Sample records for a-type hecke algebras

  1. On boundary fusion and functional relations in the Baxterized affine Hecke algebra

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

    Babichenko, A., E-mail: babichen@weizmann.ac.il; Regelskis, V., E-mail: v.regelskis@surrey.ac.uk

    2014-04-15

    We construct boundary type operators satisfying fused reflection equation for arbitrary representations of the Baxterized affine Hecke algebra. These operators are analogues of the fused reflection matrices in solvable half-line spin chain models. We show that these operators lead to a family of commuting transfer matrices of Sklyanin type. We derive fusion type functional relations for these operators for two families of representations.

  2. Eisenstein Hecke algebras and Iwasawa theory

    NASA Astrophysics Data System (ADS)

    Wake, Preston

    We show that if an Eisenstein component of the p-adic Hecke algebra associated to modular forms is Gorenstein, then it is necessary that the plus-part of a certain ideal class group is trivial. We also show that this condition is sufficient whenever a conjecture of Sharifi holds. We also formulate a weaker Gorenstein property, and show that this weak Gorenstein property holds if and only if a weak form of Sharifi's conjecture and a weak form of Greenberg's conjecture hold.

  3. Two-rowed Hecke algebra representations at roots of unity

    NASA Astrophysics Data System (ADS)

    Welsh, Trevor Alan

    1996-02-01

    In this paper, we initiate a study into the explicit construction of irreducible representations of the Hecke algebraH n (q) of typeA n-1 in the non-generic case whereq is a root of unity. The approach is via the Specht modules ofH n (q) which are irreducible in the generic case, and possess a natural basis indexed by Young tableaux. The general framework in which the irreducible non-genericH n (q)-modules are to be constructed is set up and, in particular, the full set of modules corresponding to two-part partitions is described. Plentiful examples are given.

  4. Spherical Hecke algebra in the Nekrasov-Shatashvili limit

    NASA Astrophysics Data System (ADS)

    Bourgine, Jean-Emile

    2015-01-01

    The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal variations of the roots. These operators obey the commutation relations defining the SHc algebra at first order in the equivariant parameter ɛ 2. Furthermore, their action on the bifundamental contributions reproduces the Kanno-Matsuo-Zhang transformation. We also discuss the connections with the Mayer cluster expansion approach that leads to TBA-like equations.

  5. A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank

    NASA Astrophysics Data System (ADS)

    Fu, Yuchen; Shelley-Abrahamson, Seth

    2016-06-01

    We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using R-matrices for U_q(sl_N). Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.

  6. Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation

    NASA Astrophysics Data System (ADS)

    Rim, Chaiho; Zhang, Hong

    2017-06-01

    AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with intertwiners we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state, with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conformal states corresponding to Argyres-Douglas theory, which involves summation of functions over Young diagrams.

  7. A simple and facile Heck-type arylation of alkenes with diaryliodonium salts using magnetically recoverable Pd-catalyst

    EPA Science Inventory

    The Heck-type arylation of alkenes was achieved in aqueous polyethylene glycol using a magnetically recoverable heterogenized palladium catalyst employing diaryliodonium salts under ambient conditions. The benign reaction medium and the stability of the catalyst are the salient f...

  8. Palladium-catalyzed Heck-type cross-couplings of unactivated alkyl iodides.

    PubMed

    McMahon, Caitlin M; Alexanian, Erik J

    2014-06-02

    A palladium-catalyzed, intermolecular Heck-type coupling of alkyl iodides and alkenes is described. This process is successful with a variety of primary and secondary unactivated alkyl iodides as reaction partners, including those with hydrogen atoms in the β position. The mild catalytic conditions enable intermolecular C-C bond formations with a diverse set of alkyl iodides and alkenes, including substrates containing base- or nucleophile-sensitive functionality. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

  9. A non-symmetric Yang-Baxter algebra for the quantum nonlinear Schrödinger model

    NASA Astrophysics Data System (ADS)

    Vlaar, Bart

    2013-06-01

    We study certain non-symmetric wavefunctions associated with the quantum nonlinear Schrödinger model, introduced by Komori and Hikami using Gutkin’s propagation operator, which involves representations of the degenerate affine Hecke algebra. We highlight how these functions can be generated using a vertex-type operator formalism similar to the recursion defining the symmetric (Bethe) wavefunction in the quantum inverse scattering method. Furthermore, some of the commutation relations encoded in the Yang-Baxter equation for the relevant monodromy matrix are generalized to the non-symmetric case.

  10. Copper-catalyzed oxidative Heck reactions between alkyltrifluoroborates and vinyl arenes.

    PubMed

    Liwosz, Timothy W; Chemler, Sherry R

    2013-06-21

    We report herein that potassium alkyltrifluoroborates can be utilized in oxidative Heck-type reactions with vinyl arenes. The reaction is catalyzed by a Cu(OTf)2/1,10-phenanthroline with MnO2 as the stoichiometric oxidant. In addition to the alkyl Heck, amination, esterification, and dimerization reactions of alkyltrifluoroborates are demonstrated under analogous reaction conditions. Evidence for an alkyl radical intermediate is presented.

  11. Drinfeld-Sokolov reduction in quantum algebras: canonical form of generating matrices

    NASA Astrophysics Data System (ADS)

    Gurevich, Dimitri; Saponov, Pavel; Talalaev, Dmitry

    2018-04-01

    We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley-Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904-911, 2008).

  12. Remarks towards the spectrum of the Heisenberg spin chain type models

    NASA Astrophysics Data System (ADS)

    Burdík, Č.; Fuksa, J.; Isaev, A. P.; Krivonos, S. O.; Navrátil, O.

    2015-05-01

    The integrable close and open chain models can be formulated in terms of generators of the Hecke algebras. In this review paper, we describe in detail the Bethe ansatz for the XXX and the XXZ integrable close chain models. We find the Bethe vectors for two-component and inhomogeneous models. We also find the Bethe vectors for the fermionic realization of the integrable XXX and XXZ close chain models by means of the algebraic and coordinate Bethe ansatz. Special modification of the XXZ closed spin chain model ("small polaron model") is considered. Finally, we discuss some questions relating to the general open Hecke chain models.

  13. Quantum Torus Algebras and B(C)-Type Toda Systems

    NASA Astrophysics Data System (ADS)

    Wang, Na; Li, Chuanzhong

    2017-12-01

    In this paper, we construct a new even constrained B(C)-type Toda hierarchy and derive its B(C)-type Block-type additional symmetry. Also we generalize the B(C)-type Toda hierarchy to the N-component B(C)-type Toda hierarchy which is proved to have symmetries of a coupled \\bigotimes ^NQT_+ algebra ( N-fold direct product of the positive half of the quantum torus algebra QT).

  14. Block algebra in two-component BKP and D type Drinfeld-Sokolov hierarchies

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

    Li, Chuanzhong, E-mail: lichuanzhong@nbu.edu.cn; He, Jingsong, E-mail: hejingsong@nbu.edu.cn

    We construct generalized additional symmetries of a two-component BKP hierarchy defined by two pseudo-differential Lax operators. These additional symmetry flows form a Block type algebra with some modified (or additional) terms because of a B type reduction condition of this integrable hierarchy. Further we show that the D type Drinfeld-Sokolov hierarchy, which is a reduction of the two-component BKP hierarchy, possess a complete Block type additional symmetry algebra. That D type Drinfeld-Sokolov hierarchy has a similar algebraic structure as the bigraded Toda hierarchy which is a differential-discrete integrable system.

  15. Generic, Type-Safe and Object Oriented Computer Algebra Software

    NASA Astrophysics Data System (ADS)

    Kredel, Heinz; Jolly, Raphael

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

  16. Ni-Catalyzed Regioselective 1,2-Dicarbofunctionalization of Olefins by Intercepting Heck Intermediates as Imine-Stabilized Transient Metallacycles

    DOE PAGES

    Shrestha, Bijay; Basnet, Prakash; Dhungana, Roshan K.; ...

    2017-07-24

    We disclose a strategy for Ni-catalyzed regioselective dicarbofunctionalization of olefins in styrene derivatives by intercepting Heck C(sp 3)-NiX intermediates with arylzinc reagents. This approach utilizes a readily removable imine as a coordinating group that plays a dual role of intercepting oxidative addition species derived from aryl halides and triflates to promote Heck carbometallation, and stabilizing the Heck C(sp 3)-NiX intermediates as transient metallacycles to suppress β-hydride elimination and facilitate transmetalation/reductive elimination steps. This method affords diversely-substituted 1,1,2-riarylethyl products that occur as structural motifs in various natural products.

  17. Higher Inductive Types as Homotopy-Initial Algebras

    DTIC Science & Technology

    2016-08-01

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

  18. Relationships between Classroom Schedule Types and Performance on the Algebra I Criterion-Referenced Test

    ERIC Educational Resources Information Center

    Murray, Gregory V.; Moyer-Packenham, Patricia S.

    2014-01-01

    One option for length of individual mathematics class periods is the schedule type selected for Algebra I classes. This study examined the relationship between student achievement, as indicated by Algebra I Criterion-Referenced Test scores, and the schedule type for Algebra I classes. Data obtained from the Utah State Office of Education included…

  19. Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type

    NASA Astrophysics Data System (ADS)

    Dey, Sanjib; Fring, Andreas; Mathanaranjan, Thilagarajah

    2015-11-01

    We propose a noncommutative version of the Euclidean Lie algebra E 2. Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.

  20. The Heck Reaction: A Microscale Synthesis Using a Palladium Catalyst

    NASA Astrophysics Data System (ADS)

    Martin, William B.; Kateley, Laura J.

    2000-06-01

    Palladium catalysts are central to a large variety of modern organic syntheses. Heck reactions use palladium acetate as the preferred precatalyst to effect vinylic substitutions involving haloarenes and haloalkenes. The microscale synthesis described uses a reaction between a bromoiodobenzene and acrylic acid to produce a bromocinnamic acid. Structure verification for the product uses IR and 1H NMR spectroscopy. This experiment is appropriate for a second-semester introductory organic chemistry laboratory or an intermediate-level organic synthesis laboratory. It could be adapted as a project for two or three students, with each member of the group preparing a different isomer or using a different catalyst source.

  1. Magnetically Separable Fe3O4@DOPA-Pd: A Heterogeneous Catalyst for Aqueous Heck Reaction

    EPA Science Inventory

    Magnetically separable Fe3O4@DOPA-Pd catalyst has been synthesized via anchoring of palladium over dopamine-coated magnetite via non-covalent interaction and the catalyst is utilized for expeditious Heck coupling in aqueous media.

  2. Kinetic Study of the Heck Reaction: An Interdisciplinary Experiment

    ERIC Educational Resources Information Center

    Gozzi, Christel; Bouzidi, Naoual

    2008-01-01

    The aim of this experiment is to study and calculate the kinetic constant of a Heck reaction: the arylation of but-3-en-2-ol by iodobenzene catalyzed by palladium acetate in presence of triethylamine in DMF. The reaction leads to a mixture of two ketones. Students use GC analysis to quantify reagents and products of reaction. They control the…

  3. The behaviour of resonances in Hecke triangular billiards under deformation

    NASA Astrophysics Data System (ADS)

    Howard, P. J.; O'Mahony, P. F.

    2007-08-01

    The right-hand boundary of Artin's billiard on the Poincaré half-plane is continuously deformed to generate a class of chaotic billiards which includes fundamental domains of the Hecke groups Γ(2, n) at certain values of the deformation parameter. The quantum scattering problem in these open chaotic billiards is described and the distributions of both real and imaginary parts of the resonant eigenvalues are investigated. The transitions to arithmetic chaos in the cases n ∈ {4, 6} are closely examined and the explicit analytic form for the scattering matrix is given together with the Fourier coefficients for the scattered wavefunction. The n = 4 and 6 cases have an additional set of regular equally spaced resonances compared to Artin's billiard (n = 3). For a general deformation, a numerical procedure is presented which generates the resonance eigenvalues and the evolution of the eigenvalues is followed as the boundary is varied continuously which leads to dramatic changes in their distribution. For deformations away from the non-generic arithmetic cases, including that of the tiling Hecke triangular billiard n = 5, the distributions of the positions and widths of the resonances are consistent with the predictions of a random matrix theory.

  4. Breaking the regioselectivity rule for acrylate insertion in the Mizoroki-Heck reaction.

    PubMed

    Wucher, Philipp; Caporaso, Lucia; Roesle, Philipp; Ragone, Francesco; Cavallo, Luigi; Mecking, Stefan; Göttker-Schnetmann, Inigo

    2011-05-31

    In modern methods for the preparation of small molecules and polymers, the insertion of substrate carbon-carbon double bonds into metal-carbon bonds is a fundamental step of paramount importance. This issue is illustrated by Mizoroki-Heck coupling as the most prominent example in organic synthesis and also by catalytic insertion polymerization. For unsymmetric substrates H(2)C = CHX the regioselectivity of insertion is decisive for the nature of the product formed. Electron-deficient olefins insert selectively in a 2,1-fashion for electronic reasons. A means for controlling this regioselectivity is lacking to date. In a combined experimental and theoretical study, we now report that, by destabilizing the transition state of 2,1-insertion via steric interactions, the regioselectivity of methyl acrylate insertion into palladium-methyl and phenyl bonds can be inverted entirely to yield the opposite "regioirregular" products in stoichiometric reactions. Insights from these experiments will aid the rational design of complexes which enable a catalytic and regioirregular Mizoroki-Heck reaction of electron-deficient olefins.

  5. Quantum cluster algebras and quantum nilpotent algebras.

    PubMed

    Goodearl, Kenneth R; Yakimov, Milen T

    2014-07-08

    A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein-Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405-455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337-397] for the case of symmetric Kac-Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1-52] associated with double Bruhat cells coincide with the corresponding cluster algebras.

  6. Quantum cluster algebras and quantum nilpotent algebras

    PubMed Central

    Goodearl, Kenneth R.; Yakimov, Milen T.

    2014-01-01

    A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197

  7. Enacted Types of Algebraic Activity in Different Classes Taught by the Same Teacher

    ERIC Educational Resources Information Center

    Eisenmann, Tammy; Even, Ruhama

    2011-01-01

    The aim of this study was to examine how teachers enact the same written algebra curriculum materials in different classes. The study addresses this issue by comparing the types of algebraic activity (Kieran, 2004) enacted in two 7th grade classes taught by the same teacher, using the same textbook. Data sources include lesson observations and an…

  8. 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.

  9. Dynamical systems defined on infinite dimensional lie algebras of the ''current algebra'' or ''Kac-Moody'' type

    NASA Astrophysics Data System (ADS)

    Hermann, Robert

    1982-07-01

    Recent work by Morrison, Marsden, and Weinstein has drawn attention to the possibility of utilizing the cosymplectic structure of the dual of the Lie algebra of certain infinite dimensional Lie groups to study hydrodynamical and plasma systems. This paper treats certain models arising in elementary particle physics, considered by Lee, Weinberg, and Zumino; Sugawara; Bardacki, Halpern, and Frishman; Hermann; and Dolan. The lie algebras involved are associated with the ''current algebras'' of Gell-Mann. This class of Lie algebras contains certain of the algebras that are called ''Kac-Moody algebras'' in the recent mathematics and mathematical physics literature.

  10. Affine cellularity of finite type KLR algebras, and homomorphisms between Specht modules for KLR algebras in affine type A

    NASA Astrophysics Data System (ADS)

    Loubert, Joseph William

    This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras Ralpha of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in Ralpha are generated by idempotents. This in particular implies the (known) result that the global dimension of Ralpha is finite. In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material.

  11. Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces

    NASA Astrophysics Data System (ADS)

    Meljanac, Stjepan; Krešić–Jurić, Saša; Martinić, Tea

    2017-07-01

    This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra g0, we construct a Lie superalgebra g =g0⊕g1 containing noncommutative coordinates and one-forms. We show that g can be extended by a set of generators TAB whose action on the enveloping algebra U (g ) gives the commutation relations between monomials in U (g0 ) and one-forms. Realizations of noncommutative coordinates, one-forms, and the generators TAB as formal power series in a semicompleted Weyl superalgebra are found. In the special case dim(g0 ) =dim(g1 ) , we also find a realization of the exterior derivative on U (g0 ) . The realizations of these geometric objects yield a bicovariant differential calculus on U (g0 ) as a deformation of the standard calculus on the Euclidean space.

  12. Reactions of nitroxides 15. Cinnamates bearing a nitroxyl moiety synthesized using a Mizoroki–Heck cross-coupling reaction

    PubMed Central

    Huras, Bogumiła

    2015-01-01

    Summary Cinnamic acid derivatives bearing a nitroxyl moiety (2,2,6,6-tetramethyl-1-oxyl-4-piperidyl 3-E-aryl acrylates) were synthesized in 30–100% yield using a Mizoroki–Heck cross-coupling reaction between 4-acryloyloxy-2,2,6,6-tetramethylpiperidine-1-oxyl and iodobenzene derivatives in the presence of palladium(II) acetate coordinated with a tri(o-tolyl)phosphine ligand immobilized in a polyurea matrix. PMID:26199672

  13. A note on derivations of Murray–von Neumann algebras

    PubMed Central

    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

  14. A note on derivations of Murray-von Neumann algebras.

    PubMed

    Kadison, Richard V; Liu, Zhe

    2014-02-11

    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.

  15. Regularized inner products and weakly holomorphic Hecke eigenforms

    NASA Astrophysics Data System (ADS)

    Bringmann, Kathrin; Kane, Ben

    2018-01-01

    We show that the image of repeated differentiation on weak cusp forms is precisely the subspace which is orthogonal to the space of weakly holomorphic modular forms. This gives a new interpretation of weakly holomorphic Hecke eigenforms. The research of the first author is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220—AQSER. The research of the second author was supported by grants from the Research Grants Council of the Hong Kong SAR, China (project numbers HKU 27300314, 17302515, and 17316416).

  16. Classification of commutator algebras leading to the new type of closed Baker-Campbell-Hausdorff formulas

    NASA Astrophysics Data System (ADS)

    Matone, Marco

    2015-11-01

    We show that there are {\\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\\exp(X)\\exp(Y)\\exp(Z)=\\exp({AX+BZ+CY+DI}) \\ , $$ derived in arXiv:1502.06589, JHEP {\\bf 1505} (2015) 113. This includes, as a particular case, $\\exp(X) \\exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\\exp(X)\\exp(Y)\\exp(Z)=\\exp(X)\\exp({\\alpha Y}) \\exp({(1-\\alpha) Y}) \\exp(Z)$, with $\\alpha$ fixed in such a way that it reduces to $\\exp({\\tilde X})\\exp({\\tilde Y})$, with $\\tilde X$ and $\\tilde Y$ satisfying the Van-Brunt and Visser condition $[\\tilde X,\\tilde Y]=\\tilde u\\tilde X+\\tilde v\\tilde Y+\\tilde cI$. It turns out that $e^\\alpha$ satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine {\\it types} of commutator algebras, such an equation leads to rational solutions for $\\alpha$. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity.

  17. A non-commutative *-algebra of Borel functions

    NASA Astrophysics Data System (ADS)

    Hart, Robert

    To the pair (E, sigma), where E is a countable Borel equivalence relation on a standard Borel space ( X, A ) and sigma a normalized Borel T -valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra B*r (E, sigma), contained in the bounded linear operators on ℓ2(E). Associated to B*r (E, sigma) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L( Bo (X)) isomorphic to the bounded Borel functions on X. Then L( Bo (X)) and its normalizer (the set of the unitaries u ∈ B*r (E, sigma) such that u* fu ∈ L( Bo (X)), f ∈ L( Bo (X))) countably generates the Borel *-algebra B*r (E, sigma). In this thesis, we study B*r (E, sigma) and in particular prove that: i) If E is smooth, then B*r (E, sigma) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then B*r (E, sigma) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist Gamma over E and its associated sequentially closed Borel *-algebra B*r (Gamma). iv) Let a Borel Cartan pair ( B,B0 ) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra B0 , where B is countably B0 -generated. Generalizing Feldman-Moore's result, we prove that any pair ( B,B0 ) can be realized uniquely as a pair ( B*r (E, sigma), L( Bo (X))). Moreover, we show that the pair ( B*r (E), L( Bo (X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras B*r (E1) and B*r (E2) are isomorphic.

  18. Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver

    NASA Astrophysics Data System (ADS)

    Bourgine, J.-E.; Fukuda, M.; Matsuo, Y.; Zhu, R.-D.

    2017-12-01

    Reflection states are introduced in the vertical and horizontal modules of the Ding-Iohara-Miki (DIM) algebra (quantum toroidal gl_1 ). Webs of DIM representations are in correspondence with ( p, q)-web diagrams of type IIB string theory, under the identification of the algebraic intertwiner of Awata, Feigin and Shiraishi with the refined topological vertex. Extending the correspondence to the vertical reflection states, it is possible to engineer the N=1 quiver gauge theory of D-type (with unitary gauge groups). In this way, the Nekrasov instanton partition function is reproduced from the evaluation of expectation values of intertwiners. This computation leads to the identification of the vertical reflection state with the orientifold plane of string theory. We also provide a translation of this construction in the Iqbal-Kozcaz-Vafa refined topological vertex formalism.

  19. Polyhedral realizations of crystal bases for quantum algebras of classical affine types

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

    Hoshino, A.

    2013-05-15

    We give the explicit forms of the crystal bases B({infinity}) for the quantum affine algebras of types A{sub 2n-1}{sup (2)}, A{sub 2n}{sup (2)}, B{sub n}{sup (1)}, C{sub n}{sup (1)}, D{sub n}{sup (1)}, and D{sub n+1}{sup (2)} by using the method of polyhedral realizations of crystal bases.

  20. Case Report of Focal Epithelial Hyperplasia (Heck's Disease) with Polymerase Chain Reaction Detection of Human Papillomavirus 13.

    PubMed

    Brehm, Mary A; Gordon, Katie; Firan, Miahil; Rady, Peter; Agim, Nnenna

    2016-05-01

    Focal epithelial hyperplasia (FEH), or Heck's disease, is an uncommon benign proliferation of oral mucosa caused by the human papillomavirus (HPV), particularly subtypes 13 and 32. The disease typically presents in young Native American patients and is characterized by multiple asymptomatic papules and nodules on the oral mucosa, lips, tongue, and gingiva. The factors that determine susceptibility to FEH are unknown, but the ethnic and geographic distribution of FEH suggests that genetic predisposition, particularly having the human lymphocytic antigen DR4 type, may be involved in pathogenesis. We report a case of FEH with polymerase chain reaction detection of HPV13 in a healthy 11-year-old Hispanic girl and discuss the current understanding of disease pathogenesis, susceptibility, and treatment. © 2016 Wiley Periodicals, Inc.

  1. A Categorification of the Crystal Isomorphism B 1,1 B + B(Lambda i) = B(Lambdasigma (i) and a Graphical Calculus for the Shifted Symmetric Functions

    NASA Astrophysics Data System (ADS)

    Kvinge, Henry

    We prove two results at the intersection of Lie theory and the representation theory of symmetric groups, Hecke algebras, and their generalizations. The first is a categorification of the crystal isomorphism B. (1,1) tensor B1,1 ⊕ B(Lambdai ) ≅ B(Lambdasigma (i)). Here B(Lambdai and B(Lambda sigma(i)) are two affine type highest weight crystals of weight Lambdai and Lambdasigma (i) respectively, sigma is a specific map from the Dynkin indexing set I to itself, and B1,1 is a Kirillov-Reshetikhin crystal. We show that this crystal isomorphism is in fact the shadow of a richer module-theoretic phenomenon in the representation theory of Khovanov-Lauda-Rouquier algebras of classical affine type. Our second result identifies the center EndH'( 1) of Khovanov's Heisenberg category H', as the algebra of shifted symmetric functions Lambda* of Okounkov and Olshanski, i.e. End H'(1) ≅ Lambda*. This isomorphism provides us with a graphical calculus for Lambda*. It also allows us to describe EndH'(1) in terms of the transition and co-transition measure of Kerov and the noncommutative probability spaces of Biane.

  2. Classical Affine W-Algebras and the Associated Integrable Hamiltonian Hierarchies for Classical Lie Algebras

    NASA Astrophysics Data System (ADS)

    De Sole, Alberto; Kac, Victor G.; Valeri, Daniele

    2018-06-01

    We prove that any classical affine W-algebra W (g, f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of generalized Adler type operators and of generalized quasideterminants, which we develop in the paper. Moreover, we show that under certain conditions, the product of two generalized Adler type operators is a Lax type operator. We use this fact to construct a large number of integrable Hamiltonian systems, recovering, as a special case, all KdV type hierarchies constructed by Drinfeld and Sokolov.

  3. The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.

    PubMed

    Yu, Zhang; Zhang, Yufeng

    2009-01-15

    With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.

  4. Macdonald index and chiral algebra

    NASA Astrophysics Data System (ADS)

    Song, Jaewon

    2017-08-01

    For any 4d N = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type ( A 1 , A 2 n ) and ( A 1 , D 2 n+1) where the chiral algebras are given by Virasoro and \\widehat{su}(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.

  5. Macdonald index and chiral algebra

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

    Song, Jaewon

    For any 4dN = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. Here, we conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type (A 1, A 2n) and (A 1, D 2n+1) where the chiral algebras are given by Virasoro andmore » $$ˆ\\atop{su}$$(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.« less

  6. Macdonald index and chiral algebra

    DOE PAGES

    Song, Jaewon

    2017-08-10

    For any 4dN = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. Here, we conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type (A 1, A 2n) and (A 1, D 2n+1) where the chiral algebras are given by Virasoro andmore » $$ˆ\\atop{su}$$(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.« less

  7. Enhancement in the catalytic activity of Pd/USY in the heck reaction induced by H2 bubbling.

    PubMed

    Okumura, Kazu; Tomiyama, Takuya; Moriyama, Sayaka; Nakamichi, Ayaka; Niwa, Miki

    2010-12-24

    Pd was loaded on ultra stable Y (USY) zeolites prepared by steaming NH(4)-Y zeolite under different conditions. Heck reactions were carried out over the prepared Pd/USY. We found that H₂ bubbling was effective in improving not only the catalytic activity of Pd/USY, but also that of other supported Pd catalysts and Pd(OAc)₂. Moreover, the catalytic activity of Pd/USY could be optimized by choosing appropriate steaming conditions for the preparation of the USY zeolites; Pd loaded on USY prepared at 873 K with 100% H₂O gave the highest activity (TOF = 61,000 h⁻¹), which was higher than that of Pd loaded on other kinds of supports. The prepared Pd/USY catalysts were applicable to the Heck reactions using various kinds of substrates including bromo- and chloro-substituted aromatic and heteroaromatic compounds. Characterization of the acid properties of the USY zeolites revealed that the strong acid site (OH(strong)) generated as a result of steaming had a profound effect on the catalytic activity of Pd.

  8. The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

    PubMed Central

    Yu, Zhang; Zhang, Yufeng

    2009-01-01

    With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings. PMID:20084092

  9. Pd loaded amphiphilic COF as catalyst for multi-fold Heck reactions, C-C couplings and CO oxidation

    PubMed Central

    Mullangi, Dinesh; Nandi, Shyamapada; Shalini, Sorout; Sreedhala, Sheshadri; Vinod, Chathakudath P.; Vaidhyanathan, Ramanathan

    2015-01-01

    COFs represent a class of polymers with designable crystalline structures capable of interacting with active metal nanoparticles to form excellent heterogeneous catalysts. Many valuable ligands/monomers employed in making coordination/organic polymers are prepared via Heck and C-C couplings. Here, we report an amphiphilic triazine COF and the facile single-step loading of Pd0 nanoparticles into it. An 18–20% nano-Pd loading gives highly active composite working in open air at low concentrations (Conc. Pd(0) <0.05 mol%, average TON 1500) catalyzing simultaneous multiple site Heck couplings and C-C couplings using ‘non-boronic acid’ substrates, and exhibits good recyclability with no sign of catalyst leaching. As an oxidation catalyst, it shows 100% conversion of CO to CO2 at 150 °C with no loss of activity with time and between cycles. Both vapor sorptions and contact angle measurements confirm the amphiphilic character of the COF. DFT-TB studies showed the presence of Pd-triazine and Pd-Schiff bond interactions as being favorable. PMID:26057044

  10. Synthesis of Pd-N-heterocyclic carbene Pd-catalyst and its application in MW-assisted Heck and Suzuki reaction

    EPA Science Inventory

    The first Pd-N-heterocyclic carbene (NHC) complex in the form of organic silica is prepared using sol-gel method and its application in Heck and Suzuki reactions are demonstrated. These C-C coupling reactions proceeded efficiently under the influence of microwave irradiation, wit...

  11. Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

    NASA Astrophysics Data System (ADS)

    Saveliev, M. V.; Vershik, A. M.

    1989-12-01

    We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations of Z-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras “continuum Lie algebras.” The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.

  12. Derivation of rigorous conditions for high cell-type diversity by algebraic approach.

    PubMed

    Yoshida, Hiroshi; Anai, Hirokazu; Horimoto, Katsuhisa

    2007-01-01

    The development of a multicellular organism is a dynamic process. Starting with one or a few cells, the organism develops into different types of cells with distinct functions. We have constructed a simple model by considering the cell number increase and the cell-type order conservation, and have assessed conditions for cell-type diversity. This model is based on a stochastic Lindenmayer system with cell-to-cell interactions for three types of cells. In the present model, we have successfully derived complex but rigorous algebraic relations between the proliferation and transition rates for cell-type diversity by using a symbolic method: quantifier elimination (QE). Surprisingly, three modes for the proliferation and transition rates have emerged for large ratios of the initial cells to the developed cells. The three modes have revealed that the equality between the development rates for the highest cell-type diversity is reduced during the development process of multicellular organisms. Furthermore, we have found that the highest cell-type diversity originates from order conservation.

  13. Filiform Lie algebras of order 3

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

    Navarro, R. M., E-mail: rnavarro@unex.es

    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 lamore » 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.« less

  14. 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,…

  15. The quantum holonomy-diffeomorphism algebra and quantum gravity

    NASA Astrophysics Data System (ADS)

    Aastrup, Johannes; Grimstrup, Jesper Møller

    2016-03-01

    We introduce the quantum holonomy-diffeomorphism ∗-algebra, which is generated by holonomy-diffeomorphisms on a three-dimensional manifold and translations on a space of SU(2)-connections. We show that this algebra encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Furthermore, we show that semiclassical states exist on the holonomy-diffeomorphism part of the algebra but that these states cannot be extended to the full algebra. Via a Dirac-type operator we derive a certain class of unbounded operators that act in the GNS construction of the semiclassical states. These unbounded operators are the type of operators, which we have previously shown to entail the spatial three-dimensional Dirac operator and Dirac-Hamiltonian in a semiclassical limit. Finally, we show that the structure of the Hamilton constraint emerges from a Yang-Mills-type operator over the space of SU(2)-connections.

