Numerical algebraic geometry and algebraic kinematics
NASA Astrophysics Data System (ADS)
Wampler, Charles W.; Sommese, Andrew J.
In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.
Math Sense: Algebra and Geometry.
ERIC Educational Resources Information Center
Howett, Jerry
This book is designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four strands: (1) skill lessons…
Mathematics: Algebra and Geometry. GED Scoreboost.
ERIC Educational Resources Information Center
Hoyt, Cathy
GED "Scoreboost" materials target exactly the skills one needs to pass the General Educational Development (GED) tests. This book focuses on the GED Mathematics test. To prepare for the test, the test taker needs to learn skills in number and operation sense, data and statistics, geometry and measurement, and algebra. To pass the test, the test…
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
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.
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
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.
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.
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.
Enumerative Geometry, Tau-Functions and Heisenberg-Virasoro Algebra
NASA Astrophysics Data System (ADS)
Alexandrov, A.
2015-08-01
In this paper we establish relations between three enumerative geometry tau-functions, namely the Kontsevich-Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe the tau-functions in terms of matrix integrals, Virasoro constraints and Kac-Schwarz operators. All constructed operators belong to the algebra (or group) of symmetries of the KP hierarchy.
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.
Math Sense: Algebra and Geometry. Teacher's Resource Guide.
ERIC Educational Resources Information Center
Phillips, Jan; Osmus, Kathy
This book is a teacher's resource guide designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four…
Leikin, Mark; Waisman, Ilana; Shaul, Shelley; Leikin, Roza
2014-03-01
This paper presents a small part of a larger interdisciplinary study that investigates brain activity (using event related potential methodology) of male adolescents when solving mathematical problems of different types. The study design links mathematics education research with neurocognitive studies. In this paper we performed a comparative analysis of brain activity associated with the translation from visual to symbolic representations of mathematical objects in algebra and geometry. Algebraic tasks require translation from graphical to symbolic representation of a function, whereas tasks in geometry require translation from a drawing of a geometric figure to a symbolic representation of its property. The findings demonstrate that electrical activity associated with the performance of geometrical tasks is stronger than that associated with solving algebraic tasks. Additionally, we found different scalp topography of the brain activity associated with algebraic and geometric tasks. Based on these results, we argue that problem solving in algebra and geometry is associated with different patterns of brain activity.
Numerical algebraic geometry: a new perspective on gauge and string theories
NASA Astrophysics Data System (ADS)
Mehta, Dhagash; He, Yang-Hui; Hauensteine, Jonathan D.
2012-07-01
There is a rich interplay between algebraic geometry and string and gauge theories which has been recently aided immensely by advances in computational algebra. However, symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these shortcomings. The so-called `embarrassing parallelizability' allows us to solve many problems and extract physical information which elude symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.
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.
A Linear Algebra Measure of Cluster Quality.
ERIC Educational Resources Information Center
Mather, Laura A.
2000-01-01
Discussion of models for information retrieval focuses on an application of linear algebra to text clustering, namely, a metric for measuring cluster quality based on the theory that cluster quality is proportional to the number of terms that are disjoint across the clusters. Explains term-document matrices and clustering algorithms. (Author/LRW)
Quantum error-correcting codes from algebraic geometry codes of Castle type
NASA Astrophysics Data System (ADS)
Munuera, Carlos; Tenório, Wanderson; Torres, Fernando
2016-10-01
We study algebraic geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.
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.
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…
Degeneracy measures for the algebraic classification of numerical spacetimes
NASA Astrophysics Data System (ADS)
Owen, Robert
2010-06-01
We study the issue of algebraic classification of the Weyl curvature tensor, with a particular focus on numerical relativity simulations. The spacetimes of interest in this context, binary black hole mergers, and the ringdowns that follow them, present subtleties in that they are generically, strictly speaking, type I, but in many regions approximately, in some sense, type D. To provide meaning to any claims of “approximate” Petrov class, one must define a measure of degeneracy on the space of null rays at a point. We will investigate such a measure, used recently to argue that certain binary black hole merger simulations ring down to the Kerr geometry, after hanging up for some time in Petrov type II. In particular, we argue that this hangup in Petrov type II is an artefact of the particular measure being used, and that a geometrically better-motivated measure shows a black hole merger produced by our group settling directly to Petrov type D.
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.
Handy elementary algebraic properties of the geometry of entanglement
NASA Astrophysics Data System (ADS)
Blair, Howard A.; Alsing, Paul M.
2013-05-01
The space of separable states of a quantum system is a hyperbolic surface in a high dimensional linear space, which we call the separation surface, within the exponentially high dimensional linear space containing the quantum states of an n component multipartite quantum system. A vector in the linear space is representable as an n-dimensional hypermatrix with respect to bases of the component linear spaces. A vector will be on the separation surface iff every determinant of every 2-dimensional, 2-by-2 submatrix of the hypermatrix vanishes. This highly rigid constraint can be tested merely in time asymptotically proportional to d, where d is the dimension of the state space of the system due to the extreme interdependence of the 2-by-2 submatrices. The constraint on 2-by-2 determinants entails an elementary closed formformula for a parametric characterization of the entire separation surface with d-1 parameters in the char- acterization. The state of a factor of a partially separable state can be calculated in time asymptotically proportional to the dimension of the state space of the component. If all components of the system have approximately the same dimension, the time complexity of calculating a component state as a function of the parameters is asymptotically pro- portional to the time required to sort the basis. Metric-based entanglement measures of pure states are characterized in terms of the separation hypersurface.
Measuring the Readability of Elementary Algebra Using the Cloze Technique.
ERIC Educational Resources Information Center
Kulm, Gerald
The relationship to readability of ten variables characterizing structural properties of mathematical prose was investigated in elementary algebra textbooks. Readability was measured by algebra student's responses to two forms of cloze tests. Linear and currilinear correlations were calculated between each structural variable and the cloze test.…
Non-separability of the Gelfand space of measure algebras
NASA Astrophysics Data System (ADS)
Ohrysko, Przemysław; Wojciechowski, Michał; Graham, Colin C.
2016-10-01
We prove that there exists uncountably many pairwise disjoint open subsets of the Gelfand space of the measure algebra on any locally compact non-discrete abelian group which shows that this space is not separable (in fact, we prove this assertion for the ideal M0(G) consisting of measures with Fourier-Stieltjes transforms vanishing at infinity which is a stronger statement). As a corollary, we obtain that the spectras of elements in the algebra of measures cannot be recovered from the image of one countable subset of the Gelfand space under Gelfand transform, common for all elements in the algebra.
An analytical approach to bistable biological circuit discrimination using real algebraic geometry.
Siegal-Gaskins, Dan; Franco, Elisa; Zhou, Tiffany; Murray, Richard M
2015-07-01
Biomolecular circuits with two distinct and stable steady states have been identified as essential components in a wide range of biological networks, with a variety of mechanisms and topologies giving rise to their important bistable property. Understanding the differences between circuit implementations is an important question, particularly for the synthetic biologist faced with determining which bistable circuit design out of many is best for their specific application. In this work we explore the applicability of Sturm's theorem--a tool from nineteenth-century real algebraic geometry--to comparing 'functionally equivalent' bistable circuits without the need for numerical simulation. We first consider two genetic toggle variants and two different positive feedback circuits, and show how specific topological properties present in each type of circuit can serve to increase the size of the regions of parameter space in which they function as switches. We then demonstrate that a single competitive monomeric activator added to a purely monomeric (and otherwise monostable) mutual repressor circuit is sufficient for bistability. Finally, we compare our approach with the Routh-Hurwitz method and derive consistent, yet more powerful, parametric conditions. The predictive power and ease of use of Sturm's theorem demonstrated in this work suggest that algebraic geometric techniques may be underused in biomolecular circuit analysis.
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…
Tool 3D geometry measurement system
NASA Astrophysics Data System (ADS)
Zhao, Huijie; Ni, Jun; Sun, Yi; Lin, Xuewen
2001-10-01
A new non-contact tool 3D geometry measurement system based on machine vision is described. In this system, analytical and optimization methods are used respectively to achieve system calibration, which can determine the rotation center of the drill. The data merging method is fully studied which can translate the scattered different groups of raw data in sensor coordinates into drill coordinates and get 3-D topography of the drill body. Corresponding data processing methods for drill geometry are also studied. Statistical methods are used to remove the outliers. Laplacian of Gaussian operator are used to detect the boundary on drill cross-section and drill tip profile. The arithmetic method for calculating the parameters is introduced. The initial measurement results are presented. The cross-section profile, drill tips geometry are shown. Pictures of drill wear on drill tip are given. Parameters extracted from the cross-section are listed. Compared with the measurement results using CMM, the difference between this drill geometry measurement system and CMM is, Radius of drill: 0.020mm, Helix angle: 1.310, Web thickness: 0.034mm.
Measurement of quantum fluctuations in geometry
Hogan, Craig J.
2008-05-15
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.
A Visual Approach to Simplifying Square Roots Applied in Geometry and Advanced Algebra Courses.
ERIC Educational Resources Information Center
Walz, Kevin
This report describes a program for increasing students' ability to simplify square roots. The targeted population consisted of high school students in a rural community in a midwestern plains state. The problem of the ability to understand the abstract algebraic process of simplifying square roots was documented through teacher-made tests,…
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
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.
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.
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.
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.
Quantum Algorithms for Problems in Number Theory, Algebraic Geometry, and Group Theory
NASA Astrophysics Data System (ADS)
van Dam, Wim; Sasaki, Yoshitaka
2013-09-01
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.
An analytical approach to bistable biological circuit discrimination using real algebraic geometry
Siegal-Gaskins, Dan; Franco, Elisa; Zhou, Tiffany; Murray, Richard M.
2015-01-01
Biomolecular circuits with two distinct and stable steady states have been identified as essential components in a wide range of biological networks, with a variety of mechanisms and topologies giving rise to their important bistable property. Understanding the differences between circuit implementations is an important question, particularly for the synthetic biologist faced with determining which bistable circuit design out of many is best for their specific application. In this work we explore the applicability of Sturm's theorem—a tool from nineteenth-century real algebraic geometry—to comparing ‘functionally equivalent’ bistable circuits without the need for numerical simulation. We first consider two genetic toggle variants and two different positive feedback circuits, and show how specific topological properties present in each type of circuit can serve to increase the size of the regions of parameter space in which they function as switches. We then demonstrate that a single competitive monomeric activator added to a purely monomeric (and otherwise monostable) mutual repressor circuit is sufficient for bistability. Finally, we compare our approach with the Routh–Hurwitz method and derive consistent, yet more powerful, parametric conditions. The predictive power and ease of use of Sturm's theorem demonstrated in this work suggest that algebraic geometric techniques may be underused in biomolecular circuit analysis. PMID:26109633
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…
ERIC Educational Resources Information Center
Mohr, Doris J.
2008-01-01
In a geometry content course for pre-service elementary teachers, technology was utilized to assist students in making sense of shapes. They learned to write simple procedures in Logo that would program a turtle to draw various quadrilaterals. In the context of writing these procedures, the pre-service teachers used variables to represent the…
Using Dynamic Geometry and Computer Algebra Systems in Problem Based Courses for Future Engineers
ERIC Educational Resources Information Center
Tomiczková, Svetlana; Lávicka, Miroslav
2015-01-01
It is a modern trend today when formulating the curriculum of a geometric course at the technical universities to start from a real-life problem originated in technical praxis and subsequently to define which geometric theories and which skills are necessary for its solving. Nowadays, interactive and dynamic geometry software plays a more and more…
Lee, N Y; Jung, S H; Kim, J B
2009-01-01
In this paper, we evaluated the measurement geometries and data processing algorithms for industrial gamma tomography technology. Several phantoms simulating industrial objects were tested in various conditions with the gamma-ray CT system developed in KAERI (Korea Atomic Energy Research Institute). Radiation was measured with lead shielded 24 1x1in Nal detectors. Regarding the parallel beam geometry, the EM algorithm showed the best resolution among the algebraic reconstruction technique (ART), simultaneous iterative reconstructive technique (SIRT) and expectation maximization (EM). However, the fan beam scanning was more time efficient than the parallel projection for the similar quality of reconstructed image. Future developments of the industrial gamma ray CT will be focused on a large-scale application which is more practical for a diagnosis in the petrochemical industry. PMID:19376727
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.
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.
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
Interpretation for scales of measurement linking with abstract algebra.
Sawamura, Jitsuki; Morishita, Shigeru; Ishigooka, Jun
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.
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.
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.
Stratospheric Aerosol Extinction Retrieval for SCIAMACHY Measurements in Limb Geometry
NASA Astrophysics Data System (ADS)
Dörner, S.; Pukite, J.; Penning de Vries, M.; Beirle, S.; Wagner, T.
2015-12-01
Techniques for retrieving height resolved information on stratospheric aerosol improved significantly in the past decade with the availability of satellites 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 between 2002 and 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 potential uncertainties especially for wavelengths below 700 nm 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. Comparisons with SAGE II measurement in occultation geometry and balloon borne measurements with an optical particle counter confirm the viability of our retrieval algorithm.
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.
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.
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.
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.
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…
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.
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.
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.
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…
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…
Discrimination in a General Algebraic Setting.
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.
Discrimination in a General Algebraic Setting
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras. PMID:26171421
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.
NASA Astrophysics Data System (ADS)
Levin, A. M.; Olshanetsky, M. A.; Zotov, A. V.
2016-08-01
We construct twisted Calogero-Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D'Hoker-Phong and Bordner-Corrigan-Sasaki-Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of Lie algebras.
Measuring in Dynamic Geometry Environments as a Tool for Conjecturing and Proving
ERIC Educational Resources Information Center
Olivero, Federica; Robutti, Ornella
2007-01-01
This paper sits within the research on the affordances of new technologies in the mathematics classroom and focuses on a specific feature that is available in dynamic geometry environments, i.e. measuring tools, within the context of conjecturing and proving in open geometry problems. We develop a classification of different modalities of…
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.
