; Jullo, Eric; Koekemoer, Anton M.
2012-04-20
Gravitational lensing can provide pure geometric tests of the structure of spacetime, for instance by determining empirically the angular diameter distance-redshift relation. This geometric test has been demonstrated several times using massive clusters which produce a large lensing signal. In this case, matter at a single redshift dominates the lensing signal, so the analysis is straightforward. It is less clear how weaker signals from multiple sources at different redshifts can be stacked to demonstrate the geometric dependence. We introduce a simple measure of relative shear which for flat cosmologies separates the effect of lens and source positions into multiplicative terms, allowing signals from many different source-lens pairs to be combined. Applying this technique to a sample of groups and low-mass clusters in the COSMOS survey, we detect a clear variation of shear with distance behind the lens. This represents the first detection of the geometric effect using weak lensing by multiple, low-mass groups. The variation of distance with redshift is measured with sufficient precision to constrain the equation of state of the universe under the assumption of flatness, equivalent to a detection of a dark energy component {Omega}{sub X} at greater than 99% confidence for an equation-of-state parameter -2.5 {<=} w {<=} -0.1. For the case w = -1, we find a value for the cosmological constant density parameter {Omega}{sub {Lambda}} = 0.85{sup +0.044}{sub -}0{sub .19} (68% CL) and detect cosmic acceleration (q{sub 0} < 0) at the 98% CL. We consider the systematic uncertainties associated with this technique and discuss the prospects for applying it in forthcoming weak-lensing surveys.
Hopf algebras and Dyson-Schwinger equations
NASA Astrophysics Data System (ADS)
Weinzierl, Stefan
2016-06-01
In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
Students' misconceptions and errors in transformation geometry
NASA Astrophysics Data System (ADS)
Ada, Tuba; Kurtuluş, Aytaç
2010-10-01
This study analyses the students' performances in two-dimensional transformation geometry and explores the mistakes made by the students taking the analytic geometry course given by researchers. An examination was given to students of Education Faculties who have taken the analytic geometry course at Eskisehir Osmangazi University in Turkey. The subject of this study included 126 third-year students in the Department of Mathematics Education. Data were collected from a seven questions exam. This exam consisted of three procedural questions, two conceptual questions and two procedural-conceptual questions. In data analysis, a descriptor code key was used. When the students' overall performances were considered for all seven questions, the results showed that they did not understand how to apply rotation transformation. The mostly observed mistakes showed that the students seemed to know the algebraic meaning of translation and also rotation but they did not seem to understand the geometric meaning of them.
Berzins, A; Shah, B; Weinans, H; Sumner, D R
1997-03-01
Push-out and pull-out tests are used for destructive evaluation of implant-bone interface strength. Because nondestructive mechanical tests would allow maintenance of an intact interface for subsequent morphological study, we developed such a test to determine the shear modulus of the interface by measuring the shear deformation of a thin layer adjacent to the implant. A polyurethane foam model was used to test the experimental setup on a group of nine cylindrical implants with three different lengths (15-48 mm) and three different diameters (5-9.7 mm). The shear modulus of the interface, as calculated from the pull-out test, was validated against the shear modulus of the foam derived from tensile tests. The two values of shear modulus were well correlated (R2 = 0.8, p < 0.001), thus encouraging further application of the setup for tests of implant-bone interface mechanics. In addition, we also examined the effects of implant length and diameter. The length of the implants had a significant influence on the interface shear modulus (p < 0.05), indicating that comparisons of the variable should only be made of implants with the same length. The length and diameter of the implants were not critical parameters for the ultimate fixation strength. PMID:9086403
NASA Astrophysics Data System (ADS)
Bassel, Léna; Tauzin, Xavier; Queffelec, Alain; Ferrier, Catherine; Lacanette, Delphine; Chapoulie, Rémy; Bousquet, Bruno
2016-04-01
The diameter of an X-ray beam was determined, using the knife-edge technique, widely applied for beam profiling, by taking advantage of the fluorescence emission generated by the X-ray beam. The knife-edge has to be appropriate to the configuration of the device, in our case a double-material target made of plastic and cardboard was scanned in a transversal plane compared to the beam propagation direction. Along the scanning axis, for each position, the intensity of the Kα line of chlorine was recorded. The first derivative of the intensity evolution as a function of the edge position, fitted by a Gaussian function, makes it possible to obtain the beam diameter along the scan direction. We measured a slightly elliptic diameter close to 3 mm. In this note we underline the significance of the knife-edge technique which represents a useful tool, easy to be set up, to control X-ray beam dimensions in portable devices often routinely used by non-specialists.
Maximizing algebraic connectivity in air transportation networks
NASA Astrophysics Data System (ADS)
Wei, Peng
In air transportation networks the robustness of a network regarding node and link failures is a key factor for its design. An experiment based on the real air transportation network is performed to show that the algebraic connectivity is a good measure for network robustness. Three optimization problems of algebraic connectivity maximization are then formulated in order to find the most robust network design under different constraints. The algebraic connectivity maximization problem with flight routes addition or deletion is first formulated. Three methods to optimize and analyze the network algebraic connectivity are proposed. The Modified Greedy Perturbation Algorithm (MGP) provides a sub-optimal solution in a fast iterative manner. The Weighted Tabu Search (WTS) is designed to offer a near optimal solution with longer running time. The relaxed semi-definite programming (SDP) is used to set a performance upper bound and three rounding techniques are discussed to find the feasible solution. The simulation results present the trade-off among the three methods. The case study on two air transportation networks of Virgin America and Southwest Airlines show that the developed methods can be applied in real world large scale networks. The algebraic connectivity maximization problem is extended by adding the leg number constraint, which considers the traveler's tolerance for the total connecting stops. The Binary Semi-Definite Programming (BSDP) with cutting plane method provides the optimal solution. The tabu search and 2-opt search heuristics can find the optimal solution in small scale networks and the near optimal solution in large scale networks. The third algebraic connectivity maximization problem with operating cost constraint is formulated. When the total operating cost budget is given, the number of the edges to be added is not fixed. Each edge weight needs to be calculated instead of being pre-determined. It is illustrated that the edge addition and the
Curvature calculations with spacetime algebra
Hestenes, D.
1986-06-01
A new method for calculating the curvature tensor is developed and applied to the Scharzschild case. The method employs Clifford algebra and has definite advantages over conventional methods using differential forms or tensor analysis.
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)
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.
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.
Semiclassical states on Lie algebras
Tsobanjan, Artur
2015-03-15
The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.
NASA Astrophysics Data System (ADS)
Lannes, A.; Teunissen, P. J. G.