  16. Multicomponent, flow diazotization/Mizoroki-Heck coupling protocol: dispelling myths about working with diazonium salts.

    PubMed

    Nalivela, Kumara S; Tilley, Michael; McGuire, Michael A; Organ, Michael G

    2014-05-26

    A single pass flow diazotization/Mizoroki-Heck protocol has been developed for the production of cinnimoyl and styryl products. The factors that govern aryl diazonium salt stability have been examined in detail leading to the development of a MeOH/DMF co-solvent system in which the diazonium salts can be generated in the presence of all other reaction components and then coupled selectively to give the desired products. Finally the key role of the reaction quench for flow reactions has been demonstrated. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

  17. Classification and Realizations of Type III Factor Representations of Cuntz-Krieger Algebras Associated with Quasi-Free States

    NASA Astrophysics Data System (ADS)

    Kawamura, Katsunori

    2009-03-01

    We completely classify type III factor representations of Cuntz-Krieger algebras associated with quasi-free states up to unitary equivalence. Furthermore, we realize these representations on concrete Hilbert spaces without using GNS construction. Free groups and their type II1 factor representations are used in these realizations.

  18. Scope and limitations of the Heck-Matsuda-coupling of phenol diazonium salts and styrenes: a protecting-group economic synthesis of phenolic stilbenes.

    PubMed

    Schmidt, Bernd; Elizarov, Nelli; Berger, René; Hölter, Frank

    2013-06-14

    4-Phenol diazonium salts undergo Pd-catalyzed Heck reactions with various styrenes to 4'-hydroxy stilbenes. In almost all cases higher yields and fewer side products were observed, compared to the analogous 4-methoxy benzene diazonium salts. In contrast, the reaction fails completely with 2- and 3-phenol diazonium salts. For these substitution patterns the methoxy-substituted derivatives are superior.

  19. The hopf algebra of vector fields on complex quantum groups

    NASA Astrophysics Data System (ADS)

    Drabant, Bernhard; Jurčo, Branislav; Schlieker, Michael; Weich, Wolfgang; Zumino, Bruno

    1992-10-01

    We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types A n , B n , C n , and D n by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.

  20. Rational design of cyclopropane-based chiral PHOX ligands for intermolecular asymmetric Heck reaction

    PubMed Central

    Rubina, Marina; Sherrill, William M; Barkov, Alexey Yu

    2014-01-01

    Summary A novel class of chiral phosphanyl-oxazoline (PHOX) ligands with a conformationally rigid cyclopropyl backbone was synthesized and tested in the intermolecular asymmetric Heck reaction. Mechanistic modelling and crystallographic studies were used to predict the optimal ligand structure and helped to design a very efficient and highly selective catalytic system. Employment of the optimized ligands in the asymmetric arylation of cyclic olefins allowed for achieving high enantioselectivities and significantly suppressing product isomerization. Factors affecting the selectivity and the rate of the isomerization were identified. It was shown that the nature of this isomerization is different from that demonstrated previously using chiral diphosphine ligands. PMID:25161709

  1. Algebraic integrability: a survey.

    PubMed

    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.

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

    PubMed

    Verburgt, Lukas M

    2016-01-01

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

  3. 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…

  4. Discrimination in a General Algebraic Setting

    PubMed Central

    Fine, Benjamin; Lipschutz, Seymour; Spellman, Dennis

    2015-01-01

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

  5. Derive Workshop Matrix Algebra and Linear Algebra.

    ERIC Educational Resources Information Center

    Townsley Kulich, Lisa; Victor, Barbara

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

  6. Generalized EMV-Effect Algebras

    NASA Astrophysics Data System (ADS)

    Borzooei, R. A.; Dvurečenskij, A.; Sharafi, A. H.

    2018-04-01

    Recently in Dvurečenskij and Zahiri (2017), new algebraic structures, called EMV-algebras which generalize both MV-algebras and generalized Boolean algebras, were introduced. We present equivalent conditions for EMV-algebras. In addition, we define a partial algebraic structure, called a generalized EMV-effect algebra, which is close to generalized MV-effect algebras. Finally, we show that every generalized EMV-effect algebra is either an MV-effect algebra or can be embedded into an MV-effect algebra as a maximal ideal.

  7. Banach Synaptic Algebras

    NASA Astrophysics Data System (ADS)

    Foulis, David J.; Pulmannov, Sylvia

    2018-04-01

    Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Størmer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C∗-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW∗-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.

  8. Homomorphisms in C*-ternary algebras and JB*-triples

    NASA Astrophysics Data System (ADS)

    Park, Choonkil; Rassias, Themistocles M.

    2008-01-01

    In this paper, we investigate homomorphisms between C*-ternary algebras and derivations on C*-ternary algebras, and homomorphisms between JB*-triples and derivations on JB*-triples, associated with the following Apollonius type additive functional equation

  9. Single-step synthesis of styryl phosphonic acids via palladium-catalyzed Heck coupling of vinyl phosphonic acid with aryl halides

    DOE PAGES

    McNichols, Brett W.; Koubek, Joshua T.; Sellinger, Alan

    2017-10-27

    Here, we have developed a single step palladium-catalyzed Heck coupling of aryl halides with vinyl phosphonic acid to produce functionalized (E)-styryl phosphonic acids. This pathway utilizes a variety of commercially available aryl halides, vinyl phosphonic acid and Pd(P(tBu) 3) 2 as catalyst. These conditions produce a wide range of styryl phosphonic acids with high purities and good to excellent yields (31–80%).

  10. Single-step synthesis of styryl phosphonic acids via palladium-catalyzed Heck coupling of vinyl phosphonic acid with aryl halides

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

    McNichols, Brett W.; Koubek, Joshua T.; Sellinger, Alan

    Here, we have developed a single step palladium-catalyzed Heck coupling of aryl halides with vinyl phosphonic acid to produce functionalized (E)-styryl phosphonic acids. This pathway utilizes a variety of commercially available aryl halides, vinyl phosphonic acid and Pd(P(tBu) 3) 2 as catalyst. These conditions produce a wide range of styryl phosphonic acids with high purities and good to excellent yields (31–80%).

  11. A calculus based on a q-deformed Heisenberg algebra

    DOE PAGES

    Cerchiai, B. L.; Hinterding, R.; Madore, J.; ...

    1999-04-27

    We show how one can construct a differential calculus over an algebra where position variables $x$ and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by cursive Greek chi and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on thismore » derivative differential forms and an exterior differential calculus can be constructed.« less

  12. 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…

  13. 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.

  14. Early Algebra with Graphics Software as a Type II Application of Technology

    ERIC Educational Resources Information Center

    Abramovich, Sergei

    2006-01-01

    This paper describes the use of Kid Pix-graphics software for creative activities of young children--in the context of early algebra as determined by the mathematics core curriculum of New York state. It shows how grade-two appropriate pedagogy makes it possible to bring about a qualitative change in the learning process of those commonly…

  15. A Study of Students' Achievement in Algebra: Considering the Effect of Gender and Types of Schools

    ERIC Educational Resources Information Center

    Saleh, Salmiza; Rahman, Muhamad Asyrah Abdul

    2016-01-01

    Algebra is a branch of mathematics that deals with symbols and also the rules for manipulating those symbols, which are used to represent numbers and quantities in mathematical formulae and equations. It is needed and used in our everyday lives. The purpose of this study was to examine students' performance in algebra related to gender and types…

  16. Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories

    NASA Astrophysics Data System (ADS)

    Cheng, Tao; Huang, Hua-Lin; Yang, Yuping

    2016-01-01

    By viewing Clifford algebras as algebras in some suitable symmetric Gr-categories, Albuquerque and Majid were able to give a new derivation of some well known results about Clifford algebras and to generalize them. Along the same line, Bulacu observed that Clifford algebras are weak Hopf algebras in the aforementioned categories and obtained other interesting properties. The aim of this paper is to study generalized Clifford algebras in a similar manner and extend the results of Albuquerque, Majid and Bulacu to the generalized setting. In particular, by taking full advantage of the gauge transformations in symmetric linear Gr-categories, we derive the decomposition theorem and provide categorical weak Hopf structures for generalized Clifford algebras in a conceptual and simpler manner.

  17. Algebraic K-theory, K-regularity, and -duality of -stable C ∗-algebras

    NASA Astrophysics Data System (ADS)

    Mahanta, Snigdhayan

    2015-12-01

    We develop an algebraic formalism for topological -duality. More precisely, we show that topological -duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C ∗-algebra . Then we show that there is a functorial topological K-theory symmetric spectrum construction on the category of separable C ∗-algebras, such that is an algebra over this operad; moreover, is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C ∗-algebras. We also show that -stable C ∗-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of a x+ b-semigroup C ∗-algebras coming from number theory and that of -stabilized noncommutative tori.

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

    PubMed

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

    2014-03-01

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

  19. A double commutant theorem for Murray–von Neumann algebras

    PubMed Central

    Liu, Zhe

    2012-01-01

    Murray–von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra of the Murray–von Neumann algebra associated with a finite von Neumann algebra is the Murray–von Neumann algebra , where is a maximal abelian self-adjoint subalgebra of and, in addition, is . We also prove that the Murray–von Neumann algebra with the center of is the center of the Murray–von Neumann algebra . Von Neumann’s celebrated double commutant theorem characterizes von Neumann algebras as those for which , where , the commutant of , is the set of bounded operators on the Hilbert space that commute with all operators in . At the end of this article, we present a double commutant theorem for Murray–von Neumann algebras. PMID:22543165

  20. Understanding Anionic "Ligandless" Palladium Species in the Mizoroki-Heck Reaction.

    PubMed

    Schroeter, Felix; Strassner, Thomas

    2018-05-07

    The anionic complex [NBu 4 ][Pd(DMSO)Cl 3 ], as a "ligandless" system, was shown to be an active catalyst in the Mizoroki-Heck coupling of aryl chlorides in the absence of strongly σ-donating ligands. To investigate the experimentally observed influence of halides and the amount of water on the catalytic activity, we employed a combination of experiments and theoretical calculations. The presence of water was shown to be critical for the formation of the active palladium(0) species by oxidation of in situ generated tributylamine. Oxidative addition to an anionic palladium(0) species was found to be the rate-determining step of the catalytic cycle. For the ensuing steps, both neutral and anionic pathways were considered. It was shown that, in the absence of strongly σ-donating neutral ligands, chloride ions stabilize the catalytic intermediates. Therefore, an anionic pathway is preferred, which explains the need for tetrabutylammonium chloride as an additive. The study of the influence of bromide ions on the catalytic activity revealed that the strongly exergonic displacement of the neutral substrates by bromide ions lowers the catalytic activity.

  1. Algebra for Everyone.

    ERIC Educational Resources Information Center

    Edwards, Edgar L., Jr., Ed.

    The fundamentals of algebra and algebraic thinking should be a part of the background of all citizens in society. The vast increase in the use of technology requires that school mathematics ensure the teaching of algebraic thinking as well as its use at both the elementary and secondary school levels. Algebra is a universal theme that runs through…

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

    ERIC Educational Resources Information Center

    Gonzalez-Vega, Laureano

    1999-01-01

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

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

    NASA Astrophysics Data System (ADS)

    Coffield, Patrick Cyrus

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

  4. A mechanized process algebra for verification of device synchronization protocols

    NASA Technical Reports Server (NTRS)

    Schubert, E. Thomas

    1992-01-01

    We describe the formalization of a process algebra based on CCS within the Higher Order Logic (HOL) theorem-proving system. The representation of four types of device interactions and a correctness proof of the communication between a microprocessor and MMU is presented.

  5. Continuous-flow Heck synthesis of 4-methoxybiphenyl and methyl 4-methoxycinnamate in supercritical carbon dioxide expanded solvent solutions

    PubMed Central

    Lau, Phei Li; Allen, Ray W K

    2013-01-01

    Summary The palladium metal catalysed Heck reaction of 4-iodoanisole with styrene or methyl acrylate has been studied in a continuous plug flow reactor (PFR) using supercritical carbon dioxide (scCO2) as the solvent, with THF and methanol as modifiers. The catalyst was 2% palladium on silica and the base was diisopropylethylamine due to its solubility in the reaction solvent. No phosphine co-catalysts were used so the work-up procedure was simplified and the green credentials of the reaction were enhanced. The reactions were studied as a function of temperature, pressure and flow rate and in the case of the reaction with styrene compared against a standard, stirred autoclave reaction. Conversion was determined and, in the case of the reaction with styrene, the isomeric product distribution was monitored by GC. In the case of the reaction with methyl acrylate the reactor was scaled from a 1.0 mm to 3.9 mm internal diameter and the conversion and turnover frequency determined. The results show that the Heck reaction can be effectively performed in scCO2 under continuous flow conditions with a palladium metal, phosphine-free catalyst, but care must be taken when selecting the reaction temperature in order to ensure the appropriate isomer distribution is achieved. Higher reaction temperatures were found to enhance formation of the branched terminal alkene isomer as opposed to the linear trans-isomer. PMID:24367454

  6. 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…

  7. Algebraic reasoning and bat-and-ball problem variants: Solving isomorphic algebra first facilitates problem solving later.

    PubMed

    Hoover, Jerome D; Healy, Alice F

    2017-12-01

    The classic bat-and-ball problem is used widely to measure biased and correct reasoning in decision-making. University students overwhelmingly tend to provide the biased answer to this problem. To what extent might reasoners be led to modify their judgement, and, more specifically, is it possible to facilitate problem solution by prompting participants to consider the problem from an algebraic perspective? One hundred ninety-seven participants were recruited to investigate the effect of algebraic cueing as a debiasing strategy on variants of the bat-and-ball problem. Participants who were cued to consider the problem algebraically were significantly more likely to answer correctly relative to control participants. Most of this cueing effect was confined to a condition that required participants to solve isomorphic algebra equations corresponding to the structure of bat-and-ball question types. On a subsequent critical question with differing item and dollar amounts presented without a cue, participants were able to generalize the learned information to significantly reduce overall bias. Math anxiety was also found to be significantly related to bat-and-ball problem accuracy. These results suggest that, under specific conditions, algebraic reasoning is an effective debiasing strategy on bat-and-ball problem variants, and provide the first documented evidence for the influence of math anxiety on Cognitive Reflection Test performance.

  8. Virasoro algebra in the KN algebra; Bosonic string with fermionic ghosts on Riemann surfaces

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

    Koibuchi, H.

    1991-10-10

    In this paper the bosonic string model with fermionic ghosts is considered in the framework of the KN algebra. The authors' attentions are paid to representations of KN algebra and a Clifford algebra of the ghosts. The authors show that a Virasoro-like algebra is obtained from KN algebra when KN algebra has certain antilinear anti-involution, and that it is isomorphic to the usual Virasoro algebra. The authors show that there is an expected relation between a central charge of this Virasoro-like algebra and an anomaly of the combined system.

  9. A Method for the Microanalysis of Pre-Algebra Transfer

    ERIC Educational Resources Information Center

    Pavlik, Philip I., Jr.; Yudelson, Michael; Koedinger, Kenneth R.

    2011-01-01

    The objective of this research was to better understand the transfer of learning between different variations of pre-algebra problems. While the authors could have addressed a specific variation that might address transfer, they were interested in developing a general model of transfer, so we gathered data from multiple problem types and their…

  10. Commutative Algebras of Toeplitz Operators in Action

    NASA Astrophysics Data System (ADS)

    Vasilevski, Nikolai

    2011-09-01

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

  11. 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…

  12. SATA II - Stochastic Algebraic Topology and Applications

    DTIC Science & Technology

    2017-01-30

    AFRL-AFOSR-UK-TR-2017-0018 SATA II - Stochastic Algebraic Topology and Applications 150032 Robert Adler TECHNION ISRAEL INSTITUTE OF TECHNOLOGY Final...REPORT TYPE Final 3. DATES COVERED (From - To) 15 Dec 2014 to 14 Dec 2016 4. TITLE AND SUBTITLE SATA II - Stochastic Algebraic Topology and Applications ...has recently been submitted to AFOSR. 15. SUBJECT TERMS Network Theory, Sensor Technology, Mathematical Modeling, EOARD 16. SECURITY CLASSIFICATION OF

  13. 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,…

  14. Quiver W-algebras

    NASA Astrophysics Data System (ADS)

    Kimura, Taro; Pestun, Vasily

    2018-06-01

    For a quiver with weighted arrows, we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al. and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds-Kac-Moody Lie algebras, their quantum affinizations and associated W-algebras.

  15. a Triangular Deformation of the Two-Dimensional POINCARÉ Algebra

    NASA Astrophysics Data System (ADS)

    Khorrami, M.; Shariati, A.; Abolhassani, M. R.; Aghamohammadi, A.

    Contracting the h-deformation of SL(2, ℝ), we construct a new deformation of two-dimensional Poincaré's algebra, the algebra of functions on its group and its differential structure. It is seen that these dual Hopf algebras are isomorphic to each other. It is also shown that the Hopf algebra is triangular, and its universal R-matrix is also constructed explicitly. We then find a deformation map for the universal enveloping algebra, and at the end, give the deformed mass shells and Lorentz transformation.

  16. 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…

  17. Locally Compact Quantum Groups. A von Neumann Algebra Approach

    NASA Astrophysics Data System (ADS)

    Van Daele, Alfons

    2014-08-01

    In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We start with a von Neumann algebra and a comultiplication on this von Neumann algebra. We assume that there exist faithful left and right Haar weights. Then we develop the theory within this von Neumann algebra setting. In [Math. Scand. 92 (2003), 68-92] locally compact quantum groups are also studied in the von Neumann algebraic context. This approach is independent of the original C^*-algebraic approach in the sense that the earlier results are not used. However, this paper is not really independent because for many proofs, the reader is referred to the original paper where the C^*-version is developed. In this paper, we give a completely self-contained approach. Moreover, at various points, we do things differently. We have a different treatment of the antipode. It is similar to the original treatment in [Ann. Sci. & #201;cole Norm. Sup. (4) 33 (2000), 837-934]. But together with the fact that we work in the von Neumann algebra framework, it allows us to use an idea from [Rev. Roumaine Math. Pures Appl. 21 (1976), 1411-1449] to obtain the uniqueness of the Haar weights in an early stage. We take advantage of this fact when deriving the other main results in the theory. We also give a slightly different approach to duality. Finally, we collect, in a systematic way, several important formulas. In an appendix, we indicate very briefly how the C^*-approach and the von Neumann algebra approach eventually yield the same objects. The passage from the von Neumann algebra setting to the C^*-algebra setting is more or less standard. For the other direction, we use a new method. It is based on the observation that the Haar weights on the C^*-algebra extend to weights on the double dual with central support and that all these supports are the same. Of course, we get the von Neumann algebra by cutting down the double dual with this unique

  18. Making Algebra Work: Instructional Strategies that Deepen Student Understanding, within and between Algebraic Representations

    ERIC Educational Resources Information Center

    Star, Jon R.; Rittle-Johnson, Bethany

    2009-01-01

    Competence in algebra is increasingly recognized as a critical milestone in students' middle and high school years. The transition from arithmetic to algebra is a notoriously difficult one, and improvements in algebra instruction are greatly needed (National Research Council, 2001). Algebra historically has represented students' first sustained…

  19. 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.

  20. THE RADICAL OF A JORDAN ALGEBRA

    PubMed Central

    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

  1. From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets

    NASA Astrophysics Data System (ADS)

    Jurčo, Branislav

    2012-12-01

    Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, such that each Gn is simply connected. We use the 1-jet of the classifying space W¯ G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a differential graded Lie algebra. The differential and the brackets are explicitly described in terms (of a part) of the corresponding k-hypercrossed complex structure of Ng. The result can be seen as a geometric interpretation of Quillen's (purely algebraic) construction of the adjunction between simplicial Lie algebras and dg-Lie algebras.

  2. Line defect Schur indices, Verlinde algebras and U(1) r fixed points

    NASA Astrophysics Data System (ADS)

    Neitzke, Andrew; Yan, Fei

    2017-11-01

    Given an N=2 superconformal field theory, we reconsider the Schur index ℐ L ( q) in the presence of a half line defect L. Recently Cordova-Gaiotto-Shao found that ℐ L ( q) admits an expansion in terms of characters of the chiral algebra A introduced by Beem et al., with simple coefficients υ L, β ( q). We report a puzzling new feature of this expansion: the q → 1 limit of the coefficients υ L, β ( q) is linearly related to the vacuum expectation values 〈 L〉 in U(1) r -invariant vacua of the theory compactified on S 1. This relation can be expressed algebraically as a commutative diagram involving three algebras: the algebra generated by line defects, the algebra of functions on U(1) r -invariant vacua, and a Verlindelike algebra associated to A . Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type ( A 1, A 2), ( A 1, A 4), ( A 1, A 6), ( A 1, D 3) and ( A 1, D 5). In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.

  3. Derivation in INK-algebras

    NASA Astrophysics Data System (ADS)

    Kaviyarasu, M.; Indhira, K.

    2018-04-01

    In 2017 we introduced a new notion of algebra called IKN-algebra. Motivated by some result on derivations (rightleft)-derivation and (leftright)- derivation in ring. In this paper we introduce derivation in INK-Algebras and investigate some important result.

  4. Elliptic biquaternion algebra

    NASA Astrophysics Data System (ADS)

    Özen, Kahraman Esen; Tosun, Murat

    2018-01-01

    In this study, we define the elliptic biquaternions and construct the algebra of elliptic biquaternions over the elliptic number field. Also we give basic properties of elliptic biquaternions. An elliptic biquaternion is in the form A0 + A1i + A2j + A3k which is a linear combination of {1, i, j, k} where the four components A0, A1, A2 and A3 are elliptic numbers. Here, 1, i, j, k are the quaternion basis of the elliptic biquaternion algebra and satisfy the same multiplication rules which are satisfied in both real quaternion algebra and complex quaternion algebra. In addition, we discuss the terms; conjugate, inner product, semi-norm, modulus and inverse for elliptic biquaternions.

  5. Universal vertex-IRF transformation for quantum affine algebras

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

    Buffenoir, E.; Roche, Ph.; Terras, V.

    2012-10-15

    We construct a universal solution of the generalized coboundary equation in the case of quantum affine algebras, which is an extension of our previous work to U{sub q}(A{sub r}{sup (1)}). This universal solution has a simple Gauss decomposition which is constructed using Sevostyanov's characters of twisted quantum Borel algebras. We show that in the evaluation representations it gives a vertex-face transformation between a vertex type solution and a face type solution of the quantum dynamical Yang-Baxter equation. In particular, in the evaluation representation of U{sub q}(A{sub 1}{sup (1)}), it gives Baxter's well-known transformation between the 8-vertex model and the interaction-round-facesmore » (IRF) height model.« less

  6. Color Algebras

    NASA Technical Reports Server (NTRS)

    Mulligan, Jeffrey B.

    2017-01-01

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

  7. (Fuzzy) Ideals of BN-Algebras

    PubMed Central

    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

  8. Generalizing the bms3 and 2D-conformal algebras by expanding the Virasoro algebra

    NASA Astrophysics Data System (ADS)

    Caroca, Ricardo; Concha, Patrick; Rodríguez, Evelyn; Salgado-Rebolledo, Patricio

    2018-03-01

    By means of the Lie algebra expansion method, the centrally extended conformal algebra in two dimensions and the bms3 algebra are obtained from the Virasoro algebra. We extend this result to construct new families of expanded Virasoro algebras that turn out to be infinite-dimensional lifts of the so-called Bk, Ck and Dk algebras recently introduced in the literature in the context of (super)gravity. We also show how some of these new infinite-dimensional symmetries can be obtained from expanded Kač-Moody algebras using modified Sugawara constructions. Applications in the context of three-dimensional gravity are briefly discussed.

  9. Synthesis of hybrid interfacial silica-based nanospheres composite as a support for ultra-small palladium nanoparticle and application of PdNPs/HSN in Mizoroki-Heck reaction

    NASA Astrophysics Data System (ADS)

    Rostamnia, Sadegh; Kholdi, Saba

    2017-12-01

    The silica based hollow nanosphere (silica-HNS) containing polymer of polyaniline was synthesized and chosen as a promising support for PdNPs. Then it was applied as a green catalyst in the reaction of Heck coupling with high yield. TEM and SEM-EDX/mapping images were used to study the structure and morphology. FT-IR spectroscopy, Thermal gravimetry analysis (TGA), and BET were used to characterize and investigate the catalyst. Also, the amounts of Pd loading were characterized by ICP-AES technique. Catalyst recyclability showed 5 successful runs for the reaction.

  10. The Unitality of Quantum B-algebras

    NASA Astrophysics Data System (ADS)

    Han, Shengwei; Xu, Xiaoting; Qin, Feng

    2018-02-01

    Quantum B-algebras as a generalization of quantales were introduced by Rump and Yang, which cover the majority of implicational algebras and provide a unified semantic for a wide class of substructural logics. Unital quantum B-algebras play an important role in the classification of implicational algebras. The main purpose of this paper is to construct unital quantum B-algebras from non-unital quantum B-algebras.

  11. A Relational Algebra Query Language for Programming Relational Databases

    ERIC Educational Resources Information Center

    McMaster, Kirby; Sambasivam, Samuel; Anderson, Nicole

    2011-01-01

    In this paper, we describe a Relational Algebra Query Language (RAQL) and Relational Algebra Query (RAQ) software product we have developed that allows database instructors to teach relational algebra through programming. Instead of defining query operations using mathematical notation (the approach commonly taken in database textbooks), students…

  12. 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.…

  13. Intuitionistic fuzzy n-fold KU-ideal of KU-algebra

    NASA Astrophysics Data System (ADS)

    Mostafa, Samy M.; Kareem, Fatema F.

    2018-05-01

    In this paper, we apply the notion of intuitionistic fuzzy n-fold KU-ideal of KU-algebra. Some types of ideals such as intuitionistic fuzzy KU-ideal, intuitionistic fuzzy closed ideal and intuitionistic fuzzy n-fold KU-ideal are studied. Also, the relations between intuitionistic fuzzy n-fold KU-ideal and intuitionistic fuzzy KU-ideal are discussed. Furthermore, a few results of intuitionistic fuzzy n-fold KU-ideals of a KU-algebra under homomorphism are discussed.

  14. Spontaneous Meta-Arithmetic as a First Step toward School Algebra

    ERIC Educational Resources Information Center

    Caspi, Shai; Sfard, Anna

    2012-01-01

    Taking as the point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following five pairs of 7th grade students as they progress in algebraic discourse during 24 months, from their informal algebraic talk to the formal algebraic discourse, as taught in school. Our analysis follows changes that…

  15. 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.

  16. Fractional quiver W-algebras

    NASA Astrophysics Data System (ADS)

    Kimura, Taro; Pestun, Vasily

    2018-04-01

    We introduce quiver gauge theory associated with the non-simply laced type fractional quiver and define fractional quiver W-algebras by using construction of Kimura and Pestun (Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1072-1; Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1073-0) with representation of fractional quivers.

  17. Paper 3: Content and Rigor of Algebra Credit Recovery Courses

    ERIC Educational Resources Information Center

    Walters, Kirk; Stachel, Suzanne

    2014-01-01

    This paper describes the content, organization and rigor of the f2f and online summer algebra courses that were delivered in summers 2011 and 2012. Examining the content of both types of courses is important because research suggests that algebra courses with certain features may be better than others in promoting success for struggling students.…

  18. On special Lie algebras having a faithful module with Krull dimension

    NASA Astrophysics Data System (ADS)

    Pikhtilkova, O. A.; Pikhtilkov, S. A.

    2017-02-01

    For special Lie algebras we prove an analogue of Markov's theorem on {PI}-algebras having a faithful module with Krull dimension: the solubility of the prime radical. We give an example of a semiprime Lie algebra that has a faithful module with Krull dimension but cannot be represented as a subdirect product of finitely many prime Lie algebras. We prove a criterion for a semiprime Lie algebra to be representable as such a subdirect product.

  19. A set for relational reasoning: Facilitation of algebraic modeling by a fraction task.

    PubMed

    DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J

    2016-12-01

    Recent work has identified correlations between early mastery of fractions and later math achievement, especially in algebra. However, causal connections between aspects of reasoning with fractions and improved algebra performance have yet to be established. The current study investigated whether relational reasoning with fractions facilitates subsequent algebraic reasoning using both pre-algebra students and adult college students. Participants were first given either a relational reasoning fractions task or a fraction algebra procedures control task. Then, all participants solved word problems and constructed algebraic equations in either multiplication or division format. The word problems and the equation construction tasks involved simple multiplicative comparison statements such as "There are 4 times as many students as teachers in a classroom." Performance on the algebraic equation construction task was enhanced for participants who had previously completed the relational fractions task compared with those who completed the fraction algebra procedures task. This finding suggests that relational reasoning with fractions can establish a relational set that promotes students' tendency to model relations using algebraic expressions. Copyright © 2016 Elsevier Inc. All rights reserved.

  20. A facile one-pot solvothermal method for synthesis of magnetically recoverable Pd-Fe3O4 hybrid nanocatalysts for the Mizoroki-Heck reaction

    NASA Astrophysics Data System (ADS)

    Zhen, Fangchen; Ran, Maofei; Chu, Wei; Jiang, Chengfa; Sun, Wenjing

    2018-03-01

    Pd-Fe3O4 hybrid nanostructures were prepared using a simple one-pot hydrothermal method. The prepared materials were characterized by Fourier transform-infrared spectroscopy, X-ray diffraction, X-ray photoelectron spectroscopy, scanning electron microscopy, transmission electron microscopy, inductively coupled plasma, N2 adsorption-desorption, and vibrating sample magnetometry. This self-assembled nanosystem acted as an efficient magnetically recyclable noble metal-based multi-functional nanocatalyst. It showed excellent catalytic activity and stability for the Heck reaction of iodobenzene and styrene under mild conditions. The methods used to prepare the Pd-Fe3O4 catalysts were simple and low-cost, which will be useful for the large-scale development and application of a magnetically recoverable Pd catalyst.

  1. Situating the Debate on "Geometrical Algebra" within the Framework of Premodern Algebra.

    PubMed

    Sialaros, Michalis; Christianidis, Jean

    2016-06-01

    Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.

  2. Investigating the nature of palladium chain-walking in the enantioselective redox-relay Heck reaction of alkenyl alcohols.