NASA Astrophysics Data System (ADS)
Zwinkels, Joanne; Neil, William; Noël, Mario
2016-10-01
For highest accuracy fluorescence colorimetry, standardizing organizations recommend the use of a two-monochromator method with a bidirectional illumination and viewing geometry (45:0 or 0:45). For this reason, reference fluorescence instruments developed by National Measurement Institutes (NMIs) have largely conformed to this bidirectional geometry. However, for many practical applications in colorimetry where the samples exhibit texture, surface roughness or other spatial non-uniformities, the relevant standard test methods specify a sphere geometry with diffuse illumination or viewing (e.g. d:8 or 8:d) which gives improved measurement precision. This difference in the measurement geometry between the primary instrument used to realize the fluorescence scale and the secondary testing instruments used for practical measurements, compromises the traceability of these fluorescence calibrations. To address this metrology issue, a two-monochromator goniospectrofluorimeter instrument has been developed at the National Research Council of Canada (NRC). This instrument can be configured for different illumination and viewing geometries to conform with international standards for different colorimetric applications. To improve the traceability chain for measurements using different geometries, the instrument has been thoroughly characterized and validated by means of comparison measurements with NRC’s other spectrophotometric and fluorescence reference instruments. This uncertainty analysis has been carried out in a step-wise manner; first, for a bidirectional geometry (45:0) and then for a sphere geometry (8:d) to provide an uninterrupted traceability to primary radiometric scales. The first paper in this two paper series reviews the background to this work and provides details of the basic design of the new instrument and its characterization for measurements using a bidirectional geometry (45:0), including a representative uncertainty budget. In part 2, the major
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
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…
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…
Geometry of Logarithmic Strain Measures in Solid Mechanics
NASA Astrophysics Data System (ADS)
Neff, Patrizio; Eidel, Bernhard; Martin, Robert J.
2016-11-01
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 {log 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 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.
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
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.
Liouville’s theorem and the canonical measure for nonconservative systems from contact geometry
NASA Astrophysics Data System (ADS)
Bravetti, A.; Tapias, D.
2015-06-01
Standard statistical mechanics of conservative systems relies on the symplectic geometry of the phase space. This is exploited to derive Hamilton’s equations, Liouville’s theorem and to find the canonical invariant measure. In this work we analyze the statistical mechanics of a class of nonconservative systems stemming from contact geometry. In particular, we find out the generalized Hamilton’s equations, Liouville’s theorem and the microcanonical and canonical measures invariant under the contact flow. Remarkably, the latter measure has a power law density distribution with respect to the standard contact volume form. Finally, we argue on the several possible applications of our results.
ERIC Educational Resources Information Center
Pavelle, Richard; And Others
1981-01-01
Describes the nature and use of computer algebra and its applications to various physical sciences. Includes diagrams illustrating, among others, a computer algebra system and flow chart of operation of the Euclidean algorithm. (SK)
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…
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.
Ferrero, Alejandro; Rabal, Ana; Campos, Joaquín; Martínez-Verdú, Francisco; Chorro, Elísabet; Perales, Esther; Pons, Alicia; Hernanz, María Luisa
2013-02-01
A reduced set of measurement geometries allows the spectral reflectance of special effect coatings to be predicted for any other geometry. A physical model based on flake-related parameters has been used to determine nonredundant measurement geometries for the complete description of the spectral bidirectional reflectance distribution function (BRDF). The analysis of experimental spectral BRDF was carried out by means of principal component analysis. From this analysis, a set of nine measurement geometries was proposed to characterize special effect coatings. It was shown that, for two different special effect coatings, these geometries provide a good prediction of their complete color shift.
The Impact of Challenging Geometry and Measurement Units on the Achievement of Grade 2 Students
ERIC Educational Resources Information Center
Gavin, M. Katherine; Casa, Tutita M.; Adelson, Jill L.; Firmender, Janine M.
2013-01-01
The primary goal of Project M[superscript 2] was to develop and field-test challenging geometry and measurement units for all K-2 students. This article reports on the achievement results for students in Grade 2 at 12 urban and suburban sites in 4 states using the Iowa Tests of Basic Skills (ITBS) mathematics concepts subtest and an open-response…
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
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.
Tsui, T.Y.; Pharr, G.M.; Oliver, W.C.
1996-05-01
The measurement of mechanical properties by nanoindentation methods is most often conducted using indenters with the Berkovich geometry (a triangular pyramid) or with a sphere. These indenters provide a wealth of information, but there are certain circumstances in which it would be useful to make measurements with indenters of other geometries. We have recently explored how the measurement of hardness and elastic modulus can be achieved using sharp indenters other than the Berkovich. Systematic studies in several materials were conducted with a Vickers indenter, a conical indenter with a half-included tip angle of 70.3{degrees}, and the standard Berkovich indenter. All three indenters are geometrically similar and have nominally the same area-to-depth relationship, but there are distinct differences in the behavior of each. Here, we report on the application of these indenters in the measurement of hardness and elastic modulus by nanoindentation methods and some of the difficulties that occur.
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.
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.
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
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.…
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
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.
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.
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…
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
NASA Astrophysics Data System (ADS)
Aiken, Brenda L.
The Commonwealth of Virginia requires high school students to receive a passing grade in core courses and a passing score on End-of-Course Standards of Learning (EOC SOL) tests to receive verified credits that lead to a Virginia high school diploma. These tests are believed to accurately reflect what students should know and be able to do in order to experience success in their endeavors beyond high school. For some students remediation is required to experience success on EOC SOL tests. This study sought to determine the effect of a County's public high school summer remediation program on student gains on EOC SOL tests in Algebra I, Biology, Chemistry, Geometry, and World History and Geography II. Specifically, the purpose of the study sought to determine the following: (a) If significant gains were made by students who attended the summer remediation program; (b) If significant gains were made by students who did not attend the summer remediation program; (c) If there were differences in gain scores of students who attended and those who did not attend the summer remediation program; and (d) If there were differences in gain scores among students who attended the summer remediation program related to school site, gender, ethnicity, learning ability group, socioeconomic status, and level of English proficiency. The results of the study indicate that students who attended and those who did not attend the summer remediation program made significant gains. However, the gains for students who attended the summer remediation program were significantly greater than the gains made by students who did not attend. The study also found that there were no significant differences in gain scores among students who attended the summer remediation program related to gender, ethnicity, learning ability group, socioeconomic status, and level of English proficiency. There were significant differences in Algebra I gain scores related to school site. Recommendations for
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.
Models to relate wafer geometry measurements to in-plane distortion of wafers
NASA Astrophysics Data System (ADS)
Turner, Kevin T.; Vukkadala, Pradeep; Sinha, Jaydeep K.
2016-04-01
Achieving satisfactory overlay is increasingly challenging as feature sizes are reduced and allowable overlay budgets shrink to several nanometers and below. Overlay errors induced by wafer processing, such as film deposition and etching, constitute a meaningful fraction of overlay budgets. Wafer geometry measurements provide the opportunity to quantify stress-induced distortions at the wafer level and provide information that can be used in a feedback mode to alter wafer processing or in a feed-forward mode to set wafer-specific corrections in the lithography tool. In order for such feed-forward schemes based on wafer geometry to be realized, there is a need for mechanics models that relate in-plane distortion of a chucked wafer to the out-of-plane distortion of a wafer in a free state. Here, a simple analytical model is presented that shows the stress-induced component of overlay is correlated to a corrected local wafer slope metric for a wide range of cases. The analytical model is validated via finite element (FE) simulations of wafers with nonuniform stress distributions. Furthermore, FE modeling is used here to examine the effect of the spatial wavelength of stress variation on the connection between slope and the wafer stress-induced component of overlay.
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
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}.
ERIC Educational Resources Information Center
Benjamin, Carl; And Others
Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra II. Topics covered include: differencing and complements; real numbers; factoring; fractions; linear equations; exponents and radicals; complex numbers,…
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.…
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.…
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…
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.
NASA Technical Reports Server (NTRS)
Ruge, J. W.; Stueben, K.
1987-01-01
The state of the art in algebraic multgrid (AMG) methods is discussed. The interaction between the relaxation process and the coarse grid correction necessary for proper behavior of the solution probes is discussed in detail. Sufficient conditions on relaxation and interpolation for the convergence of the V-cycle are given. The relaxation used in AMG, what smoothing means in an algebraic setting, and how it relates to the existing theory are considered. Some properties of the coarse grid operator are discussed, and results on the convergence of two-level and multilevel convergence are given. Details of an algorithm particularly studied for problems obtained by discretizing a single elliptic, second order partial differential equation are given. Results of experiments with such problems using both finite difference and finite element discretizations are presented.
NASA Astrophysics Data System (ADS)
Velicu, R.; Bobancu, S.; Popa, S.
2016-08-01
The paper presents theoretical bases and experimental test on the pin on disk module of the UMT tribometer. In order to determine the friction coefficient between a chain and a guide, the rotational pin on disk module of the UMT tribometer has been adapted using a plate of the chain (instead of the pin) pushed against a rotating disk made from the guide material. In this case the contact surface between plate and disk is a rectangle. In comparison with the pin on disk case, the differences between sliding velocities on the rectangular contact surface of the plate on disk case may be considerable bigger. Study of kinematics shows the maximum and minimum sliding velocities and friction forces. The relative extreme sliding velocities and friction forces are expressed depending on geometrical inputs. The study continues with the measurements of friction coefficients maintaining the same couple of materials, surface dimension, normal force and sliding velocity at the centre of the rectangle with variation of the radius. Conclusion is drawn on the influence of the geometry and kinematics of the plate on disk measured friction.
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…
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.
Influence of probe geometry on measurement results of non-ideal thermal conductivity sensors
NASA Astrophysics Data System (ADS)
Tiefenbacher, Patrick; Kömle, Norbert I.; Macher, Wolfgang; Kargl, Günter
2016-09-01
The thermal properties of the surface and subsurface layers of planets and planetary objects yield important information that allows us to better understand the thermal evolution of the body itself and its interactions with the environment. Various planetary bodies of our Solar System are covered by so-called regolith, a granular and porous material. On such planetary bodies the dominant heat transfer mechanism is heat conduction via IR radiation and contact points between particles. In this case the energy balance is mainly controlled by the effective thermal conductivity of the top surface layers, which can be directly measured by thermal conductivity probes. A traditionally used method for measuring the thermal conductivity of solid materials is the needle-probe method. Such probes consist of thin steel needles with an embedded heating wire and temperature sensors. For the evaluation of the thermal conductivity of a specific material the temperature change with time is determined by heating a resistance wire with a well-defined electrical current flowing through it and simultaneously measuring the temperature increase inside the probe over a certain time. For thin needle probes with a large length-to-diameter ratio it is mathematically easy to derive the thermal conductivity, while this is not so straightforward for more rugged probes with a larger diameter and thus a smaller length-to-diameter ratio. Due to the geometry of the standard thin needle probes they are mechanically weak and subject to bending when driven into a soil. Therefore, using them for planetary missions can be problematic. In this paper the thermal conductivity values determined by measurements with two non-ideal, ruggedized thermal conductivity sensors, which only differ in length, are compared to each other. Since the theory describing the temperature response of non-ideal sensors is highly complicated, those sensors were calibrated with an ideal reference sensor in various solid and
Moving frames and prolongation algebras
NASA Technical Reports Server (NTRS)
Estabrook, F. B.
1982-01-01
Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.
NASA 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.
NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
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
Gavin, M. Katherine; Casa, Tutita M.; Firmender, Janine M.; Carroll, Susan R.
2013-01-01
The goal of Project M2 was to develop and field-test challenging geometry and measurement units for K-2 students. The units were developed using recommendations from gifted, mathematics, and early childhood education. This article reports on achievement results for students in Grade 1 at 12 diverse sites in four states using the Iowa Tests of…
Broy, Susan B; Cauley, Jane A; Lewiecki, Michael E; Schousboe, John T; Shepherd, John A; Leslie, William D
2015-01-01
Bone mineral density (BMD) measured by dual-energy X-ray absorptiometry is the current imaging procedure of choice to assess fracture risk. However, BMD is only one of the factors that explain bone strength or resistance to fracture. Other factors include bone microarchitecture and macroarchitecture. We now have the ability to assess some of these non-BMD parameters from a dual-energy X-ray absorptiometry image. Available measurements include various measurements of hip geometry including hip structural analysis, hip axis length, and neck-shaft angle. At the 2015 Position Development Conference, the International Society of Clinical Densitometry established official positions for the clinical utility of measurements of hip geometry. We present the official positions approved by an expert panel after careful review of the recommendations and evidence prepared by an independent task force. Each question addressed by the task force is presented followed by the official position with the associated medical evidence and rationale. PMID:26277848
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
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
NASA Astrophysics Data System (ADS)
Lobato, Justo; Cañizares, Pablo; Rodrigo, Manuel A.; Pinar, F. Javier; Úbeda, Diego
To improve fuel cell design and performance, research studies supported by a wide variety of physical and electrochemical methods have to be carried out. Among the different techniques, current distribution measurement owns the desired feature that can be performed during operation, revealing information about internal phenomena when the fuel cell is working. Moreover, short durability is one of the main problems that is hindering fuel cell wide implementation and it is known to be related to current density heterogeneities over the electrode surface. A good flow channel geometry design can favor a uniform current density profile, hence hypothetically extending fuel cell life. With this, it was thought that a study on the influence of flow channel geometry on the performance of a high temperature polymer electrolyte membrane (PEM) fuel cell using current distribution measurement should be a very solid work to optimize flow field design. Results demonstrate that the 4 step serpentine and pin-type geometries distribute the reactants more effectively, obtaining a relatively flat current density map at higher current densities than parallel or interdigitated ones and yielding maximum powers up to 25% higher when using oxygen as comburent. If air is the oxidant chosen, interdigitated flow channels perform almost as well as serpentine or pin-type due to that the flow conditions are very important for this geometry.
A Taxonomic Approach to Measuring Achievement in Mathematics 223 - Geometry for Elementary Teachers.
ERIC Educational Resources Information Center
Little, Richard A.
The purpose was to evaluate the effectiveness of a geometry course for preservice elementary teachers and, at the same time, to validate the assumptions that the arrangement of categories in Bloom's "Taxonomy" is hierarchical and cumulative. Sixty-two preservice elementary education majors took an investigator-constructed achievement test after…
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.
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.
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.
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.
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…
Quantization of Algebraic Reduction
Sniatycki, Jeodrzej
2007-11-14
For a Poisson algebra obtained by algebraic reduction of symmetries of a quantizable system we develop an analogue of geometric quantization based on the quantization structure of the original system.
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.
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…
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.
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…
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…
Exceptional geometry and Borcherds superalgebras
NASA Astrophysics Data System (ADS)
Palmkvist, Jakob
2015-11-01
We study generalized diffeomorphisms in exceptional geometry with U-duality group E n( n) from an algebraic point of view. By extending the Lie algebra {e}_n to an infinite-dimensional Borcherds superalgebra, involving also the extension to {e}_{n+1} , the generalized Lie derivatives can be expressed in a simple way, and the expressions take the same form for any n ≤ 7. The closure of the transformations then follows from the Jacobi identity and the grading of {e}_{n+1} with respect to {e}_n.