2011-05-01
The first objective of this paper is to show that some basic concepts used in global navigation satellite systems (GNSS) are similar to those introduced in Fourier synthesis for handling some phase calibration problems. In experimental astronomy, the latter are at the heart of what is called `phase closure imaging.' In both cases, the analysis of the related structures appeals to the algebraic graph theory and the algebraic number theory. For example, the estimable functions of carrier-phase ambiguities, which were introduced in GNSS to correct some rank defects of the undifferenced equations, prove to be `closure-phase ambiguities:' the so-called `closure-delay' (CD) ambiguities. The notion of closure delay thus generalizes that of double difference (DD). The other estimable functional variables involved in the phase and code undifferenced equations are the receiver and satellite pseudo-clock biases. A related application, which corresponds to the second objective of this paper, concerns the definition of the clock information to be broadcasted to the network users for their precise point positioning (PPP). It is shown that this positioning can be achieved by simply having access to the satellite pseudo-clock biases. For simplicity, the study is restricted to relatively small networks. Concerning the phase for example, these biases then include five components: a frequency-dependent satellite-clock error, a tropospheric satellite delay, an ionospheric satellite delay, an initial satellite phase, and an integer satellite ambiguity. The form of the PPP equations to be solved by the network user is then similar to that of the traditional PPP equations. As soon as the CD ambiguities are fixed and validated, an operation which can be performed in real time via appropriate decorrelation techniques, estimates of these float biases can be immediately obtained. No other ambiguity is to be fixed. The satellite pseudo-clock biases can thus be obtained in real time. This is
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).
2005-04-11
The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.
NASA Astrophysics Data System (ADS)
Bircher, Chad; Shao, Yiping
2012-11-01
Depth of Interaction (DOI) information can improve quality of reconstructed images acquired from Positron Emission Tomography (PET), especially in high resolution and compact scanners dedicated for breast, brain, or small animal imaging applications. Additionally, clinical scanners with time of flight capability can also benefit from DOI information. One of the most promising methods of determining DOI in a crystal involves reading the signal from two ends of a scintillation crystal, and calculating the signal ratio between the two detectors. This method is known to deliver a better DOI resolution with rough crystals compared to highly polished crystals. However, what is still not well studied is how much of a tradeoff is involved between spatial, energy, temporal, and DOI resolutions as a function of the crystal surface treatment and geometry with the use of Silicon Photomultipliers (SiPM) as the photo detectors. This study investigates the effects of different crystal surface finishes and geometries on energy, timing and DOI resolutions at different crystal depths. The results show that for LYSO scintillators of 1.5×1.5×20 mm3 and 2×2×20 mm3 with their surfaces finished from 0.5 to 30 μm roughness, almost the same energy and coincidence timing resolutions were maintained, around 15% and 2.4 ns, respectively across different crystal depths, while the DOI resolutions were steadily improved from worse than 5 mm to better than 2 mm. They demonstrate that crystal roughness, with proper surface preparing, does not have a significant effect on the energy and coincidence timing resolutions in the crystals examined, and there does not appear to be a tradeoff between improving DOI resolution and degrading other detector performances. These results will be valuable to guide the selection of crystal surface conditions for developing a DOI measurable PET detector with a full array of LYSO scintillators coupled to SiPM arrays.
Bircher, Chad
2012-01-01
Depth of Interaction (DOI) information can improve quality of reconstructed images acquired from Positron Emission Tomography (PET), especially in high resolution and compact scanners dedicated for breast, brain, or small animal imaging applications. Additionally, clinical scanners with time of flight capability can also benefit from DOI information. One of the most promising methods of determining DOI in a crystal involves reading the signal from two ends of a scintillation crystal, and calculating the signal ratio between the two detectors. This method is known to deliver a better DOI resolution with rough crystals compared to highly polished crystals. However, what is still not well studied is how much of a tradeoff is involved between spatial, energy, temporal, and DOI resolutions as a function of the crystal surface treatment and geometry with the use of Silicon Photomultipliers (SiPM) as the photo detectors. This study investigates the effects of different crystal surface finishes and geometries on energy, timing and DOI resolutions at different crystal depths. The results show that for LYSO scintillators of 1.5×1.5×20 mm3 and 2×2×20 mm3 with their surfaces finished from 0.5 to 30 micron roughness, almost the same energy and coincidence timing resolutions were maintained, around 15% and 2.4 ns respectively across different crystal depths, while the DOI resolutions were steadily improved from worse than 5 mm to better than 2 mm. They demonstrate that crystal roughness, with proper surface preparing, does not have a significant effect on the energy and coincidence timing resolutions in the crystals examined, and there does not appear to be a tradeoff between improving DOI resolution and degrading other detector performances. These results will be valuable to guide the selection of crystal surface conditions for developing a DOI measurable PET detector with a full array of LYSO scintillators coupled to SiPM arrays. PMID:23087497
Heat transfer predictions for two turbine nozzle geometries at high Reynolds and Mach numbers
NASA Technical Reports Server (NTRS)
Boyle, R. J.; Jackson, R.
1995-01-01
Predictions of turbine vane and endwall heat transfer and pressure distributions are compared with experimental measurements for two vane geometries. The differences in geometries were due to differences in the hub profile, and both geometries were derived from the design of a high rim speed turbine (HRST). The experiments were conducted in the Isentropic Light Piston Facility (ILPF) at Pyestock at a Reynolds number of 5.3 x 10(exp 6), a Mach number of 1.2, and a wall-to-gas temperature ratio of 0.66. Predictions are given for two different steady-state three-dimensional Navier-Stokes computational analyses. C-type meshes were used, and algebraic models were employed to calculate the turbulent eddy viscosity. The effects of different turbulence modeling assumptions on the predicted results are examined. Comparisons are also given between predicted and measured total pressure distributions behind the vane. The combination of realistic engine geometries and flow conditions proved to be quite demanding in terms of the convergence of the CFD solutions. An appropriate method of grid generation, which resulted in consistently converged CFD solutions, was identified.
Heat transfer predictions for two turbine nozzle geometries at high Reynolds and Mach numbers
NASA Astrophysics Data System (ADS)
Boyle, R. J.; Jackson, R.
1995-09-01
Predictions of turbine vane and endwall heat transfer and pressure distributions are compared with experimental measurements for two vane geometries. The differences in geometries were due to differences in the hub profile, and both geometries were derived from the design of a high rim speed turbine (HRST). The experiments were conducted in the Isentropic Light Piston Facility (ILPF) at Pyestock at a Reynolds number of 5.3 x 10(exp 6), a Mach number of 1.2, and a wall-to-gas temperature ratio of 0.66. Predictions are given for two different steady-state three-dimensional Navier-Stokes computational analyses. C-type meshes were used, and algebraic models were employed to calculate the turbulent eddy viscosity. The effects of different turbulence modeling assumptions on the predicted results are examined. Comparisons are also given between predicted and measured total pressure distributions behind the vane. The combination of realistic engine geometries and flow conditions proved to be quite demanding in terms of the convergence of the CFD solutions. An appropriate method of grid generation, which resulted in consistently converged CFD solutions, was identified.
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.