    PubMed

    Hilton, Margaret J; Xu, Li-Ping; Norrby, Per-Ola; Wu, Yun-Dong; Wiest, Olaf; Sigman, Matthew S

    2014-12-19

    The mechanism of the redox-relay Heck reaction was investigated using deuterium-labeled substrates. Results support a pathway through a low energy palladium-alkyl intermediate that immediately precedes product formation, ruling out a tautomerization mechanism. DFT calculations of the relevant transition structures at the M06/LAN2DZ+f/6-31+G* level of theory show that the former pathway is favored by 5.8 kcal/mol. Palladium chain-walking toward the alcohol, following successive β-hydride eliminations and migratory insertions, is also supported in this study. The stereochemistry of deuterium labels is determined, lending support that the catalyst remains bound to the substrate during the relay process and that both cis- and trans-alkenes form from β-hydride elimination.

  3. A spatial operator algebra for manipulator modeling and control

    NASA Technical Reports Server (NTRS)

    Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.

    1991-01-01

    A recently developed spatial operator algebra for manipulator modeling, control, and trajectory design is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and for control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics.

  4. Conditions for the Effectiveness of a Tablet-Based Algebra Program

    ERIC Educational Resources Information Center

    Jaciw, Andrew P.; Toby, Megan; Ma, Boya

    2012-01-01

    Tablets such as the iPad represent the next iteration of technologies that hold promise to facilitate learning, particularly in mathematics. In the case of algebra, tablets have the potential to bring the curriculum to life by easily linking to supporting materials and they allow an interactive experience whereby manipulation of one type of…

  5. The Development of Children's Algebraic Thinking: The Impact of a Comprehensive Early Algebra Intervention in Third Grade

    ERIC Educational Resources Information Center

    Blanton, Maria; Stephens, Ana; Knuth, Eric; Gardiner, Angela Murphy; Isler, Isil; Kim, Jee-Seon

    2015-01-01

    This article reports results from a study investigating the impact of a sustained, comprehensive early algebra intervention in third grade. Participants included 106 students; 39 received the early algebra intervention, and 67 received their district's regularly planned mathematics instruction. We share and discuss students' responses to a written…

  6. A Geometric Construction of Cyclic Cocycles on Twisted Convolution Algebras

    NASA Astrophysics Data System (ADS)

    Angel, Eitan

    2010-09-01

    In this thesis we give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. In his seminal book, Connes constructs a map from the equivariant cohomology of a manifold carrying the action of a discrete group into the periodic cyclic cohomology of the associated convolution algebra. Furthermore, for proper étale groupoids, J.-L. Tu and P. Xu provide a map between the periodic cyclic cohomology of a gerbe twisted convolution algebra and twisted cohomology groups. Our focus will be the convolution algebra with a product defined by a gerbe over a discrete translation groupoid. When the action is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial notions related to ideas of J. Dupont to construct a simplicial form representing the Dixmier-Douady class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial Dixmier-Douady form to the mixed bicomplex of certain matrix algebras. Finally, we define a morphism from this complex to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.

  7. Hilbert space structure in quantum gravity: an algebraic perspective

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

    Giddings, Steven B.

    If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime. Here, this viewpoint is supported by difficulties of such quantization, and by the apparent lack of a fundamental role for locality. In finite or discrete quantum systems, important structure is provided by tensor factorizations of the Hilbert space. However, even in local quantum field theory properties of the generic type III von Neumann algebras and of long range gauge fields indicate that factorization of themore » Hilbert space is problematic. Instead it is better to focus on the structure of the algebra of observables, and in particular on its subalgebras corresponding to regions. This paper suggests that study of analogous algebraic structure in gravity gives an important perspective on the nature of the quantum theory. Significant departures from the subalgebra structure of local quantum field theory are found, working in the correspondence limit of long-distances/low-energies. Particularly, there are obstacles to identifying commuting algebras of localized operators. In addition to suggesting important properties of the algebraic structure, this and related observations pose challenges to proposals of a fundamental role for entanglement.« less

  8. Hilbert space structure in quantum gravity: an algebraic perspective

    DOE PAGES

    Giddings, Steven B.

    2015-12-16

    If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime. Here, this viewpoint is supported by difficulties of such quantization, and by the apparent lack of a fundamental role for locality. In finite or discrete quantum systems, important structure is provided by tensor factorizations of the Hilbert space. However, even in local quantum field theory properties of the generic type III von Neumann algebras and of long range gauge fields indicate that factorization of themore » Hilbert space is problematic. Instead it is better to focus on the structure of the algebra of observables, and in particular on its subalgebras corresponding to regions. This paper suggests that study of analogous algebraic structure in gravity gives an important perspective on the nature of the quantum theory. Significant departures from the subalgebra structure of local quantum field theory are found, working in the correspondence limit of long-distances/low-energies. Particularly, there are obstacles to identifying commuting algebras of localized operators. In addition to suggesting important properties of the algebraic structure, this and related observations pose challenges to proposals of a fundamental role for entanglement.« less

  9. A spatial operator algebra for manipulator modeling and control

    NASA Technical Reports Server (NTRS)

    Rodriguez, G.; Kreutz, Kenneth; Jain, Abhinandan

    1989-01-01

    A recently developed spatial operator algebra, useful for modeling, control, and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanizaton of specific algorithms is greatly simplified. The analytical formulation of the operator algebra, as well as its implementation in the Ada programming language are discussed.

  10. A Research on Future Mathematics Teachers' Instructional Explanations: The case of Algebra

    ERIC Educational Resources Information Center

    Guler, Mustafa; Celik, Derya

    2016-01-01

    In this study, explanations of future mathematics teachers about algebra were analysed according to the levels of understanding used by Kinach (2002). The participants for the study were 101 teacher candidates attending the final semester of a teacher training program. For data collection, a form containing four scenario-type items were…

  11. 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…

  12. A description of pseudo-bosons in terms of nilpotent Lie algebras

    NASA Astrophysics Data System (ADS)

    Bagarello, Fabio; Russo, Francesco G.

    2018-02-01

    We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we do not find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed into the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behavior of pseudo-bosonic operators in many quantum models.

  13. Algebraic theory of molecules

    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.

  14. Symmetries and Invariants of Twisted Quantum Algebras and Associated Poisson Algebras

    NASA Astrophysics Data System (ADS)

    Molev, A. I.; Ragoucy, E.

    We construct an action of the braid group BN on the twisted quantized enveloping algebra U q'( {o}N) where the elements of BN act as automorphisms. In the classical limit q → 1, we recover the action of BN on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra U q'( {sp}2n). We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras.

  15. Characterizations of Some Fuzzy Prefilters (Filters) in EQ-Algebras

    PubMed Central

    Xin, Xiao Long; Yang, Yong Wei

    2014-01-01

    We introduce and study some types of fuzzy prefilters (filters) in EQ-algebras. First, we present several characterizations of fuzzy positive implicative prefilters (filters), fuzzy implicative prefilters (filters), and fuzzy fantastic prefilters (filters). Next, using their characterizations, we mainly consider the relationships among these special fuzzy filters. Particularly, we find some conditions under which a fuzzy implicative prefilter (filter) is equivalent to a fuzzy positive implicative prefilter (filter). As applications, we obtain some new results about classical filters in EQ-algebras and some related results about fuzzy filters in residuated lattices. PMID:24892096

  16. On Weak-BCC-Algebras

    PubMed Central

    Thomys, Janus; Zhang, Xiaohong

    2013-01-01

    We describe weak-BCC-algebras (also called BZ-algebras) in which the condition (x∗y)∗z = (x∗z)∗y is satisfied only in the case when elements x, y belong to the same branch. We also characterize ideals, nilradicals, and nilpotent elements of such algebras. PMID:24311983

  17. Earth Algebra.

    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…

  18. Roughness in Lattice Ordered Effect Algebras

    PubMed Central

    Xin, Xiao Long; Hua, Xiu Juan; Zhu, Xi

    2014-01-01

    Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice ordered effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a function e(a, b) in a lattice ordered effect algebra E and build a relationship between it and congruence classes. Then we study some properties about approximation of lattice ordered effect algebras. PMID:25170523

  19. The Structural Algebra Option: A Discussion Paper.

    ERIC Educational Resources Information Center

    Kirshner, David

    The goal of this paper is to renew interest in the structural option to algebra instruction. Concern for the usual secondary school algebra curriculum related to simplifying expressions, solving equations, and rationalizing numerators and denominators is viewed from three pedagogical approaches: (1) structural approach, (2) empirical approach, and…

  20. A Proposed Algebra Assessment for Use in a Problem-Analysis Framework

    ERIC Educational Resources Information Center

    Walick, Christopher M.; Burns, Matthew K.

    2017-01-01

    Algebra is critical to high school graduation and college success, but student achievement in algebra frequently falls significantly below expected proficiency levels. While existing research emphasizes the importance of quality algebra instruction, there is little research about how to conduct problem analysis for struggling secondary students.…

  1. A Loomis-Sikorski theorem and functional calculus for a generalized Hermitian algebra

    NASA Astrophysics Data System (ADS)

    Foulis, David J.; Jenčová, Anna; Pulmannová, Sylvia

    2017-10-01

    A generalized Hermitian (GH-) algebra is a generalization of the partially ordered Jordan algebra of all Hermitian operators on a Hilbert space. We introduce the notion of a gh-tribe, which is a commutative GH-algebra of functions on a nonempty set X with pointwise partial order and operations, and we prove that every commutative GH-algebra is the image of a gh-tribe under a surjective GH-morphism. Using this result, we prove that each element a of a GH-algebra A corresponds to a real observable ξa on the σ-orthomodular lattice of projections in A and that ξa determines the spectral resolution of a. Also, if f is a continuous function defined on the spectrum of a, we formulate a definition of f (a), thus obtaining a continuous functional calculus for A.

  2. N-heterocycle carbene (NHC)-ligated cyclopalladated N,N-dimethylbenzylamine: a highly active, practical and versatile catalyst for the Heck-Mizoroki reaction.

    PubMed

    Peh, Guang-Rong; Kantchev, Eric Assen B; Zhang, Chi; Ying, Jackie Y

    2009-05-21

    The wide dissemination of catalytic protocols in academic and industrial laboratories is facilitated by the development of catalysts that are not only highly active but also user-friendly, stable to moisture, air and long term storage and easy to prepare on a large scale. Herein we describe a protocol for the Heck-Mizoroki reaction mediated by cyclopalladated N,N-dimethylbenzylamine (dmba) ligated with a N-heterocyclic carbene, 1,3-bis(mesityl)imidazol-2-ylidene (IMes), that fulfils these criteria. The precatalyst can be synthesized on approximately 100 g scale by a tri-component, sequential, one-pot reaction of N,N-dimethylbenzylamine, PdCl2 and IMes.HCl in refluxing acetonitrile in air in the presence of K2CO3. This single component catalyst is stable to air, moisture and long term storage and can be conveniently dispensed as a stock solution in NMP. It mediates the Heck-Mizoroki reaction of a range of aryl- and heteroaryl bromides in reagent grade NMP at the 0.1-2 mol% range without the need for rigorous anhydrous techniques or a glovebox, and is active even in air. The catalyst is capable of achieving very high levels of catalytic activity (TON of up to 5.22 x 10(5)) for the coupling of a deactivated arylbromide, p-bromoanisole, with tBu acrylate as a benchmark substrate pair. A wide range of aryl bromides, iodides and, for the first time with a NHC-Pd catalyst, a triflate was coupled with diverse acrylate derivatives (nitrile, tert-butyl ester and amides) and styrene derivatives. The use of excess (>2 equiv.) of the aryl bromide and tert-butyl acrylate leads to mixture of tert-butyl beta,beta-diarylacrylate and tert-butyl cinnamate derivatives depending on the substitution pattern of the aryl bromide. Electron rich m- and p-substituted arylbromides give the diarylated products exclusively, whereas electron-poor aryl bromides give predominantly mono-arylated products. For o-substituted aryl bromides, no doubly arylated products could be obtained under any

  3. Pre-Algebra Lexicon.

    ERIC Educational Resources Information Center

    Hayden, Dunstan; Cuevas, Gilberto

    The pre-algebra lexicon is a set of classroom exercises designed to teach the technical words and phrases of pre-algebra mathematics, and includes the terms most commonly found in related mathematics courses. The lexicon has three parts, each with its own introduction. The first introduces vocabulary items in three groups forming a learning…

  4. A Process Algebra Approach to Quantum Electrodynamics

    NASA Astrophysics Data System (ADS)

    Sulis, William

    2017-12-01

    The process algebra program is directed towards developing a realist model of quantum mechanics free of paradoxes, divergences and conceptual confusions. From this perspective, fundamental phenomena are viewed as emerging from primitive informational elements generated by processes. The process algebra has been shown to successfully reproduce scalar non-relativistic quantum mechanics (NRQM) without the usual paradoxes and dualities. NRQM appears as an effective theory which emerges under specific asymptotic limits. Space-time, scalar particle wave functions and the Born rule are all emergent in this framework. In this paper, the process algebra model is reviewed, extended to the relativistic setting, and then applied to the problem of electrodynamics. A semiclassical version is presented in which a Minkowski-like space-time emerges as well as a vector potential that is discrete and photon-like at small scales and near-continuous and wave-like at large scales. QED is viewed as an effective theory at small scales while Maxwell theory becomes an effective theory at large scales. The process algebra version of quantum electrodynamics is intuitive and realist, free from divergences and eliminates the distinction between particle, field and wave. Computations are carried out using the configuration space process covering map, although the connection to second quantization has not been fully explored.

  5. Representing k-graphs as Matrix Algebras

    NASA Astrophysics Data System (ADS)

    Rosjanuardi, R.

    2018-05-01

    For any commutative unital ring R and finitely aligned k-graph Λ with |Λ| < ∞ without cycles, we can realise Kumjian-Pask algebra KP R (Λ) as a direct sum of of matrix algebra over some vertices v with properties ν = νΛ, i.e: ⊕ νΛ=ν M |Λv|(R). When there is only a single vertex ν ∈ Λ° such that ν = νΛ, we can realise the Kumjian-Pask algebra as the matrix algebra M |ΛV|(R). Hence the matrix algebra M |vΛ|(R) can be regarded as a representation of the k-graph Λ. In this talk we will figure out the relation between finitely aligned k-graph and matrix algebra.

  6. 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…

  7. Hurwitz Algebras and the Octonion Algebra

    NASA Astrophysics Data System (ADS)

    Burdik, Čestmir; Catto, Sultan

    2018-02-01

    We explore some consequences of a theory of internal symmetries for elementary particles constructed on exceptional quantum mechanical spaces based on Jordan algebra formulation that admit exceptional groups as gauge groups.

  8. 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.

  9. Synthesis and characterization of Heck-cross coupling 4-(4-nitrostyryl) aniline as potential precursor in Schiff-base synthesis

    NASA Astrophysics Data System (ADS)

    Hassan, Norhafiefa; Izwani, Fatin; Yusoff, H. M.

    2017-09-01

    This is a preliminary study of alkoxy substituted Heck-schiff base compound as recognition layer in electrochemical DNA sensor for liver cancer. 4-(4-nitrostyryl)aniline was synthesized by bis (triphenylphosphine) palladium (II) dichloride as catalyst and has been characterized by using Fourier transform-infrared spectrometer (FTIR), UV-Vis spectrophotometer and Nuclear Magnetic Resonance (NMR) spectra. The result obtained from FTIR show that there are formation of N=O (NO2) asymmetric stretching vibrations 1340 cm-1. In UV-vis, absorbance of NO2 can be observed at peak 410 nm. While in the 1H NMR and 13C NMR the peak of C-C coupling was found in the range of δH 6.84-7.09 ppm and δC 125.23 ppm.

  10. An Arithmetic-Algebraic Work Space for the Promotion of Arithmetic and Algebraic Thinking: Triangular Numbers

    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…

  11. A spatial operator algebra for manipulator modeling and control

    NASA Technical Reports Server (NTRS)

    Rodriguez, G.; Kreutz, K.; Milman, M.

    1988-01-01

    A powerful new spatial operator algebra for modeling, control, and trajectory design of manipulators is discussed along with its implementation in the Ada programming language. Applications of this algebra to robotics include an operator representation of the manipulator Jacobian matrix; the robot dynamical equations formulated in terms of the spatial algebra, showing the complete equivalence between the recursive Newton-Euler formulations to robot dynamics; the operator factorization and inversion of the manipulator mass matrix which immediately results in O(N) recursive forward dynamics algorithms; the joint accelerations of a manipulator due to a tip contact force; the recursive computation of the equivalent mass matrix as seen at the tip of a manipulator; and recursive forward dynamics of a closed chain system. Finally, additional applications and current research involving the use of the spatial operator algebra are discussed in general terms.

  12. Deformed twistors and higher spin conformal (super-)algebras in four dimensions

    DOE PAGES

    Govil, Karan; Gunaydin, Murat

    2015-03-05

    Massless conformal scalar field in d = 4 corresponds to the minimal unitary representation (minrep) of the conformal group SU(2, 2) which admits a one-parameter family of deformations that describe massless fields of arbitrary helicity. The minrep and its deformations were obtained by quantization of the nonlinear realization of SU(2, 2) as a quasiconformal group in arXiv:0908.3624. We show that the generators of SU(2,2) for these unitary irreducible representations can be written as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group and apply them to define and study higher spin algebras and superalgebras in AdS 5.more » The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS 5 is simply the enveloping algebra of SU(2, 2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS 5. Furthermore, the enveloping algebras of the deformations of the minrep define a one parameter family of HS algebras in AdS 5 for which certain 4d covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras SU(2, 2|N) and we find a one parameter family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a family of (supersymmetric) HS theories in AdS 5 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 4d. We also discuss the corresponding picture in HS algebras in AdS 4 where the corresponding 3d conformal group Sp(4,R) admits only two massless representations (minreps), namely the scalar and spinor singletons.« less

  13. Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

    NASA Astrophysics Data System (ADS)

    Hoque, Md. Fazlul; Marquette, Ian; Post, Sarah; Zhang, Yao-Zhong

    2018-04-01

    We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schrödinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.

  14. Highest-weight representations of Brocherd`s algebras

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

    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.

  15. Symmetry algebra of a generalized anisotropic harmonic oscillator

    NASA Technical Reports Server (NTRS)

    Castanos, O.; Lopez-Pena, R.

    1993-01-01

    It is shown that the symmetry Lie algebra of a quantum system with accidental degeneracy can be obtained by means of the Noether's theorem. The procedure is illustrated by considering a generalized anisotropic two dimensional harmonic oscillator, which can have an infinite set of states with the same energy characterized by an u(1,1) Lie algebra.

  16. 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.

  17. Synthesis and characterization of para-pyridine linked NHC palladium complexes and their studies for the Heck-Mizoroki coupling reaction.

    PubMed

    Liu, Ya-Ming; Lin, Yi-Chun; Chen, Wen-Ching; Cheng, Jen-Hao; Chen, Yi-Lin; Yap, Glenn P A; Sun, Shih-Sheng; Ong, Tiow-Gan

    2012-06-28

    This paper describes the synthesis of 1-(pyridine-4-ylmethyl) NHC and their Pd(II) and Ag(I) complexes, which are fully characterized. Interestingly, we have also synthesized a Pd complex 3a-CO(3) using a more direct treatment of K(2)CO(3) with PdCl(2). 3a-CO(3) represents the first reported solid structure of a Pd η(2)-carbonato complex stabilized by an NHC framework. 3a-CO(3) can be easily converted to a PdCl(2) derivative by treating it with chloroform. We have found these palladium complexes mediate the Heck-Mizoroki coupling with a low catalyst loading. Furthermore, we also expand such catalytic manifold toward constructing fused polyaromatic substrates, a highly useful class of compounds in optoelectronic chemistry.

  18. 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…

  19. Asymptotic aspect of derivations in Banach algebras.

    PubMed

    Roh, Jaiok; Chang, Ick-Soon

    2017-01-01

    We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.

  20. A tensor Banach algebra approach to abstract kinetic equations

    NASA Astrophysics Data System (ADS)

    Greenberg, W.; van der Mee, C. V. M.

    The study deals with a concrete algebraic construction providing the existence theory for abstract kinetic equation boundary-value problems, when the collision operator A is an accretive finite-rank perturbation of the identity operator in a Hilbert space H. An algebraic generalization of the Bochner-Phillips theorem is utilized to study solvability of the abstract boundary-value problem without any regulatory condition. A Banach algebra in which the convolution kernel acts is obtained explicitly, and this result is used to prove a perturbation theorem for bisemigroups, which then plays a vital role in solving the initial equations.

  1. An Uncommon Approach to a Common Algebraic Error

    ERIC Educational Resources Information Center

    Rossi, Paul S.

    2008-01-01

    The basic rules of elementary algebra can often appear beyond the grasp of many students. Even though most subjects, including calculus, prove to be more difficult, it is the simple rules of algebra that continue to be the "thorn in the side" of many mathematics students. In this paper we present a result intended to help students achieve a…

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

    NASA Astrophysics Data System (ADS)

    Senovilla, José M. M.

    2010-11-01

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

  3. Linear-Algebra Programs

    NASA Technical Reports Server (NTRS)

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

    1982-01-01

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

  4. A new family of N dimensional superintegrable double singular oscillators and quadratic algebra Q(3) ⨁ so(n) ⨁ so(N-n)

    NASA Astrophysics Data System (ADS)

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

    2015-11-01

    We introduce a new family of N dimensional quantum superintegrable models consisting of double singular oscillators of type (n, N-n). The special cases (2,2) and (4,4) have previously been identified as the duals of 3- and 5-dimensional deformed Kepler-Coulomb systems with u(1) and su(2) monopoles, respectively. The models are multiseparable and their wave functions are obtained in (n, N-n) double-hyperspherical coordinates. We obtain the integrals of motion and construct the finitely generated polynomial algebra that is the direct sum of a quadratic algebra Q(3) involving three generators, so(n), so(N-n) (i.e. Q(3) ⨁ so(n) ⨁ so(N-n)). The structure constants of the quadratic algebra itself involve the Casimir operators of the two Lie algebras so(n) and so(N-n). Moreover, we obtain the finite dimensional unitary representations (unirreps) of the quadratic algebra and present an algebraic derivation of the degenerate energy spectrum of the superintegrable model.

  5. Lie algebra of conformal Killing-Yano forms

    NASA Astrophysics Data System (ADS)

    Ertem, Ümit

    2016-06-01

    We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms. A new Lie bracket for conformal Killing-Yano forms that corresponds to slightly modified Schouten-Nijenhuis bracket of differential forms is proposed. We show that conformal Killing-Yano forms satisfy a graded Lie algebra in constant curvature manifolds. It is also proven that normal conformal Killing-Yano forms in Einstein manifolds also satisfy a graded Lie algebra. The constructed graded Lie algebras reduce to the graded Lie algebra of Killing-Yano forms and the Lie algebras of conformal Killing and Killing vector fields in special cases.

  6. Quantum superintegrable system with a novel chain structure of quadratic algebras

    NASA Astrophysics Data System (ADS)

    Liao, Yidong; Marquette, Ian; Zhang, Yao-Zhong

    2018-06-01

    We analyse the n-dimensional superintegrable Kepler–Coulomb system with non-central terms. We find a novel underlying chain structure of quadratic algebras formed by the integrals of motion. We identify the elements for each sub-structure and obtain the algebra relations satisfied by them and the corresponding Casimir operators. These quadratic sub-algebras are realized in terms of a chain of deformed oscillators with factorized structure functions. We construct the finite-dimensional unitary representations of the deformed oscillators, and give an algebraic derivation of the energy spectrum of the superintegrable system.

  7. Comparison of the Effectiveness of a Traditional Intermediate Algebra Course With That of a Less Rigorous Intermediate Algebra Course in Preparing Students for Success in a Subsequent Mathematics Course

    ERIC Educational Resources Information Center

    Sworder, Steven C.

    2007-01-01

    An experimental two-track intermediate algebra course was offered at Saddleback College, Mission Viejo, CA, between the Fall, 2002 and Fall, 2005 semesters. One track was modeled after the existing traditional California community college intermediate algebra course and the other track was a less rigorous intermediate algebra course in which the…

  8. Particle-like structure of coaxial Lie algebras

    NASA Astrophysics Data System (ADS)

    Vinogradov, A. M.

    2018-01-01

    This paper is a natural continuation of Vinogradov [J. Math. Phys. 58, 071703 (2017)] where we proved that any Lie algebra over an algebraically closed field or over R can be assembled in a number of steps from two elementary constituents, called dyons and triadons. Here we consider the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons. We describe the maximal families of pairwise compatible base dyons and triadons called clusters, and, as a consequence, we give a complete description of the coaxial Lie algebras. The remarkable fact is that dyons and triadons in clusters are self-organised in structural groups which are surrounded by casings and linked by connectives. We discuss generalisations and applications to the theory of deformations of Lie algebras.

  9. On the structure of quantum L∞ algebras

    NASA Astrophysics Data System (ADS)

    Blumenhagen, Ralph; Fuchs, Michael; Traube, Matthias

    2017-10-01

    It is believed that any classical gauge symmetry gives rise to an L∞ algebra. Based on the recently realized relation between classical W algebras and L∞ algebras, we analyze how this generalizes to the quantum case. Guided by the existence of quantum W algebras, we provide a physically well motivated definition of quantum L∞ algebras describing the consistency of global symmetries in quantum field theories. In this case we are restricted to only two non-trivial graded vector spaces X 0 and X -1 containing the symmetry variations and the symmetry generators. This quantum L∞ algebra structure is explicitly exemplified for the quantum W_3 algebra. The natural quantum product between fields is the normal ordered one so that, due to contractions between quantum fields, the higher L∞ relations receive off-diagonal quantum corrections. Curiously, these are not present in the loop L∞ algebra of closed string field theory.

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

    PubMed

    Li, Ming; Zhao, Wei

    2012-01-01

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

  11. An algebra of reversible computation.

    PubMed

    Wang, Yong

    2016-01-01

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

  12. Elliptic complexes over C∗-algebras of compact operators

    NASA Astrophysics Data System (ADS)

    Krýsl, Svatopluk

    2016-03-01

    For a C∗-algebra A of compact operators and a compact manifold M, we prove that the Hodge theory holds for A-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective A-Hilbert bundles over M. For these C∗-algebras and manifolds, we get a topological isomorphism between the cohomology groups of an A-elliptic complex and the space of harmonic elements of the complex. Consequently, the cohomology groups appear to be finitely generated projective C∗-Hilbert modules and especially, Banach spaces. We also prove that in the category of Hilbert A-modules and continuous adjointable Hilbert A-module homomorphisms, the property of a complex of being self-adjoint parametrix possessing characterizes the complexes of Hodge type.

  13. Meta-Representation in an Algebra I Classroom

    ERIC Educational Resources Information Center

    Izsak, Andrew; Caglayan, Gunhan; Olive, John

    2009-01-01

    We describe how 1 Algebra I teacher and her 8th-grade students used meta-representational knowledge when generating and evaluating equations to solve word problems. Analyzing data from a sequence of 4 lessons, we found that the teacher and her students used criteria for evaluating equations, in addition to other types of knowledge (e.g., different…

  14. Aspects of QCD current algebra on a null plane

    NASA Astrophysics Data System (ADS)

    Beane, S. R.; Hobbs, T. J.

    2016-09-01

    Consequences of QCD current algebra formulated on a light-like hyperplane are derived for the forward scattering of vector and axial-vector currents on an arbitrary hadronic target. It is shown that current algebra gives rise to a special class of sum rules that are direct consequences of the independent chiral symmetry that exists at every point on the two-dimensional transverse plane orthogonal to the lightlike direction. These sum rules are obtained by exploiting the closed, infinite-dimensional algebra satisfied by the transverse moments of null-plane axial-vector and vector charge distributions. In the special case of a nucleon target, this procedure leads to the Adler-Weisberger, Gerasimov-Drell-Hearn, Cabibbo-Radicati and Fubini-Furlan-Rossetti sum rules. Matching to the dispersion-theoretic language which is usually invoked in deriving these sum rules, the moment sum rules are shown to be equivalent to algebraic constraints on forward S-matrix elements in the Regge limit.

  15. FRT presentation of the Onsager algebras

    NASA Astrophysics Data System (ADS)

    Baseilhac, Pascal; Belliard, Samuel; Crampé, Nicolas

    2018-03-01

    A presentation à la Faddeev-Reshetikhin-Takhtajan (FRT) of the Onsager, augmented Onsager and sl_2 -invariant Onsager algebras is given, using the framework of the nonstandard classical Yang-Baxter algebras. Associated current algebras are identified, and generating functions of mutually commuting quantities are obtained.

  16. Constraint-Referenced Analytics of Algebra Learning

    ERIC Educational Resources Information Center

    Sutherland, Scot M.; White, Tobin F.

    2016-01-01

    The development of the constraint-referenced analytics tool for monitoring algebra learning activities presented here came from the desire to firstly, take a more quantitative look at student responses in collaborative algebra activities, and secondly, to situate those activities in a more traditional introductory algebra setting focusing on…

  17. Post-Lie algebras and factorization theorems

    NASA Astrophysics Data System (ADS)

    Ebrahimi-Fard, Kurusch; Mencattini, Igor; Munthe-Kaas, Hans

    2017-09-01

    In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions of) those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang-Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.

  18. Inverse Modelling Problems in Linear Algebra Undergraduate Courses

    ERIC Educational Resources Information Center

    Martinez-Luaces, Victor E.

    2013-01-01

    This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…

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

    ERIC Educational Resources Information Center

    Ozgun-Koca, S. Ash

    2010-01-01

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

  20. Exploring Teacher Noticing of Student Algebraic Thinking in a Video Club

    ERIC Educational Resources Information Center

    Walkoe, Janet

    2015-01-01

    Learning algebra is critical for students in the USA today, yet many students in the USA struggle in algebra classes. Researchers claim that one reason for these difficulties is that algebra classes often focus on symbol manipulation and procedures above, and many times at the expense of, a more conceptual understanding of the content. Teaching…

  1. q-Derivatives, quantization methods and q-algebras

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

    Twarock, Reidun

    1998-12-15

    Using the example of Borel quantization on S{sup 1}, we discuss the relation between quantization methods and q-algebras. In particular, it is shown that a q-deformation of the Witt algebra with generators labeled by Z is realized by q-difference operators. This leads to a discrete quantum mechanics. Because of Z, the discretization is equidistant. As an approach to a non-equidistant discretization of quantum mechanics one can change the Witt algebra using not the number field Z as labels but a quadratic extension of Z characterized by an irrational number {tau}. This extension is denoted as quasi-crystal Lie algebra, because thismore » is a relation to one-dimensional quasicrystals. The q-deformation of this quasicrystal Lie algebra is discussed. It is pointed out that quasicrystal Lie algebras can be considered also as a 'deformed' Witt algebra with a 'deformation' of the labeling number field. Their application to the theory is discussed.« less

  2. Visual Salience of Algebraic Transformations

    ERIC Educational Resources Information Center

    Kirshner, David; Awtry, Thomas

    2004-01-01

    Information processing researchers have assumed that algebra symbol skills depend on mastery of the abstract rules presented in the curriculum (Matz, 1980; Sleeman, 1986). Thus, students' ubiquitous algebra errors have been taken as indicating the need to embed algebra in rich contextual settings (Kaput, 1995; National Council of Teachers of…

  3. Numerical Linear Algebra.

    DTIC Science & Technology

    1980-09-08

    February 1979 through 31 March 1980 Title of Research: NUMERICAL LINEAR ALGEBRA Principal Investigators: Gene H. Golub James H. Wilkinson Research...BEFORE COMPLETING FORM 2 OTAgSSION NO. 3. RECIPIENT’S CATALOG NUMBER ITE~ btitle) ~qEE NUMERICAL LINEAR ALGEBRA #I ~ f#7&/8 PER.ORMING ORG. REPORT NUM 27R 7

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

    PubMed Central

    Li, Ming; Zhao, Wei

    2012-01-01

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

  5. Predicting NonInertial Effects with Algebraic Stress Models which Account for Dissipation Rate Anisotropies

    NASA Technical Reports Server (NTRS)

    Jongen, T.; Machiels, L.; Gatski, T. B.