Form-Profiling of Optics Using the Geometry Measuring Machine and the M-48 CMM at NIST
Machkour-Deshayes, Nadia; Stoup, John; Lu, Z. Q. John; Soons, Johannes; Griesmann, Ulf; Polvani, Robert
2006-01-01
We are developing an instrument, the Geometry Measuring Machine (GEMM), to measure the profile errors of aspheric and free form optical surfaces, with measurement uncertainties near 1 nm. Using GEMM, an optical profile is reconstructed from local curvatures of a surface, which are measured at points on the optic’s surface. We will describe a prototype version of GEMM, its repeatability with time, a measurements registry practice, and the calibration practice needed to make nanometer resolution comparisons with other instruments. Over three months, the repeatability of GEMM is 3 nm rms, and is based on the constancy of the measured profile of an elliptical mirror with a radius of curvature of about 83 m. As a demonstration of GEMM’s capabilities for curvature measurement, profiles of that same mirror were measured with GEMM and the NIST Moore M-48 coordinate measuring machine. Although the methods are far different, two reconstructed profiles differ by 22 nm peak-to-valley, or 6 nm rms. This comparability clearly demonstrates that with appropriate calibration, our prototype of the GEMM can measure complex-shaped optics. PMID:27274939
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.
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.
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.
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.
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.
Discrete Minimal Surface Algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Hoppe, Jens
2010-05-01
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
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.
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).
NASA Astrophysics Data System (ADS)
Bloshanskiĭ, I. L.
1984-02-01
The precise geometry is found of measurable sets in N-dimensional Euclidean space on which generalized localization almost everywhere holds for multiple Fourier series which are rectangularly summable.Bibliography: 14 titles.
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.
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.
Hertrick, A.K.; Bryan, W.J.; Dorogy, G.M.; Hopkins, R.J.; Riddell, R.A.; Schwirian, E.R.
1984-11-01
Comparisons of analytical and experimental results are presented for the fluid jetting resulting from the existence of small gaps between parallel flow regions with dissimilar hydraulic characteristics. The experiment simulates the baffle gaps between a nuclear reactor core and the peripheral region around it, called the barrel-baffle region. Baffle gap fluid velocities are measured by a technique in which the only disturbance to the gap flow is a small pressure tap in the gap wall. The analysis uses an iterative, hydraulic network approach and is shown to yield good results when compared to the measured gap jet velocity and pressure drop distributions.
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.
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…
NASA Astrophysics Data System (ADS)
Marsella, Maria; Junior Valentino D'Aranno, Peppe; Martino, Michele; Nardinocchi, Carla; Scifoni, Silvia; Scutti, Marianna; Sonnessa, Alberico; Coltelli, Mauro; Proietti, Cristina
2016-04-01
The lava flow propagation can pose a high risk both for the territory and the infrastructures located on the flanks of an active volcano. The presented study is aimed at analyzing different methodology for measuring the evolution of a flow field by combining ground and aerial observations, as well as satellite data. Data acquired by ground sensors must be opportunely processed to extract orthorectified images. In addition during volcanic crisis, the frequent acquisition of oblique digital images from helicopter allows for quasi-real-time monitoring to support mitigation actions by civil protection. These images can be processed through a straightforward photogrammetric approach to generate digital orthophotos from single-view oblique images. Moreover, airborne remote sensing systems, such as those based on digital photogrammetry and laser scanner sensors, can be also adopted to monitor the lava emplacement processes in active volcanic areas. The capability of extracting accurate topographic data from ground and aerial acquisitions, allow to use these methods to constrain the regular and more frequent measurements derived from satellite observations. The presented work analyses the discrepancy among the different datasets in terms of accuracy and resolution and will attempt to provide an approach for combining the different datasets. The measured evolution of a flow field can be used to evaluate the related hazard by applying a numerical model for evaluating the areas potentially at risk of lava invasion. Numerical simulations can also be aimed at evaluating the reduction of the hazard by simulating the application of different containment systems for mitigating the effect of lava flow propagation.
NASA Astrophysics Data System (ADS)
Sakaamini, Ahmad; Amami, Sadek; Murray, Andrew James; Ning, Chuangang; Madison, Don
2016-10-01
Ionisation triple differential cross sections have been determined experimentally and theoretically for the neutral molecule N2 over a range of geometries from coplanar to the perpendicular plane. Data were obtained at incident electron energies ∼10 and ∼20 eV above the ionisation potential of the 3σ g, 1π u and 2σ g states, using both equal and non-equal outgoing electron energies. The data were taken with the incident electron beam in the scattering plane (ψ = 0°), at 45° to this plane and orthogonal to the plane (ψ = 90°). The set of nine measured differential cross sections at a given energy were then inter-normalised to each other. The data are compared to new calculations using various distorted wave methods, and differences between theory and experiment are discussed.
Working memory, worry, and algebraic ability.
Trezise, Kelly; Reeve, Robert A
2014-05-01
Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.
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)
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
The Krichever map, vector bundles over algebraic curves, and Heisenberg algebras
NASA Astrophysics Data System (ADS)
Adams, M. R.; Bergvelt, M. J.
1993-06-01
We study the Grassmannian Gr {/x n } consisting of equivalence classes of rank n algebraic vector bundles over a Riemann surface X with an holomorphic trivialization at a fixed point p. Commutative subalgebras of gl(n, H λ), H λ being the ring of functions holomorphic on a punctured disc about p, define flows on the Grassmannian, giving rise to classes of solutions to multi-component KP hierarchies. These commutative subalgebras correspond to Heisenberg algebras in the Kac-Moody algebra associated to gl(n, H λ). One can obtain, by the Krichever map, points of Gr {/x n } (and solutions of mcKP) from coverings f: Y→X and other geometric data. Conversely for every point of Gr {/x n } and for every choice of Heisenberg algebra we construct, using the cotangent bundle of Gr {/x n }, an algebraic curve covering X and other data, thus inverting the Krichever map. We show the explicit relation between the choice of Heisenberg algebra and the geometry of the covering space.
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
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…
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.
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.
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.
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.
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.
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.
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.
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
Differential Geometry and Lie Groups for Physicists
NASA Astrophysics Data System (ADS)
Fecko, Marián.
2006-10-01
Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.
Differential Geometry and Lie Groups for Physicists
NASA Astrophysics Data System (ADS)
Fecko, Marián.
2011-03-01
Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.
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.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl( N;?)-case is discussed.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.
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.
University of Chicago School Mathematics Project (UCSMP) Algebra. WWC Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2009
2009-01-01
University of Chicago School Mathematics Project (UCSMP) Algebra is a one-year course covering three primary topics: (1) linear and quadratic expressions, sentences, and functions; (2) exponential expressions and functions; and (3) linear systems. Topics from geometry, probability, and statistics are integrated with the appropriate algebra.…
Authorized Course of Instruction for the Quinmester Program, Mathematics: Survey of Algebra 1.
ERIC Educational Resources Information Center
Moore, Mary N.; Rose, Patricia
Outlined are the minimum requirements for a quinmester course intended to strengthen a student's experience in a first algebra course, prior to entry to high school geometry and the second algebra course. After a brief description of overall goals and strategies, further details are presented in eight sections. Each section gives performance…
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)
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
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.
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.
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.
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.
Ulanovsky, A V; Minenko, V F; Korneev, S V
1997-01-01
An approach for evaluating the influence of measurement geometry on estimates of 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.
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…
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…
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…
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.
Algebra for Gifted Third Graders.
ERIC Educational Resources Information Center
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
Vortex lattice theory: A linear algebra approach
NASA Astrophysics Data System (ADS)
Chamoun, George C.
Vortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics-based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we require nullity( A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A 'Brownian ratchet' construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value of A. Distances between configurations and equilibria were measured using the Frobenius norm ||·||F and 2-norm ||·||2, and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves.
Pseudo Algebraically Closed Extensions
NASA Astrophysics Data System (ADS)
Bary-Soroker, Lior
2009-07-01
This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that although there are many PAC extensions, the Galois closure of a proper PAC extension is separably closed. The dissertation also contains the following subjects. The group theoretical counterpart of pseudo algebraically closed extensions, the so-called projective pairs. Applications to seemingly unrelated subjects, e.g., an analog of Dirichlet's theorem about primes in arithmetic progression for polynomial rings in one variable over infinite fields.
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…
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.
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,…
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
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
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…
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
Zhao, Qile; Wang, Guangxing; Liu, Zhizhao; Hu, Zhigang; Dai, Zhiqiang; Liu, Jingnan
2016-01-20
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.
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
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…
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.
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
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…
Teaching Geometry with Tangrams.
ERIC Educational Resources Information Center
Russell, Dorothy S.; Bologna, Elaine M.
1982-01-01
Geometry is viewed as the most neglected area of the elementary school mathematics curriculum. Tangram activities provide numerous worthwhile mathematical experiences for children. A method of constructing tangrams through paper folding is followed by suggested spatial visualization, measurement, and additional activities. (MP)
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.
NASA Astrophysics Data System (ADS)
Pommerol, A.; Schmitt, B.
Near-IR reflectance spectroscopy is widely used to detect mineral hydration on Solar System surfaces by the observation of absorption bands at 1.9 and 3 µm. Recent studies established empirical relationships between the strength of the 3 µm band and the water content of the studied minerals (Milliken et al., 2005). These results have especially been applied to the OMEGA dataset to derive global maps of the Martian regolith water content (Jouglet et al., 2006 and Milliken et al., 2006). However, parameters such as surface texture and measurement geometry are known to have a strong effect on reflectance spectra but their influence on the hydration bands is poorly documented. The aim of this work is the determination of the quantitative effects of particle size, mixing between materials with different albedo and measurement geometry on the absorption bands at 1.9 and 3 µm. We used both an experimental and a modeling approach to study these effects. Bidirectional reflectance spectra were measured for series of well characterized samples (smectite, volcanic tuff and coals, pure and mixed) and modeled with optical constants of a smectite (Roush, 2005). Criteria commonly used to estimate the strength of the bands were then calculated on these spectra. We show that particle size has a strong effect on the 1.9 and 3 µm bands strength, especially for the finest particles (less than 200 µm). Mixing between a fine smectite powder and anthracite powders with various particle sizes (modeled by a synthetic neutral material) highlights the strong effect of the materials albedo on the hydration band estimation criteria. Measurement geometry has a significant effect on the bands strength for high phase angles. Furthermore, the relative variations of band strength with measurement geometry appear very dependent on the surface texture. We will present in details the relationships between these physical parameters and various criteria chosen to estimate the hydration bands
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…
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®,…
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
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.
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.
Tyagi, Neelam; Yang, Kai; Yan, Di
2014-07-08
The purpose of this study was to compare the measurement-derived (3DVH) dose reconstruction method with machine log file-derived dose reconstruction method in patient geometries for VMAT delivery. A total of ten patient plans were selected from a regular fractionation plan to complex SBRT plans. Treatment sites in the lung and abdomen were chosen to explore the effects of tissue heterogeneity on the respective dose reconstruction algorithms. Single- and multiple-arc VMAT plans were generated to achieve the desired target objectives. Delivered plan in the patient geometry was reconstructed by using ArcCHECK Planned Dose Perturbation (ACPDP) within 3DVH software, and by converting the machine log file to Pinnacle3 9.0 treatment plan format and recalculating dose with CVSP algorithm. In addition, delivered gantry angles between machine log file and 3DVH 4D measurement were also compared to evaluate the accuracy of the virtual inclinometer within the 3DVH. Measured ion chamber and 3DVH-derived isocenter dose agreed with planned dose within 0.4% ± 1.2% and -1.0% ± 1.6%, respectively. 3D gamma analysis showed greater than 98% between log files and 3DVH reconstructed dose. Machine log file reconstructed doses and TPS dose agreed to within 2% in PTV and OARs over the entire treatment. 3DVH reconstructed dose showed an average maximum dose difference of 3% ± 1.2% in PTV, and an average mean difference of -4.5% ± 10.5% in OAR doses. The average virtual inclinometer error (VIE) was -0.65° ± 1.6° for all patients, with a maximum error of -5.16° ± 4.54° for an SRS case. The time averaged VIE was within 1°-2°, and did not have a large impact on the overall accuracy of the estimated patient dose from ACPDP algorithm. In this study, we have compared two independent dose reconstruction methods for VMAT QA. Both methods are capable of taking into account the measurement and delivery parameter discrepancy, and display the delivered dose in CT patient geometry rather than
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…
Second-Order Algebraic Theories
NASA Astrophysics Data System (ADS)
Fiore, Marcelo; Mahmoud, Ola
Fiore and Hur [10] recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.
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
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
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.
NASA Astrophysics Data System (ADS)
Bengtsson, Ingemar; Zyczkowski, Karol
2007-12-01
Preface; 1. Convexity, colours and statistics; 2. Geometry of probability distributions; 3. Much ado about spheres; 4. Complex projective spaces; 5. Outline of quantum mechanics; 6. Coherent states and group actions; 7. The stellar representation; 8. The space of density matrices; 9. Purification of mixed quantum states; 10. Quantum operations; 11. Duality: maps versus states; 12. Density matrices and entropies; 13. Distinguishability measures; 14. Monotone metrics and measures; 15. Quantum entanglement; Epilogue; Appendices; References; Index.
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…
2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras
NASA Astrophysics Data System (ADS)
Ayupov, Shavkat; Kudaybergenov, Karimbergen
2016-03-01
The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2n (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation.
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.
Plethystic algebras and vector symmetric functions.
Rota, G C; Stein, J A
1994-01-01
An isomorphism is established between the plethystic Hopf algebra Pleth(Super[L]) and the algebra of vector symmetric functions. The Hall inner product of symmetric function theory is extended to the Hopf algebra Pleth(Super[L]). PMID:11607504
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
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…
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
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)
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.
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.)
Linear algebra and image processing
NASA Astrophysics Data System (ADS)
Allali, Mohamed
2010-09-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty.
A Programmed Course in Algebra.
ERIC Educational Resources Information Center
Mewborn, Ancel C.; Hively, Wells II
This programed textbook consists of short sections of text interspersed with questions designed to aid the student in understanding the material. The course is designed to increase the student's understanding of some of the basic ideas of algebra. Some general experience and manipulative skill with respect to high school algebra is assumed.…
ERIC Educational Resources Information Center
1997
Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…
Gamow functionals on operator algebras
NASA Astrophysics Data System (ADS)
Castagnino, M.; Gadella, M.; Betán, R. Id; Laura, R.