NASA Astrophysics Data System (ADS)
Kuzmin, Dmitri; Möller, Matthias; Gurris, Marcel
Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCT-like algebraic flux correction schemes for the Euler equations of gas dynamics. In particular, we discuss the construction of artificial viscosity operators, the choice of variables to be limited, and the transformation of antidiffusive fluxes. An a posteriori control mechanism is implemented to make the limiter failsafe. The numerical treatment of initial and boundary conditions is discussed in some detail. The initialization is performed using an FCT-constrained L 2 projection. The characteristic boundary conditions are imposed in a weak sense, and an approximate Riemann solver is used to evaluate the fluxes on the boundary. We also present an unconditionally stable semi-implicit time-stepping scheme and an iterative solver for the fully discrete problem. The results of a numerical study indicate that the nonlinearity and non-differentiability of the flux limiter do not inhibit steady state convergence even in the case of strongly varying Mach numbers. Moreover, the convergence rates improve as the pseudo-time step is increased.
Analytical solution using computer algebra of a biosensor for detecting toxic substances in water
NASA Astrophysics Data System (ADS)
Rúa Taborda, María. Isabel
2014-05-01
In a relatively recent paper an electrochemical biosensor for water toxicity detection based on a bio-chip as a whole cell was proposed and numerically solved and analyzed. In such paper the kinetic processes in a miniaturized electrochemical biosensor system was described using the equations for specific enzymatic reaction and the diffusion equation. The numerical solution shown excellent agreement with the measured data but such numerical solution is not enough to design efficiently the corresponding bio-chip. For this reason an analytical solution is demanded. The object of the present work is to provide such analytical solution and then to give algebraic guides to design the bio-sensor. The analytical solution is obtained using computer algebra software, specifically Maple. The method of solution is the Laplace transform, with Bromwich integral and residue theorem. The final solution is given as a series of Bessel functions and the effective time for the bio-sensor is computed. It is claimed that the analytical solutions that were obtained will be very useful to predict further current variations in similar systems with different geometries, materials and biological components. Beside of this the analytical solution that we provide is very useful to investigate the relationship between different chamber parameters such as cell radius and height; and electrode radius.
Quanta of Geometry: Noncommutative Aspects
NASA Astrophysics Data System (ADS)
Chamseddine, Ali H.; Connes, Alain; Mukhanov, Viatcheslav
2015-03-01
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 M2(H ) and M4(C ) are obtained, which are the exact constituents of the standard model. Using the two maps from M4 to S4 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.
Quanta of geometry: noncommutative aspects.
Chamseddine, Ali H; Connes, Alain; Mukhanov, Viatcheslav
2015-03-01
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. PMID:25793795
Nonnumeric Computer Applications to Algebra, Trigonometry and Calculus.
ERIC Educational Resources Information Center
Stoutemyer, David R.
1983-01-01
Described are computer program packages requiring little or no knowledge of computer programing for college algebra, calculus, and abstract algebra. Widely available computer algebra systems are listed. (MNS)
Virasoro algebra in the KN algebra; Bosonic string with fermionic ghosts on Riemann surfaces
Koibuchi, H. )
1991-10-10
In this paper the bosonic string model with fermionic ghosts is considered in the framework of the KN algebra. The authors' attentions are paid to representations of KN algebra and a Clifford algebra of the ghosts. The authors show that a Virasoro-like algebra is obtained from KN algebra when KN algebra has certain antilinear anti-involution, and that it is isomorphic to the usual Virasoro algebra. The authors show that there is an expected relation between a central charge of this Virasoro-like algebra and an anomaly of the combined system.
Invertible linear transformations and the Lie algebras
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Tam, Honwah; Guo, Fukui
2008-07-01
With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.
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…
ERIC Educational Resources Information Center
Kuntz, Gilles
The first section of this paper on World Wide Web applications related to dynamic geometry addresses dynamic geometry and teaching, including the relationship between dynamic geometry and direct manipulation, key features of dynamic geometry environments, the importance of direct engagement of the learner using construction software for…
Erlangen Program at Large-1: Geometry of Invariants
NASA Astrophysics Data System (ADS)
Kisil, Vladimir V.
2010-09-01
This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of SL2(R) group. We describe here geometries of corresponding domains. The principal rôle is played by Clifford algebras of matching types. In this paper we also generalise the Fillmore-Springer-Cnops construction which describes cycles as points in the extended space. This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach.
Noncommutative spectral geometry, Bogoliubov transformations and neutrino oscillations
NASA Astrophysics Data System (ADS)
Vittoria Gargiulo, Maria; Sakellariadou, Mairi; Vitiello, Giuseppe
2015-07-01
In this report we show that neutrino mixing is intrinsically contained in Connes’ noncommutatives pectral geometry construction, thanks to the introduction of the doubling of algebra, which is connected to the Bogoliubov transformation. It is known indeed that these transformations are responsible for the mixing, turning the mass vacuum state into the flavor vacuum state, in such a way that mass and flavor vacuum states are not unitary equivalent. There is thus a red thread that binds the doubling of algebra of Connes’ model to the neutrino mixing.
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.
Ternary generalization of Heisenberg's algebra
NASA Astrophysics Data System (ADS)
Kerner, Richard
2015-06-01
A concise study of ternary and cubic algebras with Z3 grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, S3, and its abelian subgroup Z3 may play an important role in quantum physics. We show then how most of important algebras with Z2 grading can be generalized with ternary composition laws combined with a Z3 grading. We investigate in particular a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, , one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra. An analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator, and some properties of its eigenfunctions are briefly discussed.
Beyond Dirac - a Unified Algebra
NASA Astrophysics Data System (ADS)
Lundberg, Wayne R.
2001-10-01
A introductory insight will be shared regarding a 'separation of variables' approach to understanding the relationship between QCD and the origins of cosmological and particle mass. The discussion will then build upon work presented at DFP 2000, focussing on the formal basis for using 3x3x3 matrix algebra as it underlies and extends Dirac notation. A set of restrictions are established which break the multiple symmetries of the 3x3x3 matrix algebra, yielding Standard Model QCD objects and interactions. It will be shown that the 3x3x3 matrix representation unifies the algebra of strong and weak (and by extension, electromagnetic) interactions. A direct correspondence to string theoretic objects is established by considering the string to be partitioned in thirds. Rubik's cube is used as a graphical means of handling algebraic manipulation of 3x3x3 algebra. Further, its potential utility for advancing pedagogical methods through active engagement is discussed. A simulated classroom exercize will be conducted.
Causal structure and algebraic classification of non-dissipative linear optical media
Schuller, Frederic P.; Witte, Christof; Wohlfarth, Mattias N.R.
2010-09-15
In crystal optics and quantum electrodynamics in gravitational vacua, the propagation of light is not described by a metric, but an area metric geometry. In this article, this prompts us to study conditions for linear electrodynamics on area metric manifolds to be well-posed. This includes an identification of the timelike future cones and their duals associated to an area metric geometry, and thus paves the ground for a discussion of the related local and global causal structures in standard fashion. In order to provide simple algebraic criteria for an area metric manifold to present a consistent spacetime structure, we develop a complete algebraic classification of area metric tensors up to general transformations of frame. This classification, valuable in its own right, is then employed to prove a theorem excluding the majority of algebraic classes of area metrics as viable spacetimes. Physically, these results classify and drastically restrict the viable constitutive tensors of non-dissipative linear optical media.