    1997-01-01

    Three types of turbulence models which account for rotational effects in noninertial frames of reference are evaluated for the case of incompressible, fully developed rotating turbulent channel flow. The different types of models are a Coriolis-modified eddy-viscosity model, a realizable algebraic stress model, and an algebraic stress model which accounts for dissipation rate anisotropies. A direct numerical simulation of a rotating channel flow is used for the turbulent model validation. This simulation differs from previous studies in that significantly higher rotation numbers are investigated. Flows at these higher rotation numbers are characterized by a relaminarization on the cyclonic or suction side of the channel, and a linear velocity profile on the anticyclonic or pressure side of the channel. The predictive performance of the three types of models are examined in detail, and formulation deficiencies are identified which cause poor predictive performance for some of the models. Criteria are identified which allow for accurate prediction of such flows by algebraic stress models and their corresponding Reynolds stress formulations.

  6. Generalized conformal realizations of Kac-Moody algebras

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

    Palmkvist, Jakob

    2009-01-15

    We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n=1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3x3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f{sub 4}, e{sub 6}, e{sub 7}, e{sub 8} for n=2. Moreover, we obtain their infinite-dimensional extensions for n{>=}3. In the casemore » of 2x2 matrices, the resulting Lie algebras are of the form so(p+n,q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q)« less

  7. Quiver elliptic W-algebras

    NASA Astrophysics Data System (ADS)

    Kimura, Taro; Pestun, Vasily

    2018-06-01

    We define elliptic generalization of W-algebras associated with arbitrary quiver using our construction (Kimura and Pestun in Quiver W-algebras, 2015. arXiv:1512.08533 [hep-th]) with six-dimensional gauge theory.

  8. Learning Activity Package, Algebra.

    ERIC Educational Resources Information Center

    Evans, Diane

    A set of ten teacher-prepared Learning Activity Packages (LAPs) in beginning algebra and nine in intermediate algebra, these units cover sets, properties of operations, number systems, open expressions, solution sets of equations and inequalities in one and two variables, exponents, factoring and polynomials, relations and functions, radicals,…

  9. Algebraic special functions and SO(3,2)

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

    Celeghini, E., E-mail: celeghini@fi.infn.it; Olmo, M.A. del, E-mail: olmo@fta.uva.es

    2013-06-15

    A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L{sup 2} functions defined on (−1,1)×Z and on the sphere S{sup 2}, respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining inmore » this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L{sup 2} functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2). -- Highlights: •The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP). •ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation. •A ladder structure is the condition to get a Lie group representation defining “algebraic special functions”. •The “algebraic special functions” connect Lie algebras and L{sup 2} functions.« less

  10. 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)

  11. Algebra for All: The Effect of Algebra Coursework and Classroom Peer Academic Composition on Low-Achieving Students

    ERIC Educational Resources Information Center

    Nomi, Takako; Raudenbush, Stephen W.

    2014-01-01

    Algebra is often considered as a gateway for later achievement. A recent report by the Mathematics Advisory Panel (2008) underscores the importance of improving algebra learning in secondary school. Today, a growing number of states and districts require algebra for all students in ninth grade or earlier. Chicago is at the forefront of this…

  12. Ready, Set, Algebra?

    ERIC Educational Resources Information Center

    Levy, Alissa Beth

    2012-01-01

    The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…

  13. Algorithms for computations of Loday algebras' invariants

    NASA Astrophysics Data System (ADS)

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

    2017-04-01

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

  14. Astro Algebra [CD-ROM].

    ERIC Educational Resources Information Center

    1997

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

  15. Semiclassical states on Lie algebras

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

    Tsobanjan, Artur, E-mail: artur.tsobanjan@gmail.com

    2015-03-15

    The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following themore » methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.« less

  16. A Regression Analysis of South Carolina Algebra I End-of-Course Exam Scores by Schedule Type

    ERIC Educational Resources Information Center

    Smith, Dawn M.

    2017-01-01

    The purpose of this study was to examine the relationship between scheduling and first-year-high-school students' exam scores on the South Carolina Algebra I End-of-Course (EOC) assessment. The study compared existing empirical data from two southeastern high schools from the same school district using 4 X 4 block schedules from 2011-2014 and…

  17. On character amenability of Banach algebras

    NASA Astrophysics Data System (ADS)

    Kaniuth, E.; Lau, A. T.; Pym, J.

    2008-08-01

    We continue our work [E. Kaniuth, A.T. Lau, J. Pym, On [phi]-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 85-96] in the study of amenability of a Banach algebra A defined with respect to a character [phi] of A. Various necessary and sufficient conditions of a global and a pointwise nature are found for a Banach algebra to possess a [phi]-mean of norm 1. We also completely determine the size of the set of [phi]-means for a separable weakly sequentially complete Banach algebra A with no [phi]-mean in A itself. A number of illustrative examples are discussed.

  18. 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.

  19. 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.

  20. A Cohomological Perspective on Algebraic Quantum Field Theory

    NASA Astrophysics Data System (ADS)

    Hawkins, Eli

    2018-05-01

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

  1. Assessing non-uniqueness: An algebraic approach

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

    Vasco, Don W.

    Geophysical inverse problems are endowed with a rich mathematical structure. When discretized, most differential and integral equations of interest are algebraic (polynomial) in form. Techniques from algebraic geometry and computational algebra provide a means to address questions of existence and uniqueness for both linear and non-linear inverse problem. In a sense, the methods extend ideas which have proven fruitful in treating linear inverse problems.

  2. Working memory, worry, and algebraic ability.

    PubMed

    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. Copyright © 2013 Elsevier Inc. All rights reserved.

  3. Proof and Reasoning in Secondary School Algebra Textbooks

    ERIC Educational Resources Information Center

    Dituri, Philip

    2013-01-01

    The purpose of this study was to determine the extent to which the modeling of deductive reasoning and proof-type thinking occurs in a mathematics course in which students are not explicitly preparing to write formal mathematical proofs. Algebra was chosen because it is the course that typically directly precedes a student's first formal…

  4. Contractions and deformations of quasiclassical Lie algebras preserving a nondegenerate quadratic Casimir operator

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

    Campoamor-Stursberg, R., E-mail: rutwig@mat.ucm.e

    2008-05-15

    By means of contractions of Lie algebras, we obtain new classes of indecomposable quasiclassical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from noncompact real simple algebras with nonsimple complexification, where we impose that a nondegenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem and obtain sufficient conditions on integrable cocycles of quasiclassical Lie algebras in order to preserve nondegenerate quadratic Casimir operators by the associated linear deformations.

  5. Super-Lie n-algebra extensions, higher WZW models and super-p-branes with tensor multiplet fields

    NASA Astrophysics Data System (ADS)

    Fiorenza, Domenico; Sati, Hisham; Schreiber, Urs

    2015-12-01

    We formalize higher-dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L∞-extension theory and how the resulting "extended (super-)space-times" formalize spacetimes containing σ-model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super-p-brane spectrum of superstring/M-theory is realized this way, including the pure σ-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional (11D) spacetime with an M2-brane condensate turns out to be the "M-theory super-Lie algebra". We also observe that in this formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11D sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string theory. Finally we give the non-perturbative description of all this by higher WZW-type σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology.

  6. Catching Up on Algebra

    ERIC Educational Resources Information Center

    Cavanagh, Sean

    2008-01-01

    A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…

  7. Algebra. A Guidebook for Teaching.

    ERIC Educational Resources Information Center

    Goodman, Terry A.; And Others

    This book provides a general framework for organizing the instructional program in algebra. For each topic, a general approach for instruction, together with specific strategies, activities, and resources that can be used daily, are provided. Also included are worksheet pages that can be used with students. Most activities provide for student…

  8. Teaching Structure in Algebra

    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…

  9. The BMS4 algebra at spatial infinity

    NASA Astrophysics Data System (ADS)

    Troessaert, Cédric

    2018-04-01

    We show how a global BMS4 algebra appears as part of the asymptotic symmetry algebra at spatial infinity. Using linearised theory, we then show that this global BMS4 algebra is the one introduced by Strominger as a symmetry of the S-matrix.

  10. Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds

    NASA Astrophysics Data System (ADS)

    Liu, Chiu-Chu Melissa; Sheshmani, Artan

    2017-07-01

    An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.

  11. Constructing Meanings and Utilities within Algebraic Tasks

    ERIC Educational Resources Information Center

    Ainley, Janet; Bills, Liz; Wilson, Kirsty

    2004-01-01

    The Purposeful Algebraic Activity project aims to explore the potential of spreadsheets in the introduction to algebra and algebraic thinking. We discuss two sub-themes within the project: tracing the development of pupils' construction of meaning for variable from arithmetic-based activity, through use of spreadsheets, and into formal algebra,…

  12. A Learning Progressions Approach to Early Algebra Research and Practice

    ERIC Educational Resources Information Center

    Fonger, Nicole L.; Stephens, Ana; Blanton, Maria; Knuth, Eric

    2015-01-01

    We detail a learning progressions approach to early algebra research and how existing work around learning progressions and trajectories in mathematics and science education has informed our development of a four-component theoretical framework consisting of: a curricular progression of learning goals across big algebraic ideas; an instructional…

  13. The connection-set algebra--a novel formalism for the representation of connectivity structure in neuronal network models.

    PubMed

    Djurfeldt, Mikael

    2012-07-01

    The connection-set algebra (CSA) is a novel and general formalism for the description of connectivity in neuronal network models, from small-scale to large-scale structure. The algebra provides operators to form more complex sets of connections from simpler ones and also provides parameterization of such sets. CSA is expressive enough to describe a wide range of connection patterns, including multiple types of random and/or geometrically dependent connectivity, and can serve as a concise notation for network structure in scientific writing. CSA implementations allow for scalable and efficient representation of connectivity in parallel neuronal network simulators and could even allow for avoiding explicit representation of connections in computer memory. The expressiveness of CSA makes prototyping of network structure easy. A C+ + version of the algebra has been implemented and used in a large-scale neuronal network simulation (Djurfeldt et al., IBM J Res Dev 52(1/2):31-42, 2008b) and an implementation in Python has been publicly released.

  14. 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.

  15. 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"…

  16. A natural history of mathematics: George Peacock and the making of English algebra.

    PubMed

    Lambert, Kevin

    2013-06-01

    In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic would suggest arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's "principle of equivalent forms," which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra.

  17. 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…

  18. Algebraic Algorithm Design and Local Search

    DTIC Science & Technology

    1996-12-01

    method for performing algorithm design that is more purely algebraic than that of KIDS. This method is then applied to local search. Local search is a...synthesis. Our approach was to follow KIDS in spirit, but to adopt a pure algebraic formalism, supported by Kestrel’s SPECWARE environment (79), that...design was developed that is more purely algebraic than that of KIDS. This method was then applied to local search. A general theory of local search was

  19. 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.

  20. A New Age for Algebra

    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…

  1. Metric 3-Leibniz algebras and M2-branes

    NASA Astrophysics Data System (ADS)

    Méndez-Escobar, Elena

    2010-08-01

    This thesis is concerned with superconformal Chern-Simons theories with matter in 3 dimensions. The interest in these theories is two-fold. On the one hand, it is a new family of theories in which to test the AdS/CFT correspondence and on the other, they are important to study one of the main objects of M-theory (M2-branes). All these theories have something in common: they can be written in terms of 3-Leibniz algebras. Here we study the structure theory of such algebras, paying special attention to a subclass of them that gives rise to maximal supersymmetry and that was the first to appear in this context: 3-Lie algebras. In chapter 2, we review the structure theory of metric Lie algebras and their unitary representations. In chapter 3, we study metric 3-Leibniz algebras and show, by specialising a construction originally due to Faulkner, that they are in one to one correspondence with pairs of real metric Lie algebras and unitary representations of them. We also show a third characterisation for six extreme cases of 3-Leibniz algebras as graded Lie (super)algebras. In chapter 4, we study metric 3-Lie algebras in detail. We prove a structural result and also classify those with a maximally isotropic centre, which is the requirement that ensures unitarity of the corresponding conformal field theory. Finally, in chapter 5, we study the universal structure of superpotentials in this class of superconformal Chern-Simons theories with matter in three dimensions. We provide a uniform formulation for all these theories and establish the connection between the amount of supersymmetry preserved and the gauge Lie algebra and the appropriate unitary representation to be used to write down the Lagrangian. The conditions for supersymmetry enhancement are then expressed equivalently in the language of representation theory of Lie algebras or the language of 3-Leibniz algebras.

  2. Identities of Finitely Generated Algebras Over AN Infinite Field

    NASA Astrophysics Data System (ADS)

    Kemer, A. R.

    1991-02-01

    It is proved that for each finitely generated associative PI-algebra U over an infinite field F, there is a finite-dimensional F-algebra C such that the ideals of identities of the algebras U and C coincide. This yields a positive solution to the local problem of Specht for algebras over an infinite field: A finitely generated free associative algebra satisfies the maximum condition for T-ideals.

  3. A Meta-Analysis of Algebra Interventions for Learners with Disabilities and Struggling Learners

    ERIC Educational Resources Information Center

    Hughes, Elizabeth M.; Witzel, Bradley S.; Riccomini, Paul J.; Fries, Karen M.; Kanyongo, Gibbs Y.

    2014-01-01

    The need for global competence in mathematics is apparent. Algebra is considered a gateway course to prepare students for the demands of a competitive global market. Many students demonstrate low performance in algebra; this is especially true for students with disabilities. Effective algebra instruction is essential to increase algebra…

  4. Vertex Algebras W(p)Am and W(p)Dm and Constant Term Identities

    NASA Astrophysics Data System (ADS)

    Adamović, Dražen; Lin, Xianzu; Milas, Antun

    2015-03-01

    We consider AD-type orbifolds of the triplet vertex algebras W(p) extending the well-known c=1 orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras A(W(p)^{A_m}) and A(W(p)^{D_m}), where A_m and D_m are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible W(p)^Γ-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of m and p with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages; Internat. J. Math. 25 (2014), 1450001, 34 pages] provide a complete list of irreducible W(p)^{A_m} and W(p)^{D_m}-modules. This paper is a continuation of our previous work on the ADE subalgebras of the triplet vertex algebra W(p).

  5. Teacher Actions to Facilitate Early Algebraic Reasoning

    ERIC Educational Resources Information Center

    Hunter, Jodie

    2015-01-01

    In recent years there has been an increased emphasis on integrating the teaching of arithmetic and algebra in primary school classrooms. This requires teachers to develop links between arithmetic and algebra and use pedagogical actions that facilitate algebraic reasoning. Drawing on findings from a classroom-based study, this paper provides an…

  6. Teachers' Understanding of Algebraic Generalization

    NASA Astrophysics Data System (ADS)

    Hawthorne, Casey Wayne

    Generalization has been identified as a cornerstone of algebraic thinking (e.g., Lee, 1996; Sfard, 1995) and is at the center of a rich conceptualization of K-8 algebra (Kaput, 2008; Smith, 2003). Moreover, mathematics teachers are being encouraged to use figural-pattern generalizing tasks as a basis of student-centered instruction, whereby teachers respond to and build upon the ideas that arise from students' explorations of these activities. Although more and more teachers are engaging their students in such generalizing tasks, little is known about teachers' understanding of generalization and their understanding of students' mathematical thinking in this domain. In this work, I addressed this gap, exploring the understanding of algebraic generalization of 4 exemplary 8th-grade teachers from multiple perspectives. A significant feature of this investigation is an examination of teachers' understanding of the generalization process, including the use of algebraic symbols. The research consisted of two phases. Phase I was an examination of the teachers' understandings of the underlying quantities and quantitative relationships represented by algebraic notation. In Phase II, I observed the instruction of 2 of these teachers. Using the lens of professional noticing of students' mathematical thinking, I explored the teachers' enacted knowledge of algebraic generalization, characterizing how it supported them to effectively respond to the needs and queries of their students. Results indicated that teachers predominantly see these figural patterns as enrichment activities, disconnected from course content. Furthermore, in my analysis, I identified conceptual difficulties teachers experienced when solving generalization tasks, in particular, connecting multiple symbolic representations with the quantities in the figures. Moreover, while the teachers strived to overcome the challenges of connecting different representations, they invoked both productive and unproductive

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

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

    Kadets, Boris; Karolinsky, Eugene; Pop, Iulia

    2016-05-15

    In this paper we continue to study Belavin–Drinfeld cohomology introduced in Kadets et al., Commun. Math. Phys. 344(1), 1-24 (2016) and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra #Mathematical Fraktur Small G#. Here we compute Belavin–Drinfeld cohomology for all non-skewsymmetric r-matrices on the Belavin–Drinfeld list for simple Lie algebras of type B, C, and D.

  8. Linear {GLP}-algebras and their elementary theories

    NASA Astrophysics Data System (ADS)

    Pakhomov, F. N.

    2016-12-01

    The polymodal provability logic {GLP} was introduced by Japaridze in 1986. It is the provability logic of certain chains of provability predicates of increasing strength. Every polymodal logic corresponds to a variety of polymodal algebras. Beklemishev and Visser asked whether the elementary theory of the free {GLP}-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable [1]. For every positive integer n we solve the corresponding question for the logics {GLP}_n that are the fragments of {GLP} with n modalities. We prove that the elementary theory of the free {GLP}_n-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable for all n. We introduce the notion of a linear {GLP}_n-algebra and prove that all free {GLP}_n-algebras generated by the constants \\mathbf{0}, \\mathbf{1} are linear. We also consider the more general case of the logics {GLP}_α whose modalities are indexed by the elements of a linearly ordered set α: we define the notion of a linear algebra and prove the latter result in this case.

  9. The Dixmier Map for Nilpotent Super Lie Algebras

    NASA Astrophysics Data System (ADS)

    Herscovich, Estanislao

    2012-07-01

    In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by {Prim({U}({g}))} the set of (graded) primitive ideals of the enveloping algebra {{U}({g})} of a nilpotent Lie superalgebra {{g}} and {{A}d0} the adjoint group of {{g}0}, we prove that the usual Dixmier map for nilpotent Lie algebras can be naturally extended to the context of nilpotent super Lie algebras, i.e. there exists a bijective map I : {g}0^{*}/{A}d0 rightarrow Prim({U}({g})) defined by sending the equivalence class [ λ] of a functional λ to a primitive ideal I( λ) of {{U}({g})}, and which coincides with the Dixmier map in the case of nilpotent Lie algebras. Moreover, the construction of the previous map is explicit, and more or less parallel to the one for Lie algebras, a major difference with a previous approach ( cf. [18]). One key fact in the construction is the existence of polarizations for super Lie algebras, generalizing the concept defined for Lie algebras. As a corollary of the previous description, we obtain the isomorphism {{U}({g})/I(λ) ˜eq Cliffq(k) ⊗ Ap(k)}, where {(p,q) = (dim({g}0/{g}0^{λ})/2,dim({g}1/{g}1^{λ}))}, we get a direct construction of the maximal ideals of the underlying algebra of {{U}({g})} and also some properties of the stabilizers of the primitive ideals of {{U}({g})}.

  10. 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.

  11. 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…

  12. Who Takes College Algebra?

    ERIC Educational Resources Information Center

    Herriott, Scott R.; Dunbar, Steven R.

    2009-01-01

    The common understanding within the mathematics community is that the role of the college algebra course is to prepare students for calculus. Though exceptions are emerging, the curriculum of most college algebra courses and the content of most textbooks on the market both reflect that assumption. This article calls that assumption into question…

  13. The Effects of Representations, Constructivist Approaches, and Engagement on Middle School Students' Algebraic Procedure and Conceptual Understanding

    ERIC Educational Resources Information Center

    Ross, Amanda; Willson, Victor

    2012-01-01

    This study examined the effects of types of representations, constructivist teaching approaches, and student engagement on middle school algebra students' procedural knowledge and conceptual understanding. Data gathered from 16 video lessons and algebra pretest/posttests were used to run three multilevel structural equation models. Symbolic…

  14. Schwarz maps of algebraic linear ordinary differential equations

    NASA Astrophysics Data System (ADS)

    Sanabria Malagón, Camilo

    2017-12-01

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

  15. Learning Algebra from Worked Examples

    ERIC Educational Resources Information Center

    Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.

    2014-01-01

    For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…

  16. Color Algebras

    NASA Technical Reports Server (NTRS)

    Mulligan, Jeffrey B.

    2017-01-01

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

  17. Generalized Quantum Field Theory Based on a Nonlinear Deformed Heisenberg Algebra

    NASA Astrophysics Data System (ADS)

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

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

  18. 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.

  19. A new general access to either type of Securinega alkaloids: synthesis of securinine and (-)-allonorsecurinine.

    PubMed

    Alibés, Ramon; Ballbé, Marta; Busqué, Félix; de March, Pedro; Elias, Laia; Figueredo, Marta; Font, Josep

    2004-05-27

    The syntheses of securinine and (-)-allonorsecurinine have been achieved starting from easily available alpha-amino acid derivatives and using as key steps a RCM and a Heck reaction for the formation of rings D and C, respectively. [reaction: see text

  20. On Some Nonclassical Algebraic Properties of Interval-Valued Fuzzy Soft Sets

    PubMed Central

    2014-01-01

    Interval-valued fuzzy soft sets realize a hybrid soft computing model in a general framework. Both Molodtsov's soft sets and interval-valued fuzzy sets can be seen as special cases of interval-valued fuzzy soft sets. In this study, we first compare four different types of interval-valued fuzzy soft subsets and reveal the relations among them. Then we concentrate on investigating some nonclassical algebraic properties of interval-valued fuzzy soft sets under the soft product operations. We show that some fundamental algebraic properties including the commutative and associative laws do not hold in the conventional sense, but hold in weaker forms characterized in terms of the relation =L. We obtain a number of algebraic inequalities of interval-valued fuzzy soft sets characterized by interval-valued fuzzy soft inclusions. We also establish the weak idempotent law and the weak absorptive law of interval-valued fuzzy soft sets using interval-valued fuzzy soft J-equal relations. It is revealed that the soft product operations ∧ and ∨ of interval-valued fuzzy soft sets do not always have similar algebraic properties. Moreover, we find that only distributive inequalities described by the interval-valued fuzzy soft L-inclusions hold for interval-valued fuzzy soft sets. PMID:25143964

  1. On some nonclassical algebraic properties of interval-valued fuzzy soft sets.

    PubMed

    Liu, Xiaoyan; Feng, Feng; Zhang, Hui

    2014-01-01

    Interval-valued fuzzy soft sets realize a hybrid soft computing model in a general framework. Both Molodtsov's soft sets and interval-valued fuzzy sets can be seen as special cases of interval-valued fuzzy soft sets. In this study, we first compare four different types of interval-valued fuzzy soft subsets and reveal the relations among them. Then we concentrate on investigating some nonclassical algebraic properties of interval-valued fuzzy soft sets under the soft product operations. We show that some fundamental algebraic properties including the commutative and associative laws do not hold in the conventional sense, but hold in weaker forms characterized in terms of the relation = L . We obtain a number of algebraic inequalities of interval-valued fuzzy soft sets characterized by interval-valued fuzzy soft inclusions. We also establish the weak idempotent law and the weak absorptive law of interval-valued fuzzy soft sets using interval-valued fuzzy soft J-equal relations. It is revealed that the soft product operations ∧ and ∨ of interval-valued fuzzy soft sets do not always have similar algebraic properties. Moreover, we find that only distributive inequalities described by the interval-valued fuzzy soft L-inclusions hold for interval-valued fuzzy soft sets.

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

    DTIC Science & Technology

    2014-11-01

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

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

    ERIC Educational Resources Information Center

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

    2009-01-01

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

  4. A comparison of three algebraic stress closures for combustor flow calculations

    NASA Technical Reports Server (NTRS)

    Nikjooy, M.; So, R. M. C.; Hwang, B. C.

    1985-01-01

    A comparison is made of the performance of two locally nonequilibrium and one equilibrium algebraic stress closures in calculating combustor flows. Effects of four different pressure-strain models on these closure models are also analyzed. The results show that the pressure-strain models have a much greater influence on the calculated mean velocity and turbulence field than the algebraic stress closures, and that the best mean strain model for the pressure-strain terms is that proposed by Launder, Reece and Rodi (1975). However, the equilibrium algebraic stress closure with the Rotta return-to-isotropy model (1951) for the pressure-strain terms gives as good a correlation with measurements as when the Launder et al. mean strain model is included in the pressure-strain model. Finally, comparison of the calculations with the standard k-epsilon closure results show that the algebraic stress closures are better suited for simple turbulent flow calculations.

  5. Student and Instructor Perceptions of a Flipped College Algebra Classroom

    ERIC Educational Resources Information Center

    Jaster, Robert W.

    2017-01-01

    Each year about half a million students fail to make planned academic progress due to college algebra, hence the need for researchers to find ways of improving the quality of instruction in the course. Recent research suggests that flipping college algebra to allow time for active learning in the classroom may improve student performance. Also,…

  6. Developing "Algebraic Thinking": Two Key Ways to Establish Some Early Algebraic Ideas in Primary Classrooms

    ERIC Educational Resources Information Center

    Ormond, Christine

    2012-01-01

    Primary teachers play a key role in their students' future mathematical success in the early secondary years. While the word "algebra" may make some primary teachers feel uncomfortable or worried, the basic arithmetic ideas underlying algebra are vitally important for older primary students as they are increasingly required to use "algebraic…

  7. Quantum Observables and Effect Algebras

    NASA Astrophysics Data System (ADS)

    Dvurečenskij, Anatolij

    2018-03-01

    We study observables on monotone σ-complete effect algebras. We find conditions when a spectral resolution implies existence of the corresponding observable. We characterize sharp observables of a monotone σ-complete homogeneous effect algebra using its orthoalgebraic skeleton. In addition, we study compatibility in orthoalgebras and we show that every orthoalgebra satisfying RIP is an orthomodular poset.

  8. Clifford Algebra Implying Three Fermion Generations Revisited

    NASA Astrophysics Data System (ADS)

    Krolikowski, Wojciech

    2002-09-01

    The author's idea of algebraic compositeness of fundamental particles, allowing to understand the existence in Nature of three fermion generations, is revisited. It is based on two postulates. Primo, for all fundamental particles of matter the Dirac square-root procedure √ {p2} → {Γ }(N)p works, leading to a sequence N = 1,2,3, ... of Dirac-type equations, where four Dirac-type matrices {Γ }(N)μ are embedded into a Clifford algebra via a Jacobi definition introducing four ``centre-of-mass'' and (N-1)× four ``relative'' Dirac-type matrices. These define one ``centre-of-mass'' and (N-1) ``relative'' Dirac bispinor indices. Secundo, the ``centre-of-mass'' Dirac bispinor index is coupled to the Standard Model gauge fields, while (N-1) ``relative'' Dirac bispinor indices are all free indistinguishable physical objects obeying Fermi statistics along with the Pauli principle which requires the full antisymmetry with respect to ``relative'' Dirac indices. This allows only for three Dirac-type equations with N = 1,3,5 in the case of N odd, and two with N = 2,4 in the case of N even. The first of these results implies unavoidably the existence of three and only three generations of fundamental fermions, namely leptons and quarks, as labelled by the Standard Model signature. At the end, a comment is added on the possible shape of Dirac 3x3 mass matrices for four sorts of spin-1/2 fundamental fermions appearing in three generations. For charged leptons a prediction is mτ = 1776.80 MeV, when the input of experimental me and mμ is used.

  9. Conformal Galilei algebras, symmetric polynomials and singular vectors

    NASA Astrophysics Data System (ADS)

    Křižka, Libor; Somberg, Petr

    2018-01-01

    We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras cga_ℓ (d,C) with d=1 for any integer value ℓ \\in N. The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.

  10. pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

    DOE PAGES

    Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul; ...

    2017-12-20

    We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

  11. pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

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

    Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul

    We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

  12. Singular vectors for the WN algebras

    NASA Astrophysics Data System (ADS)

    Ridout, David; Siu, Steve; Wood, Simon

    2018-03-01

    In this paper, we use free field realisations of the A-type principal, or Casimir, WN algebras to derive explicit formulae for singular vectors in Fock modules. These singular vectors are constructed by applying screening operators to Fock module highest weight vectors. The action of the screening operators is then explicitly evaluated in terms of Jack symmetric functions and their skew analogues. The resulting formulae depend on sequences of pairs of integers that completely determine the Fock module as well as the Jack symmetric functions.

  13. The Algebra of Complex Numbers.

    ERIC Educational Resources Information Center

    LePage, Wilbur R.

    This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…

  14. Application of geometric algebra for the description of polymer conformations.

    PubMed

    Chys, Pieter

    2008-03-14

    In this paper a Clifford algebra-based method is applied to calculate polymer chain conformations. The approach enables the calculation of the position of an atom in space with the knowledge of the bond length (l), valence angle (theta), and rotation angle (phi) of each of the preceding bonds in the chain. Hence, the set of geometrical parameters {l(i),theta(i),phi(i)} yields all the position coordinates p(i) of the main chain atoms. Moreover, the method allows the calculation of side chain conformations and the computation of rotations of chain segments. With these features it is, in principle, possible to generate conformations of any type of chemical structure. This method is proposed as an alternative for the classical approach by matrix algebra. It is more straightforward and its final symbolic representation considerably simpler than that of matrix algebra. Approaches for realistic modeling by means of incorporation of energetic considerations can be combined with it. This article, however, is entirely focused at showing the suitable mathematical framework on which further developments and applications can be built.