2001-11-01
We obtain the precise form of two Gamow functionals representing the exponentially decaying part of a quantum resonance and its mirror image that grows exponentially, as a linear, positive and continuous functional on an algebra containing observables. These functionals do not admit normalization and, with an appropriate choice of the algebra, are time reversal of each other.
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.)
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
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…
Elementary maps on nest algebras
NASA Astrophysics Data System (ADS)
Li, Pengtong
2006-08-01
Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.
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…
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…
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)
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.
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.
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.
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.
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.
Constraint algebra in bigravity
NASA Astrophysics Data System (ADS)
Soloviev, V. O.
2015-07-01
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.
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.
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.
Readiness and Preparation for Beginning Algebra.
ERIC Educational Resources Information Center
Rotman, Jack W.
Drawing from experience at Lansing Community College (LCC), this paper discusses how to best prepare students for success in a beginning algebra course. First, an overview is presented of LCC's developmental math sequence, which includes Basic Arithmetic (MTH 008), Pre-Algebra (MTH 009), Beginning Algebra (MTH 012), and Intermediate Algebra (MTH…
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.
Two-parameter twisted quantum affine algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Zhang, Honglian
2016-09-01
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.
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.
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.
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
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.
Towards an invariant geometry of double field theory
NASA Astrophysics Data System (ADS)
Hohm, Olaf; Zwiebach, Barton
2013-03-01
We introduce a geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis. This invariant geometry provides a unifying framework for the frame-like and metric-like formulations developed before. We discuss the relation to generalized geometry and give an "index-free" proof of the algebraic Bianchi identity. Finally, we analyze to what extent the generalized Riemann tensor encodes the curvatures of Riemannian geometry. We show that it contains the conventional Ricci tensor and scalar curvature but not the full Riemann tensor, suggesting the possibility of a further extension of this framework.
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.
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)
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
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).
Sjaardema, G.; Gilkey, A.; Smith, M.; Forsythe, C.
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.
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.
Algebraic Systems and Pushdown Automata
NASA Astrophysics Data System (ADS)
Petre, Ion; Salomaa, Arto
We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.
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…
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.
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.
Causal structure and algebraic classification of non-dissipative linear optical media
NASA Astrophysics Data System (ADS)
Schuller, Frederic P.; Witte, Christof; Wohlfarth, Mattias N. R.
2010-09-01
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.
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.
BRST charges for finite nonlinear algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2010-07-01
Some ingredients of the BRST construction for quantum Lie algebras are applied to a wider class of quadratic algebras of constraints. We build the BRST charge for a quantum Lie algebra with three generators and ghost-anti-ghosts commuting with constraints. We consider a one-parametric family of quadratic algebras with three generators and show that the BRST charge acquires the conventional form after a redefinition of ghosts. The modified ghosts form a quadratic algebra. The family possesses a nonlinear involution, which implies the existence of two independent BRST charges for each algebra in the family. These BRST charges anticommute and form a double BRST complex.
Some Remarks on Kite Pseudo Effect Algebras
NASA Astrophysics Data System (ADS)
Dvurečenskij, Anatolij; Holland, W. Charles
2014-05-01
Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group G, a set I and with two bijections λ, ρ: I→ I. Using a clever construction on the ordinal sum of ( G +) I and ( G -) I , we can define a pseudo effect algebra which can be non-commutative even if G is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop.
Operator product expansion algebra
Holland, Jan; Hollands, Stefan
2013-07-15
We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ{sup 4}-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hep-th/1105.3375, that the 3-point OPE,
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…
Spectral geometry of {kappa}-Minkowski space
D'Andrea, Francesco
2006-06-15
After recalling Snyder's idea [Phys. Rev. 71, 38 (1947)] of using vector fields over a smooth manifold as 'coordinates on a noncommutative space', we discuss a two-dimensional toy-model whose 'dual' noncommutative coordinates form a Lie algebra: this is the well-known {kappa}-Minkowski space [Phys. Lett. B 334, 348 (1994)]. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of {kappa}-Minkowski as linear operators on an Hilbert space (a major problem in the construction of a physical theory), study its 'spectral properties', and discuss how to obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of Dimitrijevic et al. [Eur. Phys. J. C 31, 129 (2003)] can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.
Constructing a parasupersymmetric Virasoro algebra
NASA Astrophysics Data System (ADS)
Kuwata, S.
2011-03-01
We construct a para SUSY Virasoro algebra by generalizing the ordinary fermion in SUSY Virasoro algebra (Ramond or Neveu-Schwarz algebra) to the parafermion. First, we obtain a polynomial relation (PR) between different-mode parafermion fi's by generalizing the corresponding single-mode PR to such that is invariant under the unitary transformation of fi (Green's condition). Differently from a usual context, where the Green's condition is imposed only on the defining relation of fi (degree three with respect to fi and fi†), we impose it on any degree of PR. For the case of order-two parafermion (the simplest case of para SUSY), we calculate a PR between the parasupercharge G0, the bosonic hamiltonian LB0 and parafermionic one LF0, although it is difficult to obtain a PR between G0 and the total hamiltonian L0 (= LB0 + LF0). Finally, we construct a para SUSY Virasoro algebra by generalizing L0 to the Ln's such that form a Virasoro algebra.
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…
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,…
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…
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.
The local geometry of compact homogeneous Lorentz spaces
NASA Astrophysics Data System (ADS)
Günther, Felix
2015-03-01
In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.
A Metric Conceptual Space Algebra
NASA Astrophysics Data System (ADS)
Adams, Benjamin; Raubal, Martin
The modeling of concepts from a cognitive perspective is important for designing spatial information systems that interoperate with human users. Concept representations that are built using geometric and topological conceptual space structures are well suited for semantic similarity and concept combination operations. In addition, concepts that are more closely grounded in the physical world, such as many spatial concepts, have a natural fit with the geometric structure of conceptual spaces. Despite these apparent advantages, conceptual spaces are underutilized because existing formalizations of conceptual space theory have focused on individual aspects of the theory rather than the creation of a comprehensive algebra. In this paper we present a metric conceptual space algebra that is designed to facilitate the creation of conceptual space knowledge bases and inferencing systems. Conceptual regions are represented as convex polytopes and context is built in as a fundamental element. We demonstrate the applicability of the algebra to spatial information systems with a proof-of-concept application.
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.
Colored Quantum Algebra and Its Bethe State
NASA Astrophysics Data System (ADS)
Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun
2014-12-01
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.
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.
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.
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.
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.
Characteristic Numbers of Matrix Lie Algebras
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Fan, En-Gui
2008-04-01
A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.
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.
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…
Ju, Dehao; Shrimpton, John; Bowdrey, Moira; Hearn, Alex
2012-08-01
A breath activated, pressurized metered dose inhaler (pMDI) device (Oxette(®)) has been developed to replace the traditional cigarette. In this paper, internal and external spray characters are measured by high speed imaging along with sizing the residual droplets at the distance from the discharge orifice where the human oropharynx locates. Two different formulations with 95% and 98% mass fraction of HFA 134a and two prototype cigarette alternatives with different expansion chamber volumes have been analyzed. The internal and external flows issuing from early stage prototype Oxette(®) are discussed along with boiling and evaporation phenomena. The expansion and entrainment regions of the jet are observed and discussed with comparison to the turbulent round jet of a single phase. From the visualizations of internal flows in the earlier design, a small expansion chamber can hardly generate small bubbles, which is difficult to produce fine sprays. The larger the expansion chamber volume, the more room for the propellant evaporation, recirculation, bubble generation and growth, all of which produces finer sprays. Therefore the later prototype of Oxette(®) 2 made a significant improvement to produce fine sprays and facilitated development of the cigarette alternative. Furthermore, the characters of the spray generated by Oxette(®) are compared to that issuing from a pMDI by previous researchers, where the residual MMD is larger than that of a pMDI, because the Oxette(®) has a smaller expansion chamber and the geometry provides less opportunity for the recirculation due to restrictions of the design space. Although the formulation with higher mass fraction of HFA 134a can generate smaller droplets, it cannot produce steady puffs with relatively low mass flow rate.
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
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.
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…
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…
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…
A Balancing Act: Making Sense of Algebra
ERIC Educational Resources Information Center
Gavin, M. Katherine; Sheffield, Linda Jensen
2015-01-01
For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…
Algebra? A Gate! A Barrier! A Mystery!
ERIC Educational Resources Information Center
Mathematics Educatio Dialogues, 2000
2000-01-01
This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…
Computer Algebra Systems, Pedagogy, and Epistemology
ERIC Educational Resources Information Center
Bosse, Michael J.; Nandakumar, N. R.
2004-01-01
The advent of powerful Computer Algebra Systems (CAS) continues to dramatically affect curricula, pedagogy, and epistemology in secondary and college algebra classrooms. However, epistemological and pedagogical research regarding the role and effectiveness of CAS in the learning of algebra lags behind. This paper investigates concerns regarding…
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"…
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.
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.
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
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.
Algebraic Legendrian Varieties
NASA Astrophysics Data System (ADS)
Buczyński, Jarosław
2008-05-01
Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space and prior to the author's research only few smooth examples of these were known. The first series of results of this thesis is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable minor assumptions) is completely determined by the sections of the distinguished line bundle on the contact manifold vanishing on the Legendrian variety. Moreover its action preserves the contact structure. The second series of results is devoted to finding new examples of smooth Legendrian subvarieties of projective space. The contribution of this thesis is in three steps: First we find an example of a smooth toric surface. Next we find a smooth quasihomogeneous Fano 8-fold that admits a Legendrian embedding. Finally, we realise that both of these are special cases of a very general construction: a general hyperplane section of a smooth Legendrian variety, after a suitable projection, is a smooth Legendrian variety of smaller dimension. By applying this result to known examples and decomposable Legendrian varieties, we construct infinitely many new examples in every dimension, with various Picard rank, canonical degree, Kodaira dimension and other invariants.
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…
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…
Dimension independence in exterior algebra.
Hawrylycz, M
1995-01-01
The identities between homogeneous expressions in rank 1 vectors and rank n - 1 covectors in a Grassmann-Cayley algebra of rank n, in which one set occurs multilinearly, are shown to represent a set of dimension-independent identities. The theorem yields an infinite set of nontrivial geometric identities from a given identity. PMID:11607520
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…
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…
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.
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.
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…
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…
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…
ERIC Educational Resources Information Center
Deakin, Michael A. B.
1974-01-01
Euler's famous formula, e to the (i, pi) power equals -1, is developed by a purely algebraic method that avoids the use of both trigonometry and calculus. A heuristic outline is given followed by the rigorous theory. Pedagogical considerations for classroom presentation are suggested. (LS)
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…
Math for All Learners: Algebra.
ERIC Educational Resources Information Center
Meader, Pam; Storer, Judy
This book consists of a series of activities aimed at providing a problem solving, hands-on approach so that students can experience concepts in algebra. Topics include ratio and proportion, patterns and formulas, integers, polynomials, linear equations, graphs, and probability. The activities come in the form of reproducible blackline masters…
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…
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…
NASA Astrophysics Data System (ADS)
Kuipers, J.
2012-06-01
New features of the symbolic algebra package Form 4 are discussed. Most importantly, these features include polynomial factorization and polynomial gcd computation. Examples of their use are shown. One of them is an exact version of Mincer which gives answers in terms of rational polynomials and 5 master integrals.
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)
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.
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.
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.
Formal scattering theory by an algebraic approach
NASA Astrophysics Data System (ADS)
Alhassid, Y.; Levine, R. D.
1985-02-01
Formal scattering theory is recast in a Lie-algebraic form. The central result is an algebraic Lippmann-Schwinger equation for the wave operator from which an algebraic form of the Born series (containing only linked terms) is obtained. When a finite Lie algebra is sufficient, The Mo/ller wave operator, on the energy shell, can be solved for explicitly as an element of the corresponding group. The method is illustrated for the separable potential whose relevant algebra is found to be U(1,1).
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.
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.
Mathematical aspects of molecular replacement. II. Geometry of motion spaces.
Chirikjian, Gregory S; Yan, Yan
2012-03-01
Molecular replacement (MR) is a well established computational method for phasing in macromolecular crystallography. In MR searches, spaces of motions are explored for determining the appropriate placement of rigid models of macromolecules in crystallographic asymmetric units. In the first paper of this series, it was shown that this space of motions, when endowed with an appropriate composition operator, forms an algebraic structure called a quasigroup. In this second paper, the geometric properties of these MR search spaces are explored and analyzed. This analysis includes the local differential geometry, global geometry and symmetry properties of these spaces.
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.
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.
Two and three dimensional grid generation by an algebraic homotopy procedure
NASA Technical Reports Server (NTRS)
Moitra, Anutosh
1990-01-01
An algebraic method for generating two- and three-dimensional grid systems for aerospace vehicles is presented. The method is based on algebraic procedures derived from homotopic relations for blending between inner and outer boundaries of any given configuration. Stable properties of homotopic maps have been exploited to provide near-orthogonality and specified constant spacing at the inner boundary. The method has been successfully applied to analytically generated blended wing-body configurations as well as discretely defined geometries such as the High-Speed Civil Transport Aircraft. Grid examples representative of the capabilities of the method are presented.
HOMAR: A computer code for generating homotopic grids using algebraic relations: User's manual
NASA Technical Reports Server (NTRS)
Moitra, Anutosh
1989-01-01
A computer code for fast automatic generation of quasi-three-dimensional grid systems for aerospace configurations is described. The code employs a homotopic method to algebraically generate two-dimensional grids in cross-sectional planes, which are stacked to produce a three-dimensional grid system. Implementation of the algebraic equivalents of the homotopic relations for generating body geometries and grids are explained. Procedures for controlling grid orthogonality and distortion are described. Test cases with description and specification of inputs are presented in detail. The FORTRAN computer program and notes on implementation and use are included.
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.
Linear algebra algorithms for divisors on an algebraic curve
NASA Astrophysics Data System (ADS)
Khuri-Makdisi, Kamal
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to work with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Hess' 1999 Ph.D. thesis, which works with function fields as extensions of $k[x]$. However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor), superelliptic curves (Galbraith, Paulus, and Smart), and $C_{ab}$ curves (Harasawa and Suzuki); in all those cases, one can attain a complexity of $O(g^2)$.
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.
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.