Non-simply laced Lie algebras via F theory strings
NASA Astrophysics Data System (ADS)
Bonora, L.; Savelli, R.
2010-11-01
In order to describe the appearance in F theory of the non-simply-laced Lie algebras, we use the representation of symmetry enhancements by means of string junctions. After an introduction to the techniques used to describe symmetry enhancement, that is algebraic geometry, BPS states analysis and string junctions, we concentrate on the latter. We give an explicit description of the folding of D 2n to B n , of the folding of E 6 to F 4 and that of D 4 to G 2 in terms of junctions and Jordan strings. We also discuss the case of C n , but we are unable in this case to provide a string interpretation.
Quantum groups: Geometry and applications
Chu, C.S.
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.
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…
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…
Working Memory and Literacy as Predictors of Performance on Algebraic Word Problems
ERIC Educational Resources Information Center
Lee, Kerry; Ng, Swee-Fong; Ng, Ee-Lynn; Lim, Zee-Ying
2004-01-01
Previous studies on individual differences in mathematical abilities have shown that working memory contributes to early arithmetic performance. In this study, we extended the investigation to algebraic word problem solving. A total of 151 10-year-olds were administered algebraic word problems and measures of working memory, intelligence quotient…
Is Arithmetic Really Necessary for Algebra? A Case for an Integrated Curriculum.
ERIC Educational Resources Information Center
Palow, William P.
By measuring the performance of 62 students enrolled in a community college introductory algebra course, this study challenges the generally accepted assumption among mathematics instructors that mastery of arithmetic is necessary for the learning of algebra. Study subjects were 35% male, 74% Hispanic, 16% Black, 8% white, and 2% other. A pretest,…
ERIC Educational Resources Information Center
Panasuk, Regina M.
2010-01-01
Algebra students may often demonstrate a certain degree of proficiency when manipulating algebraic expressions and verbalizing their behaviors. Do these abilities imply conceptual understanding? What is a reliable indicator that would provide educators with a relatively trustworthy and consistent measure to identify whether students learn…
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-04-01
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.
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.
T-duality and exceptional generalized geometry through symmetries of dg-manifolds
NASA Astrophysics Data System (ADS)
Lupercio, Ernesto; Rengifo, Camilo; Uribe, Bernardo
2014-09-01
We study dg-manifolds which are R[2]-bundles over R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of T-dual dg-manifolds. Within this framework, we construct the T-duality map as a degree -1 map between the cohomologies of the T-dual dg-manifolds and we show an explicit isomorphism between the differential graded algebra of the symmetries of the T-dual dg-manifolds. We, furthermore, show how the algebraic structure underlying Bn generalized geometry could be recovered as derived dg-Leibniz algebra of the fixed points of the T-dual automorphism acting on the symmetries of a self T-dual dg-manifold, and we show how other types of algebraic structures underlying exceptional generalized geometry could be obtained as derived symmetries of certain dg-manifolds.
NASA Astrophysics Data System (ADS)
Castro, Carlos
2006-11-01
A novel Weyl-Heisenberg algebra in Clifford spaces is constructed that is based on a matrix-valued {\\cal H}^{AB} extension of Planck's constant. As a result of this modified Weyl-Heisenberg algebra one will no longer be able to measure, simultaneously, the pairs of variables (x, px), (x, py), (x, pz), (y, px), ... with absolute precision. New Klein-Gordon and Dirac wave equations and dispersion relations in Clifford spaces are presented. The latter Dirac equation is a generalization of the Dirac-Lanczos-Barut-Hestenes equation. We display the explicit isomorphism between Yang's noncommutative spacetime algebra and the area-coordinates algebra associated with Clifford spaces. The former Yang's algebra involves noncommuting coordinates and momenta with a minimum Planck scale λ (ultraviolet cutoff) and a minimum momentum p = planck/R (maximal length R, infrared cutoff). The double-scaling limit of Yang's algebra λ → 0, R → ∞, in conjunction with the large n → ∞ limit, leads naturally to the area quantization condition λR = L2 = nλ2 (in Planck area units) given in terms of the discrete angular-momentum eigenvalues n. It is shown how modified Newtonian dynamics is also a consequence of Yang's algebra resulting from the modified Poisson brackets. Finally, another noncommutative algebra which differs from Yang's algebra and related to the minimal length uncertainty relations is presented. We conclude with a discussion of the implications of noncommutative QM and QFT's in Clifford spaces.
Generalized Galilean algebras and Newtonian gravity
NASA Astrophysics Data System (ADS)
González, N.; Rubio, G.; Salgado, P.; Salgado, S.
2016-04-01
The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.
Computer Algebra Systems in Undergraduate Instruction.
ERIC Educational Resources Information Center
Small, Don; And Others
1986-01-01
Computer algebra systems (such as MACSYMA and muMath) can carry out many of the operations of calculus, linear algebra, and differential equations. Use of them with sketching graphs of rational functions and with other topics is discussed. (MNS)
Motivating Activities that Lead to Algebra
ERIC Educational Resources Information Center
Menon, Ramakrishnan
2004-01-01
Four activities consisting of puzzles are introduced, which help students to recognize the strength of algebraic generalizations. They also assist them to comprehend algebraic concepts, and enable them to develop their individual puzzles and games.
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.
Computational triadic algebras of signs
Zadrozny, W.
1996-12-31
We present a finite model of Peirce`s ten classes of signs. We briefly describe Peirce`s taxonomy of signs; we prove that any finite collection of signs can be extended to a finite algebra of signs in which all interpretants are themselves being interpreted; and we argue that Peirce`s ten classes of signs can be defined using constraints on algebras of signs. The paper opens the possibility of defining multimodal cognitive agents using Peirce`s classes of signs, and is a first step towards building a computational logic of signs based on Peirce`s taxonomies.
ERIC Educational Resources Information Center
Star, Jon R.; Rittle-Johnson, Bethany
2009-01-01
Competence in algebra is increasingly recognized as a critical milestone in students' middle and high school years. The transition from arithmetic to algebra is a notoriously difficult one, and improvements in algebra instruction are greatly needed (National Research Council, 2001). Algebra historically has represented students' first sustained…
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.
The weak Hopf algebras related to generalized Kac-Moody algebra
Wu Zhixiang
2006-06-15
We define a kind of quantized enveloping algebra of a generalized Kac-Moody algebra G by adding a generator J satisfying J{sup m}=J{sup m-1} for some integer m. We denote this algebra by wU{sub q}{sup {tau}}(G). This algebra is a weak Hopf algebra if and only if m=2. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usually quantum envelope algebra U{sub q}(G) of a generalized Kac-Moody algebra G.