  15. Quantum teleportation and Birman-Murakami-Wenzl algebra

    NASA Astrophysics Data System (ADS)

    Zhang, Kun; Zhang, Yong

    2017-02-01

    In this paper, we investigate the relationship of quantum teleportation in quantum information science and the Birman-Murakami-Wenzl (BMW) algebra in low-dimensional topology. For simplicity, we focus on the two spin-1/2 representation of the BMW algebra, which is generated by both the Temperley-Lieb projector and the Yang-Baxter gate. We describe quantum teleportation using the Temperley-Lieb projector and the Yang-Baxter gate, respectively, and study teleportation-based quantum computation using the Yang-Baxter gate. On the other hand, we exploit the extended Temperley-Lieb diagrammatical approach to clearly show that the tangle relations of the BMW algebra have a natural interpretation of quantum teleportation. Inspired by this interpretation, we construct a general representation of the tangle relations of the BMW algebra and obtain interesting representations of the BMW algebra. Therefore, our research sheds a light on a link between quantum information science and low-dimensional topology.

  16. Category-theoretic models of algebraic computer systems

    NASA Astrophysics Data System (ADS)

    Kovalyov, S. P.

    2016-01-01

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

  17. Enhanced asymptotic BMS3 algebra of the flat spacetime solutions of generalized minimal massive gravity

    NASA Astrophysics Data System (ADS)

    Setare, M. R.; Adami, H.

    2018-01-01

    We apply the new fall of conditions presented in the paper [1] on asymptotically flat spacetime solutions of Chern-Simons-like theories of gravity. We show that the considered fall of conditions asymptotically solve equations of motion of generalized minimal massive gravity. We demonstrate that there exist two type of solutions, one of those is trivial and the others are non-trivial. By looking at non-trivial solutions, for asymptotically flat spacetimes in the generalized minimal massive gravity, in contrast to Einstein gravity, cosmological parameter can be non-zero. We obtain the conserved charges of the asymptotically flat spacetimes in generalized minimal massive gravity, and by introducing Fourier modes we show that the asymptotic symmetry algebra is a semidirect product of a BMS3 algebra and two U (1) current algebras. Also we verify that the BMS3 algebra can be obtained by a contraction of the AdS3 asymptotic symmetry algebra when the AdS3 radius tends to infinity in the flat-space limit. Finally we find energy, angular momentum and entropy for a particular case and deduce that these quantities satisfy the first law of flat space cosmologies.

  18. 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.

  19. Minimal unitary representation of 5d superconformal algebra F(4) and AdS 6/CFT 5 higher spin (super)-algebras

    DOE PAGES

    Fernando, Sudarshan; Günaydin, Murat

    2014-11-28

    We study the minimal unitary representation (minrep) of SO(5, 2), obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO(5, 2) describes a massless conformal scalar field in five dimensions and admits a unique “deformation” which describes a massless conformal spinor. Scalar and spinor minreps of SO(5, 2) are the 5d analogs of Dirac’s singletons of SO(3, 2). We then construct the minimal unitary representation of the unique 5d supercon-formal algebra F(4) with the even subalgebra SO(5, 2) ×SU(2). The minrep of F(4) describes a massless conformal supermultiplet consisting of two scalar andmore » one spinor fields. We then extend our results to the construction of higher spin AdS 6/CFT 5 (super)-algebras. The Joseph ideal of the minrep of SO(5, 2) vanishes identically as operators and hence its enveloping algebra yields the AdS 6/CFT 5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a “deformed” higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS 6/CFT 5 superalgebra as the enveloping algebra of the minimal unitary realization of F(4) obtained by the quasiconformal methods.« less

  20. On the homotopy equivalence of simple AI-algebras

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

    Aristov, O Yu

    1999-02-28

    Let A and B be simple unital AI-algebras (an AI-algebra is an inductive limit of C*-algebras of the form BigOplus{sub i}{sup k}C([0,1],M{sub N{sub i}}). It is proved that two arbitrary unital homomorphisms from A into B such that the corresponding maps K{sub 0}A{yields}K{sub 0}B coincide are homotopic. Necessary and sufficient conditions on the Elliott invariant for A and B to be homotopy equivalent are indicated. Moreover, two algebras in the above class having the same K-theory but not homotopy equivalent are constructed. A theorem on the homotopy of approximately unitarily equivalent homomorphisms between AI-algebras is used in the proof, whichmore » is deduced in its turn from a generalization to the case of AI-algebras of a theorem of Manuilov stating that a unitary matrix almost commuting with a self-adjoint matrix h can be joined to 1 by a continuous path consisting of unitary matrices almost commuting with h.« less

  1. 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.

  2. The effects of an integrated Algebra 1/physical science curriculum on student achievement in Algebra 1, proportional reasoning and graphing abilities

    NASA Astrophysics Data System (ADS)

    Lawrence, Lettie Carol

    1997-08-01

    The purpose of this investigation was to determine if an integrated curriculum in algebra 1/physical science facilitates acquisition of proportional reasoning and graphing abilities better than a non-integrated, traditional, algebra 1 curriculum. Also, this study was to ascertain if the integrated algebra 1/physical science curriculum resulted in greater student achievement in algebra 1. The curriculum used in the experimental class was SAM 9 (Science and Mathematics 9), an investigation-based curriculum that was written to integrate physical science and basic algebra content. The experiment was conducted over one school year. The subjects in the study were 61 ninth grade students. The experimental group consisted of one class taught concurrently by a mathematics teacher and a physical science teacher. The control group consisted of three classes of algebra 1 students taught by one mathematics teacher and taking physical science with other teachers in the school who were not participating in the SAM 9 program. This study utilized a quasi-experimental non-randomized control group pretest-posttest design. The investigator obtained end-of-algebra 1 scores from student records. The written open-ended graphing instruments and the proportional reasoning instrument were administered to both groups as pretests and posttests. The graphing instruments were also administered as a midtest. A two sample t-test for independent means was used to determine significant differences in achievement on the end-of-course algebra 1 test. Quantitative data from the proportional reasoning and graphing instruments were analyzed using a repeated measures analysis of variance to determine differences in scores over time for the experimental and control groups. The findings indicate no significant difference between the experimental and control groups on the end-of-course algebra 1 test. Results also indicate no significant differences in proportional reasoning and graphing abilities between

  3. UCSMP Algebra. What Works Clearinghouse Intervention Report

    ERIC Educational Resources Information Center

    What Works Clearinghouse, 2007

    2007-01-01

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

  4. 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…

  5. The tensor hierarchy algebra

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

    Palmkvist, Jakob, E-mail: palmkvist@ihes.fr

    We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of ourmore » Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.« less

  6. Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions

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

    Marquette, Ian, E-mail: i.marquette@uq.edu.au; Quesne, Christiane, E-mail: cquesne@ulb.ac.be

    2015-06-15

    We extend the construction of 2D superintegrable Hamiltonians with separation of variables in spherical coordinates using combinations of shift, ladder, and supercharge operators to models involving rational extensions of the two-parameter Lissajous systems on the sphere. These new families of superintegrable systems with integrals of arbitrary order are connected with Jacobi exceptional orthogonal polynomials of type I (or II) and supersymmetric quantum mechanics. Moreover, we present an algebraic derivation of the degenerate energy spectrum for the one- and two-parameter Lissajous systems and the rationally extended models. These results are based on finitely generated polynomial algebras, Casimir operators, realizations as deformedmore » oscillator algebras, and finite-dimensional unitary representations. Such results have only been established so far for 2D superintegrable systems separable in Cartesian coordinates, which are related to a class of polynomial algebras that display a simpler structure. We also point out how the structure function of these deformed oscillator algebras is directly related with the generalized Heisenberg algebras spanned by the nonpolynomial integrals.« less

  7. The Growing Importance of Linear Algebra in Undergraduate Mathematics.

    ERIC Educational Resources Information Center

    Tucker, Alan

    1993-01-01

    Discusses the theoretical and practical importance of linear algebra. Presents a brief history of linear algebra and matrix theory and describes the place of linear algebra in the undergraduate curriculum. (MDH)

  8. Teaching Linear Algebra: Proceeding More Efficiently by Staying Comfortably within Z

    ERIC Educational Resources Information Center

    Beaver, Scott

    2015-01-01

    For efficiency in a linear algebra course the instructor may wish to avoid the undue arithmetical distractions of rational arithmetic. In this paper we explore how to write fraction-free problems of various types including elimination, matrix inverses, orthogonality, and the (non-normalizing) Gram-Schmidt process.

  9. 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.

  10. 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…

  11. Effectiveness of Cognitive Tutor Algebra I at Scale

    ERIC Educational Resources Information Center

    Pane, John F.; Griffin, Beth Ann; McCaffrey, Daniel F.; Karam, Rita

    2014-01-01

    This article examines the effectiveness of a technology-based algebra curriculum in a wide variety of middle schools and high schools in seven states. Participating schools were matched into similar pairs and randomly assigned to either continue with the current algebra curriculum for 2 years or to adopt Cognitive Tutor Algebra I (CTAI), which…

  12. A Computer Algebra Approach to Solving Chemical Equilibria in General Chemistry

    ERIC Educational Resources Information Center

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

    2012-01-01

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

  13. Classical versus Computer Algebra Methods in Elementary Geometry

    ERIC Educational Resources Information Center

    Pech, Pavel

    2005-01-01

    Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…

  14. Using Homemade Algebra Tiles To Develop Algebra and Prealgebra Concepts.

    ERIC Educational Resources Information Center

    Leitze, Annette Ricks; Kitt, Nancy A.

    2000-01-01

    Describes how to use homemade tiles, sketches, and the box method to reach a broader group of students for successful algebra learning. Provides a list of concepts appropriate for such an approach. (KHR)

  15. A simplified formalism of the algebra of partially transposed permutation operators with applications

    NASA Astrophysics Data System (ADS)

    Mozrzymas, Marek; Studziński, Michał; Horodecki, Michał

    2018-03-01

    Herein we continue the study of the representation theory of the algebra of permutation operators acting on the n -fold tensor product space, partially transposed on the last subsystem. We develop the concept of partially reduced irreducible representations, which allows us to significantly simplify previously proved theorems and, most importantly, derive new results for irreducible representations of the mentioned algebra. In our analysis we are able to reduce the complexity of the central expressions by getting rid of sums over all permutations from the symmetric group, obtaining equations which are much more handy in practical applications. We also find relatively simple matrix representations for the generators of the underlying algebra. The obtained simplifications and developments are applied to derive the characteristics of a deterministic port-based teleportation scheme written purely in terms of irreducible representations of the studied algebra. We solve an eigenproblem for the generators of the algebra, which is the first step towards a hybrid port-based teleportation scheme and gives us new proofs of the asymptotic behaviour of teleportation fidelity. We also show a connection between the density operator characterising port-based teleportation and a particular matrix composed of an irreducible representation of the symmetric group, which encodes properties of the investigated algebra.

  16. Simple nuclear C*-algebras not isomorphic to their opposites

    PubMed Central

    Hirshberg, Ilan

    2017-01-01

    We show that it is consistent with Zermelo–Fraenkel set theory with the axiom of choice (ZFC) that there is a simple nuclear nonseparable C∗-algebra, which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O2 or of the canonical anticommutation relations (CAR) algebra. PMID:28559339

  17. Internally connected graphs and the Kashiwara-Vergne Lie algebra

    NASA Astrophysics Data System (ADS)

    Felder, Matteo

    2018-06-01

    It is conjectured that the Kashiwara-Vergne Lie algebra \\widehat{krv}_2 is isomorphic to the direct sum of the Grothendieck-Teichmüller Lie algebra grt_1 and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of \\widehat{krv}_2 whose intersection is grt_1, thus giving a way to interpolate between these two Lie algebras.

  18. Structure of Lie point and variational symmetry algebras for a class of odes

    NASA Astrophysics Data System (ADS)

    Ndogmo, J. C.

    2018-04-01

    It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation y(n)=0. For arbitrary systems of ordinary differential equations of order n ≥ 3 reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.

  19. How Middle Grade Teachers Think about Algebraic Reasoning

    ERIC Educational Resources Information Center

    Glassmeyer, David; Edwards, Belinda

    2016-01-01

    Algebraic reasoning is an essential habit of mind for building conceptual knowledge in K-12 mathematics, yet little is known about how middle school mathematics teachers think about algebraic reasoning. In this article we describe a research project examining how algebraic reasoning was considered by grades 6, 7, or 8 mathematics teachers in a…

  20. Quantum walled Brauer algebra: commuting families, Baxterization, and representations

    NASA Astrophysics Data System (ADS)

    Semikhatov, A. M.; Tipunin, I. Yu

    2017-02-01

    For the quantum walled Brauer algebra, we construct its Specht modules and (for generic parameters of the algebra) seminormal modules. The latter construction yields the spectrum of a commuting family of Jucys-Murphy elements. We also propose a Baxterization prescription; it involves representing the quantum walled Brauer algebra in terms of morphisms in a braided monoidal category and introducing parameters into these morphisms, which allows constructing a ‘universal transfer matrix’ that generates commuting elements of the algebra.

  1. Card Games and Algebra Tic Tacmatics on Achievement of Junior Secondary II Students in Algebraic Expressions

    ERIC Educational Resources Information Center

    Okpube, Nnaemeka Michael; Anugwo, M. N.

    2016-01-01

    This study investigated the Card Games and Algebra tic-Tacmatics on Junior Secondary II Students' Achievement in Algebraic Expressions. Three research questions and three null hypotheses guided the study. The study adopted the pre-test, post-test control group design. A total of two hundred and forty (240) Junior Secondary School II students were…

  2. On Maximal Subalgebras and the Hypercentre of Lie Algebras.

    ERIC Educational Resources Information Center

    Honda, Masanobu

    1997-01-01

    Derives two sufficient conditions for a finitely generated Lie algebra to have the nilpotent hypercenter. Presents a relatively large class of generalized soluble Lie algebras. Proves that if a finitely generated Lie algebra has a nilpotent maximal subalgebra, the Fitting radical is nilpotent. (DDR)

  3. 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…

  4. Continuous family of finite-dimensional representations of a solvable Lie algebra arising from singularities

    PubMed Central

    Yau, Stephen S.-T.

    1983-01-01

    A natural mapping from the set of complex analytic isolated hypersurface singularities to the set of finite dimensional Lie algebras is first defined. It is proven that the image under this natural mapping is contained in the set of solvable Lie algebras. This approach gives rise to a continuous inequivalent family of finite dimensional representations of a solvable Lie algebra. PMID:16593401

  5. Vector fields and nilpotent Lie algebras

    NASA Technical Reports Server (NTRS)

    Grayson, Matthew; Grossman, Robert

    1987-01-01

    An infinite-dimensional family of flows E is described with the property that the associated dynamical system: x(t) = E(x(t)), where x(0) is a member of the set R to the Nth power, is explicitly integrable in closed form. These flows E are of the form E = E1 + E2, where E1 and E2 are the generators of a nilpotent Lie algebra, which is either free, or satisfies some relations at a point. These flows can then be used to approximate the flows of more general types of dynamical systems.

  6. Spatial-Operator Algebra For Flexible-Link Manipulators

    NASA Technical Reports Server (NTRS)

    Jain, Abhinandan; Rodriguez, Guillermo

    1994-01-01

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

  7. Algebra for All: California's Eighth-Grade Algebra Initiative as Constrained Curricula.

    PubMed

    Domina, Thurston; Penner, Andrew M; Penner, Emily K; Conley, Annemarie

    2014-08-01

    Across the United States, secondary school curricula are intensifying as a growing proportion of students enroll in high-level academic math courses. In many districts, this intensification process occurs as early as eighth grade, where schools are effectively constraining their mathematics curricula by restricting course offerings and placing more students into Algebra I. This paper provides a quantitative single-case research study of policy-driven curricular intensification in one California school district. (1a) What effect did 8th eighth grade curricular intensification have on mathematics course enrollment patterns in Towering Pines Unified schools? (2b) How did the distribution of prior achievement in Towering Pines math classrooms change as the district constrained the curriculum by universalizing 8th eighth grade Algebra? (3c) Did 8th eighth grade curricular intensification improve students' mathematics achievement? Towering Pines is an immigrant enclave in the inner-ring suburbs of a major metropolitan area. The district's 10 middle schools together enroll approximately 4,000 eighth graders each year. The districts' students are ethnically diverse and largely economically disadvantaged. The study draws upon administrative data describing 8th eighth graders in the district in the 2004-20-05 through 2007-20-08 school years. During the study period, Towering Pines dramatically intensified middle school students' math curricula: In the 2004-20-05 school year 32% of the district's 8th eighth graders enrolled in Algebra or a higher- level mathematics course; by the 2007-20-08 school year that proportion had increased to 84%. We use an interrupted time-series design, comparing students' 8th eighth grade math course enrollments, 10th grade math course enrollments, and 10th grade math test scores across the four cohorts, controlling for demographics and prior achievement. We find that students' odds of taking higher level mathematics courses increased as this

  8. Curriculum Reform in Irish Secondary Schools--A Focus on Algebra

    ERIC Educational Resources Information Center

    Prendergast, Mark; Treacy, Paraic

    2018-01-01

    Algebra has long been identified as an area of difficulty in the teaching and learning of mathematics. Evidence of this difficulty can be found in Irish secondary-level classrooms. Chief Examiner Reports have consistently identified algebra as an area of student weakness in State examinations. In light of poor student performance, and as part of a…

  9. Conceptualizing Routines of Practice That Support Algebraic Reasoning in Elementary Schools: A Constructivist Grounded Theory

    ERIC Educational Resources Information Center

    Store, Jessie Chitsanzo

    2012-01-01

    There is ample literature documenting that, for many decades, high school students view algebra as difficult and do not demonstrate understanding of algebraic concepts. Algebraic reasoning in elementary school aims at meaningfully introducing algebra to elementary school students in preparation for higher-level mathematics. While there is research…

  10. Formal mechanization of device interactions with a process algebra

    NASA Technical Reports Server (NTRS)

    Schubert, E. Thomas; Levitt, Karl; Cohen, Gerald C.

    1992-01-01

    The principle emphasis is to develop a methodology to formally verify correct synchronization communication of devices in a composed hardware system. Previous system integration efforts have focused on vertical integration of one layer on top of another. This task examines 'horizontal' integration of peer devices. To formally reason about communication, we mechanize a process algebra in the Higher Order Logic (HOL) theorem proving system. Using this formalization we show how four types of device interactions can be represented and verified to behave as specified. The report also describes the specification of a system consisting of an AVM-1 microprocessor and a memory management unit which were verified in previous work. A proof of correct communication is presented, and the extensions to the system specification to add a direct memory device are discussed.

  11. 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…

  12. The Standard Model Algebra - a summary

    NASA Astrophysics Data System (ADS)

    Cristinel Stoica, Ovidiu

    2017-08-01

    A generation of leptons and quarks and the gauge symmetries of the Standard Model can be obtained from the Clifford algebra ℂℓ 6. An instance of ℂℓ 6 is implicitly generated by the Dirac algebra combined with the electroweak symmetry, while the color symmetry gives another instance of ℂℓ 6 with a Witt decomposition. The minimal mathematical model proposed here results by identifying the two instances of ℂℓ 6. The left ideal decomposition generated by the Witt decomposition represents the leptons and quarks, and their antiparticles. The SU(3)c and U(1)em symmetries of the SM are the symmetries of this ideal decomposition. The patterns of electric charges, colors, chirality, weak isospins, and hypercharges, follow from this, without predicting additional particles or forces, or proton decay. The electroweak symmetry is present in its broken form, due to the geometry. The predicted Weinberg angle is given by sin2 W = 0.25. The model shares common features with previously known models, particularly with Chisholm and Farwell, 1996, Trayling and Baylis, 2004, and Furey, 2016.

  13. Lattices of Varieties of Algebras

    NASA Astrophysics Data System (ADS)

    Volkov, M. V.

    1980-02-01

    Let A be an associative and commutative ring with 1, S a subsemigroup of the multiplicative semigroup of A, not containing divisors of zero, and \\mathfrak{X} some variety of A-algebras. A study is made of the homomorphism from the lattice L(\\mathfrak{X}) of all subvarieties of \\mathfrak{X} into the lattice of all varieties of S^{-1} A-algebras, which is induced in a certain natural sense by the functor S^{-1}. Under one weak restriction on \\mathfrak{X} a description is given of the kernel of this homomorphism, and this makes it possible to establish a good interrelation between the properties of the lattice L(\\mathfrak{X}) and the lattice of varieties of S^{-1} A-algebras. These results are applied to prove that a number of varieties of associative and Lie rings have the Specht property.Bibliography: 18 titles.

  14. Gender differences in algebraic thinking ability to solve mathematics problems

    NASA Astrophysics Data System (ADS)

    Kusumaningsih, W.; Darhim; Herman, T.; Turmudi

    2018-05-01

    This study aimed to conduct a gender study on students' algebraic thinking ability in solving a mathematics problem, polyhedron concept, for grade VIII. This research used a qualitative method. The data was collected using: test and interview methods. The subjects in this study were eight male and female students with different level of abilities. It was found that the algebraic thinking skills of male students reached high group of five categories. They were superior in terms of reasoning and quick understanding in solving problems. Algebraic thinking ability of high-achieving group of female students also met five categories of algebraic thinking indicators. They were more diligent, tenacious and thorough in solving problems. Algebraic thinking ability of male students in medium category only satisfied three categories of algebraic thinking indicators. They were sufficient in terms of reasoning and understanding in solving problems. Algebraic thinking ability group of female students in medium group also satisfied three categories of algebraic thinking indicators. They were fairly diligent, tenacious and meticulous on working on the problems.

  15. Yang-Baxter algebras, integrable theories and Bethe Ansatz

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

    De Vega, H.J.

    1990-03-10

    This paper presents the Yang-Baxter algebras (YBA) in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Behe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. This approachmore » permit to define and solve rigorously massive QFT as an appropriate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underlay the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized.« less

  16. A systematic investigation of the link between rational number processing and algebra ability.

    PubMed

    Hurst, Michelle; Cordes, Sara

    2018-02-01

    Recent research suggests that fraction understanding is predictive of algebra ability; however, the relative contributions of various aspects of rational number knowledge are unclear. Furthermore, whether this relationship is notation-dependent or rather relies upon a general understanding of rational numbers (independent of notation) is an open question. In this study, college students completed a rational number magnitude task, procedural arithmetic tasks in fraction and decimal notation, and an algebra assessment. Using these tasks, we measured three different aspects of rational number ability in both fraction and decimal notation: (1) acuity of underlying magnitude representations, (2) fluency with which symbols are mapped to the underlying magnitudes, and (3) fluency with arithmetic procedures. Analyses reveal that when looking at the measures of magnitude understanding, the relationship between adults' rational number magnitude performance and algebra ability is dependent upon notation. However, once performance on arithmetic measures is included in the relationship, individual measures of magnitude understanding are no longer unique predictors of algebra performance. Furthermore, when including all measures simultaneously, results revealed that arithmetic fluency in both fraction and decimal notation each uniquely predicted algebra ability. Findings are the first to demonstrate a relationship between rational number understanding and algebra ability in adults while providing a clearer picture of the nature of this relationship. © 2017 The British Psychological Society.

  17. Chinese Algebra: Using Historical Problems to Think about Current Curricula

    ERIC Educational Resources Information Center

    Tillema, Erik

    2005-01-01

    The Chinese used the idea of generating equivalent expressions for solving problems where the problems from a historical Chinese text are studied to understand the ways in which the ideas can lead into algebraic calculations and help students to learn algebra. The texts unify algebraic problem solving through complex algebraic thought and afford…

  18. 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…

  19. An Algebraic Formulation of Level One Wess-Zumino Models

    NASA Astrophysics Data System (ADS)

    Böckenhauer, Jens

    The highest weight modules of the chiral algebra of orthogonal WZW models at level one possess a realization in fermionic representation spaces; the Kac-Moody and Virasoro generators are represented as unbounded limits of even CAR algebras. It is shown that the representation theory of the underlying even CAR algebras reproduces precisely the sectors of the chiral algebra. This fact allows to develop a theory of local von Neumann algebras on the punctured circle, fitting nicely in the Doplicher-Haag-Roberts framework. The relevant localized endomorphisms which generate the charged sectors are explicitly constructed by means of Bogoliubov transformations. Using CAR theory, the fusion rules in terms of sector equivalence classes are proven.

  20. Closed form of the Baker-Campbell-Hausdorff formula for the generators of semisimple complex Lie algebras

    NASA Astrophysics Data System (ADS)

    Matone, Marco

    2016-11-01

    Recently it has been introduced an algorithm for the Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are 13 types of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, X, Y of the Cartan-Weyl basis, we find W, linear combination of X, Y, such that exp (X) exp (Y)=exp (W). The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of the root system. Furthermore, if X, Y, and Z are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, W, a linear combination of X, Y and Z, such that exp (X) exp (Y) exp (Z)=exp (W). It turns out that the relevant commutator algebras are type 1c-i, type 4 and type 5. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper.

  1. 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.)

  2. The Algebra of the Arches

    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…

  3. A Study of Improving Eighth Graders' Learning Deficiency in Algebra by Applying a Realistic Context Instructional Design

    ERIC Educational Resources Information Center

    Chang, Yu-Liang; Huang, Yu-I

    2014-01-01

    The intention of this study was to improve the learning deficiency in algebraic learning and to enhance Taiwanese middle students' learning achievement and interest in algebra. By using a grade skipping experimental design, the research team intended to find out an effective way to benefit these students' leaning in abstract algebraic concepts.…

  4. A Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies

    NASA Astrophysics Data System (ADS)

    Lu, Wei

    2017-09-01

    We propose a Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies in the context of composite Higgs bosons. Standard model fermions are represented by algebraic spinors of six-dimensional binary Clifford algebra, while ternary Clifford algebra-related flavor projection operators control allowable flavor-mixing interactions. There are three composite electroweak Higgs bosons resulted from top quark, tau neutrino, and tau lepton condensations. Each of the three condensations gives rise to masses of four different fermions. The fermion mass hierarchies within these three groups are determined by four-fermion condensations, which break two global chiral symmetries. The four-fermion condensations induce axion-like pseudo-Nambu-Goldstone bosons and can be dark matter candidates. In addition to the 125 GeV Higgs boson observed at the Large Hadron Collider, we anticipate detection of tau neutrino composite Higgs boson via the charm quark decay channel.

  5. Computer Algebra Systems in Undergraduate Instruction.

    ERIC Educational Resources Information Center

    Small, Don; And Others

    1986-01-01

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

  6. Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis

    NASA Astrophysics Data System (ADS)

    Martin, J.; Shore, B. W.; Bergmann, K.

    1995-07-01

    We extend previous theoretical work on coherent population transfer by stimulated Raman adiabatic passage for states involving nonzero angular momentum. The pump and Stokes fields are either copropagating or counterpropagating with the corresponding linearly polarized electric-field vectors lying in a common plane with the magnetic-field direction. Zeeman splitting lifts the magnetic sublevel degeneracy. We present an algebraic analysis of dressed-state properties to explain the behavior noted in numerical studies. In particular, we discuss conditions which are likely to lead to a failure of complete population transfer. The applied strategy, based on simple methods of linear algebra, will also be successful for other types of discrete multilevel systems, provided the rotating-wave and adiabatic approximation are valid.

  7. Just-in-Time Algebra: A Problem Solving Approach Including Multimedia and Animation.

    ERIC Educational Resources Information Center

    Hofmann, Roseanne S.; Hunter, Walter R.

    2003-01-01

    Describes a beginning algebra course that places stronger emphasis on learning to solve problems and introduces topics using real world applications. Students learn estimating, graphing, and algebraic algorithms for the purpose of solving problems. Indicates that applications motivate students by appearing to be a more relevant topic as well as…

  8. AN ADA LINEAR ALGEBRA PACKAGE MODELED AFTER HAL/S

    NASA Technical Reports Server (NTRS)

    Klumpp, A. R.

    1994-01-01

    This package extends the Ada programming language to include linear algebra capabilities similar to those of the HAL/S programming language. The package is designed for avionics applications such as Space Station flight software. In addition to the HAL/S built-in functions, the package incorporates the quaternion functions used in the Shuttle and Galileo projects, and routines from LINPAK that solve systems of equations involving general square matrices. Language conventions in this package follow those of HAL/S to the maximum extent practical and minimize the effort required for writing new avionics software and translating existent software into Ada. Valid numeric types in this package include scalar, vector, matrix, and quaternion declarations. (Quaternions are fourcomponent vectors used in representing motion between two coordinate frames). Single precision and double precision floating point arithmetic is available in addition to the standard double precision integer manipulation. Infix operators are used instead of function calls to define dot products, cross products, quaternion products, and mixed scalar-vector, scalar-matrix, and vector-matrix products. The package contains two generic programs: one for floating point, and one for integer. The actual component type is passed as a formal parameter to the generic linear algebra package. The procedures for solving systems of linear equations defined by general matrices include GEFA, GECO, GESL, and GIDI. The HAL/S functions include ABVAL, UNIT, TRACE, DET, INVERSE, TRANSPOSE, GET, PUT, FETCH, PLACE, and IDENTITY. This package is written in Ada (Version 1.2) for batch execution and is machine independent. The linear algebra software depends on nothing outside the Ada language except for a call to a square root function for floating point scalars (such as SQRT in the DEC VAX MATHLIB library). This program was developed in 1989, and is a copyrighted work with all copyright vested in NASA.

  9. Generalized Lotka—Volterra systems connected with simple Lie algebras

    NASA Astrophysics Data System (ADS)

    Charalambides, Stelios A.; Damianou, Pantelis A.; Evripidou, Charalambos A.

    2015-06-01

    We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type An for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type An, we produce new integrable Hamiltonian systems.

  10. General algebraic method applied to control analysis of complex engine types

    NASA Technical Reports Server (NTRS)

    Boksenbom, Aaron S; Hood, Richard

    1950-01-01

    A general algebraic method of attack on the problem of controlling gas-turbine engines having any number of independent variables was utilized employing operational functions to describe the assumed linear characteristics for the engine, the control, and the other units in the system. Matrices were used to describe the various units of the system, to form a combined system showing all effects, and to form a single condensed matrix showing the principal effects. This method directly led to the conditions on the control system for noninteraction so that any setting disturbance would affect only its corresponding controlled variable. The response-action characteristics were expressed in terms of the control system and the engine characteristics. The ideal control-system characteristics were explicitly determined in terms of any desired response action.

  11. Invariant algebraic surfaces for a virus dynamics

    NASA Astrophysics Data System (ADS)

    Valls, Claudia

    2015-08-01

    In this paper, we provide a complete classification of the invariant algebraic surfaces and of the rational first integrals for a well-known virus system. In the proofs, we use the weight-homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.