Geometric and algebraic properties of minimal bases of singular systems
NASA Astrophysics Data System (ADS)
Karcanias, Nicos
2013-11-01
For a general singular system ? with an associated pencil T(S), a complete classification of the right polynomial vector pairs ?, connected with the ? rational vector space, is given according to the proper-nonproper property, characterising the relationship of the degrees of those two vectors. An integral part of the classification of right pairs is the development of the notions of canonical and normal minimal bases for ? and ? rational vector spaces, where R(s) is the state restriction pencil of ?. It is shown that the notions of canonical and normal minimal bases are equivalent; the first notion characterises the pure algebraic aspect of the classification, whereas the second is intimately connected to the real geometry properties and the underlying generation mechanism of the proper and nonproper state vectors ?. The results describe the algebraic and geometric dimensions of the invariant partitioning of the set of reachability indices of singular systems. The classification of all proper and nonproper polynomial vectors ? induces a corresponding classification for the reachability spaces to proper-nonproper and results related to the possible dimensions feedback-spectra assignment properties of them are also given. The classification of minimal bases introduces new feedback invariants for singular systems, based on the real geometry of polynomial minimal bases, and provides an extension of the standard theory for proper systems (Warren, M.E., & Eckenberg, A.E. (1975).
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 code 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.
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
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
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.
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…
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.
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.
Introduction to Image Algebra Ada
NASA Astrophysics Data System (ADS)
Wilson, Joseph N.
1991-07-01
Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.
Hidden geometry of traffic jamming
NASA Astrophysics Data System (ADS)
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.
Physiological optics and physical geometry.
Hyder, D J
2001-09-01
Hermann von Helmholtz's distinction between "pure intuitive" and "physical" geometry must be counted as the most influential of his many contributions to the philosophy of science. In a series of papers from the 1860s and 70s, Helmholtz argued against Kant's claim that our knowledge of Euclidean geometry was an a priori condition for empirical knowledge. He claimed that geometrical propositions could be meaningful only if they were taken to concern the behaviors of physical bodies used in measurement, from which it followed that it was posterior to our acquaintance with this behavior. This paper argues that Helmholtz's understanding of geometry was fundamentally shaped by his work in sense-physiology, above all on the continuum of colors. For in the course of that research, Helmholtz was forced to realize that the color-space had no inherent metrical structure. The latter was a product of axiomatic definitions of color-addition and the empirical results of such additions. Helmholtz's development of these views is explained with detailed reference to the competing work of the mathematician Hermann Grassmann and that of the young James Clerk Maxwell. It is this separation between 1) essential properties of a continuum, 2) supplementary axioms concerning distance-measurement, and 3) the behaviors of the physical apparatus used to realize the axioms, which is definitive of Helmholtz's arguments concerning geometry.
ERIC Educational Resources Information Center
Lyublinskaya, Irina; Funsch, Dan
2012-01-01
Several interactive geometry software packages are available today to secondary school teachers. An example is The Geometer's Sketchpad[R] (GSP), also known as Dynamic Geometry[R] software, developed by Key Curriculum Press. This numeric based technology has been widely adopted in the last twenty years, and a vast amount of creativity has been…
Euclidean Geometry via Programming.
ERIC Educational Resources Information Center
Filimonov, Rossen; Kreith, Kurt
1992-01-01
Describes the Plane Geometry System computer software developed at the Educational Computer Systems laboratory in Sofia, Bulgaria. The system enables students to use the concept of "algorithm" to correspond to the process of "deductive proof" in the development of plane geometry. Provides an example of the software's capability and compares it to…
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.…
Frolov, Vadim A; Escalada, Artur; Akimov, Sergey A; Shnyrova, Anna V
2015-01-01
Cellular membranes define the functional geometry of intracellular space. Formation of new membrane compartments and maintenance of complex organelles require division and disconnection of cellular membranes, a process termed membrane fission. Peripheral membrane proteins generally control membrane remodeling during fission. Local membrane stresses, reflecting molecular geometry of membrane-interacting parts of these proteins, sum up to produce the key membrane geometries of fission: the saddle-shaped neck and hour-glass hemifission intermediate. Here, we review the fundamental principles behind the translation of molecular geometry into membrane shape and topology during fission. We emphasize the central role the membrane insertion of specialized protein domains plays in orchestrating fission in vitro and in cells. We further compare individual to synergistic action of the membrane insertion during fission mediated by individual protein species, proteins complexes or membrane domains. Finally, we describe how local geometry of fission intermediates defines the functional design of the protein complexes catalyzing fission of cellular membranes. PMID:25062896
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.
Quantum gravity and causal structures: Second quantization of conformal Dirac algebras
NASA Astrophysics Data System (ADS)
Bonezzi, R.; Corradini, O.; Latini, E.; Waldron, A.
2015-06-01
It is postulated that quantum gravity is a sum over causal structures coupled to matter via scale evolution. Quantized causal structures can be described by studying simple matrix models where matrices are replaced by an algebra of quantum mechanical observables. In particular, previous studies constructed quantum gravity models by quantizing the moduli of Laplace, weight, and defining-function operators on Fefferman-Graham ambient spaces. The algebra of these operators underlies conformal geometries. We extend those results to include fermions by taking an o s p (1 |2 ) "Dirac square root" of these algebras. The theory is a simple, Grassmann, two-matrix model. Its quantum action is a Chern-Simons theory whose differential is a first-quantized, quantum mechanical Becchi-Rouet-Stora-Tyutin operator. The theory is a basic ingredient for building fundamental theories of physical observables.
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
Kumjian-Pask algebras of desourcification
NASA Astrophysics Data System (ADS)
Rosjanuardi, Rizky; Yusnitha, Isnie
2016-02-01
Kumjian-Pask algebra which was introduced by Pino, Clark, an Huef and Raeburn [1] in 2013, gives a purely algebraic version of a k-graph algebra. Rosjanuardi [2] gave necessary and sufficient condition of finitely dimensional complex Kumjian-Pask algebra of row-finite k-graph without sources. We will improve the previous results which allows us to deal with sources. We will consider Kumjian-Pask algebra for locally convex row-finite k-graph which was introduced by Clark, Flynn and an Huef [3], and use the desourcification of the graph to get conditions which characterise when the complex Kumjian-Pask algebra of locally convex row-finite k-graph is finite dimensional.
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
NASA Astrophysics Data System (ADS)
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
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].
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.
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.
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.
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.
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.
Entanglement and algebraic independence in fermion systems
NASA Astrophysics Data System (ADS)
Benatti, Fabio; Floreanini, Roberto
2014-04-01
In the case of systems composed of identical particles, a typical instance in quantum statistical mechanics, the standard approach to separability and entanglement ought to be reformulated and rephrased in terms of correlations between operators from subalgebras localized in spatially disjoint regions. While this algebraic approach is straightforward for bosons, in the case of fermions it is subtler since one has to distinguish between micro-causality, that is the anti-commutativity of the basic creation and annihilation operators, and algebraic independence that is the commutativity of local observables. We argue that a consistent algebraic formulation of separability and entanglement should be compatible with micro-causality rather than with algebraic independence.
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.
ERIC Educational Resources Information Center
Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos
2016-01-01
This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…
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.
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…
Algebraic approach to solve tt dilepton equations
Sonnenschein, Lars
2005-11-01
The set of nonlinear equations describing the standard model kinematics of the top quark antiquark production system in the dilepton decay channel has at most a fourfold ambiguity due to two not fully reconstructed neutrinos. Its most precise solution is of major importance for measurements of top quark properties like the top quark mass and tt spin correlations. Simple algebraic operations allow one to transform the nonlinear equations into a system of two polynomial equations with two unknowns. These two polynomials of multidegree eight can in turn be analytically reduced to one polynomial with one unknown by means of resultants. The obtained univariate polynomial is of degree 16. The number of its real solutions is determined analytically by means of Sturm's theorem, which is as well used to isolate each real solution into a unique pairwise disjoint interval. The solutions are polished by seeking the sign change of the polynomial in a given interval through binary bracketing.
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.
ERIC Educational Resources Information Center
Stanford Univ., CA. School Mathematics Study Group.
The first chapter, Perpendiculars and Parallels (II), of the ninth unit in this SMSG series includes a discussion of the properties of triangles, circles and perpendiculars, parallels in space, perpendicular lines and planes, and parallel planes. The next chapter, on coordinate geometry, covers distance; midpoints; algebraic descriptions of…
Cluster automorphism groups of cluster algebras with coefficients
NASA Astrophysics Data System (ADS)
Chang, Wen; Zhu, Bin
2016-10-01
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e. the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geometric coefficients, and cluster algebras from surfaces (except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.
The Structure of Parafermion Vertex Operator Algebras: General Case
NASA Astrophysics Data System (ADS)
Dong, Chongying; Wang, Qing
2010-11-01
The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.
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…
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)
Gene algebra from a genetic code algebraic structure.
Sanchez, R; Morgado, E; Grau, R
2005-10-01
By considering two important factors involved in the codon-anticodon interactions, the hydrogen bond number and the chemical type of bases, a codon array of the genetic code table as an increasing code scale of interaction energies of amino acids in proteins was obtained. Next, in order to consecutively obtain all codons from the codon AAC, a sum operation has been introduced in the set of codons. The group obtained over the set of codons is isomorphic to the group (Z(64), +) of the integer module 64. On the Z(64)-algebra of the set of 64(N) codon sequences of length N, gene mutations are described by means of endomorphisms f:(Z(64))(N)-->(Z(64))(N). Endomorphisms and automorphisms helped us describe the gene mutation pathways. For instance, 77.7% mutations in 749 HIV protease gene sequences correspond to unique diagonal endomorphisms of the wild type strain HXB2. In particular, most of the reported mutations that confer drug resistance to the HIV protease gene correspond to diagonal automorphisms of the wild type. What is more, in the human beta-globin gene a similar situation appears where most of the single codon mutations correspond to automorphisms. Hence, in the analyses of molecular evolution process on the DNA sequence set of length N, the Z(64)-algebra will help us explain the quantitative relationships between genes.
Noncommutative Geometry and Physics
Connes, Alain
2006-11-03
In this very short essay we shall describe a 'spectral' point of view on geometry which allows to start taking into account the lessons from both renormalization and of general relativity. We shall first do that for renormalization and explain in rough outline the content of our recent collaborations with Dirk Kreimer and Matilde Marcolli leading to the universal Galois symmetry of renormalizable quantum field theories provided by the renormalization group in its cosmic Galois group incarnation. As far as general relativity is concerned, since the functional integral cannot be treated in the traditional perturbative manner, it relies heavily as a 'sum over geometries' on the chosen paradigm of geometric space. This will give us the occasion to discuss, in the light of noncommutative geometry, the issue of 'observables' in gravity and our joint work with Ali Chamseddine on the spectral action, with a first attempt to write down a functional integral on the space of noncommutative geometries.
ERIC Educational Resources Information Center
Chern, Shiing-Shen
1990-01-01
Discussed are the major historical developments of geometry. Euclid, Descartes, Klein's Erlanger Program, Gaus and Riemann, globalization, topology, Elie Cartan, and an application to molecular biology are included as topics. (KR)
Proof in Transformation Geometry
ERIC Educational Resources Information Center
Bell, A. W.
1971-01-01
The first of three articles showing how inductively-obtained results in transformation geometry may be organized into a deductive system. This article discusses two approaches to enlargement (dilatation), one using coordinates and the other using synthetic methods. (MM)
ERIC Educational Resources Information Center
Emenaker, Charles E.
1999-01-01
Describes a sixth-grade interdisciplinary geometry unit based on Charles Dickens's "A Christmas Carol". Focuses on finding area, volume, and perimeter, and working with estimation, decimals, and fractions in the context of making gingerbread houses. (ASK)
Dirac matrices as elements of a superalgebraic matrix algebra
NASA Astrophysics Data System (ADS)
Monakhov, V. V.
2016-08-01
The paper considers a Clifford extension of the Grassmann algebra, in which operators are built from Grassmann variables and by the derivatives with respect to them. It is shown that a subalgebra which is isomorphic to the usual matrix algebra exists in this algebra, the Clifford exten-sion of the Grassmann algebra is a generalization of the matrix algebra and contains superalgebraic operators expanding matrix algebra and produces supersymmetric transformations.
Automated Angular Momentum Recoupling Algebra
NASA Astrophysics Data System (ADS)
Williams, H. T.; Silbar, Richard R.
1992-04-01
We present a set of heuristic rules for algebraic solution of angular momentum recoupling problems. The general problem reduces to that of finding an optimal path from one binary tree (representing the angular momentum coupling scheme for the reduced matrix element) to another (representing the sub-integrals and spin sums to be done). The method lends itself to implementation on a microcomputer, and we have developed such an implementation using a dialect of LISP. We describe both how our code, called RACAH, works and how it appears to the user. We illustrate the use of RACAH for several transition and scattering amplitude matrix elements occurring in atomic, nuclear, and particle physics.
Induced geometry from disformal transformation
NASA Astrophysics Data System (ADS)
Yuan, Fang-Fang; Huang, Peng
2015-05-01
In this note, we use the disformal transformation to induce a geometry from the manifold which is originally Riemannian. The new geometry obtained here can be considered as a generalization of Weyl integrable geometry. Based on these results, we further propose a geometry which is naturally a generalization of Weyl geometry.
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.
Software Geometry in Simulations
NASA Astrophysics Data System (ADS)
Alion, Tyler; Viren, Brett; Junk, Tom
2015-04-01
The Long Baseline Neutrino Experiment (LBNE) involves many detectors. The experiment's near detector (ND) facility, may ultimately involve several detectors. The far detector (FD) will be significantly larger than any other Liquid Argon (LAr) detector yet constructed; many prototype detectors are being constructed and studied to motivate a plethora of proposed FD designs. Whether it be a constructed prototype or a proposed ND/FD design, every design must be simulated and analyzed. This presents a considerable challenge to LBNE software experts; each detector geometry must be described to the simulation software in an efficient way which allows for multiple authors to easily collaborate. Furthermore, different geometry versions must be tracked throughout their use. We present a framework called General Geometry Description (GGD), written and developed by LBNE software collaborators for managing software to generate geometries. Though GGD is flexible enough to be used by any experiment working with detectors, we present it's first use in generating Geometry Description Markup Language (GDML) files to interface with LArSoft, a framework of detector simulations, event reconstruction, and data analyses written for all LAr technology users at Fermilab. Brett is the other of the framework discussed here, the General Geometry Description (GGD).