Conformal invariance in noncommutative geometry and mutually interacting Snyder particles
NASA Astrophysics Data System (ADS)
Pramanik, Souvik; Ghosh, Subir; Pal, Probir
2014-11-01
A system of relativistic Snyder particles with mutual two-body interaction that lives in a noncommutative Snyder geometry is studied. The underlying novel symplectic structure is a coupled and extended version of (single-particle) Snyder algebra. In a recent work by Casalbuoni and Gomis [Phys. Rev. D 90, 026001 (2014)], a system of interacting conventional particles (in commutative spacetime) was studied with special emphasis on its conformal invariance. Proceeding along the same lines, we have shown that our interacting Snyder particle model is also conformally invariant. Moreover, the conformal Killing vectors have been constructed. Our main emphasis is on the Hamiltonian analysis of the conformal symmetry generators. We demonstrate that the Lorentz algebra remains undeformed, but validity of the full conformal algebra requires further restrictions.
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…
Graphing Calculator Use in Algebra Teaching
ERIC Educational Resources Information Center
Dewey, Brenda L.; Singletary, Ted J.; Kinzel, Margaret T.
2009-01-01
This study examines graphing calculator technology availability, characteristics of teachers who use it, teacher attitudes, and how use reflects changes to algebra curriculum and instructional practices. Algebra I and Algebra II teachers in 75 high school and junior high/middle schools in a diverse region of a northwestern state were surveyed.…
New family of Maxwell like algebras
NASA Astrophysics Data System (ADS)
Concha, P. K.; Durka, R.; Merino, N.; Rodríguez, E. K.
2016-08-01
We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.
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…
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…
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…
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"…
Lessons for Algebraic Thinking. Grades K-2.
ERIC Educational Resources Information Center
von Rotz, Leyani; Burns, Marilyn
Algebra is one of the top priorities of mathematics instruction for the elementary and middle grades. This book is designed to help K-2 teachers meet the challenge of making algebra an integral part of their mathematics instruction and realize both what to teach and how to teach central algebraic concepts. Classroom-tested lessons help teachers…
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…
Combinatorial Geometry Printer Plotting.
1987-01-05
Picture generates plots of two-dimensional slices through the three-dimensional geometry described by the combinatorial geometry (CG) package used in such codes as MORSE and QAD-CG. These plots are printed on a standard line printer.
NASA Astrophysics Data System (ADS)
Taylor, Marika
2006-03-01
Two charge BPS horizon free supergravity geometries are important in proposals for understanding black hole microstates. In this paper we construct a new class of geometries in the NS1-P system, corresponding to solitonic strings carrying fermionic as well as bosonic condensates. Such geometries are required to account for the full microscopic entropy of the NS1-P system. We then briefly discuss the properties of the corresponding geometries in the dual D1-D5 system.
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…
ERIC Educational Resources Information Center
Bosse, Michael J.; Ries, Heather; Chandler, Kayla
2012-01-01
Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…
Implementing Change in College Algebra
ERIC Educational Resources Information Center
Haver, William E.
2007-01-01
In this paper, departments are urged to consider implementing the type of changes proposed in Beyond Crossroads in College Algebra. The author of this paper is chair of the Curriculum Renewal Across the First Two Years (CRAFTY) Committee of the Mathematical Association of America. The committee has members from two-year colleges, four-year…
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…
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 national…
ERIC Educational Resources Information Center
Oishi, Lindsay
2011-01-01
"Solve for x." While many people first encountered this enigmatic instruction in high school, the last 20 years have seen a strong push to get students to take algebra in eighth grade or even before. Today, concerns about the economy highlight a familiar worry: American eighth-graders trailed their peers in five Asian countries on the 2007 TIMSS…
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…
An Introduction to Algebraic Multigrid
Falgout, R D
2006-04-25
Algebraic multigrid (AMG) solves linear systems based on multigrid principles, but in a way that only depends on the coefficients in the underlying matrix. The author begins with a basic introduction to AMG methods, and then describes some more recent advances and theoretical developments
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…
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…
Kinds of Knowledge in Algebra.
ERIC Educational Resources Information Center
Lewis, Clayton
Solving equations in elementary algebra requires knowledge of the permitted operations, and knowledge of what operation to use at a given point in the solution process. While just these kinds of knowledge would be adequate for an ideal solver, human solvers appear to need and use other kinds of knowledge. First, many errors seem to indicate that…
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…
Algebra from Chips and Chopsticks
ERIC Educational Resources Information Center
Yun, Jeong Oak; Flores, Alfinio
2012-01-01
Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…
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 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.
Principals + Algebra (- Fear) = Instructional Leadership
ERIC Educational Resources Information Center
Carver, Cynthia L.
2010-01-01
Recent state legislation in Michigan mandates that all graduating seniors successfully pass algebra I and II. Numerous initiatives have been enacted to help mathematics teachers meet this challenge, yet school principals have had little preparation for the necessary curricular and instructional changes. To address this unmet need, university-based…
Experts Question California's Algebra Edict
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
Business leaders from important sectors of the American economy have been urging schools to set higher standards in math and science--and California officials, in mandating that 8th graders be tested in introductory algebra, have responded with one of the highest such standards in the land. Still, many California educators and school…
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.
NASA Astrophysics Data System (ADS)
Putnam, Ian F.
2010-03-01
We investigate the C*-algebras associated to aperiodic structures called model sets obtained by the cut-and-project method. These C*-algebras are Morita equivalent to crossed product C*-algebras obtained from dynamics on a disconnected version of the internal space. This construction may be made from more general data, which we call a hyperplane system. From a hyperplane system, others may be constructed by a process of reduction and we show how the C*-algebras involved are related to each other. In particular, there are natural elements in the Kasparov KK-groups for the C*-algebra of a hyperplane system and that of its reduction. The induced map on K-theory fits in a six-term exact sequence. This provides a new method of the computation of the K-theory of such C*-algebras which is done completely in the setting of non-commutative geometry.
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…
Algebraic approach to small-world network models
NASA Astrophysics Data System (ADS)
Rudolph-Lilith, Michelle; Muller, Lyle E.
2014-01-01
We introduce an analytic model for directed Watts-Strogatz small-world graphs and deduce an algebraic expression of its defining adjacency matrix. The latter is then used to calculate the small-world digraph's asymmetry index and clustering coefficient in an analytically exact fashion, valid nonasymptotically for all graph sizes. The proposed approach is general and can be applied to all algebraically well-defined graph-theoretical measures, thus allowing for an analytical investigation of finite-size small-world graphs.
NASA Astrophysics Data System (ADS)
Bengtsson, Ingemar; Zyczkowski, Karol
2006-05-01
Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied. The first book to focus on the geometry of quantum states Stresses the similarities and differences between classical and quantum theory Uses a non-technical style and numerous figures to make the book accessible to non-specialists
Scaling Linear Algebra Kernels using Remote Memory Access
Krishnan, Manoj Kumar; Lewis, Robert R.; Vishnu, Abhinav
2010-09-13
This paper describes the scalability of linear algebra kernels based on remote memory access approach. The current approach differs from the other linear algebra algorithms by the explicit use of shared memory and remote memory access (RMA) communication rather than message passing. It is suitable for clusters and scalable shared memory systems. The experimental results on large scale systems (Linux-Infiniband cluster, Cray XT) demonstrate consistent performance advantages over ScaLAPACK suite, the leading implementation of parallel linear algebra algorithms used today. For example, on a Cray XT4 for a matrix size of 102400, our RMA-based matrix multiplication achieved over 55 teraflops while ScaLAPACK’s pdgemm measured close to 42 teraflops on 10000 processes.