  12. Preparing Elementary Prospective Teachers to Teach Early Algebra

    ERIC Educational Resources Information Center

    Hohensee, Charles

    2017-01-01

    Researchers have argued that integrating early algebra into elementary grades will better prepare students for algebra. However, currently little research exists to guide teacher preparation programs on how to prepare prospective elementary teachers to teach early algebra. This study examines the insights and challenges that prospective teachers…

  13. D{sub {infinity}}-differential A{sub {infinity}}-algebras and spectral sequences

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

    Lapin, S V

    2002-02-28

    In the present paper the construction of a D{sub {infinity}}-differential A{sub {infinity}}-(co)algebra is introduced and basic homotopy properties of this construction are studied. The connection between D{sub {infinity}}-differential A{sub {infinity}}-(co)algebras and spectral sequences is established, which enables us to construct the structure of an A{sub {infinity}} -coalgebra on the Milnor coalgebra directly from the differentials of the Adams spectral sequence.

  14. Deriving the Regression Line with Algebra

    ERIC Educational Resources Information Center

    Quintanilla, John A.

    2017-01-01

    Exploration with spreadsheets and reliance on previous skills can lead students to determine the line of best fit. To perform linear regression on a set of data, students in Algebra 2 (or, in principle, Algebra 1) do not have to settle for using the mysterious "black box" of their graphing calculators (or other classroom technologies).…

  15. A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras

    NASA Astrophysics Data System (ADS)

    Alshammari, Fahad; Isaac, Phillip S.; Marquette, Ian

    2018-02-01

    We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and (2) the Schrödinger Lie algebras. We find that an abstract form of dimensional analysis assists us in our algorithm, and greatly reduces the complexity of the problem.

  16. Hom Gel'fand-Dorfman bialgebras and Hom-Lie conformal algebras

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

    Yuan, Lamei, E-mail: lmyuan@hit.edu.cn

    2014-04-15

    The aim of this paper is to introduce the notions of Hom Gel'fand-Dorfman bialgebra and Hom-Lie conformal algebra. In this paper, we give four constructions of Hom Gel'fand-Dorfman bialgebras. Also, we provide a general construction of Hom-Lie conformal algebras from Hom-Lie algebras. Finally, we prove that a Hom Gel'fand-Dorfman bialgebra is equivalent to a Hom-Lie conformal algebra of degree 2.

  17. A Modeling-Based College Algebra Course and Its Effect on Student Achievement

    ERIC Educational Resources Information Center

    Ellington, Aimee J.

    2005-01-01

    In Fall 2004, Virginia Commonwealth University (VCU) piloted a modeling-based approach to college algebra. This paper describes the course and an assessment that was conducted to determine the effect of this approach on student achievement in comparison to a traditional approach to college algebra. The results show that compared with their…

  18. Absolute continuity for operator valued completely positive maps on C∗-algebras

    NASA Astrophysics Data System (ADS)

    Gheondea, Aurelian; Kavruk, Ali Şamil

    2009-02-01

    Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C∗-algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34-54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon-Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon-Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi's matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained.

  19. School Algebra Reform: Meeting the Grade?

    ERIC Educational Resources Information Center

    Telese, James A.

    This paper reports on a case study that was conducted at five high schools from a large, urban school district located in South Texas. The purpose of the study was to gain an understanding of Algebra 1 teaching strategies. The research questions were: (1) What is the predominant mode of instruction for Algebra 1? and (2) What is the level of…

  20. Quantum Sets and Clifford Algebras

    NASA Astrophysics Data System (ADS)

    Finkelstein, David

    1982-06-01

    The mathematical language presently used for quantum physics is a high-level language. As a lowest-level or basic language I construct a quantum set theory in three stages: (1) Classical set theory, formulated as a Clifford algebra of “ S numbers” generated by a single monadic operation, “bracing,” Br = {…}. (2) Indefinite set theory, a modification of set theory dealing with the modal logical concept of possibility. (3) Quantum set theory. The quantum set is constructed from the null set by the familiar quantum techniques of tensor product and antisymmetrization. There are both a Clifford and a Grassmann algebra with sets as basis elements. Rank and cardinality operators are analogous to Schroedinger coordinates of the theory, in that they are multiplication or “ Q-type” operators. “ P-type” operators analogous to Schroedinger momenta, in that they transform the Q-type quantities, are bracing (Br), Clifford multiplication by a set X, and the creator of X, represented by Grassmann multiplication c( X) by the set X. Br and its adjoint Br* form a Bose-Einstein canonical pair, and c( X) and its adjoint c( X)* form a Fermi-Dirac or anticanonical pair. Many coefficient number systems can be employed in this quantization. I use the integers for a discrete quantum theory, with the usual complex quantum theory as limit. Quantum set theory may be applied to a quantum time space and a quantum automaton.

  1. A Concurrent Support Course for Intermediate Algebra

    ERIC Educational Resources Information Center

    Cooper, Cameron I.

    2011-01-01

    This article summarizes the creation and implementation of a concurrent support class for TRS 92--Intermediate Algebra, a developmental mathematics course at Fort Lewis College in Durango, Colorado. The concurrent course outlined in this article demonstrates a statistically significant increase in student success rates since its inception.…

  2. Attributions for Success and Failure in Algebra of Samoan Community College Students: A Profile Analysis.

    ERIC Educational Resources Information Center

    Powers, Stephen; And Others

    Sex differences in attributions for success and failure in algebra of Samoan community college students were examined and compared with attributions of a large group of mainland U.S. students. study included the Mathematics Attribution Scale: Algebra Version (MAS), which assessed students' attributions of achievement in algebra to their effort,…

  3. Connections between Kac-Moody algebras and M-theory

    NASA Astrophysics Data System (ADS)

    Cook, Paul P.

    2007-11-01

    We investigate some of the motivations and consequences of the conjecture that the Kac-Moody algebra E11 is the symmetry algebra of M-theory, and we develop methods to aid the further investigation of this idea. The definitions required to work with abstract root systems of Lie algebras are given in review leading up to the definition of a Kac-Moody algebra. The motivations for the E11 conjecture are presented and the nonlinear realisation of gravity relevant to the conjecture is described. We give a beginner's guide to producing the algebras of E11, relevant to M-theory, and K27, relevant to the bosonic string theory, along with their l1 representations are constructed. Reference tables of low level roots are produced for both the adjoint and l1 representations of these algebras. In addition a particular group element, having a generic form for all G+++ algebras, is shown to encode all the half-BPS brane solutions of the maximally oxidised supergravities. Special analysis is given to the role of space-time signature in the context of this group element and subsequent to this analysis spacelike brane solutions are derived from the same solution generating group element. Finally the appearance of U-duality charge multiplets from E11 is reviewed. General formulae for finding the content of arbitrary brane charge multiplets are given and the content of the particle and string multiplets in dimensions 4,5,6,7 and 8 is shown to be contained in the l1 representation of E11.

  4. Solving the Unknown with Algebra: Poster/Teaching Guide for Pre-Algebra Students. Expect the Unexpected with Math[R

    ERIC Educational Resources Information Center

    Actuarial Foundation, 2013

    2013-01-01

    "Solving the Unknown with Algebra" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards and designed to help students practice pre-algebra skills including using formulas, solving for unknowns, and manipulating equations. Developed by The Actuarial Foundation with Scholastic, this program provides…

  5. Computing Gröbner Bases within Linear Algebra

    NASA Astrophysics Data System (ADS)

    Suzuki, Akira

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

  6. Quantum walks, deformed relativity and Hopf algebra symmetries.

    PubMed

    Bisio, Alessandro; D'Ariano, Giacomo Mauro; Perinotti, Paolo

    2016-05-28

    We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014Phys. Rev. A90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras-the usual Poincaré and theκ-Poincaré algebras. © 2016 The Author(s).

  7. Mobile Learning: Integrating Text Messaging into a Community College Pre-Algebra Course

    ERIC Educational Resources Information Center

    Bull, Prince; McCormick, Carlos

    2012-01-01

    This study investigated the use of text messaging as an educational tool in a pre-algebra course at a community college in the central region of North Carolina. The research was conducted in two pre-algebra classes with thirty-three students and one instructor. Data were gathered using qualitative and quantitative methods. A mixed method design…

  8. An Intensification Approach to Double-Block Algebra: A Pilot Implementation of Intensified Algebra in A Large Urban School District

    ERIC Educational Resources Information Center

    Tidd, Simon T.; Stoelinga, Timothy M.; Bush-Richards, Angela M.; De Sena, Donna L.; Dwyer, Theodore J.

    2018-01-01

    Double-block instruction has become a popular strategy for supporting struggling mathematics students in algebra I. Despite its widespread adoption, little consistent evidence supports the attributes of a successful double-block design or the effectiveness of this instructional strategy. In this study, the authors examine a pilot implementation of…

  9. Genetic hotels for the standard genetic code: evolutionary analysis based upon novel three-dimensional algebraic models.

    PubMed

    José, Marco V; Morgado, Eberto R; Govezensky, Tzipe

    2011-07-01

    Herein, we rigorously develop novel 3-dimensional algebraic models called Genetic Hotels of the Standard Genetic Code (SGC). We start by considering the primeval RNA genetic code which consists of the 16 codons of type RNY (purine-any base-pyrimidine). Using simple algebraic operations, we show how the RNA code could have evolved toward the current SGC via two different intermediate evolutionary stages called Extended RNA code type I and II. By rotations or translations of the subset RNY, we arrive at the SGC via the former (type I) or via the latter (type II), respectively. Biologically, the Extended RNA code type I, consists of all codons of the type RNY plus codons obtained by considering the RNA code but in the second (NYR type) and third (YRN type) reading frames. The Extended RNA code type II, comprises all codons of the type RNY plus codons that arise from transversions of the RNA code in the first (YNY type) and third (RNR) nucleotide bases. Since the dimensions of remarkable subsets of the Genetic Hotels are not necessarily integer numbers, we also introduce the concept of algebraic fractal dimension. A general decoding function which maps each codon to its corresponding amino acid or the stop signals is also derived. The Phenotypic Hotel of amino acids is also illustrated. The proposed evolutionary paths are discussed in terms of the existing theories of the evolution of the SGC. The adoption of 3-dimensional models of the Genetic and Phenotypic Hotels will facilitate the understanding of the biological properties of the SGC.

  10. Meanings Given to Algebraic Symbolism in Problem-Posing

    ERIC Educational Resources Information Center

    Cañadas, María C.; Molina, Marta; del Río, Aurora

    2018-01-01

    Some errors in the learning of algebra suggest that students might have difficulties giving meaning to algebraic symbolism. In this paper, we use problem posing to analyze the students' capacity to assign meaning to algebraic symbolism and the difficulties that students encounter in this process, depending on the characteristics of the algebraic…

  11. Cubic map algebra functions for spatio-temporal analysis

    USGS Publications Warehouse

    Mennis, J.; Viger, R.; Tomlin, C.D.

    2005-01-01

    We propose an extension of map algebra to three dimensions for spatio-temporal data handling. This approach yields a new class of map algebra functions that we call "cube functions." Whereas conventional map algebra functions operate on data layers representing two-dimensional space, cube functions operate on data cubes representing two-dimensional space over a third-dimensional period of time. We describe the prototype implementation of a spatio-temporal data structure and selected cube function versions of conventional local, focal, and zonal map algebra functions. The utility of cube functions is demonstrated through a case study analyzing the spatio-temporal variability of remotely sensed, southeastern U.S. vegetation character over various land covers and during different El Nin??o/Southern Oscillation (ENSO) phases. Like conventional map algebra, the application of cube functions may demand significant data preprocessing when integrating diverse data sets, and are subject to limitations related to data storage and algorithm performance. Solutions to these issues include extending data compression and computing strategies for calculations on very large data volumes to spatio-temporal data handling.

  12. Tensor Algebra Library for NVidia Graphics Processing Units

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

    Liakh, Dmitry

    This is a general purpose math library implementing basic tensor algebra operations on NVidia GPU accelerators. This software is a tensor algebra library that can perform basic tensor algebra operations, including tensor contractions, tensor products, tensor additions, etc., on NVidia GPU accelerators, asynchronously with respect to the CPU host. It supports a simultaneous use of multiple NVidia GPUs. Each asynchronous API function returns a handle which can later be used for querying the completion of the corresponding tensor algebra operation on a specific GPU. The tensors participating in a particular tensor operation are assumed to be stored in local RAMmore » of a node or GPU RAM. The main research area where this library can be utilized is the quantum many-body theory (e.g., in electronic structure theory).« less

  13. Assessment of polytechnic students' understanding of basic algebra

    NASA Astrophysics Data System (ADS)

    Mokmin, Nur Azlina Mohamed; Masood, Mona

    2015-12-01

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

  14. 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…

  15. Relational Algebra and SQL: Better Together

    ERIC Educational Resources Information Center

    McMaster, Kirby; Sambasivam, Samuel; Hadfield, Steven; Wolthuis, Stuart

    2013-01-01

    In this paper, we describe how database instructors can teach Relational Algebra and Structured Query Language together through programming. Students write query programs consisting of sequences of Relational Algebra operations vs. Structured Query Language SELECT statements. The query programs can then be run interactively, allowing students to…

  16. Chiropractic biophysics technique: a linear algebra approach to posture in chiropractic.

    PubMed

    Harrison, D D; Janik, T J; Harrison, G R; Troyanovich, S; Harrison, D E; Harrison, S O

    1996-10-01

    This paper discusses linear algebra as applied to human posture in chiropractic, specifically chiropractic biophysics technique (CBP). Rotations, reflections and translations are geometric functions studied in vector spaces in linear algebra. These mathematical functions are termed rigid body transformations and are applied to segmental spinal movement in the literature. Review of the literature indicates that these linear algebra concepts have been used to describe vertebral motion. However, these rigid body movers are presented here as applying to the global postural movements of the head, thoracic cage and pelvis. The unique inverse functions of rotations, reflections and translations provide a theoretical basis for making postural corrections in neutral static resting posture. Chiropractic biophysics technique (CBP) uses these concepts in examination procedures, manual spinal manipulation, instrument assisted spinal manipulation, postural exercises, extension traction and clinical outcome measures.

  17. The algebra of supertraces for 2+1 super de Sitter gravity

    NASA Technical Reports Server (NTRS)

    Urrutia, L. F.; Waelbroeck, H.; Zertuche, F.

    1993-01-01

    The algebra of the observables for 2+1 super de Sitter gravity, for one genus of the spatial surface is calculated. The algebra turns out to be an infinite Lie algebra subject to non-linear constraints. The constraints are solved explicitly in terms of five independent complex supertraces. These variables are the true degrees of freedom of the system and their quantized algebra generates a new structure which is referred to as a 'central extension' of the quantum algebra SU(2)q.

  18. Bootstrapping non-commutative gauge theories from L∞ algebras

    NASA Astrophysics Data System (ADS)

    Blumenhagen, Ralph; Brunner, Ilka; Kupriyanov, Vladislav; Lüst, Dieter

    2018-05-01

    Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying L∞ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. We recall that such field theories arise in the context of branes in WZW models and briefly comment on its appearance for integrable deformations of AdS5 sigma models. For the SU(2) WZW model, we show that the earlier proposed matrix valued gauge theory on the fuzzy 2-sphere can be bootstrapped via an L∞ algebra. We then apply this approach to the construction of non-commutative Chern-Simons and Yang-Mills theories on flat and curved backgrounds with non-constant NC-structure. More concretely, up to the second order, we demonstrate how derivative and curvature corrections to the equations of motion can be bootstrapped in an algebraic way from the L∞ algebra. The appearance of a non-trivial A∞ algebra is discussed, as well.

  19. Matrix De Rham Complex and Quantum A-infinity algebras

    NASA Astrophysics Data System (ADS)

    Barannikov, S.

    2014-04-01

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

  20. A Linear Algebraic Approach to Teaching Interpolation

    ERIC Educational Resources Information Center

    Tassa, Tamir

    2007-01-01

    A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…

  1. 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.

  2. Algebra 1r, Mathematics (Experimental): 5215.13.

    ERIC Educational Resources Information Center

    Strachan, Florence

    This third of six guidebooks on minimum course content for first-year algebra includes work with laws of exponents; multiplication, division, and factoring of polynomials; and fundamental operations with rational algebraic expressions. Course goals are stated, performance objectives listed, a course outline provided, testbook references specified…

  3. The Ideas of Algebra, K-12. 1988 Yearbook.

    ERIC Educational Resources Information Center

    Coxford, Arthur F., Ed.; Shulte, Albert P., Ed.

    This volume is organized into six parts. Chapters 1-5, which make up Part 1, first discuss the forces impinging on algebra in the curriculum and suggest possible directions for change. Chapters 6-8, Part 2, concentrate on concepts and teaching possibilities available prior to the formal introduction of algebra. The notion that algebraic ideas are…

  4. On squares of representations of compact Lie algebras

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

    Zeier, Robert, E-mail: robert.zeier@ch.tum.de; Zimborás, Zoltán, E-mail: zimboras@gmail.com

    We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the summore » of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.« less

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

    ERIC Educational Resources Information Center

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

    2005-01-01

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

  6. Playing Your Cards Right: Integers for Algebra

    ERIC Educational Resources Information Center

    Tillema, Erik; Gatza, Andrew; Ulrich, Catherine

    2017-01-01

    The number and algebra strand of the "Australian Curriculum: Mathematics" (2015) advocates for holding together the study of number and algebra across years K-8--a position that mathematics educators have endorsed in many countries. This recommendation along with the report "Shape of the Australian Curriculum: Mathematics"…

  7. Classical affine W-algebras associated to Lie superalgebras

    NASA Astrophysics Data System (ADS)

    Suh, Uhi Rinn

    2016-02-01

    In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras), which can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum affine W-superalgebras. Also, we show that a classical finite W-superalgebra can be obtained by a Zhu algebra of a classical affine W-superalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical W-superalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical W-algebra associated to spo(2|3) and its principal nilpotent. In the last part of this paper, we introduce a generalization of classical affine W-superalgebras called classical affine fractional W-superalgebras. We show these have Poisson vertex algebra structures and find generators of a fractional W-superalgebra associated to a minimal nilpotent.

  8. Students’ Algebraic Reasonsing In Solving Mathematical Problems With Adversity Quotient

    NASA Astrophysics Data System (ADS)

    Aryani, F.; Amin, S. M.; Sulaiman, R.

    2018-01-01

    Algebraic reasoning is a process in which students generalize mathematical ideas from a set of particular instances and express them in increasingly formal and age-appropriate ways. Using problem solving approach to develop algebraic reasoning of mathematics may enhace the long-term learning trajectory of the majority students. The purpose of this research was to describe the algebraic reasoning of quitter, camper, and climber junior high school students in solving mathematical problems. This research used qualitative descriptive method. Subjects were determined by purposive sampling. The technique of collecting data was done by task-based interviews.The results showed that the algebraic reasoning of three students in the process of pattern seeking by identifying the things that are known and asked in a similar way. But three students found the elements of pattern recognition in different ways or method. So, they are generalize the problem of pattern formation with different ways. The study of algebraic reasoning and problem solving can be a learning paradigm in the improve students’ knowledge and skills in algebra work. The goal is to help students’ improve academic competence, develop algebraic reasoning in problem solving.

  9. The State of the Gate: A Description of Instructional Practice in Algebra in Five Urban Districts

    ERIC Educational Resources Information Center

    Litke, Erica G.

    2015-01-01

    Algebra is considered a linchpin for success in secondary mathematics, serving as a gatekeeper to higher-level courses. Access to algebra is also considered an important lever for educational equity. Yet despite its prominence, large-scale examinations of algebra instruction are rare. In my dissertation, I endeavor to better understand what…

  10. Designing Spreadsheet-Based Tasks for Purposeful Algebra

    ERIC Educational Resources Information Center

    Ainley, Janet; Bills, Liz; Wilson, Kirsty

    2005-01-01

    We describe the design of a sequence of spreadsheet-based pedagogic tasks for the introduction of algebra in the early years of secondary schooling within the Purposeful Algebraic Activity project. This design combines two relatively novel features to bring a different perspective to research in the use of spreadsheets for the learning and…

  11. On Non-Abelian Extensions of 3-Lie Algebras

    NASA Astrophysics Data System (ADS)

    Song, Li-Na; Makhlouf, Abdenacer; Tang, Rong

    2018-04-01

    In this paper, we study non-abelian extensions of 3-Lie algebras through Maurer-Cartan elements. We show that there is a one-to-one correspondence between isomorphism classes of non-abelian extensions of 3-Lie algebras and equivalence classes of Maurer-Cartan elements in a DGLA. The structure of the Leibniz algebra on the space of fundamental objects is also analyzed. Supported by National Natural Science Foundation of China under Grant No. 11471139 and National Natural Science Foundation of Jilin Province under Grant No. 20170101050JC

  12. Principles of Stagewise Separation Process Calculations: A Simple Algebraic Approach Using Solvent Extraction.

    ERIC Educational Resources Information Center

    Crittenden, Barry D.

    1991-01-01

    A simple liquid-liquid equilibrium (LLE) system involving a constant partition coefficient based on solute ratios is used to develop an algebraic understanding of multistage contacting in a first-year separation processes course. This algebraic approach to the LLE system is shown to be operable for the introduction of graphical techniques…

  13. 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)

  14. 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…

  15. The general symmetry algebra structure of the underdetermined equation ux=(vxx)2

    NASA Astrophysics Data System (ADS)

    Kersten, Paul H. M.

    1991-08-01

    In a recent paper, Anderson, Kamran, and Olver [``Interior, exterior, and generalized symmetries,'' preprint (1990)] obtained the first- and second-order generalized symmetry algebra for the system ux=(vxx)2, leading to the noncompact real form of the exceptional Lie algebra G2. Here, the structure of the general higher-order symmetry algebra is obtained. Moreover, the Lie algebra G2 is obtained as ordinary symmetry algebra of the associated first-order system. The general symmetry algebra for ux=f(u,v,vx,...,) is established also.

  16. An algebra of discrete event processes

    NASA Technical Reports Server (NTRS)

    Heymann, Michael; Meyer, George

    1991-01-01

    This report deals with an algebraic framework for modeling and control of discrete event processes. The report consists of two parts. The first part is introductory, and consists of a tutorial survey of the theory of concurrency in the spirit of Hoare's CSP, and an examination of the suitability of such an algebraic framework for dealing with various aspects of discrete event control. To this end a new concurrency operator is introduced and it is shown how the resulting framework can be applied. It is further shown that a suitable theory that deals with the new concurrency operator must be developed. In the second part of the report the formal algebra of discrete event control is developed. At the present time the second part of the report is still an incomplete and occasionally tentative working paper.

  17. Visualizing the inner product space ℝm×n in a MATLAB-assisted linear algebra classroom

    NASA Astrophysics Data System (ADS)

    Caglayan, Günhan

    2018-05-01

    This linear algebra note offers teaching and learning ideas in the treatment of the inner product space ? in a technology-supported learning environment. Classroom activities proposed in this note demonstrate creative ways of integrating MATLAB technology into various properties of Frobenius inner product as visualization tools that complement the algebraic approach. As implemented in linear algebra lessons in a university in the Unites States, the article also incorporates algebraic and visual work of students who experienced these activities with MATLAB software. The connection between the Frobenius norm and the Euclidean norm is also emphasized.

  18. 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…

  19. 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…

  20. Geometric interpretation of vertex operator algebras.

    PubMed Central

    Huang, Y Z

    1991-01-01

    In this paper, Vafa's approach to the formulation of conformal field theories is combined with the formal calculus developed in Frenkel, Lepowsky, and Meurman's work on the vertex operator construction of the Monster to give a geometric definition of vertex operator algebras. The main result announced is the equivalence between this definition and the algebraic one in the sense that the categories determined by these definitions are isomorphic. PMID:11607240

  1. A Framework for Understanding and Cultivating the Transition from Arithmetic to Algebraic Reasoning

    ERIC Educational Resources Information Center

    Nathan, Mitchell J.; Koellner, Karen

    2007-01-01

    Algebraic reasoning stands as a formidable gatekeeper for students in their efforts to progress in mathematics and science, and to obtain economic opportunities (Ladson-Billings, 1998; RAND, 2003). Currently, mathematics education research has focused on algebra in order to provide access and opportunities for more students. There is now a growing…

  2. Kato type operators and Weyl's theorem

    NASA Astrophysics Data System (ADS)

    Duggal, B. P.; Djordjevic, S. V.; Kubrusly, Carlos

    2005-09-01

    A Banach space operator T satisfies Weyl's theorem if and only if T or T* has SVEP at all complex numbers [lambda] in the complement of the Weyl spectrum of T and T is Kato type at all [lambda] which are isolated eigenvalues of T of finite algebraic multiplicity. If T* (respectively, T) has SVEP and T is Kato type at all [lambda] which are isolated eigenvalues of T of finite algebraic multiplicity (respectively, T is Kato type at all [lambda][set membership, variant]iso[sigma](T)), then T satisfies a-Weyl's theorem (respectively, T* satisfies a-Weyl's theorem).

  3. College Algebra I.

    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 I. Topics covered include: sets; vocabulary; linear equations; inequalities; real numbers; operations; factoring; fractions; formulas; ratio, proportion, and…

  4. Classical affine W-algebras associated to Lie superalgebras

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

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

    2016-02-15

    In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras), which can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum affine W-superalgebras. Also, we show that a classical finite W-superalgebra can be obtained by a Zhu algebra of a classical affine W-superalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical W-superalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical W-algebra associated to spo(2|3) and its principal nilpotent. In the last part of this paper, we introduce a generalizationmore » of classical affine W-superalgebras called classical affine fractional W-superalgebras. We show these have Poisson vertex algebra structures and find generators of a fractional W-superalgebra associated to a minimal nilpotent.« less

  5. 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…

  6. The operator algebra approach to quantum groups

    PubMed Central

    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

  7. Algebra for All: California’s Eighth-Grade Algebra Initiative as Constrained Curricula

    PubMed Central

    Domina, Thurston; Penner, Andrew M.; Penner, Emily K.; Conley, Annemarie

    2015-01-01

    Background/Context Across the United States, secondary school curricula are intensifying as a growing proportion of students enroll in high-level academic math courses. In many districts, this intensification process occurs as early as eighth grade, where schools are effectively constraining their mathematics curricula by restricting course offerings and placing more students into Algebra I. This paper provides a quantitative single-case research study of policy-driven curricular intensification in one California school district. Research Questions (1a) What effect did 8th eighth grade curricular intensification have on mathematics course enrollment patterns in Towering Pines Unified schools? (2b) How did the distribution of prior achievement in Towering Pines math classrooms change as the district constrained the curriculum by universalizing 8th eighth grade Algebra? (3c) Did 8th eighth grade curricular intensification improve students’ mathematics achievement? Setting Towering Pines is an immigrant enclave in the inner-ring suburbs of a major metropolitan area. The district’s 10 middle schools together enroll approximately 4,000 eighth graders each year. The districts’ students are ethnically diverse and largely economically disadvantaged. The study draws upon administrative data describing 8th eighth graders in the district in the 2004–20-05 through 2007–20-08 school years. Intervention/Program/Practice During the study period, Towering Pines dramatically intensified middle school students’ math curricula: In the 2004–20-05 school year 32% of the district’s 8th eighth graders enrolled in Algebra or a higher- level mathematics course; by the 2007–20-08 school year that proportion had increased to 84%. Research Design We use an interrupted time-series design, comparing students’ 8th eighth grade math course enrollments, 10th grade math course enrollments, and 10th grade math test scores across the four cohorts, controlling for demographics and

  8. Towards classical spectrum generating algebras for f-deformations

    NASA Astrophysics Data System (ADS)

    Kullock, Ricardo; Latini, Danilo

    2016-01-01

    In this paper we revise the classical analog of f-oscillators, a generalization of q-oscillators given in Man'ko et al. (1997) [8], in the framework of classical spectrum generating algebras (SGA) introduced in Kuru and Negro (2008) [9]. We write down the deformed Poisson algebra characterizing the entire family of non-linear oscillators and construct its general solution algebraically. The latter, covering the full range of f-deformations, shows an energy dependence both in the amplitude and the frequency of the motion.

  9. Using Linguistics in the Teaching of Developmental and Remedial Algebra.

    ERIC Educational Resources Information Center

    Lesnak, Richard J.

    Basic algebra at Robert Morris College (RMC) in Pittsburgh, Pennsylvania, is a remedial course for students with virtually no algebra background, and for students whose previous experiences with algebra have created math blocks and math anxiety. A study was conducted in an effort to measure quantitatively the benefits of using linguistic methods…

  10. Abstract Numeric Relations and the Visual Structure of Algebra

    ERIC Educational Resources Information Center

    Landy, David; Brookes, David; Smout, Ryan

    2014-01-01

    Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition,…

  11. Developing learning environments which support early algebraic reasoning: a case from a New Zealand primary classroom

    NASA Astrophysics Data System (ADS)

    Hunter, Jodie

    2014-12-01

    Current reforms in mathematics education advocate the development of mathematical learning communities in which students have opportunities to engage in mathematical discourse and classroom practices which underlie algebraic reasoning. This article specifically addresses the pedagogical actions teachers take which structure student engagement in dialogical discourse and activity which facilitates early algebraic reasoning. Using videotaped recordings of classroom observations, the teacher and researcher collaboratively examined the classroom practices and modified the participatory practices to develop a learning environment which supported early algebraic reasoning. Facilitating change in the classroom environment was a lengthy process which required consistent and ongoing attention initially to the social norms and then to the socio-mathematical norms. Specific pedagogical actions such as the use of specifically designed tasks, materials and representations and a constant press for justification and generalisation were required to support students to link their numerical understandings to algebraic reasoning.

  12. Error-Detecting Identification Codes for Algebra Students.

    ERIC Educational Resources Information Center

    Sutherland, David C.

    1990-01-01

    Discusses common error-detecting identification codes using linear algebra terminology to provide an interesting application of algebra. Presents examples from the International Standard Book Number, the Universal Product Code, bank identification numbers, and the ZIP code bar code. (YP)

  13. Students' Perceptions about the Symbols, Letters and Signs in Algebra and How Do These Affect Their Learning of Algebra: A Case Study in a Government Girls Secondary School Karachi

    ERIC Educational Resources Information Center

    Samo, Mashooque Ali

    2009-01-01

    Algebra uses symbols for generalizing arithmetic. These symbols have different meanings and interpretations in different situations. Students have different perceptions about these symbols, letters and signs. Despite the vast research by on the students' difficulties in understanding letters in Algebra, the overall image that emerges from the…

  14. SD-CAS: Spin Dynamics by Computer Algebra System.

    PubMed

    Filip, Xenia; Filip, Claudiu

    2010-11-01

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

  15. Eighth Grade Algebra Placement Policies: Promoting Equity, Achievement, and Access

    ERIC Educational Resources Information Center

    Wambsgans, Cynthia

    2014-01-01

    This study was an investigation of a standardized 8th grade Algebra I placement policy across multiple educational districts. Researchers have documented benefits of students' 8th grade Algebra I education, while others have detailed the consequences of algebra enrollment without necessary prerequisite skills. The purpose of this study was to…

  16. A note on probabilistic models over strings: the linear algebra approach.

    PubMed

    Bouchard-Côté, Alexandre

    2013-12-01

    Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the "TKF91" model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems.