2005-01-01
The Common Geometry Module (CGM) is a code library which provides geometry functionality used for mesh generation and other applications. This functionality includes that commonly found in solid modeling engines, like geometry creation, query and modification; CGM also includes capabilities not commonly found in solid modeling engines, like geometry decomposition tools and support for shared material interfaces. CGM is built upon the ACIS solid modeling engine, but also includes geometry capability developed beside and onmore » top of ACIS. CGM can be used as-is to provide geometry functionality for codes needing this capability. However, CGM can also be extended using derived classes in C++, allowing the geometric model to serve as the basis for other applications, for example mesh generation. CGM is supported on Sun Solaris, SGI, HP, IBM, DEC, Linux and Windows NT platforms. CGM also indudes support for loading ACIS models on parallel computers, using MPI-based communication. Future plans for CGM are to port it to different solid modeling engines, including Pro/Engineer or SolidWorks. CGM is being released into the public domain under an LGPL license; the ACIS-based engine is available to ACIS licensees on request.« less
Algebraic Thinking through Koch Snowflake Constructions
ERIC Educational Resources Information Center
Ghosh, Jonaki B.
2016-01-01
Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…
Investigating Algebraic Procedures Using Discussion and Writing
ERIC Educational Resources Information Center
Harper, Jonathan; Ford, Jeffrey
2012-01-01
This study reports on the implementation of an intermediate algebra curriculum centered on a framework of student-centered questions designed to investigate algebraic procedures. Instructional activities were designed to build discourse in the small-group discussion meetings of the course. Students were assigned writing prompts to emphasize the…
Practicing Algebraic Skills: A Conceptual Approach
ERIC Educational Resources Information Center
Friedlander, Alex; Arcavi, Abraham
2012-01-01
Traditionally, a considerable part of teaching and learning algebra has focused on routine practice and the application of rules, procedures, and techniques. Although today's computerized environments may have decreased the need to master algebraic skills, procedural competence is still a central component in any mathematical activity. However,…
Using Students' Interests as Algebraic Models
ERIC Educational Resources Information Center
Whaley, Kenneth A.
2012-01-01
Fostering algebraic thinking is an important goal for middle-grades mathematics teachers. Developing mathematical reasoning requires that teachers cultivate students' habits of mind. Teachers develop students' understanding of algebra by engaging them in tasks that involve modeling and representation. This study was designed to investigate how…
THE RADICAL OF A JORDAN ALGEBRA
McCrimmon, Kevin
1969-01-01
In this paper we define a Jacobson radical for Jordan algebras analogous to that for associative algebras and show that it enjoys many of the properties of the associative radical. We then relate the corresponding notion of “semisimplicity” to the previously defined notion of “nondegeneracy” (Jacobson, N., these Proceedings, 55, 243-251 (1966)). PMID:16591736
The operator algebra approach to quantum groups
Kustermans, Johan; Vaes, Stefaan
2000-01-01
A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory. PMID:10639116
Situated Learning in an Abstract Algebra Classroom
ERIC Educational Resources Information Center
Ticknor, Cindy S.
2012-01-01
Advisory committees of mathematics consider abstract algebra as an essential component of the mathematical preparation of secondary teachers, yet preservice teachers find it challenging to connect the topics addressed in this advanced course with the high school algebra they must someday teach. This study analyzed the mathematical content…
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.
Predicting Turkish Ninth Grade Students' Algebra Performance
ERIC Educational Resources Information Center
Erbas, Ayhan Kursat
2005-01-01
The prediction of students' achievement in algebra in eighth and ninth grades has become a research interest for practical issues of placement. A group of simple, easily accessible variables was used to predict student performance in algebra after completion of eighth grade. The three variables of school type, grade level, and previous year…
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…
How To Prepare Students for Algebra.
ERIC Educational Resources Information Center
Wu, H.
2001-01-01
Suggests that no matter how much algebraic thinking is introduced in the early grades, and no matter how worthwhile this might be, the failure rate in algebra will continue unless the teaching of fractions and decimals is radically revamped. The proper study of fractions provides a ramp that leads students gently from whole number arithmetic up to…
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…
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.
Fourier theory and C∗-algebras
NASA Astrophysics Data System (ADS)
Bédos, Erik; Conti, Roberto
2016-07-01
We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included.
Teaching Algebra to Students with Learning Disabilities
ERIC Educational Resources Information Center
Impecoven-Lind, Linda S.; Foegen, Anne
2010-01-01
Algebra is a gateway to expanded opportunities, but it often poses difficulty for students with learning disabilities. Consequently, it is essential to identify evidence-based instructional strategies for these students. The authors begin by identifying three areas of algebra difficulty experienced by students with disabilities: cognitive…
Arithmetic and Cognitive Contributions to Algebra
ERIC Educational Resources Information Center
Cirino, Paul T.; Tolar, Tammy D.; Fuchs, Lynn S.
2013-01-01
Algebra is a prerequisite for access to STEM careers and occupational success (NMAP, 2008a), yet algebra is difficult for students through high school (US DOE, 2008). Growth in children's conceptual and procedural arithmetical knowledge is reciprocal, although conceptual knowledge has more impact on procedural knowledge than the reverse…
Just Say Yes to Early Algebra!
ERIC Educational Resources Information Center
Stephens, Ana; Blanton, Maria; Knuth, Eric; Isler, Isil; Gardiner, Angela Murphy
2015-01-01
Mathematics educators have argued for some time that elementary school students are capable of engaging in algebraic thinking and should be provided with rich opportunities to do so. Recent initiatives like the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) have taken up this call by reiterating the place of early algebra in…
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…
Parabolas: Connection between Algebraic and Geometrical Representations
ERIC Educational Resources Information Center
Shriki, Atara
2011-01-01
A parabola is an interesting curve. What makes it interesting at the secondary school level is the fact that this curve is presented in both its contexts: algebraic and geometric. Being one of Apollonius' conic sections, the parabola is basically a geometric entity. It is, however, typically known for its algebraic characteristics, in particular…
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.
ERIC Educational Resources Information Center
Rickard, Caroline
2008-01-01
Shortly after starting work for the University of Chichester in the School of Teacher Education, the author was planning a session relating to algebra and found herself inspired by an article in MT182: "Algebraic Infants" by Andrews and Sayers (2003). Based on the making of families of "Multilink" animals, Andrews and Sayers (2003) claim that…
Focus on Fractions to Scaffold Algebra
ERIC Educational Resources Information Center
Ooten, Cheryl Thomas
2013-01-01
Beginning algebra is a gatekeeper course into the pipeline to higher mathematics courses required for respected professions in engineering, science, statistics, mathematics, education, and technology. Beginning algebra can also be a perfect storm if the necessary foundational skills are not within a student's grasp. What skills ensure beginning…
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)
Modern Algebra, Mathematics: 5293.36.
ERIC Educational Resources Information Center
Edwards, Raymond J.
This guidebook covers Boolean algebra, matrices, linear transformations of the plane, characteristic values, vectors, and algebraic structures. Overall course goals and performance objectives for each unit are specified; sequencing of units and various time schedules are suggested. A sample pretest and posttest are given, and an annotated list of…
Teaching Modeling and Axiomatization with Boolean Algebra.
ERIC Educational Resources Information Center
De Villiers, Michael D.
1987-01-01
Presented is an alternative approach to the traditional teaching of Boolean algebra for secondary school mathematics. The main aim of the approach is to use Boolean algebra to teach pupils such mathematical processes as modeling and axiomatization. A course using the approach is described. (RH)
MODEL IDENTIFICATION AND COMPUTER ALGEBRA
Bollen, Kenneth A.; Bauldry, Shawn
2011-01-01
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods. PMID:21769158
MODEL IDENTIFICATION AND COMPUTER ALGEBRA.
Bollen, Kenneth A; Bauldry, Shawn
2010-10-01
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods.
Classes of Invariant Subspaces for Some Operator Algebras
NASA Astrophysics Data System (ADS)
Hamhalter, Jan; Turilova, Ekaterina
2014-10-01
New results showing connections between structural properties of von Neumann algebras and order theoretic properties of structures of invariant subspaces given by them are proved. We show that for any properly infinite von Neumann algebra M there is an affiliated subspace such that all important subspace classes living on are different. Moreover, we show that can be chosen such that the set of σ-additive measures on subspace classes of are empty. We generalize measure theoretic criterion on completeness of inner product spaces to affiliated subspaces corresponding to Type I factor with finite dimensional commutant. We summarize hitherto known results in this area, discuss their importance for mathematical foundations of quantum theory, and outline perspectives of further research.
Contact Geometry of Hyperbolic Equations of Generic Type
NASA Astrophysics Data System (ADS)
The, Dennis
2008-08-01
We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampère (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the generic case. An intriguing feature of this class of equations is that every generic hyperbolic equation admits at most a nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp: normal forms for the contact-equivalence classes of these maximally symmetric generic hyperbolic equations are derived and explicit symmetry algebras are presented. Moreover, these maximally symmetric equations are Darboux integrable. An enumeration of several submaximally symmetric (eight and seven-dimensional) generic hyperbolic structures is also given.
Generalized -deformed correlation functions as spectral functions of hyperbolic geometry
NASA Astrophysics Data System (ADS)
Bonora, L.; Bytsenko, A. A.; Guimarães, M. E. X.
2014-08-01
We analyze the role of vertex operator algebra and 2d amplitudes from the point of view of the representation theory of infinite-dimensional Lie algebras, MacMahon and Ruelle functions. By definition p-dimensional MacMahon function, with , is the generating function of p-dimensional partitions of integers. These functions can be represented as amplitudes of a two-dimensional c = 1 CFT, and, as such, they can be generalized to . With some abuse of language we call the latter amplitudes generalized MacMahon functions. In this paper we show that generalized p-dimensional MacMahon functions can be rewritten in terms of Ruelle spectral functions, whose spectrum is encoded in the Patterson-Selberg function of three-dimensional hyperbolic geometry.
Quantum entanglement and geometry of determinantal varieties
Chen Hao
2006-05-15
Quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of Einstein, Podolsky, and Rosen. Recently it has been realized that quantum entanglement is a key ingredient in quantum computation, quantum communication, and quantum cryptography. In this paper, we introduce algebraic sets, which are determinantal varieties in the complex projective spaces or the products of complex projective spaces, for the mixed states on bipartite or multipartite quantum systems as their invariants under local unitary transformations. These invariants are naturally arised from the physical consideration of measuring mixed states by separable pure states. Our construction has applications in the following important topics in quantum information theory: (1) separability criterion, it is proved that the algebraic sets must be a union of the linear subspaces if the mixed states are separable; (2) simulation of Hamiltonians, it is proved that the simulation of semipositive Hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets; (3) construction of bound entangled mixed states, examples of the entangled mixed states which are invariant under partial transpositions (thus PPT bound entanglement) are constructed systematically from our new separability criterion.
Generalization of n-ary Nambu algebras and beyond
Ataguema, H.; Makhlouf, A.; Silvestrov, S.
2009-08-15
The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.
NASA Astrophysics Data System (ADS)
Ochiai, T.; Nacher, J. C.
2011-09-01
Recently, the application of geometry and conformal mappings to artificial materials (metamaterials) has attracted the attention in various research communities. These materials, characterized by a unique man-made structure, have unusual optical properties, which materials found in nature do not exhibit. By applying the geometry and conformal mappings theory to metamaterial science, it may be possible to realize so-called "Harry Potter cloaking device". Although such a device is still in the science fiction realm, several works have shown that by using such metamaterials it may be possible to control the direction of the electromagnetic field at will. We could then make an object hidden inside of a cloaking device. Here, we will explain how to design invisibility device using differential geometry and conformal mappings.
2011-01-01
Cells are highly complex and orderly machines, with defined shapes and a startling variety of internal organizations. Complex geometry is a feature of both free-living unicellular organisms and cells inside multicellular animals. Where does the geometry of a cell come from? Many of the same questions that arise in developmental biology can also be asked of cells, but in most cases we do not know the answers. How much of cellular organization is dictated by global cell polarity cues as opposed to local interactions between cellular components? Does cellular structure persist across cell generations? What is the relationship between cell geometry and tissue organization? What ensures that intracellular structures are scaled to the overall size of the cell? Cell biology is only now beginning to come to grips with these questions. PMID:21880160
Students Discovering Spherical Geometry Using Dynamic Geometry Software
ERIC Educational Resources Information Center
Guven, Bulent; Karatas, Ilhan
2009-01-01
Dynamic geometry software (DGS) such as Cabri and Geometers' Sketchpad has been regularly used worldwide for teaching and learning Euclidean geometry for a long time. The DGS with its inductive nature allows students to learn Euclidean geometry via explorations. However, with respect to non-Euclidean geometries, do we need to introduce them to…
The local symmetries of M-theory and their formulation in generalised geometry
NASA Astrophysics Data System (ADS)
Berman, David S.; Godazgar, Hadi; Godazgar, Mahdi; Perry, Malcolm J.
2012-01-01
In the doubled field theory approach to string theory the T-duality group is promoted to a manifest symmetry at the expense of replacing ordinary Riemannian geometry with generalised geometry on a doubled space. The local symmetries are then given by a generalised Lie derivative and its associated algebra. This paper constructs an analogous structure for the extended geometry of M-theory. A crucial by-product of this construction is the derivation of the physical section condition for M-theory formulated in an extended space.
Computational synthetic geometry
Sturmfels, B. )
1988-01-01
This book deals with methods for realizing abstract geometric objects in concrete vector spaces. It considers a large class of problems from convexity and discrete geometry including constructing convex polytopes from simplicial complexes, vector geometries from incidence structures and hyperplane arrangements from oriented matroids. It appears that algorithms for these constructions exist if and only if arbitrary polynomial equations are decidable with respect to the underlying field. Besides such complexity theorems, a variety of symbolic algorithms are discussed, and the methods are applied to obtain mathematical results on convex polytopes, projective configurations and the combinatories of Grassmann varieties.
Weak homological dimensions and biflat Koethe algebras
Pirkovskii, A Yu
2008-06-30
The homological properties of metrizable Koethe algebras {lambda}(P) are studied. A criterion for an algebra A={lambda}(P) to be biflat in terms of the Koethe set P is obtained, which implies, in particular, that for such algebras the properties of being biprojective, biflat, and flat on the left are equivalent to the surjectivity of the multiplication operator A otimes-hat A{yields}A. The weak homological dimensions (the weak global dimension w.dg and the weak bidimension w.db) of biflat Koethe algebras are calculated. Namely, it is shown that the conditions w.db {lambda}(P)<=1 and w.dg {lambda}(P)<=1 are equivalent to the nuclearity of {lambda}(P); and if {lambda}(P) is non-nuclear, then w.dg {lambda}(P)=w.db {lambda}(P)=2. It is established that the nuclearity of a biflat Koethe algebra {lambda}(P), under certain additional conditions on the Koethe set P, implies the stronger estimate db {lambda}(P), where db is the (projective) bidimension. On the other hand, an example is constructed of a nuclear biflat Koethe algebra {lambda}(P) such that db {lambda}(P)=2 (while w.db {lambda}(P)=1). Finally, it is shown that many biflat Koethe algebras, while not being amenable, have trivial Hochschild homology groups in positive degrees (with arbitrary coefficients). Bibliography: 37 titles.