Categorical Formulation of Finite-Dimensional Quantum Algebras
NASA Astrophysics Data System (ADS)
Vicary, Jamie
2011-06-01
We describe how †-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional `quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of †-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos. We describe the relevance of these results for topological quantum field theory.
The Exocenter of a Generalized Effect Algebra
NASA Astrophysics Data System (ADS)
Foulis, David J.; Pulmannová, Sylvia
2011-12-01
Elements of the exocenter of a generalized effect algebra (GEA) correspond to decompositions of the GEA as a direct sum and thus the exocenter is a generalization to GEAs of the center of an effect algebra. The exocenter of a GEA is shown to be a boolean algebra, and the notion of a hull mapping for an effect algebra is generalized to a hull system for a GEA. We study Dedekind orthocompleteness of GEAs and extend to GEAs the notion of a centrally orthocomplete effect algebra.
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.
Chen, Xingyuan; Miller, Gretchen R.; Rubin, Yoram; Baldocchi, Dennis
2012-09-13
The heat pulse method is widely used to measure water flux through plants; it works by inferring the velocity of water through a porous medium from the speed at which a heat pulse is propagated through the system. No systematic, non-destructive calibration procedure exists to determine the site-specific parameters necessary for calculating sap velocity, e.g., wood thermal diffusivity and probe spacing. Such parameter calibration is crucial to obtain the correct transpiration flux density from the sap flow measurements at the plant scale; and consequently, to up-scale tree-level water fluxes to canopy and landscape scales. The purpose of this study is to present a statistical framework for estimating the wood thermal diffusivity and probe spacing simutaneously from in-situ heat response curves collected by the implanted probes of a heat ratio apparatus. Conditioned on the time traces of wood temperature following a heat pulse, the parameters are inferred using a Bayesian inversion technique, based on the Markov chain Monte Carlo sampling method. The primary advantage of the proposed methodology is that it does not require known probe spacing or any further intrusive sampling of sapwood. The Bayesian framework also enables direct quantification of uncertainty in estimated sap flow velocity. Experiments using synthetic data show that repeated tests using the same apparatus are essential to obtain reliable and accurate solutions. When applied to field conditions, these tests are conducted during different seasons and automated using the existing data logging system. The seasonality of wood thermal diffusivity is obtained as a by-product of the parameter estimation process, and it is shown to be affected by both moisture content and temperature. Empirical factors are often introduced to account for the influence of non-ideal probe geometry on the estimation of heat pulse velocity, and they are estimated in this study as well. The proposed methodology can be applied for
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.
Filiform Lie algebras of order 3
Navarro, R. M.
2014-04-15
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.
Atomic effect algebras with compression bases
Caragheorgheopol, Dan; Tkadlec, Josef
2011-01-15
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.
Atomic effect algebras with compression bases
NASA Astrophysics Data System (ADS)
Caragheorgheopol, Dan; Tkadlec, Josef
2011-01-01
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.
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 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
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.
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.
Critique of information geometry
NASA Astrophysics Data System (ADS)
Skilling, John
2014-12-01
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.
Analysis on singular spaces: Lie manifolds and operator algebras
NASA Astrophysics Data System (ADS)
Nistor, Victor
2016-07-01
We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference Noncommutative geometry and applications, Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds-called "Lie manifolds" -that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here-work that spans over close to two decades-was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.
Spacetime algebra as a powerful tool for electromagnetism
NASA Astrophysics Data System (ADS)
Dressel, Justin; Bliokh, Konstantin Y.; Nori, Franco
2015-08-01
We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field.
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)
The basics of information geometry
NASA Astrophysics Data System (ADS)
Caticha, Ariel
2015-01-01
To what extent can we distinguish one probability distribution from another? Are there quantitative measures of distinguishability? The goal of this tutorial is to approach such questions by introducing the notion of the "distance" between two probability distributions and exploring some basic ideas of such an "information geometry".
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.
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.
Riemannian geometry of fluctuation theory: An introduction
NASA Astrophysics Data System (ADS)
Velazquez, Luisberis
2016-05-01
Fluctuation geometry was recently proposed as a counterpart approach of Riemannian geometry of inference theory (information geometry), which describes the geometric features of the statistical manifold M of random events that are described by a family of continuous distributions dpξ(x|θ). This theory states a connection among geometry notions and statistical properties: separation distance as a measure of relative probabilities, curvature as a measure about the existence of irreducible statistical correlations, among others. In statistical mechanics, fluctuation geometry arises as the mathematical apparatus of a Riemannian extension of Einstein fluctuation theory, which is also closely related to Ruppeiner geometry of thermodynamics. Moreover, the curvature tensor allows to express some asymptotic formulae that account for the system fluctuating behavior beyond the gaussian approximation, while curvature scalar appears as a second-order correction of Legendre transformation between thermodynamic potentials.
Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature
NASA Astrophysics Data System (ADS)
Barnes, Gwendolyn E.; Schenkel, Alexander; Szabo, Richard J.
2016-08-01
We continue our systematic development of noncommutative and nonassociative differential geometry internal to the representation category of a quasitriangular quasi-Hopf algebra. We describe derivations, differential operators, differential calculi and connections using universal categorical constructions to capture algebraic properties such as Leibniz rules. Our main result is the construction of morphisms which provide prescriptions for lifting connections to tensor products and to internal homomorphisms. We describe the curvatures of connections within our formalism, and also the formulation of Einstein-Cartan geometry as a putative framework for a nonassociative theory of gravity.
Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type
NASA Astrophysics Data System (ADS)
Khongsap, Ta; Wang, Weiqiang
2009-01-01
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
2003-06-03
The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran codemore » is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.« less
Conventionalism and integrable Weyl geometry
NASA Astrophysics Data System (ADS)
Pucheu, M. L.
2015-03-01
Since the appearance of Einstein's general relativity, gravitation has been associated to the space-time curvature. This theory introduced a geometrodynamic language which became a convenient tool to predict matter behaviour. However, the properties of space-time itself cannot be measurable by experiments. Taking Poincaré idea that the geometry of space-time is merely a convention, we show that the general theory of relativity can be completely reformulated in a more general setting, a generalization of Riemannian geometry, namely, the Weyl integrable geometry. The choice of this new mathematical language implies, among other things, that the path of particles and light rays should now correspond to Weylian geodesies. Such modification in the dynamic of bodies brings a new perception of physical phenomena that we will explore.
Single axioms for Boolean algebra.
McCune, W.
2000-06-30
Explicit single axioms are presented for Boolean algebra in terms of (1) the Sheffer stroke; (2) disjunction and negation; (3) disjunction, conjunction, and negation; and (4) disjunction, conjunction, negation, 0, and 1. It was previously known that single axioms exist for these systems, but the procedures to generate them are exponential, producing huge equations. Automated deduction techniques were applied to find axioms of lengths 105, 131, 111, and 127, respectively, each with six variables.
The algebras of large N matrix mechanics
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
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.