  17. 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.

  18. Introducing Algebra through the Graphical Representation of Functions: A Study among LD Students

    ERIC Educational Resources Information Center

    Sauriol, Jennifer

    2013-01-01

    This longitudinal study evaluates the impact of a new Algebra 1 course at a High School for language-based learning-disabled (LD) students. The new course prioritized the teaching of relationship graphs and functions as an introduction to algebra. Across three studies, the dissertation documents and evaluates the progress made by LD high school…

  19. Real-Time Algebraic Derivative Estimations Using a Novel Low-Cost Architecture Based on Reconfigurable Logic

    PubMed Central

    Morales, Rafael; Rincón, Fernando; Gazzano, Julio Dondo; López, Juan Carlos

    2014-01-01

    Time derivative estimation of signals plays a very important role in several fields, such as signal processing and control engineering, just to name a few of them. For that purpose, a non-asymptotic algebraic procedure for the approximate estimation of the system states is used in this work. The method is based on results from differential algebra and furnishes some general formulae for the time derivatives of a measurable signal in which two algebraic derivative estimators run simultaneously, but in an overlapping fashion. The algebraic derivative algorithm presented in this paper is computed online and in real-time, offering high robustness properties with regard to corrupting noises, versatility and ease of implementation. Besides, in this work, we introduce a novel architecture to accelerate this algebraic derivative estimator using reconfigurable logic. The core of the algorithm is implemented in an FPGA, improving the speed of the system and achieving real-time performance. Finally, this work proposes a low-cost platform for the integration of hardware in the loop in MATLAB. PMID:24859033

  20. Algebraic Bethe ansatz for the two species ASEP with different hopping rates

    NASA Astrophysics Data System (ADS)

    Cantini, Luigi

    2008-03-01

    An ASEP with two species of particles and different hopping rates is considered on a ring. Its integrability is proved, and the nested algebraic Bethe ansatz is used to derive the Bethe equations for states with arbitrary numbers of particles of each type, generalizing the results of Derrida and Evans [10]. We also present formulae for the total velocity of particles of a given type and their limit given the large size of the system and the finite densities of the particles.

  1. Developing students' functional thinking in algebra through different visualisations of a growing pattern's structure

    NASA Astrophysics Data System (ADS)

    Wilkie, Karina J.; Clarke, Doug M.

    2016-06-01

    Spatial visualisation of geometric patterns and their generalisation have become a recognised pathway to developing students' functional thinking and understanding of variables in algebra. This design-based research project investigated upper primary students' development of explicit generalisation of functional relationships and their representation descriptively, graphically and symbolically. Ten teachers and their classes were involved in a sequence of tasks involving growing patterns and geometric structures over 1 year. This article focuses on two aspects of the study: visualising the structure of a geometric pattern in different ways and using this to generalise the functional relationship between two quantifiable aspects (variables). It was found that in an initial assessment task ( n = 222), students' initial visualisations could be categorised according to different types and some of these were more likely to lead either to recursive or explicit generalisation. In a later task, a small number of students demonstrated the ability to find more than one way to visualise the same geometric structure and thus represent their explicit generalisations as different but equivalent symbolic equations (using pronumerals). Implications for the teaching of functional thinking in middle-school algebra are discussed.

  2. Activities for Students: Biology as a Source for Algebra Equations--The Heart

    ERIC Educational Resources Information Center

    Horak, Virginia M.

    2005-01-01

    The high school course that integrated first year algebra with an introductory environmental biology/anatomy and physiology course, in order to solve algebra problems is discussed. Lessons and activities for the course were taken by identifying the areas where mathematics and biology content intervenes may help students understand biology concepts…

  3. Application of polynomial su(1, 1) algebra to Pöschl-Teller potentials

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

    Zhang, Hong-Biao, E-mail: zhanghb017@nenu.edu.cn; Lu, Lu

    2013-12-15

    Two novel polynomial su(1, 1) algebras for the physical systems with the first and second Pöschl-Teller (PT) potentials are constructed, and their specific representations are presented. Meanwhile, these polynomial su(1, 1) algebras are used as an algebraic technique to solve eigenvalues and eigenfunctions of the Hamiltonians associated with the first and second PT potentials. The algebraic approach explores an appropriate new pair of raising and lowing operators K-circumflex{sub ±} of polynomial su(1, 1) algebra as a pair of shift operators of our Hamiltonians. In addition, two usual su(1, 1) algebras associated with the first and second PT potentials are derivedmore » naturally from the polynomial su(1, 1) algebras built by us.« less

  4. Assessing Mathematics Automatically Using Computer Algebra and the Internet

    ERIC Educational Resources Information Center

    Sangwin, Chris

    2004-01-01

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

  5. 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)

  6. Frobenius manifolds and Frobenius algebra-valued integrable systems

    NASA Astrophysics Data System (ADS)

    Strachan, Ian A. B.; Zuo, Dafeng

    2017-06-01

    The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper, a new theory of Frobenius algebra-valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann (Int Math Res Not 19:929-952, 1996), Kontsevich and Manin (Inv Math 124: 313-339, 1996). By specializing this construction, using a fixed Frobenius algebra A, one can arrive at such a theory. More generally, one can apply the same idea to construct an A-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an A-valued modified Camassa-Holm equation is constructed.

  7. Relativity symmetries and Lie algebra contractions

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

    Cho, Dai-Ning; Kong, Otto C.W., E-mail: otto@phy.ncu.edu.tw

    We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n) symmetry as an isometry on an m+n dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n) preserving a symmetry of the same type at dimension m+n−1, e.g. a G(m,n−1), together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example ofmore » symmetries obtained from the contraction of SO(2,4), which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3) may be relevant to real physics.« less

  8. Form in Algebra: Reflecting, with Peacock, on Upper Secondary School Teaching.

    ERIC Educational Resources Information Center

    Menghini, Marta

    1994-01-01

    Discusses algebra teaching by looking back into the history of algebra and the work of George Peacock, who considered algebra from two points of view: symbolic and instrumental. Claims that, to be meaningful, algebra must be linked to real-world problems. (18 references) (MKR)

  9. Topologically massive gravity and galilean conformal algebra: a study of correlation functions

    NASA Astrophysics Data System (ADS)

    Bagchi, Arjun

    2011-02-01

    The Galilean Conformal Algebra (GCA) arises from the conformal algebra in the non-relativistic limit. In two dimensions, one can view it as a limit of linear combinations of the two copies Virasoro algebra. Recently, it has been argued that Topologically Massive Gravity (TMG) realizes the quantum 2d GCA in a particular scaling limit of the gravitational Chern-Simons term. To add strength to this claim, we demonstrate a matching of correlation functions on both sides of this correspondence. A priori looking for spatially dependent correlators seems to force us to deal with high spin operators in the bulk. We get around this difficulty by constructing the non-relativistic Energy-Momentum tensor and considering its correlation functions. On the gravity side, our analysis makes heavy use of recent results of Holographic Renormalization in Topologically Massive Gravity.

  10. A spatial operator algebra for manipulator modeling and control

    NASA Technical Reports Server (NTRS)

    Rodriguez, G.; Kreutz, K.; Jain, A.

    1989-01-01

    A spatial operator algebra for modeling the control and trajectory design of manipulation is discussed, with emphasis on its analytical formulation and implementation in the Ada programming language. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of the manipulator. Inversion is obtained using techniques of recursive filtering and smoothing. The operator alegbra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. Implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection, thus greatly simplifying the transition from an abstract problem formulation and solution to the detailed mechanization of a specific algorithm.

  11. Modern Geometric Algebra: A (Very Incomplete!) Survey

    ERIC Educational Resources Information Center

    Suzuki, Jeff

    2009-01-01

    Geometric algebra is based on two simple ideas. First, the area of a rectangle is equal to the product of the lengths of its sides. Second, if a figure is broken apart into several pieces, the sum of the areas of the pieces equals the area of the original figure. Remarkably, these two ideas provide an elegant way to introduce, connect, and…

  12. From No to Yes: The Impact of an Intervention on The Persistence of Algebraic Misconceptions among Secondary School Algebra Students

    ERIC Educational Resources Information Center

    Zielinski, Susan F.

    2017-01-01

    Many students enter high school with persistent algebraic misconceptions that limit their success in mathematics and, by extension, limit potential educational attainment and future earnings. The purpose of this study was to assess the effectiveness of a warm conceptual change based intervention on remediating algebraic misconceptions held by…

  13. Capitalizing on Basic Brain Processes in Developmental Algebra--Part 2

    ERIC Educational Resources Information Center

    Laughbaum, Edward D.

    2011-01-01

    Basic brain function is not a mystery. Given that neuroscientists understand its basic functioning processes, one wonders what their research suggests to teachers of developmental algebra. What if we knew how to teach so as to improve understanding of the algebra taught to developmental algebra students? What if we knew how the brain processes…

  14. Capitalizing on Basic Brain Processes in Developmental Algebra--Part One

    ERIC Educational Resources Information Center

    Laughbaum, Edward D.

    2011-01-01

    Basic brain function is not a mystery. Given that neuroscientists understand the brain's basic functioning processes, one wonders what their research suggests to teachers of developmental algebra. What if we knew how to teach so as to improve understanding of the algebra taught to developmental algebra students? What if we knew how the brain…

  15. Using Student Work to Develop Teachers' Knowledge of Algebra

    ERIC Educational Resources Information Center

    Herbel-Eisenmann, Beth A.; Phillips, Elizabeth Difanis

    2005-01-01

    This article describes a set of learning activities that use algebraic problems and written student work to help preservice and in-service teachers understand students' algebraic thinking. (Contains 4 figures.)

  16. Rupture or Continuity: The Arithmetico-Algebraic Thinking as an Alternative in a Modelling Process in a Paper and Pencil and Technology Environment

    ERIC Educational Resources Information Center

    Hitt, Fernando; Saboya, Mireille; Zavala, Carlos Cortés

    2017-01-01

    Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which…

  17. Opening a Gateway to College Access: Algebra at the Right Time. Research Brief

    ERIC Educational Resources Information Center

    Snipes, Jason; Finkelstein, Neal

    2015-01-01

    Four years of math in high school, with a strong foundation in algebra that builds from middle school, is key to higher education access. Therefore, ensuring that middle and high school students succeed in math--and in algebra in particular--is an important issue for policy and practice. This research brief examines three recent Regional…

  18. Underprepared Students' Performance on Algebra in a Double-Period High School Mathematics Program

    ERIC Educational Resources Information Center

    Martinez, Mara V.; Bragelman, John; Stoelinga, Timothy

    2016-01-01

    The primary goal of the Intensified Algebra I (IA) program is to enable mathematically underprepared students to successfully complete Algebra I in 9th grade and stay on track to meet increasingly rigorous high school mathematics graduation requirements. The program was designed to bring a range of both cognitive and non-cognitive supports to bear…

  19. Surface defects and chiral algebras

    NASA Astrophysics Data System (ADS)

    Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng

    2017-05-01

    We investigate superconformal surface defects in four-dimensional N=2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfeld-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. In each case we find perfect agreement with the predicted characters.

  20. Earth Algebra: Real-Life Mathematics in Navajoland.

    ERIC Educational Resources Information Center

    Schaufele, Christopher; Srivastava, Ravindra

    1995-01-01

    An algebra class at Navajo Community College (Shiprock, New Mexico) uses traditional algebra topics to study real-life situations, focuses on environmental issues, encourages collaborative learning, uses modern technology, and promotes development of critical thinking and decision-making skills. Students follow principles of Dine educational…

  1. On Orders of Observables on Effect Algebras

    NASA Astrophysics Data System (ADS)

    Dvurečenskij, Anatolij

    2017-12-01

    On the set of bounded observables on an effect algebra, the Olson order defined by spectral resolutions and the standard order defined by a system of σ-additive states are introduced. We show that sharp bounded observables form a Dedekind σ-complete sublattice of a Dedekind complete lattice under the Olson order. In addition, we compare both orders, and we illustrate them on different effect algebras.

  2. On the intersection of irreducible components of the space of finite-dimensional Lie algebras

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

    Gorbatsevich, Vladimir V

    2012-07-31

    The irreducible components of the space of n-dimensional Lie algebras are investigated. The properties of Lie algebras belonging to the intersection of all the irreducible components of this kind are studied (these Lie algebras are said to be basic or founding Lie algebras). It is proved that all Lie algebras of this kind are nilpotent and each of these Lie algebras has an Abelian ideal of codimension one. Specific examples of founding Lie algebras of arbitrary dimension are described and, to describe the Lie algebras in general, we state a conjecture. The concept of spectrum of a Lie algebra ismore » considered and some of the most elementary properties of the spectrum are studied. Bibliography: 6 titles.« less

  3. The Great Debate: Should All 8th Graders Take Algebra?

    ERIC Educational Resources Information Center

    McKibben, Sarah

    2009-01-01

    While 8th grade algebra was once reserved as a course for the gifted, today, more U.S. 8th graders take algebra than any other math course. This article discusses a report from the Brookings Institution which chronicles the history of the 8th-grade algebra surge and its impact on today's low-performing students. The report indicates that many of…

  4. Does Early Algebraic Reasoning Differ as a Function of Students’ Difficulty with Calculations versus Word Problems?

    PubMed Central

    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 2nd- 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

  5. Associative Algebraic Approach to Logarithmic CFT in the Bulk: The Continuum Limit of the {gl(1|1)} Periodic Spin Chain, Howe Duality and the Interchiral Algebra

    NASA Astrophysics Data System (ADS)

    Gainutdinov, A. M.; Read, N.; Saleur, H.

    2016-01-01

    We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed {gl(1|1)} spin-chain and its continuum limit—the {c=-2} symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245-288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289-329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL N , goes over in the continuum limit to a bigger algebra than {V}, the product of the left and right Virasoro algebras. This algebra, {S}—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field {S(z,bar{z})≡ S_{αβ} ψ^α(z)bar{ψ}^β(bar{z})}, with a symmetric form {S_{αβ}} and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL N in the {gl(1|1)} spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of {sp_{N-2}}. The semi-simple part of JTL N is represented by {U sp_{N-2}}, providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL N image represented in the spin-chain. On the continuum side, simple modules over {S} are identified with "fundamental" representations of {sp_∞}.

  6. Bisimulation equivalence of differential-algebraic systems

    NASA Astrophysics Data System (ADS)

    Megawati, Noorma Yulia; Schaft, Arjan van der

    2018-01-01

    In this paper, the notion of bisimulation relation for linear input-state-output systems is extended to general linear differential-algebraic (DAE) systems. Geometric control theory is used to derive a linear-algebraic characterisation of bisimulation relations, and an algorithm for computing the maximal bisimulation relation between two linear DAE systems. The general definition is specialised to the case where the matrix pencil sE - A is regular. Furthermore, by developing a one-sided version of bisimulation, characterisations of simulation and abstraction are obtained.

  7. Performance assessment in algebra learning process

    NASA Astrophysics Data System (ADS)

    Lestariani, Ida; Sujadi, Imam; Pramudya, Ikrar

    2017-12-01

    The purpose of research to describe the implementation of performance assessment on algebra learning process. The subject in this research is math educator of SMAN 1 Ngawi class X. This research includes descriptive qualitative research type. Techniques of data collecting are done by observation method, interview, and documentation. Data analysis technique is done by data reduction, data presentation, and conclusion. The results showed any indication that the steps taken by the educator in applying the performance assessment are 1) preparing individual worksheets and group worksheets, 2) preparing rubric assessments for independent worksheets and groups and 3) making performance assessments rubric to learners’ performance results with individual or groups task.

  8. Extended gauge theory and gauged free differential algebras

    NASA Astrophysics Data System (ADS)

    Salgado, P.; Salgado, S.

    2018-01-01

    Recently, Antoniadis, Konitopoulos and Savvidy introduced, in the context of the so-called extended gauge theory, a procedure to construct background-free gauge invariants, using non-abelian gauge potentials described by higher degree forms. In this article it is shown that the extended invariants found by Antoniadis, Konitopoulos and Savvidy can be constructed from an algebraic structure known as free differential algebra. In other words, we show that the above mentioned non-abelian gauge theory, where the gauge fields are described by p-forms with p ≥ 2, can be obtained by gauging free differential algebras.

  9. Quantization and Superselection Sectors I:. Transformation Group C*-ALGEBRAS

    NASA Astrophysics Data System (ADS)

    Landsman, N. P.

    Quantization is defined as the act of assigning an appropriate C*-algebra { A} to a given configuration space Q, along with a prescription mapping self-adjoint elements of { A} into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here { A} is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of { A}. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of { A}.

  10. A Algebraic Approach to the Quantization of Constrained Systems: Finite Dimensional Examples.

    NASA Astrophysics Data System (ADS)

    Tate, Ranjeet Shekhar

    1992-01-01

    General relativity has two features in particular, which make it difficult to apply to it existing schemes for the quantization of constrained systems. First, there is no background structure in the theory, which could be used, e.g., to regularize constraint operators, to identify a "time" or to define an inner product on physical states. Second, in the Ashtekar formulation of general relativity, which is a promising avenue to quantum gravity, the natural variables for quantization are not canonical; and, classically, there are algebraic identities between them. Existing schemes are usually not concerned with such identities. Thus, from the point of view of canonical quantum gravity, it has become imperative to find a framework for quantization which provides a general prescription to find the physical inner product, and is flexible enough to accommodate non -canonical variables. In this dissertation I present an algebraic formulation of the Dirac approach to the quantization of constrained systems. The Dirac quantization program is augmented by a general principle to find the inner product on physical states. Essentially, the Hermiticity conditions on physical operators determine this inner product. I also clarify the role in quantum theory of possible algebraic identities between the elementary variables. I use this approach to quantize various finite dimensional systems. Some of these models test the new aspects of the algebraic framework. Others bear qualitative similarities to general relativity, and may give some insight into the pitfalls lurking in quantum gravity. The previous quantizations of one such model had many surprising features. When this model is quantized using the algebraic program, there is no longer any unexpected behaviour. I also construct the complete quantum theory for a previously unsolved relativistic cosmology. All these models indicate that the algebraic formulation provides powerful new tools for quantization. In (spatially compact

  11. Constitutive relations in optics in terms of geometric algebra

    NASA Astrophysics Data System (ADS)

    Dargys, A.

    2015-11-01

    To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p-form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.

  12. Generalized derivation extensions of 3-Lie algebras and corresponding Nambu-Poisson structures

    NASA Astrophysics Data System (ADS)

    Song, Lina; Jiang, Jun

    2018-01-01

    In this paper, we introduce the notion of a generalized derivation on a 3-Lie algebra. We construct a new 3-Lie algebra using a generalized derivation and call it the generalized derivation extension. We show that the corresponding Leibniz algebra on the space of fundamental objects is the double of a matched pair of Leibniz algebras. We also determine the corresponding Nambu-Poisson structures under some conditions.

  13. 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.

  14. Tensor models, Kronecker coefficients and permutation centralizer algebras

    NASA Astrophysics Data System (ADS)

    Geloun, Joseph Ben; Ramgoolam, Sanjaye

    2017-11-01

    We show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin decompositions into matrix blocks are given in terms of Clebsch-Gordan coefficients of symmetric groups. The matrix basis for the algebras also gives an orthogonal basis for the tensor observables which diagonalizes the Gaussian two-point functions. The centres of the algebras are associated with correlators which are expressible in terms of Kronecker coefficients (Clebsch-Gordan multiplicities of symmetric groups). The color-exchange symmetry present in the Gaussian model, as well as a large class of interacting models, is used to refine the description of the permutation centralizer algebras. This discussion is extended to a general number of colors d: it is used to prove the integrality of an infinite family of number sequences related to color-symmetrizations of colored graphs, and expressible in terms of symmetric group representation theory data. Generalizing a connection between matrix models and Belyi maps, correlators in Gaussian tensor models are interpreted in terms of covers of singular 2-complexes. There is an intriguing difference, between matrix and higher rank tensor models, in the computational complexity of superficially comparable correlators of observables parametrized by Young diagrams.

  15. Anti-commutative Gröbner-Shirshov basis of a free Lie algebra

    NASA Astrophysics Data System (ADS)

    Bokut, L. A.; Chen, Yuqun; Li, Yu

    2009-03-01

    One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and then to check that the product yields the Lie identities (Marshall Hall, 1950). Here we suggest another way using the Composition-Diamond lemma for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962).

  16. Graphs in kinematics—a need for adherence to principles of algebraic functions

    NASA Astrophysics Data System (ADS)

    Sokolowski, Andrzej

    2017-11-01

    Graphs in physics are central to the analysis of phenomena and to learning about a system’s behavior. The ways students handle graphs are frequently researched. Students’ misconceptions are highlighted, and methods of improvement suggested. While kinematics graphs are to represent a real motion, they are also algebraic entities that must satisfy conditions for being algebraic functions. To be algebraic functions, they must pass certain tests before they can be used to infer more about motion. A preliminary survey of some physics resources has revealed that little attention is paid to verifying if the position, velocity and acceleration versus time graphs, that are to depict real motion, satisfy the most critical condition for being an algebraic function; the vertical line test. The lack of attention to this adherence shows as vertical segments in piecewise graphs. Such graphs generate unrealistic interpretations and may confuse students. A group of 25 college physics students was provided with such a graph and asked to analyse its adherence to reality. The majority of the students (N  =  16, 64%) questioned the graph’s validity. It is inferred that such graphs might not only jeopardize the function principles studied in mathematics but also undermine the purpose of studying these principles. The aim of this study was to bring this idea forth and suggest a better alignment of physics and mathematics methods.

  17. Teaching materials of algebraic equation

    NASA Astrophysics Data System (ADS)

    Widodo, S. A.; Prahmana, R. C. I.; Purnami, A. S.; Turmudi

    2017-12-01

    The purpose of this paper is to know the effectiveness of teaching materials algebraic equation. This type of research used experimental method. The population in this study is all students of mathematics education who take numerical method in sarjanawiyata tamansiswa of university; the sample is taken using cluster random sampling. Instrument used in this research is test and questionnaire. The test is used to know the problem solving ability and achievement, while the questionnaire is used to know the student's response on the teaching materials. Data Analysis technique of quantitative used Wilcoxon test, while the qualitative data used grounded theory. Based on the results of the test can be concluded that the development of teaching materials can improve the ability to solve problems and achievement.

  18. Current algebra, statistical mechanics and quantum models

    NASA Astrophysics Data System (ADS)

    Vilela Mendes, R.

    2017-11-01

    Results obtained in the past for free boson systems at zero and nonzero temperatures are revisited to clarify the physical meaning of current algebra reducible functionals which are associated to systems with density fluctuations, leading to observable effects on phase transitions. To use current algebra as a tool for the formulation of quantum statistical mechanics amounts to the construction of unitary representations of diffeomorphism groups. Two mathematical equivalent procedures exist for this purpose. One searches for quasi-invariant measures on configuration spaces, the other for a cyclic vector in Hilbert space. Here, one argues that the second approach is closer to the physical intuition when modelling complex systems. An example of application of the current algebra methodology to the pairing phenomenon in two-dimensional fermion systems is discussed.

  19. Students’ Algebraic Thinking Process in Context of Point and Line Properties

    NASA Astrophysics Data System (ADS)

    Nurrahmi, H.; Suryadi, D.; Fatimah, S.

    2017-09-01

    Learning of schools algebra is limited to symbols and operating procedures, so students are able to work on problems that only require the ability to operate symbols but unable to generalize a pattern as one of part of algebraic thinking. The purpose of this study is to create a didactic design that facilitates students to do algebraic thinking process through the generalization of patterns, especially in the context of the property of point and line. This study used qualitative method and includes Didactical Design Research (DDR). The result is students are able to make factual, contextual, and symbolic generalization. This happen because the generalization arises based on facts on local terms, then the generalization produced an algebraic formula that was described in the context and perspective of each student. After that, the formula uses the algebraic letter symbol from the symbol t hat uses the students’ language. It can be concluded that the design has facilitated students to do algebraic thinking process through the generalization of patterns, especially in the context of property of the point and line. The impact of this study is this design can use as one of material teaching alternative in learning of school algebra.

  20. Complex quantum enveloping algebras as twisted tensor products

    NASA Astrophysics Data System (ADS)

    Chryssomalakos, Chryssomalis; Engeldinger, Ralf A.; Jurčo, Branislav; Schlieker, Michael; Zumino, Bruno

    1994-12-01

    We introduce a *-structure on the quantum double and its dual in order to make contact with various approaches to the enveloping algebras of complex quantum groups. Furthermore, we introduce a canonical basis in the quantum double, its universal R-matrices and give its relation to subgroups in the dual Hopf algebra.

  1. The analysis of mathematics teachers' learning on algebra function limit material based on teaching experience difference

    NASA Astrophysics Data System (ADS)

    Ma'rufi, Budayasa, I. Ketut; Juniati, Dwi

    2017-08-01

    The aim of this study was to describe the analysis of mathematics teachers' learning on algebra function limit material based on teaching experience difference. The purpose of this study is to describe the analysis of mathematics teacher's learning on limit algebraic functions in terms of the differences of teaching experience. Learning analysis focused on Pedagogical Content Knowledge (PCK) of teachers in mathematics on limit algebraic functions related to the knowledge of pedagogy. PCK of teachers on limit algebraic function is a type of specialized knowledge for teachers on how to teach limit algebraic function that can be understood by students. Subjects are two high school mathematics teacher who has difference of teaching experience they are one Novice Teacher (NP) and one Experienced Teacher (ET). Data are collected through observation of learning in the class, videos of learning, and then analyzed using qualitative analysis. Teacher's knowledge of Pedagogic defined as a knowledge and understanding of teacher about planning and organizing of learning, and application of learning strategy. The research results showed that the Knowledge of Pedagogy on subject NT in mathematics learning on the material of limit function algebra showed that the subject NT tended to describe procedurally, without explaining the reasons why such steps were used, asking questions which tended to be monotonous not be guiding and digging deeper, and less varied in the use of learning strategies while subject ET gave limited guidance and opportunities to the students to find their own answers, exploit the potential of students to answer questions, provide an opportunity for students to interact and work in groups, and subject ET tended to combine conceptual and procedural explanation.

  2. Developing Meaning for Algebraic Procedures: An Exploration of the Connections Undergraduate Students Make between Algebraic Rational Expressions and Basic Number Properties

    ERIC Educational Resources Information Center

    Yantz, Jennifer

    2013-01-01

    The attainment and retention of later algebra skills in high school has been identified as a factor significantly impacting the postsecondary success of students majoring in STEM fields. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. The present study…

  3. Surface defects and chiral algebras

    DOE PAGES

    Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng

    2017-05-26

    Here, we investigate superconformal surface defects in four-dimensional N = 2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfield-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. We find perfect agreement with the predicted characters, in eachmore » case.« less

  4. Surface defects and chiral algebras

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

    Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng

    Here, we investigate superconformal surface defects in four-dimensional N = 2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfield-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. We find perfect agreement with the predicted characters, in eachmore » case.« less

  5. On explicit algebraic stress models for complex turbulent flows

    NASA Technical Reports Server (NTRS)

    Gatski, T. B.; Speziale, C. G.

    1992-01-01

    Explicit algebraic stress models that are valid for three-dimensional turbulent flows in noninertial frames are systematically derived from a hierarchy of second-order closure models. This represents a generalization of the model derived by Pope who based his analysis on the Launder, Reece, and Rodi model restricted to two-dimensional turbulent flows in an inertial frame. The relationship between the new models and traditional algebraic stress models -- as well as anistropic eddy visosity models -- is theoretically established. The need for regularization is demonstrated in an effort to explain why traditional algebraic stress models have failed in complex flows. It is also shown that these explicit algebraic stress models can shed new light on what second-order closure models predict for the equilibrium states of homogeneous turbulent flows and can serve as a useful alternative in practical computations.

  6. Modularity of logarithmic parafermion vertex algebras

    NASA Astrophysics Data System (ADS)

    Auger, Jean; Creutzig, Thomas; Ridout, David

    2018-05-01

    The parafermionic cosets Ck = {Com} ( H , Lk(sl2) ) are studied for negative admissible levels k, as are certain infinite-order simple current extensions Bk of Ck . Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to Ck , irreducible Ck - and Bk -modules are obtained from those of Lk(sl2) . Assuming the validity of a certain Verlinde-type formula likewise gives the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible Bk -modules. The irreducible Ck - and Bk -characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the Bk are C_2 -cofinite vertex operator algebras.

  7. 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.

  8. Analyzing Algebraic Thinking Using "Guess My Number" Problems

    ERIC Educational Resources Information Center

    Patton, Barba; De Los Santos, Estella

    2012-01-01

    The purpose of this study was to assess student knowledge of numeric, visual and algebraic representations. A definite gap between arithmetic and algebra has been documented in the research. The researchers' goal was to identify a link between the two. Using four "Guess My Number" problems, seventh and tenth grade students were asked to write…

  9. Emphasizing Language and Visualization in Teaching Linear Algebra

    ERIC Educational Resources Information Center

    Hannah, John; Stewart, Sepideh; Thomas, Mike

    2013-01-01

    Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his…

  10. 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.

  11. 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.

  12. The Xs and Whys of Algebra: Key Ideas and Common Misconceptions

    ERIC Educational Resources Information Center

    Collins, Anne; Dacey, Linda

    2011-01-01

    In many ways, algebra can be as challenging for teachers as it is for students. With so much emphasis placed on procedural knowledge and the manipulations of variables and symbols, it can be easy to lose sight of the key ideas that underlie algebraic thinking and the relevance algebra has to the real world. In the The Xs and Whys of Algebra: Key…

  13. Symbolic Notations and Students' Achievements in Algebra

    ERIC Educational Resources Information Center

    Peter, Ebiendele E.; Olaoye, Adetunji A.