Algebraic curves of maximal cyclicity
NASA Astrophysics Data System (ADS)
Caubergh, Magdalena; Dumortier, Freddy
2006-01-01
The paper deals with analytic families of planar vector fields, studying methods to detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit cycles that can locally bifurcate from it. It is known that this multi-parameter problem can be reduced to a single-parameter one, in the sense that there exist analytic curves in parameter space along which the maximal cyclicity can be attained. In that case one speaks about a maximal cyclicity curve (mcc) in case only the number is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is also taken into account. In view of obtaining efficient algorithms for detecting the cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear depending on certain general properties of the families or of their associated Bautin ideal. In any case by well chosen examples we show that prudence is appropriate.
Inequalities, assessment and computer algebra
NASA Astrophysics Data System (ADS)
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary curricula. We consider the formal mathematical processes by which such inequalities are solved, and we consider the notation and syntax through which solutions are expressed. We review the extent to which current CAS can accurately solve these inequalities, and the form given to the solutions by the designers of this software. Finally, we discuss the functionality needed to deal with students' answers, i.e. to establish equivalence (or otherwise) of expressions representing unions of intervals. We find that while contemporary CAS accurately solve inequalities there is a wide variety of notation used.
Local Algebras of Differential Operators
NASA Astrophysics Data System (ADS)
Church, P. T.; Timourian, J. G.
2002-05-01
There is an increasing literature devoted to the study of boundary value problems using singularity theory. The resulting differential operators are typically Fredholm with index 0, defined on infinite-dimensional spaces, and they have often led to folds, cusps, and even higher-order Morin singularities. In this paper we develop some of the local algebras of germs of such differential Fredholm operators, extending the theory of the finite-dimensional case. We apply this work to nonlinear elliptic boundary value problems: in particular, we make further progress on a question proposed and initially studied by Ruf [1999, J. Differential Equations 151, 111-133]. We also make comments on several problems raised by others.
PC Basic Linear Algebra Subroutines
1992-03-09
PC-BLAS is a highly optimized version of the Basic Linear Algebra Subprograms (BLAS), a standardized set of thirty-eight routines that perform low-level operations on vectors of numbers in single and double-precision real and complex arithmetic. Routines are included to find the index of the largest component of a vector, apply a Givens or modified Givens rotation, multiply a vector by a constant, determine the Euclidean length, perform a dot product, swap and copy vectors, andmore » find the norm of a vector. The BLAS have been carefully written to minimize numerical problems such as loss of precision and underflow and are designed so that the computation is independent of the interface with the calling program. This independence is achieved through judicious use of Assembly language macros. Interfaces are provided for Lahey Fortran 77, Microsoft Fortran 77, and Ryan-McFarland IBM Professional Fortran.« less
Powell, Sarah R; Fuchs, Lynn S
2014-08-01
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 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty.
Powell, Sarah R; Fuchs, Lynn S
2014-08-01
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 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty. PMID:25309044
Thornby, John; Landheer, Dirk; Williams, Tim; Barnes-Warden, Jane; Fenne, Paul; Norman, Daniel; Attridge, Alex; Williams, Mark A
2014-01-01
Fundamental to any ballistic armour standard is the reference projectile to be defeated. Typically, for certification purposes, a consistent and symmetrical bullet geometry is assumed, however variations in bullet jacket dimensions can have far reaching consequences. Traditionally, characteristics and internal dimensions have been analysed by physically sectioning bullets--an approach which is of restricted scope and which precludes subsequent ballistic assessment. The use of a non-destructive X-ray computed tomography (CT) method has been demonstrated and validated (Kumar et al., 2011 [15]); the authors now apply this technique to correlate bullet impact response with jacket thickness variations. A set of 20 bullets (9 mm DM11) were selected for comparison and an image-based analysis method was employed to map jacket thickness and determine the centre of gravity of each specimen. Both intra- and inter-bullet variations were investigated, with thickness variations of the order of 200 μm commonly found along the length of all bullets and angular variations of up to 50 μm in some. The bullets were subsequently impacted against a rigid flat plate under controlled conditions (observed on a high-speed video camera) and the resulting deformed projectiles were re-analysed. The results of the experiments demonstrate a marked difference in ballistic performance between bullets from different manufacturers and an asymmetric thinning of the jacket is observed in regions of pre-impact weakness. The conclusions are relevant for future soft armour standards and provide important quantitative data for numerical model correlation and development. The implications of the findings of the work on the reliability and repeatability of the industry standard V50 ballistic test are also discussed.
DeLucca, John F; Peloquin, John M; Smith, Lachlan J; Wright, Alexander C; Vresilovic, Edward J; Elliott, Dawn M
2016-08-01
Geometry is an important indicator of disc mechanical function and degeneration. While the geometry and associated degenerative changes in the nucleus pulposus and the annulus fibrosus are well-defined, the geometry of the cartilage endplate (CEP) and its relationship to disc degeneration are unknown. The objectives of this study were to quantify CEP geometry in three dimensions using an MRI FLASH imaging sequence and evaluate relationships between CEP geometry and age, degeneration, spinal level, and overall disc geometry. To do so, we assessed the MRI-based measurements for accuracy and repeatability. Next, we measured CEP geometry across a larger sample set and correlated CEP geometric parameters to age, disc degeneration, level, and disc geometry. The MRI-based measures resulted in thicknesses (0.3-1 mm) that are comparable to prior measurements of CEP thickness. CEP thickness was greatest at the anterior/posterior (A/P) margins and smallest in the center. The CEP A/P thickness, axial area, and lateral width decreased with age but were not related to disc degeneration. Age-related, but not degeneration-related, changes in geometry suggest that the CEP may not follow the progression of disc degeneration. Ultimately, if the CEP undergoes significant geometric changes with aging and if these can be related to low back pain, a clinically feasible translation of the FLASH MRI-based measurement of CEP geometry presented in this study may prove a useful diagnostic tool. © 2016 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 34:1410-1417, 2016.
Jucys-Murphy elements for Birman-Murakami-Wenzl algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Ogievetsky, O. V.
2011-05-01
The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.
Kinematical superalgebras and Lie algebras of order 3
Campoamor-Stursberg, R.; Rausch de Traubenberg, M.
2008-06-15
We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonue-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order 3.
Becchi-Rouet-Stora-Tyutin operators for W algebras
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2008-07-15
The study of quantum Lie algebras motivates a use of noncanonical ghosts and antighosts for nonlinear algebras, such as W-algebras. This leads, for the W{sub 3} and W{sub 3}{sup (2)} algebras, to the Becchi-Rouet-Stora-Tyutin operator having the conventional cubic form.
Validation of efficiency transfer for Marinelli geometries.
Ferreux, Laurent; Pierre, Sylvie; Thanh, Tran Thien; Lépy, Marie-Christine
2013-11-01
In the framework of environmental measurements by gamma-ray spectrometry, some laboratories need to characterize samples in geometries for which a calibration is not directly available. A possibility is to use an efficiency transfer code, e.g., ETNA. However, validation for large volume sources, such as Marinelli geometries, is needed. With this aim in mind, ETNA is compared, initially to a Monte Carlo simulation (PENELOPE) and subsequently to experimental data obtained with a high-purity germanium detector (HPGe).
[Image reconstruction of computerized tomography pictures using functional algebra].
Bradaczek, M; Bradaczek, H
1997-07-01
A detailed presentation of the process for calculating computed tomograms from the measured data by means of functional algebra is given and an attempt is made to demonstrate the relationships to those inexperienced in mathematics. Suggestions are also made to the manufacturers for improving tomography software although the authors cannot exclude the possibility that some of the recommendations may have already been realized. An interpolation in Fourier space to right-angled coordinates was not employed so that additional computer time and errors resulting from the interpolation are avoided. The savings in calculation time can only be estimated but should amount to about 25%. The error-correction calculation is merely a suggestion since it depends considerably on the apparatus used. Functional algebra is introduced here because it is not so well known but does provide appreciable simplifications in comparison to an explicit presentation. Didactic reasons as well as the possibility for reducing calculation time provided the foundation for this work.
Imperfect Cloning Operations in Algebraic Quantum Theory
NASA Astrophysics Data System (ADS)
Kitajima, Yuichiro
2015-01-01
No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal -imperfect cloning operation which tolerates a finite loss of fidelity in the cloned state, and show that an individual system's algebra of observables is abelian if and only if there is a universal -imperfect cloning operation in the case where the loss of fidelity is less than . Therefore in this case no universal -imperfect cloning operation is possible in algebraic quantum theory.
Advanced geometries and regimes
Bulanov, S. S.; Bulanov, S. V.; Turchetti, G.; Limpouch, J.; Klimo, O.; Psikal, J.; Margarone, D.; Korn, G.
2013-07-26
We review and discuss different schemes of laser ion acceleration as well as advanced target geometries in connection with the development of the laser-driven proton source for hadron therapy of oncological diseases, which is a part of the ELIMED project.
Spacetime and Euclidean geometry
NASA Astrophysics Data System (ADS)
Brill, Dieter; Jacobson, Ted
2006-04-01
Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the spacetime Pythagoras theorem.
ERIC Educational Resources Information Center
Hartz, Viggo
1981-01-01
Allowing students to use a polystyrene cutter to fashion their own three-dimensional models is suggested as a means of allowing individuals to experience problems and develop ideas related to solid geometry. A list of ideas that can lead to mathematical discovery is provided. (MP)
Sliding vane geometry turbines
Sun, Harold Huimin; Zhang, Jizhong; Hu, Liangjun; Hanna, Dave R
2014-12-30
Various systems and methods are described for a variable geometry turbine. In one example, a turbine nozzle comprises a central axis and a nozzle vane. The nozzle vane includes a stationary vane and a sliding vane. The sliding vane is positioned to slide in a direction substantially tangent to an inner circumference of the turbine nozzle and in contact with the stationary vane.
ERIC Educational Resources Information Center
MacKeown, P. K.
1984-01-01
Clarifies two concepts of gravity--those of a fictitious force and those of how space and time may have geometry. Reviews the position of Newton's theory of gravity in the context of special relativity and considers why gravity (as distinct from electromagnetics) lends itself to Einstein's revolutionary interpretation. (JN)
ERIC Educational Resources Information Center
Martin, John
2010-01-01
The cycloid has been called the Helen of Geometry, not only because of its beautiful properties but also because of the quarrels it provoked between famous mathematicians of the 17th century. This article surveys the history of the cycloid and its importance in the development of the calculus.
ERIC Educational Resources Information Center
Case, Christine L.
1991-01-01
Presented is an activity in which students make models of viruses, which allows them to visualize the shape of these microorganisms. Included are some background on viruses, the biology and geometry of viruses, directions for building viruses, a comparison of cells and viruses, and questions for students. (KR)
ERIC Educational Resources Information Center
KLIER, KATHERINE M.
PRESENTED IS A FUSED COURSE IN PLANE, SOLID, AND COORDINATE GEOMETRY. ELEMENTARY SET THEORY, LOGIC, AND THE PRINCIPLE OF SEPARATION PROVIDE UNIFYING THREADS THROUGHOUT THE TEXT. THE TWO CURRICULUM GUIDES HAVE BEEN PREPARED FOR USE WITH TWO DIFFERENT TEXTS. EITHER CURRICULUM GUIDE MAY BE USED DEPENDING UPON THE CHOICE OF THE TEACHER AND THE NEEDS…
NASA Astrophysics Data System (ADS)
Prástaro, Agostino
2008-02-01
Following our previous results on this subject [R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(I): Webs on PDE's and integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17 (2007) 239-266; R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(II): Webs on PDE's and integral bordism groups. Applications to Riemannian geometry PDE's, Adv. Math. Sci. Appl. 17 (2007) 267-285; A. Prástaro, Geometry of PDE's and Mechanics, World Scientific, Singapore, 1996; A. Prástaro, Quantum and integral (co)bordism in partial differential equations, Acta Appl. Math. (5) (3) (1998) 243-302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59 (2) (1999) 111-201; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co, Singapore, 2004, 500 pp.; A. Prástaro, Geometry of PDE's. I: Integral bordism groups in PDE's, J. Math. Anal. Appl. 319 (2006) 547-566; A. Prástaro, Geometry of PDE's. II: Variational PDE's and integral bordism groups, J. Math. Anal. Appl. 321 (2006) 930-948; A. Prástaro, Th.M. Rassias, Ulam stability in geometry of PDE's, Nonlinear Funct. Anal. Appl. 8 (2) (2003) 259-278; I. Stakgold, Boundary Value Problems of Mathematical Physics, I, The MacMillan Company, New York, 1967; I. Stakgold, Boundary Value Problems of Mathematical Physics, II, Collier-MacMillan, Canada, Ltd, Toronto, Ontario, 1968], integral bordism groups of the Navier-Stokes equation are calculated for smooth, singular and weak solutions, respectively. Then a characterization of global solutions is made on this ground. Enough conditions to assure existence of global smooth solutions are given and related to nullity of integral characteristic numbers of the boundaries. Stability of global solutions are related to some characteristic numbers of the space-like Cauchy dataE Global solutions of variational problems constrained by (NS) are classified by means of suitable integral bordism groups too.
A note on derivations of Murray–von Neumann algebras
Kadison, Richard V.; Liu, Zhe
2014-01-01
A Murray–von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray–von Neumann algebras. We show that the “extended derivations” of a Murray–von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray–von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer’s seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements. PMID:24469831
Global differential geometry: An introduction for control engineers
NASA Technical Reports Server (NTRS)
Doolin, B. F.; Martin, C. F.
1982-01-01
The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.
Calabi-Yau Geometries: Algorithms, Databases and Physics
NASA Astrophysics Data System (ADS)
He, Yang-Hui
2013-08-01
With a bird's-eye view, we survey the landscape of Calabi-Yau threefolds, compact and noncompact, smooth and singular. Emphasis will be placed on the algorithms and databases which have been established over the years, and how they have been useful in the interaction between the physics and the mathematics, especially in string and gauge theories. A skein which runs through this review will be algorithmic and computational algebraic geometry and how, implementing its principles on powerful computers and experimenting with the vast mathematical data, new physics can be learnt. It is hoped that this interdisciplinary glimpse will be of some use to the beginning student.