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.
DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J
2015-05-01
To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers--relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)--can be linked to early success with algebraic expressions. PMID:25744594
Huang, Jie-rong; Gentner, Martin; Vajpai, Navratna; Grzesiek, Stephan; Blackledge, Martin
2012-10-01
Many functional proteins do not have well defined folded structures. In recent years, both experimental and computational approaches have been developed to study the conformational behaviour of this type of protein. It has been shown previously that experimental RDCs (residual dipolar couplings) can be used to study the backbone sampling of disordered proteins in some detail. In these studies, the backbone structure was modelled using a common geometry for all amino acids. In the present paper, we demonstrate that experimental RDCs are also sensitive to the specific geometry of each amino acid as defined by energy-minimized internal co-ordinates. We have modified the FM (flexible-Meccano) algorithm that constructs conformational ensembles on the basis of a statistical coil model, to account for these differences. The modified algorithm inherits the advantages of the FM algorithm to efficiently sample the potential energy landscape for coil conformations. The specific geometries incorporated in the new algorithm result in a better reproduction of experimental RDCs and are generally applicable for further studies to characterize the conformational properties of intrinsically disordered proteins. In addition, the internal-co-ordinate-based algorithm is an order of magnitude more efficient, and facilitates side-chain construction, surface osmolyte simulation, spin-label distribution sampling and proline cis/trans isomer simulation. PMID:22988852
Walendziak, Andrzej
2015-01-01
The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained. PMID:26125050
Lax operator algebras and integrable systems
NASA Astrophysics Data System (ADS)
Sheinman, O. K.
2016-02-01
A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. Bibliography: 51 titles.
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.…
Hidden geometry of traffic jamming.
Andjelković, Miroslav; Gupte, Neelima; Tadić, Bosiljka
2015-05-01
We introduce an approach based on algebraic topological methods that allow an accurate characterization of jamming in dynamical systems with queues. As a prototype system, we analyze the traffic of information packets with navigation and queuing at nodes on a network substrate in distinct dynamical regimes. A temporal sequence of traffic density fluctuations is mapped onto a mathematical graph in which each vertex denotes one dynamical state of the system. The coupling complexity between these states is revealed by classifying agglomerates of high-dimensional cliques that are intermingled at different topological levels and quantified by a set of geometrical and entropy measures. The free-flow, jamming, and congested traffic regimes result in graphs of different structure, while the largest geometrical complexity and minimum entropy mark the edge of the jamming region. PMID:26066222
Geometry, Student's Text, Part II, Unit 14.
ERIC Educational Resources Information Center
Allen, Frank B.; And Others
Unit 14 in the SMSG secondary school mathematics series is a student text covering the following topics in geometry: areas of polygonal regions, similarity, circles and spheres, characterization of sets, constructions, areas of circles and sectors, volumes of solids, and plane coordinate geometry. Appendices cover Eratosthenes' measurement of the…
Coverings of topological semi-abelian algebras
NASA Astrophysics Data System (ADS)
Mucuk, Osman; Demir, Serap
2016-08-01
In this work, we study on a category of topological semi-abelian algebras which are topological models of given an algebraic theory T whose category of models is semi-abelian; and investigate some results on the coverings of topological models of such theories yielding semi-abelian categories. We also consider the internal groupoid structure in the semi-abelian category of T-algebras, and give a criteria for the lifting of internal groupoid structure to the covering groupoids.
Stability of algebraically unstable dispersive flows
NASA Astrophysics Data System (ADS)
King, Kristina; Zaretzky, Paula; Weinstein, Steven; Cromer, Michael; Barlow, Nathaniel
2015-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to a class of partial differential equations describing wave propagation in dispersive media. There are several morphological differences between algebraically growing disturbances and the exponentially growing wave packets inherent to classical linear stability analysis, and these are elucidated in this study.
Explicit travelling waves and invariant algebraic curves
NASA Astrophysics Data System (ADS)
Gasull, Armengol; Giacomini, Hector
2015-06-01
We introduce a precise definition of algebraic travelling wave solution of n-th order partial differential equations and prove that the only algebraic travelling waves solutions for the celebrated Fisher-Kolmogorov equation are the ones found in 1979 by Ablowitz and Zeppetella. This question is equivalent to study when an associated one-parameter family of planar ordinary differential systems has invariant algebraic curves.
Finite-dimensional simple graded algebras
Bahturin, Yu A; Zaicev, M V; Sehgal, S K
2008-08-31
Let R be a finite-dimensional algebra over an algebraically closed field F graded by an arbitrary group G. In the paper it is proved that if the characteristic of F is zero or does not divide the order of any finite subgroup of G, then R is graded simple if and only if it is isomorphic to a matrix algebra over a finite-dimensional graded skew field. Bibliography: 24 titles.
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.
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'…
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
Kaufmann, Matthew L.; Bomer, Megan A.; Powell, Nancy Norem
2009-01-01
Students enter the geometry classroom with a strong concept of fairness and a sense of what it means to "play by the rules," yet many students have difficulty understanding the postulates, or rules, of geometry and their implications. Although they may never have articulated the properties of an axiomatic system, they have gained a practical…
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 and compares it to…
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…
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.
Representations of Super Yang-Mills Algebras
NASA Astrophysics Data System (ADS)
Herscovich, Estanislao
2013-06-01
We study in this article the representation theory of a family of super algebras, called the super Yang-Mills algebras, by exploiting the Kirillov orbit method à la Dixmier for nilpotent super Lie algebras. These super algebras are an extension of the so-called Yang-Mills algebras, introduced by A. Connes and M. Dubois-Violette in (Lett Math Phys 61(2):149-158, 2002), and in fact they appear as a "background independent" formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras {{Cliff}q(k) ⊗ Ap(k)}, for p ≥ 3, or p = 2 and q ≥ 2, appear as a quotient of all super Yang-Mills algebras, for n ≥ 3 and s ≥ 1. This provides thus a family of representations of the super Yang-Mills algebras.
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.
On Realization of Generalized Effect Algebras
NASA Astrophysics Data System (ADS)
Paseka, Jan
2012-12-01
A well-known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice L(H) of all closed subspaces of a separable complex Hilbert space. We show that a generalized effect algebra is representable in the operator generalized effect algebra G(H) of effects of a complex Hilbert space H iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Riečanová and Zajac. Further, any operator generalized effect algebra G(H) possesses an order determining set of generalized states.
Literal algebra for satellite dynamics. [perturbation analysis
NASA Technical Reports Server (NTRS)
Gaposchkin, E. M.
1975-01-01
A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.
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.
Type-Decomposition of an Effect Algebra
NASA Astrophysics Data System (ADS)
Foulis, David J.; Pulmannová, Sylvia
2010-10-01
Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.
NASA Astrophysics Data System (ADS)
Chajda, Ivan
2014-10-01
Commutative BCI-algebras can be considered as semilattices whose sections are equipped with certain involutions. A similar view can be applied to commutative BCK-algebras. However, for general BCK-algebras a certain construction was settled by the author and J. Kühr (Miskolc Math. Notes 8:11-21, 2007) showing that they can be considered as structures essentially weaker than semilattices but still with certain involutions in sections. The aim of this paper is to involve a similar approach for BCI-algebras.
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…
Results of Using Algebra Tiles as Meaningful Representations of Algebra Concepts.
ERIC Educational Resources Information Center
Sharp, Janet M.
Mathematical meanings can be developed when individuals construct translations between algebra symbol systems and physical systems that represent one another. Previous research studies indicated (1) few high school students connect whole number manipulations to algebraic manipulations and (2) students who encounter algebraic ideas through…
Some C∗-algebras which are coronas of non-C∗-Banach algebras
NASA Astrophysics Data System (ADS)
Voiculescu, Dan-Virgil
2016-07-01
We present results and motivating problems in the study of commutants of hermitian n-tuples of Hilbert space operators modulo normed ideals. In particular, the C∗-algebras which arise in this context as coronas of non-C∗-Banach algebras, the connections with normed ideal perturbations of operators, the hyponormal operators and the bidual Banach algebras one encounters are discussed.
Leibniz algebras associated with some finite-dimensional representation of Diamond Lie algebra
NASA Astrophysics Data System (ADS)
Camacho, Luisa M.; Ladra, Manuel; Karimjanov, Iqboljon A.; Omirov, Bakhrom A.
2016-03-01
In this paper we classify Leibniz algebras whose associated Lie algebra is four-dimensional Diamond Lie algebra 𝕯 and the ideal generated by squares of elements is represented by one of the finite-dimensional indecomposable D-modules Un 1, Un 2 or Wn 1 or Wn 2.
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].
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.
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.
ERIC Educational Resources Information Center
Grandau, Laura
2013-01-01
This study of fourth-grade students and teachers explores mathematics teaching and learning that focuses on discovering and modeling algebraic relationships. The study has two parts: an investigation of how students learn to construct algebraic statements and models for comparisons and measurement situations in the multiplicative domain, and an…
ERIC Educational Resources Information Center
Powell, Sarah R.; Fuchs, Lynn S.
2014-01-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 second-grade students, we administered: (1) measures of calculations and…
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.
Automorphisms and Derivations of the Insertion-Elimination Algebra and Related Graded Lie Algebras
NASA Astrophysics Data System (ADS)
Ondrus, Matthew; Wiesner, Emilie
2016-07-01
This paper addresses several structural aspects of the insertion-elimination algebra {mathfrak{g}}, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of {mathfrak{g}}, the automorphism group of {mathfrak{g}}, the derivation Lie algebra of {mathfrak{g}}, and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.
Algebras Generated by Geometric Scalar Forms and their Applications in Physics and Social Sciences
Keller, Jaime
2008-09-17
The present paper analyzes the consequences of defining that the geometric scalar form is not necessarily quadratic, but in general K-atic, that is obtained from the K{sup th} power of the linear form, requiring {l_brace}e{sub i};i = 1,...,N;(e{sub i}){sup K} = 1{r_brace} and d-vector {sigma}{sub i}x{sub i}e{sub i}. We consider the algebras which are thus generated, for positive integer K, a generalization of the geometric algebras we know under the names of Clifford or Grassmann algebras. We then obtain a set of geometric K-algebras. We also consider the generalization of special functions of geometry which corresponds to the K-order scalar forms (as trigonometric functions and other related geometric functions which are based on the use of quadratic forms). We present an overview of the use of quadratic forms in physics as in our general theory, we have called START. And, in order to give an introduction to the use of the more general K-algebras and to the possible application to sciences other than physics, the application to social sciences is considered.For the applications to physics we show that quadratic spaces are a fundamental clue to understand the structure of theoretical physics (see, for example, Keller in ICNAAM 2005 and 2006)
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…
TOPICAL REVIEW: Braided affine geometry and q-analogs of wave operators
NASA Astrophysics Data System (ADS)
Gurevich, Dimitri; Saponov, Pavel
2009-08-01
The main goal of this review is to compare different approaches to constructing the geometry associated with a Hecke type braiding (in particular, with that related to the quantum group Uq(sl(n))). We place emphasis on the affine braided geometry related to the so-called reflection equation algebra (REA). All objects of such a type of geometry are defined in the spirit of affine algebraic geometry via polynomial relations on generators. We begin by comparing the Poisson counterparts of 'quantum varieties' and describe different approaches to their quantization. Also, we exhibit two approaches to introducing q-analogs of vector bundles and defining the Chern-Connes index for them on quantum spheres. In accordance with the Serre-Swan approach, the q-vector bundles are treated as finitely generated projective modules over the corresponding quantum algebras. Besides, we describe the basic properties of the REA used in this construction and compare different ways of defining q-analogs of partial derivatives and differentials on the REA and algebras close to them. In particular, we present a way of introducing a q-differential calculus via Koszul type complexes. The elements of the q-calculus are applied to defining q-analogs of some relativistic wave operators.
Realizations of conformal current-type Lie algebras
Pei Yufeng; Bai Chengming
2010-05-15
In this paper we obtain the realizations of some infinite-dimensional Lie algebras, named 'conformal current-type Lie algebras', in terms of a two-dimensional Novikov algebra and its deformations. Furthermore, Ovsienko and Roger's loop cotangent Virasoro algebra, which can be regarded as a nice generalization of the Virasoro algebra with two space variables, is naturally realized as an affinization of the tensor product of a deformation of the two-dimensional Novikov algebra and the Laurent polynomial algebra. These realizations shed new light on various aspects of the structure and representation theory of the corresponding infinite-dimensional Lie algebras.
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…
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 solve"…
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)
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…
Algebra in the Early Years? Yes!
ERIC Educational Resources Information Center
Taylor-Cox, Jennifer
2003-01-01
Suggests ways early years educators can begin teaching young children to think algebraically and prepare them for success in algebra. Focuses on ways to promote mathematical patterns, mathematical situations and structures, models of quantitative relationship, and change. Describes how first-graders used real object representations to better…
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…
Solving Absolute Value Equations Algebraically and Geometrically
ERIC Educational Resources Information Center
Shiyuan, Wei
2005-01-01
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.
Low Performers Found Unready to Take Algebra
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
As state and school leaders across the country push to have more students take algebra in 8th grade, a new study argues that middle schoolers struggling the most in math are being enrolled in that course despite being woefully unprepared. "The Misplaced Math Student: Lost in Eighth Grade Algebra," scheduled for release by the Brookings Institution…
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…
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…
Algebraic Geodesics on Three-Dimensional Quadrics
NASA Astrophysics Data System (ADS)
Kai, Yue
2015-12-01
By Hamilton-Jacobi method, we study the problem of algebraic geodesics on the third-order surface. By the implicit function theorem, we proved the existences of the real geodesics which are the intersections of two algebraic surfaces, and we also give some numerical examples.
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)
Classical and quantum Kummer shape algebras
NASA Astrophysics Data System (ADS)
Odzijewicz, A.; Wawreniuk, E.
2016-07-01
We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras, here called Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, are also presented.