NASA Astrophysics Data System (ADS)
Rabal, A. M.; Ferrero, A.; Campos, J.; Fontecha, J. L.; Pons, A.; Rubiño, A. M.; Corróns, A.
2012-06-01
This paper presents the description and the characterization of the gonio-spectrophotometer GEFE (the acronym for 'Gonio-EspectroFotómetro Español'). This device has been designed and built for the low-uncertainty absolute measurement of the bidirectional reflectance distribution function (BRDF). It comprises a fixed, collimated and uniform light source, a six-axis robot-arm to rotate the sample and a spectroradiometer that may revolve around the sample to be able to vary the source-to-detector angular separation. This gonio-spectrophotometer makes it possible to perform spectral measurements in the visible range, both inside and outside the incidence plane, as well as measurements in retroreflection conditions. This fully automated system is able to measure autonomously a sample's complete spectral BRDF (comprising around 1000 different angular configurations) in less than 4 h.
I CAN Learn[R] Pre-Algebra and Algebra. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2009
2009-01-01
The I CAN Learn[R] Education System is an interactive, self-paced, mastery-based software system that includes the I CAN Learn[R] Fundamentals of Math (5th-6th grade math) curriculum, the I CAN Learn[R] Pre-Algebra curriculum, and the I CAN Learn[R] Algebra curriculum. College algebra credit is also available to students in participating schools…
Highest-weight representations of Brocherd`s algebras
Slansky, R.
1997-01-01
General features of highest-weight representations of Borcherd`s algebras are described. to show their typical features, several representations of Borcherd`s extensions of finite-dimensional algebras are analyzed. Then the example of the extension of affine- su(2) to a Borcherd`s algebra is examined. These algebras provide a natural way to extend a Kac-Moody algebra to include the hamiltonian and number-changing operators in a generalized symmetry structure.
On \\delta-derivations of n-ary algebras
NASA Astrophysics Data System (ADS)
Kaygorodov, Ivan B.
2012-12-01
We give a description of \\delta-derivations of (n+1)-dimensional n-ary Filippov algebras and, as a consequence, of simple finite-dimensional Filippov algebras over an algebraically closed field of characteristic zero. We also give new examples of non-trivial \\delta-derivations of Filippov algebras and show that there are no non-trivial \\delta-derivations of the simple ternary Mal'tsev algebra M_8.
Supersymmetric extension of Galilean conformal algebras
Bagchi, Arjun; Mandal, Ipsita
2009-10-15
The Galilean conformal algebra has recently been realized in the study of the nonrelativistic limit of the AdS/CFT conjecture. This was obtained by a systematic parametric group contraction of the parent relativistic conformal field theory. In this paper, we extend the analysis to include supersymmetry. We work at the level of the coordinates in superspace to construct the N=1 super-Galilean conformal algebra. One of the interesting outcomes of the analysis is that one is able to naturally extend the finite algebra to an infinite one. This looks structurally similar to the N=1 superconformal algebra in two dimensions, but is different. We also comment on the extension of our construction to cases of higher N.
Algebraic structures of sequences of numbers
NASA Astrophysics Data System (ADS)
Huang, I.-Chiau
2012-09-01
For certain sequences of numbers, commutative rings with a module structure over a non-commutative ring are constructed. Identities of these numbers are considered as realizations of algebraic relations.
Representations of filtered solvable Lie algebras
Panov, Alexander N
2012-01-31
The representation theory of filtered solvable Lie algebras is constructed. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found. Bibliography: 9 titles.
Structure of The Planar Galilean Conformal Algebra
NASA Astrophysics Data System (ADS)
Gao, Shoulan; Liu, Dong; Pei, Yufeng
2016-08-01
In this paper, we compute the low-dimensional cohomology groups of the planar Galilean conformal algebra introduced by Bagchi and Goparkumar. Consequently we determine its derivations, central extensions, and automorphisms.
Applications: Using Algebra in an Accounting Practice.
ERIC Educational Resources Information Center
Eisner, Gail A.
1994-01-01
Presents examples of algebra from the field of accounting including proportional ownership of stock, separation of a loan payment into principal and interest portions, depreciation methods, and salary withholdings computations. (MKR)
Using computer algebra and SMT solvers in algebraic biology
NASA Astrophysics Data System (ADS)
Pineda Osorio, Mateo
2014-05-01
Biologic processes are represented as Boolean networks, in a discrete time. The dynamics within these networks are approached with the help of SMT Solvers and the use of computer algebra. Software such as Maple and Z3 was used in this case. The number of stationary states for each network was calculated. The network studied here corresponds to the immune system under the effects of drastic mood changes. Mood is considered as a Boolean variable that affects the entire dynamics of the immune system, changing the Boolean satisfiability and the number of stationary states of the immune network. Results obtained show Z3's great potential as a SMT Solver. Some of these results were verified in Maple, even though it showed not to be as suitable for the problem approach. The solving code was constructed using Z3-Python and Z3-SMT-LiB. Results obtained are important in biology systems and are expected to help in the design of immune therapies. As a future line of research, more complex Boolean network representations of the immune system as well as the whole psychological apparatus are suggested.
Numerical linear algebra in data mining
NASA Astrophysics Data System (ADS)
Eldén, Lars
Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.
Vague Congruences and Quotient Lattice Implication Algebras
Qin, Xiaoyan; Xu, Yang
2014-01-01
The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given. PMID:25133207
Algebraic operator approach to gas kinetic models
NASA Astrophysics Data System (ADS)
Il'ichov, L. V.
1997-02-01
Some general properties of the linear Boltzmann kinetic equation are used to present it in the form ∂ tϕ = - Â†Âϕ with the operators ÂandÂ† possessing some nontrivial algebraic properties. When applied to the Keilson-Storer kinetic model, this method gives an example of quantum ( q-deformed) Lie algebra. This approach provides also a natural generalization of the “kangaroo model”.
Algebraic sub-structuring for electromagnetic applications
Yang, Chao; Gao, Weiguo; Bai, Zhaojun; Li, Xiaoye; Lee, Lie-Quan; Husbands, Parry; Ng, Esmond G.
2004-09-14
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, we show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Algebra and topology for applications to physics
NASA Technical Reports Server (NTRS)
Rozhkov, S. S.
1987-01-01
The principal concepts of algebra and topology are examined with emphasis on applications to physics. In particular, attention is given to sets and mapping; topological spaces and continuous mapping; manifolds; and topological groups and Lie groups. The discussion also covers the tangential spaces of the differential manifolds, including Lie algebras, vector fields, and differential forms, properties of differential forms, mapping of tangential spaces, and integration of differential forms.
Algebraic Sub-Structuring for Electromagnetic Applications
Yang, C.; Gao, W.G.; Bai, Z.J.; Li, X.Y.S.; Lee, L.Q.; Husbands, P.; Ng, E.G.; /LBL, Berkeley /UC, Davis /SLAC
2006-06-30
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, they show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Benjamin J. Crowe III
2009-09-30
Nucleon-deuteron (Nd) breakup is an important tool for obtaining a better understanding of three-nucleon (3N) dynamics and for developing meson exchange descriptions of nuclear systems. The kinematics of the nd breakup reaction enable observables to be studied in a variety of exit-channel configurations that show sensitivity to realistic nucleon-nucleon (NN) potential models and three-nucleon force (3NF) models. Rigorous 3N calculations give very good descriptions of most 3N reaction data. However, there are still some serious discrepancies between data and theory. The largest discrepancy observed between theory and data for nd breakup is for the cross section for the space-star configuration. This discrepancy is known as the “Space star Anomaly”. Several experimental groups have obtained results consistent with the “Space Star Anomaly”, but it is important to note that they all used essentially the same experimental setup and so their experimental results are subject to the same systematic errors. We propose to measure the space-star cross-section at the Triangle Universities Nuclear Laboratory (TUNL) using an experimental technique that is significantly different from the one used in previous breakup experiments. This technique has been used by a research group from the University of Bonn to measure the neutron-neutron scattering length. There are three possible scenarios for the outcome of this work: 1) the new data are consistent with previous measurements; 2) the new data are not in agreement with previous measurements, but are in agreement with theory; and 3) the new data are not in agreement with either theory or previous measurements. Any one of the three scenarios will provide valuable insight on the Space Star Anomaly.
Nebeck, H.E.
1986-08-01
The MAZE mesh generator represents an arbitrary two dimensional region of space as an ordered collection of quadrilateral elements. Each element is defined by its four corner points (nodes) and an integer material number. Models are created by subdividing the region(s) of interest into one or more PARTS and specifying the element distribution in each part. Then, parts can be merged together to form the meshed representation of the entire region. Applying boundary conditions and describing material properties completes the model construction process. This activity takes place in three distinct phases: phase I-define geometry, subdivide regions into elements; phase II-refine geometry, establish interface and boundary conditions; phase III-describe material properties. This work presents explanations and examples of the phase I commands, along with an overview of the MAZE mesh generation process.
Cylindrical geometry hall thruster
Raitses, Yevgeny; Fisch, Nathaniel J.
2002-01-01
An apparatus and method for thrusting plasma, utilizing a Hall thruster with a cylindrical geometry, wherein ions are accelerated in substantially the axial direction. The apparatus is suitable for operation at low power. It employs small size thruster components, including a ceramic channel, with the center pole piece of the conventional annular design thruster eliminated or greatly reduced. Efficient operation is accomplished through magnetic fields with a substantial radial component. The propellant gas is ionized at an optimal location in the thruster. A further improvement is accomplished by segmented electrodes, which produce localized voltage drops within the thruster at optimally prescribed locations. The apparatus differs from a conventional Hall thruster, which has an annular geometry, not well suited to scaling to small size, because the small size for an annular design has a great deal of surface area relative to the volume.
From Atiyah Classes to Homotopy Leibniz Algebras
NASA Astrophysics Data System (ADS)
Chen, Zhuo; Stiénon, Mathieu; Xu, Ping
2016-01-01
A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes T X [-1] into a Lie algebra object in D + ( X), the bounded below derived category of coherent sheaves on X. Furthermore, Kapranov proved that, for a Kähler manifold X, the Dolbeault resolution {Ω^{bullet-1}(T_X^{1, 0})} of T X [-1] is an L ∞ algebra. In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair ( L, A), i.e. a Lie algebroid L together with a Lie subalgebroid A, we define the Atiyah class α E of an A-module E as the obstruction to the existence of an A- compatible L-connection on E. We prove that the Atiyah classes α L/ A and α E respectively make L/ A[-1] and E[-1] into a Lie algebra and a Lie algebra module in the bounded below derived category {D^+(A)} , where {A} is the abelian category of left {U(A)} -modules and {U(A)} is the universal enveloping algebra of A. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of L/ A and E, and inducing the aforesaid Lie structures in {D^+(A)}.
Inflation from quantum geometry.
Bojowald, Martin
2002-12-23
Quantum geometry predicts that a universe evolves through an inflationary phase at small volume before exiting gracefully into a standard Friedmann phase. This does not require the introduction of additional matter fields with ad hoc potentials; rather, it occurs because of a quantum gravity modification of the kinetic part of ordinary matter Hamiltonians. An application of the same mechanism can explain why the present day cosmological acceleration is so tiny.
Freezing in confined geometries
NASA Technical Reports Server (NTRS)
Sokol, P. E.; Ma, W. J.; Herwig, K. W.; Snow, W. M.; Wang, Y.; Koplik, Joel; Banavar, Jayanth R.
1992-01-01
Results of detailed structural studies, using elastic neutron scattering, of the freezing of liquid O2 and D2 in porous vycor glass, are presented. The experimental studies have been complemented by computer simulations of the dynamics of freezing of a Lennard-Jones liquid in narrow channels bounded by molecular walls. Results point to a new simple physical interpretation of freezing in confined geometries.
The algebra of two dimensional generalized Chebyshev-Koornwinder oscillator
Borzov, V. V.; Damaskinsky, E. V.
2014-10-15
In the previous works of Borzov and Damaskinsky [“Chebyshev-Koornwinder oscillator,” Theor. Math. Phys. 175(3), 765–772 (2013)] and [“Ladder operators for Chebyshev-Koornwinder oscillator,” in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillator-like system that is associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev-Koornwinder oscillator. In this paper, we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev-Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space H of functions that are defined on a region which is bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev-Koornwinder polynomials and these polynomials form an orthonormalized basis in the space H. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.
Integral geometry and holography
Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel; Sully, James
2015-10-27
We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS_{3}/CFT_{2} correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS_{3} whose kinematic space is two-dimensional de Sitter space.
Integral geometry and holography
Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel; Sully, James
2015-10-27
We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS3/CFT2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulkmore » curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS3 whose kinematic space is two-dimensional de Sitter space.« less
Emergent Complex Network Geometry
NASA Astrophysics Data System (ADS)
Wu, Zhihao; Menichetti, Giulia; Rahmede, Christoph; Bianconi, Ginestra
2015-05-01
Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant for routing problems, inference and data mining. In real growing networks, topological, structural and geometrical properties emerge spontaneously from their dynamical rules. Nevertheless we still miss a model in which networks develop an emergent complex geometry. Here we show that a single two parameter network model, the growing geometrical network, can generate complex network geometries with non-trivial distribution of curvatures, combining exponential growth and small-world properties with finite spectral dimensionality. In one limit, the non-equilibrium dynamical rules of these networks can generate scale-free networks with clustering and communities, in another limit planar random geometries with non-trivial modularity. Finally we find that these properties of the geometrical growing networks are present in a large set of real networks describing biological, social and technological systems.
Instanton superpotentials, Calabi-Yau geometry, and fibrations
NASA Astrophysics Data System (ADS)
Anderson, Lara B.; Apruzzi, Fabio; Gao, Xin; Gray, James; Lee, Seung-Joo
2016-04-01
In this paper we explore contributions to nonperturbative superpotentials arising from instantons wrapping effective divisors in smooth Calabi-Yau fourfolds. We concentrate on the case of manifolds constructed as complete intersections in products of projective spaces or generalizations thereof. We systematically investigate the structure of the cone of effective (algebraic) divisors in the fourfold geometries and employ the same tools recently developed by Anderson et al. [arXiv:1507.03235] to construct more general instanton geometries than have previously been considered in the literature. We provide examples of instanton configurations on Calabi-Yau manifolds that are elliptically and K 3 fibered and explore their consequences in the context of string dualities. The examples discussed include manifolds containing infinite families of divisors with arithmetic genus, χ (D ,OD)=1 , and superpotentials exhibiting modular symmetry.