    2013-01-01

    This study focuses on symbolic notations and its impact on students' achievement in Algebra. The main reason for this study rests on the observation from personal and professional experiences on students' increasing hatred for Algebra. One hundred and fifty (150) Senior Secondary School Students (SSS) from Ojo Local Education District, Ojo, Lagos,…

  14. Eighth Grade Algebra Course Placement and Student Motivation for Mathematics

    PubMed Central

    Simzar, Rahila M.; Domina, Thurston; Tran, Cathy

    2016-01-01

    This study uses student panel data to examine the association between Algebra placement and student motivation for mathematics. Changes in achievement goals, expectancy, and task value for students in eighth grade Algebra are compared with those of peers placed in lower-level mathematics courses (N = 3,306). In our sample, students placed in Algebra reported an increase in performance-avoidance goals as well as decreases in academic self-efficacy and task value. These relations were attenuated for students who had high mathematics achievement prior to Algebra placement. Whereas all students reported an overall decline in performance-approach goals over the course of eighth grade, previously high-achieving students reported an increase in these goals. Lastly, previously high-achieving students reported an increase in mastery goals. These findings suggest that while previously high-achieving students may benefit motivationally from eighth grade Algebra placement, placing previously average- and low-performing students in Algebra can potentially undermine their motivation for mathematics. PMID:26942210

  15. Eighth Grade Algebra Course Placement and Student Motivation for Mathematics.

    PubMed

    Simzar, Rahila M; Domina, Thurston; Tran, Cathy

    2016-01-01

    This study uses student panel data to examine the association between Algebra placement and student motivation for mathematics. Changes in achievement goals, expectancy, and task value for students in eighth grade Algebra are compared with those of peers placed in lower-level mathematics courses (N = 3,306). In our sample, students placed in Algebra reported an increase in performance-avoidance goals as well as decreases in academic self-efficacy and task value. These relations were attenuated for students who had high mathematics achievement prior to Algebra placement. Whereas all students reported an overall decline in performance-approach goals over the course of eighth grade, previously high-achieving students reported an increase in these goals. Lastly, previously high-achieving students reported an increase in mastery goals. These findings suggest that while previously high-achieving students may benefit motivationally from eighth grade Algebra placement, placing previously average- and low-performing students in Algebra can potentially undermine their motivation for mathematics.

  16. Does Calculation or Word-Problem Instruction Provide A Stronger Route to Pre-Algebraic Knowledge?

    PubMed Central

    Fuchs, Lynn S.; Powell, Sarah R.; Cirino, Paul T.; Schumacher, Robin F.; Marrin, Sarah; Hamlett, Carol L.; Fuchs, Douglas; Compton, Donald L.; Changas, Paul C.

    2014-01-01

    The focus of this study was connections among 3 aspects of mathematical cognition at 2nd grade: calculations, word problems, and pre-algebraic knowledge. We extended the literature, which is dominated by correlational work, by examining whether intervention conducted on calculations or word problems contributes to improved performance in the other domain and whether intervention in either or both domains contributes to pre-algebraic knowledge. Participants were 1102 children in 127 2nd-grade classrooms in 25 schools. Teachers were randomly assigned to 3 conditions: calculation intervention, word-problem intervention, and business-as-usual control. Intervention, which lasted 17 weeks, was designed to provide research-based linkages between arithmetic calculations or arithmetic word problems (depending on condition) to pre-algebraic knowledge. Multilevel modeling suggested calculation intervention improved calculation but not word-problem outcomes; word-problem intervention enhanced word-problem but not calculation outcomes; and word-problem intervention provided a stronger route than calculation intervention to pre-algebraic knowledge. PMID:25541565

  17. Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions

    NASA Astrophysics Data System (ADS)

    Turbiner, A. V.; Escobar-Ruiz, M. A.

    2013-07-01

    The quantum mechanics of two Coulomb charges on a plane (e1, m1) and (e2, m2) subject to a constant magnetic field B perpendicular to the plane is considered. Four integrals of motion are explicitly indicated. It is shown that for two physically important particular cases, namely that of two particles of equal Larmor frequencies, {e_c} \\propto \\frac{e_1}{m_1}-\\frac{e_2}{m_2}=0 (e.g. two electrons) and one of a neutral system (e.g. the electron-positron pair, hydrogen atom) at rest (the center-of-mass momentum is zero) some outstanding properties occur. They are the most visible in double polar coordinates in CMS (R, ϕ) and relative (ρ, φ) coordinate systems: (i) eigenfunctions are factorizable, all factors except one with the explicit ρ-dependence are found analytically, they have definite relative angular momentum, (ii) dynamics in the ρ-direction is the same for both systems, it corresponds to a funnel-type potential and it has hidden sl(2) algebra, at some discrete values of dimensionless magnetic fields b ⩽ 1, (iii) particular integral(s) occur, (iv) the hidden sl(2) algebra emerges in finite-dimensional representation, thus, the system becomes quasi-exactly-solvable and (v) a finite number of polynomial eigenfunctions in ρ appear. Nine families of eigenfunctions are presented explicitly.

  18. Schroedinger operators with the q-ladder symmetry algebras

    NASA Technical Reports Server (NTRS)

    Skorik, Sergei; Spiridonov, Vyacheslav

    1994-01-01

    A class of the one-dimensional Schroedinger operators L with the symmetry algebra LB(+/-) = q(+/-2)B(+/-)L, (B(+),B(-)) = P(sub N)(L), is described. Here B(+/-) are the 'q-ladder' operators and P(sub N)(L) is a polynomial of the order N. Peculiarities of the coherent states of this algebra are briefly discussed.

  19. Lambda: A Mathematica package for operator product expansions in vertex algebras

    NASA Astrophysics Data System (ADS)

    Ekstrand, Joel

    2011-02-01

    We give an introduction to the Mathematica package Lambda, designed for calculating λ-brackets in both vertex algebras, and in SUSY vertex algebras. This is equivalent to calculating operator product expansions in two-dimensional conformal field theory. The syntax of λ-brackets is reviewed, and some simple examples are shown, both in component notation, and in N=1 superfield notation. Program summaryProgram title: Lambda Catalogue identifier: AEHF_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHF_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 18 087 No. of bytes in distributed program, including test data, etc.: 131 812 Distribution format: tar.gz Programming language: Mathematica Computer: See specifications for running Mathematica V7 or above. Operating system: See specifications for running Mathematica V7 or above. RAM: Varies greatly depending on calculation to be performed. Classification: 4.2, 5, 11.1. Nature of problem: Calculate operator product expansions (OPEs) of composite fields in 2d conformal field theory. Solution method: Implementation of the algebraic formulation of OPEs given by vertex algebras, and especially by λ-brackets. Running time: Varies greatly depending on calculation requested. The example notebook provided takes about 3 s to run.

  20. Algebraic Thinking through Origami.

    ERIC Educational Resources Information Center

    Higginson, William; Colgan, Lynda

    2001-01-01

    Describes the use of paper folding to create a rich environment for discussing algebraic concepts. Explores the effect that changing the dimensions of two-dimensional objects has on the volume of related three-dimensional objects. (Contains 13 references.) (YDS)

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

    PubMed

    Benhammouda, Brahim

    2016-01-01

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

  2. Homological dimensions and Van den Bergh isomorphisms for nuclear Fréchet algebras

    NASA Astrophysics Data System (ADS)

    Pirkovskii, Alexei Yu

    2012-08-01

    We prove the equation \\operatorname{w{.}dg} A=\\operatorname{w{.}db} A for every nuclear Fréchet-Arens-Michael algebra A of finite weak bidimension, where \\operatorname{w{.}dg} A is the weak global dimension and \\operatorname{w{.}db} A the weak bidimension of A. Assuming that A has a projective bimodule resolution of finite type, we establish the estimate \\operatorname{db}A\\le\\operatorname{dg}A+1, where \\operatorname{dg} A is the global dimension and \\operatorname{db} A the bidimension of A. We also prove that \\operatorname{dg}A=\\operatorname{db}A=\\operatorname{w{.}dg}A=\\operatorname{w{.}db} A=n for all nuclear Fréchet-Arens-Michael algebras satisfying the Van den Bergh conditions \\operatorname{VdB}(n). As an application, we calculate the homological dimensions of smooth and complex-analytic quantum tori.

  3. Partially Flipped Linear Algebra: A Team-Based Approach

    ERIC Educational Resources Information Center

    Carney, Debra; Ormes, Nicholas; Swanson, Rebecca

    2015-01-01

    In this article we describe a partially flipped Introductory Linear Algebra course developed by three faculty members at two different universities. We give motivation for our partially flipped design and describe our implementation in detail. Two main features of our course design are team-developed preview videos and related in-class activities.…

  4. Performance of a parallel algebraic multilevel preconditioner for stabilized finite element semiconductor device modeling

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

    Lin, Paul T.; Shadid, John N.; Sala, Marzio

    In this study results are presented for the large-scale parallel performance of an algebraic multilevel preconditioner for solution of the drift-diffusion model for semiconductor devices. The preconditioner is the key numerical procedure determining the robustness, efficiency and scalability of the fully-coupled Newton-Krylov based, nonlinear solution method that is employed for this system of equations. The coupled system is comprised of a source term dominated Poisson equation for the electric potential, and two convection-diffusion-reaction type equations for the electron and hole concentration. The governing PDEs are discretized in space by a stabilized finite element method. Solution of the discrete system ismore » obtained through a fully-implicit time integrator, a fully-coupled Newton-based nonlinear solver, and a restarted GMRES Krylov linear system solver. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the nonzero block structure of the Jacobian matrix. Representative performance results are presented for various choices of multigrid V-cycles and W-cycles and parameter variations for smoothers based on incomplete factorizations. Parallel scalability results are presented for solution of up to 10{sup 8} unknowns on 4096 processors of a Cray XT3/4 and an IBM POWER eServer system.« less

  5. Gauss Elimination: Workhorse of Linear Algebra.

    DTIC Science & Technology

    1995-08-05

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

  6. Systems with outer constraints. Gupta-Bleuler electromagnetism as an algebraic field theory

    NASA Astrophysics Data System (ADS)

    Grundling, Hendrik

    1988-03-01

    Since there are some important systems which have constraints not contained in their field algebras, we develop here in a C*-context the algebraic structures of these. The constraints are defined as a group G acting as outer automorphisms on the field algebra ℱ, α: G ↦ Aut ℱ, α G ⊄ Inn ℱ, and we find that the selection of G-invariant states on ℱ is the same as the selection of states ω on M( G M(Gmathop × limits_α F) ℱ) by ω( U g)=1∨ g∈ G, where U g ∈ M ( G M(Gmathop × limits_α F) ℱ)/ℱ are the canonical elements implementing α g . These states are taken as the physical states, and this specifies the resulting algebraic structure of the physics in M( G M(Gmathop × limits_α F) ℱ), and in particular the maximal constraint free physical algebra ℛ. A nontriviality condition is given for ℛ to exist, and we extend the notion of a crossed product to deal with a situation where G is not locally compact. This is necessary to deal with the field theoretical aspect of the constraints. Next the C*-algebra of the CCR is employed to define the abstract algebraic structure of Gupta-Bleuler electromagnetism in the present framework. The indefinite inner product representation structure is obtained, and this puts Gupta-Bleuler electromagnetism on a rigorous footing. Finally, as a bonus, we find that the algebraic structures just set up, provide a blueprint for constructive quadratic algebraic field theory.

  7. Digital Tools for Algebra Education: Criteria and Evaluation

    ERIC Educational Resources Information Center

    Bokhove, Christian; Drijvers, Paul

    2010-01-01

    In the Netherlands, as in many other countries, the algebraic expertise of students graduating from secondary education is an issue. The use of digital tools for algebra education is expected to change epistemologies, activity structures and student achievement. Therefore, a study was set up to investigate in what way the use of ICT in upper…

  8. Introduction to Matrix Algebra, Student's Text, Unit 23.

    ERIC Educational Resources Information Center

    Allen, Frank B.; And Others

    Unit 23 in the SMSG secondary school mathematics series is a student text covering the following topics in matrix algebra: matrix operations, the algebra of 2 X 2 matrices, matrices and linear systems, representation of column matrices as geometric vectors, and transformations of the plane. Listed in the appendix are four research exercises in…

  9. Algebra for Enterprise Ontology: towards analysis and synthesis of enterprise models

    NASA Astrophysics Data System (ADS)

    Suga, Tetsuya; Iijima, Junichi

    2018-03-01

    Enterprise modeling methodologies have made enterprises more likely to be the object of systems engineering rather than craftsmanship. However, the current state of research in enterprise modeling methodologies lacks investigations of the mathematical background embedded in these methodologies. Abstract algebra, a broad subfield of mathematics, and the study of algebraic structures may provide interesting implications in both theory and practice. Therefore, this research gives an empirical challenge to establish an algebraic structure for one aspect model proposed in Design & Engineering Methodology for Organizations (DEMO), which is a major enterprise modeling methodology in the spotlight as a modeling principle to capture the skeleton of enterprises for developing enterprise information systems. The results show that the aspect model behaves well in the sense of algebraic operations and indeed constructs a Boolean algebra. This article also discusses comparisons with other modeling languages and suggests future work.

  10. Numerical methods on some structured matrix algebra problems

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

    Jessup, E.R.

    1996-06-01

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

  11. 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

  12. General Algebraic Modeling System Tutorial | High-Performance Computing |

    Science.gov Websites

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

  13. Symmetric linear systems - An application of algebraic systems theory

    NASA Technical Reports Server (NTRS)

    Hazewinkel, M.; Martin, C.

    1983-01-01

    Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.

  14. Middle school students' reading comprehension of mathematical texts and algebraic equations

    NASA Astrophysics Data System (ADS)

    Duru, Adem; Koklu, Onder

    2011-06-01

    In this study, middle school students' abilities to translate mathematical texts into algebraic representations and vice versa were investigated. In addition, students' difficulties in making such translations and the potential sources for these difficulties were also explored. Both qualitative and quantitative methods were used to collect data for this study: questionnaire and clinical interviews. The questionnaire consisted of two general types of items: (1) selected-response (multiple-choice) items for which the respondent selects from multiple options and (2) open-ended items for which the respondent constructs a response. In order to further investigate the students' strategies while they were translating the given mathematical texts to algebraic equations and vice versa, five randomly chosen (n = 5) students were interviewed. Data were collected in the 2007-2008 school year from 185 middle-school students in five teachers' classrooms in three different schools in the city of Adıyaman, Turkey. After the analysis of data, it was found that students who participated in this study had difficulties in translating the mathematical texts into algebraic equations by using symbols. It was also observed that these students had difficulties in translating the symbolic representations into mathematical texts because of their weak reading comprehension. In addition, finding of this research revealed that students' difficulties in translating the given mathematical texts into symbolic representations or vice versa come from different sources.

  15. Directed Abelian algebras and their application to stochastic models.

    PubMed

    Alcaraz, F C; Rittenberg, V

    2008-10-01

    With each directed acyclic graph (this includes some D-dimensional lattices) one can associate some Abelian algebras that we call directed Abelian algebras (DAAs). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA, one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground-state wave functions (stationary state probability distributions) is an easy algebraic exercise. If one considers D-dimensional lattices and chooses Hamiltonians linear in the generators, in finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent z=D. One possible application of the DAA is to sandpile models. In the paper we present this application, considering one- and two-dimensional lattices. In the one-dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent sigma_(tau)=32 ). We study the local density of particles inside large avalanches, showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found sigma_(tau)=1.780+/-0.005 .

  16. Statistical Aspects of Coherent States of the Higgs Algebra

    NASA Astrophysics Data System (ADS)

    Shreecharan, T.; Kumar, M. Naveen

    2018-04-01

    We construct and study various aspects of coherent states of a polynomial angular momentum algebra. The coherent states are constructed using a new unitary representation of the nonlinear algebra. The new representation involves a parameter γ that shifts the eigenvalues of the diagonal operator J 0.

  17. D{sub {infinity}}-differential E{sub {infinity}}-algebras and spectral sequences of fibrations

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

    Lapin, Sergei V

    2007-10-31

    The notion of an E{sub {infinity}}-algebra with a filtration is introduced. The connections are established between E{sub {infinity}}-algebras with filtrations and the theory of D{sub {infinity}}-differential E{sub {infinity}}-algebras over fields. Based on the technique of D{sub {infinity}}-differential E{sub {infinity}}-algebras, the apparatus of spectral sequences is developed for E{sub {infinity}}-algebras with filtrations, and applications of this apparatus to the multiplicative cohomology spectral sequences of fibrations are given. Bibliography: 21 titles.

  18. Relations between elliptic multiple zeta values and a special derivation algebra

    NASA Astrophysics Data System (ADS)

    Broedel, Johannes; Matthes, Nils; Schlotterer, Oliver

    2016-04-01

    We investigate relations between elliptic multiple zeta values (eMZVs) and describe a method to derive the number of indecomposable elements of given weight and length. Our method is based on representing eMZVs as iterated integrals over Eisenstein series and exploiting the connection with a special derivation algebra. Its commutator relations give rise to constraints on the iterated integrals over Eisenstein series relevant for eMZVs and thereby allow to count the indecomposable representatives. Conversely, the above connection suggests apparently new relations in the derivation algebra. Under https://tools.aei.mpg.de/emzv we provide relations for eMZVs over a wide range of weights and lengths.

  19. Questions Arise about Algebra 2 for All Students

    ERIC Educational Resources Information Center

    Robelen, Erik W.

    2013-01-01

    Should all students take Algebra 2? Florida seemed to say "no" this spring with the passage of a law striking it from graduation requirements. Texas said much the same in legislation Republican Gov. Rick Perry signed this week that also backs away from Algebra 2 for all. Those steps come as the Common Core State Standards for math set…

  20. Solving Our Algebra Problem: Getting All Students through Algebra I to Improve Graduation Rates

    ERIC Educational Resources Information Center

    Schachter, Ron

    2013-01-01

    graduation as well as admission to most colleges. But taking algebra also can turn into a pathway for failure, from which some students never recover. In 2010, a national U.S. Department of Education study…

  1. A new S-type eigenvalue inclusion set for tensors and its applications.

    PubMed

    Huang, Zheng-Ge; Wang, Li-Gong; Xu, Zhong; Cui, Jing-Jing

    2016-01-01

    In this paper, a new S -type eigenvalue localization set for a tensor is derived by dividing [Formula: see text] into disjoint subsets S and its complement. It is proved that this new set is sharper than those presented by Qi (J. Symb. Comput. 40:1302-1324, 2005), Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and Li et al. (Linear Algebra Appl. 481:36-53, 2015). As applications of the results, new bounds for the spectral radius of nonnegative tensors and the minimum H -eigenvalue of strong M -tensors are established, and we prove that these bounds are tighter than those obtained by Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and He and Huang (J. Inequal. Appl. 2014:114, 2014).

  2. The Impact of New State Accountability Standards on Algebra I Students

    ERIC Educational Resources Information Center

    Heath, Kyle G.

    2013-01-01

    The purpose of this quasi-experimental quantitative study was to determine if a new Algebra I curriculum resulted in improved student performance on the state Algebra I exam. The treatment group consisted of 383 9th grade Algebra I students who received the college-ready standards-based (CRSB) curricula. The control group consisted of 338 9th…

  3. Promoting Quantitative Literacy in an Online College Algebra Course

    ERIC Educational Resources Information Center

    Tunstall, Luke; Bossé, Michael J.

    2016-01-01

    College algebra (a university freshman level algebra course) fulfills the quantitative literacy requirement of many college's general education programs and is a terminal course for most who take it. An online problem-based learning environment provides a unique means of engaging students in quantitative discussions and research. This article…

  4. The Aftermath of Accelerating Algebra: Evidence from a District Policy Initiative. Working Paper 69

    ERIC Educational Resources Information Center

    Clotfelter, Charles T.; Ladd, Helen F.; Vigdor, Jacob L.

    2012-01-01

    In 2002/03, the Charlotte-Mecklenburg Schools initiated a broad program of accelerating entry into algebra coursework. The proportion of moderately-performing students taking 8th grade algebra increased from less than half to nearly 90%, then reverted to baseline levels, in the span of just six age cohorts. We use this policy-induced variation to…

  5. The Aftermath of Accelerating Algebra: Evidence from a District Policy Initiative. Working Paper 18161

    ERIC Educational Resources Information Center

    Clotfelter, Charles T.; Ladd, Helen F.; Vigdor, Jacob L.

    2012-01-01

    In 2002/03, the Charlotte-Mecklenburg Schools in North Carolina initiated a broad program of accelerating entry into algebra coursework. The proportion of moderately-performing students taking algebra in 8th grade increased from half to 85%, then reverted to baseline levels, in the span of just five years. We use this policy-induced variation to…

  6. Enhanced asymptotic symmetry algebra of (2 +1 ) -dimensional flat space

    NASA Astrophysics Data System (ADS)

    Detournay, Stéphane; Riegler, Max

    2017-02-01

    In this paper we present a new set of asymptotic boundary conditions for Einstein gravity in (2 +1 ) -dimensions with a vanishing cosmological constant that are a generalization of the Barnich-Compère boundary conditions [G. Barnich and G. Compere, Classical Quantum Gravity 24, F15 (2007), 10.1088/0264-9381/24/5/F01]. These new boundary conditions lead to an asymptotic symmetry algebra that is generated by a bms3 algebra and two affine u ^(1 ) current algebras. We then apply these boundary conditions to topologically massive gravity (TMG) and determine how the presence of the gravitational Chern-Simons term affects the central extensions of the asymptotic symmetry algebra. We furthermore determine the thermal entropy of solutions obeying our new boundary conditions for both Einstein gravity and TMG.

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

    NASA Astrophysics Data System (ADS)

    Roussel, Marc R.

    1999-10-01

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

  8. Boolean Operations with Prism Algebraic Patches

    PubMed Central

    Bajaj, Chandrajit; Paoluzzi, Alberto; Portuesi, Simone; Lei, Na; Zhao, Wenqi

    2009-01-01

    In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C1-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation. PMID:21516262

  9. Emphasizing language and visualization in teaching linear algebra

    NASA Astrophysics Data System (ADS)

    Hannah, John; Stewart, Sepideh; Thomas, Mike

    2013-06-01

    Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his approach in both lectures and tutorials, and how he employed visualization and an emphasis on language to encourage conceptual thinking. We use Tall's framework of three worlds of mathematical thinking to reflect on the effect of these activities in students' learning. An analysis of students' attitudes to the course and their test and examination results help to answer questions about the value of such an approach, suggesting ways forward in teaching linear algebra.

  10. Vector 33: A reduce program for vector algebra and calculus in orthogonal curvilinear coordinates

    NASA Astrophysics Data System (ADS)

    Harper, David

    1989-06-01

    This paper describes a package with enables REDUCE 3.3 to perform algebra and calculus operations upon vectors. Basic algebraic operations between vectors and between scalars and vectors are provided, including scalar (dot) product and vector (cross) product. The vector differential operators curl, divergence, gradient and Laplacian are also defined, and are valid in any orthogonal curvilinear coordinate system. The package is written in RLISP to allow algebra and calculus to be performed using notation identical to that for operations. Scalars and vectors can be mixed quite freely in the same expression. The package will be of interest to mathematicians, engineers and scientists who need to perform vector calculations in orthogonal curvilinear coordinates.

  11. Graph C ∗-algebras and Z2-quotients of quantum spheres

    NASA Astrophysics Data System (ADS)

    Hajac, Piotr M.; Matthes, Rainer; Szymański, Wojciech

    2003-06-01

    We consider two Z2-actions on the Podleś generic quantum spheres. They yield, as noncommutative quotient spaces, the Klimek-Lesmewski q-disc and the quantum real projective space, respectively. The C ∗-algebas of all these quantum spaces are described as graph C ∗-algebras. The K-groups of the thus presented C ∗-algebras are then easily determined from the general theory of graph C ∗-algebas. For the quantum real projective space, we also recall the classification of the classes of irreducible ∗-representations of its algebra and give a linear basis for this algebra.

  12. Birman—Wenzl—Murakami Algebra and Topological Basis

    NASA Astrophysics Data System (ADS)

    Zhou, Cheng-Cheng; Xue, Kang; Wang, Gang-Cheng; Sun, Chun-Fang; Du, Gui-Jiao

    2012-02-01

    In this paper, we use entangled states to construct 9 × 9-matrix representations of Temperley—Lieb algebra (TLA), then a family of 9 × 9-matrix representations of Birman—Wenzl—Murakami algebra (BWMA) have been presented. Based on which, three topological basis states have been found. And we apply topological basis states to recast nine-dimensional BWMA into its three-dimensional counterpart. Finally, we find the topological basis states are spin singlet states in special case.

  13. Vague Congruences and Quotient Lattice Implication Algebras

    PubMed Central

    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

  14. A Lie algebraic condition for exponential stability of discrete hybrid systems and application to hybrid synchronization.

    PubMed

    Zhao, Shouwei

    2011-06-01

    A Lie algebraic condition for global exponential stability of linear discrete switched impulsive systems is presented in this paper. By considering a Lie algebra generated by all subsystem matrices and impulsive matrices, when not all of these matrices are Schur stable, we derive new criteria for global exponential stability of linear discrete switched impulsive systems. Moreover, simple sufficient conditions in terms of Lie algebra are established for the synchronization of nonlinear discrete systems using a hybrid switching and impulsive control. As an application, discrete chaotic system's synchronization is investigated by the proposed method.

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

    ERIC Educational Resources Information Center

    Kysh, Judith

    1991-01-01

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

  16. Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras

    NASA Astrophysics Data System (ADS)

    Grahovski, Georgi G.; Mikhailov, Alexander V.

    2013-12-01

    Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.

  17. The Algebra Artist

    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®,…

  18. Abstract Algebra for Teachers: An Evaluative Case Study

    ERIC Educational Resources Information Center

    Hoffman, Andrew Joseph

    2017-01-01

    This manuscript describes the study of an abstract algebra course for preservice secondary mathematics teachers (PSMTs). Often, courses in abstract algebra have not been viewed as productive, beneficial learning experiences for future teachers, both by researchers and PSMTs themselves. This despite calls for increased content knowledge for…

  19. The Algebra Initiative Colloquium. Volume 2: Working Group Papers.

    ERIC Educational Resources Information Center

    Lacampagne, Carole B., Ed.; And Others

    This volume presents recommendations from four working groups at a conference on reform in algebra held in Leesburg, Virginia, December 9-12, 1993. Working Group 1: Creating an Appropriate Algebra Experience for All Grades K-12 Students produced the following papers: (1) "Report" (A. H. Schoenfeld); (2) "Five Questions About Algebra…

  20. An Algebraic Approach to the Eigenstates of the Calogero Model

    NASA Astrophysics Data System (ADS)

    Ujino, Hideaki

    2002-11-01

    An algebraic treatment of the eigenstates of the (AN-1-) Calogero model is presented, which provides an algebraic construction of the nonsymmetric orthogonal eigenvectors, symmetrization, antisymmetrization and calculation of square norms in a unified way.

  1. Advanced Mathematics Online: Assessing Particularities in the Online Delivery of a Second Linear Algebra Course

    ERIC Educational Resources Information Center

    Montiel, Mariana; Bhatti, Uzma

    2010-01-01

    This article presents an overview of some issues that were confronted when delivering an online second Linear Algebra course (assuming a previous Introductory Linear Algebra course) to graduate students enrolled in a Secondary Mathematics Education program. The focus is on performance in one particular aspect of the course: "change of basis" and…

  2. Comparing the utility of image algebra operations for characterizing landscape changes: the case of the Mediterranean coast.

    PubMed

    Alphan, Hakan

    2011-11-01

    The aim of this study is to compare various image algebra procedures for their efficiency in locating and identifying different types of landscape changes on the margin of a Mediterranean coastal plain, Cukurova, Turkey. Image differencing and ratioing were applied to the reflective bands of Landsat TM datasets acquired in 1984 and 2006. Normalized Difference Vegetation index (NDVI) and Principal Component Analysis (PCA) differencing were also applied. The resulting images were tested for their capacity to detect nine change phenomena, which were a priori defined in a three-level classification scheme. These change phenomena included agricultural encroachment, sand dune afforestation, coastline changes and removal/expansion of reed beds. The percentage overall accuracies of different algebra products for each phenomenon were calculated and compared. The results showed that some of the changes such as sand dune afforestation and reed bed expansion were detected with accuracies varying between 85 and 97% by the majority of the algebra operations, while some other changes such as logging could only be detected by mid-infrared (MIR) ratioing. For optimizing change detection in similar coastal landscapes, underlying causes of these changes were discussed and the guidelines for selecting band and algebra operations were provided. Copyright © 2011 Elsevier Ltd. All rights reserved.

  3. Global identifiability of linear compartmental models--a computer algebra algorithm.

    PubMed

    Audoly, S; D'Angiò, L; Saccomani, M P; Cobelli, C

    1998-01-01

    A priori global identifiability deals with the uniqueness of the solution for the unknown parameters of a model and is, thus, a prerequisite for parameter estimation of biological dynamic models. Global identifiability is however difficult to test, since it requires solving a system of algebraic nonlinear equations which increases both in nonlinearity degree and number of terms and unknowns with increasing model order. In this paper, a computer algebra tool, GLOBI (GLOBal Identifiability) is presented, which combines the topological transfer function method with the Buchberger algorithm, to test global identifiability of linear compartmental models. GLOBI allows for the automatic testing of a priori global identifiability of general structure compartmental models from general multi input-multi output experiments. Examples of usage of GLOBI to analyze a priori global identifiability of some complex biological compartmental models are provided.

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

    ERIC Educational Resources Information Center

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

    2012-01-01

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

  5. Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*

    DOE PAGES

    Bank, R.; Falgout, R. D.; Jones, T.; ...

    2015-10-29

    In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods inmore » Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.« less

  6. Some Applications Of Semigroups And Computer Algebra In Discrete Structures

    NASA Astrophysics Data System (ADS)

    Bijev, G.

    2009-11-01

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

  7. Quantum deformations of conformal algebras with mass-like deformation parameters

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

    Frydryszak, Andrzej; Lukierski, Jerzy; Mozrzymas, Marek

    1998-12-15

    We recall the mathematical apparatus necessary for the quantum deformation of Lie algebras, namely the notions of coboundary Lie algebras, classical r-matrices, classical Yang-Baxter equations (CYBE), Froebenius algebras and parabolic subalgebras. Then we construct the quantum deformation of D=1, D=2 and D=3 conformal algebras, showing that this quantization introduce fundamental mass parameters. Finally we consider with more details the quantization of D=4 conformal algebra. We build three classes of sl(4,C) classical r-matrices, satisfying CYBE and depending respectively on 8, 10 and 12 generators of parabolic subalgebras. We show that only the 8-dimensional r-matrices allow to impose the D=4 conformal o(4,2){approx_equal}su(2,2)more » reality conditions. Weyl reflections and Dynkin diagram automorphisms for o(4,2) define the class of admissible bases for given classical r-matrices.« less

  8. Green synthesis of Pd/CuO nanoparticles by Theobroma cacao L. seeds extract and their catalytic performance for the reduction of 4-nitrophenol and phosphine-free Heck coupling reaction under aerobic conditions.

    PubMed

    Nasrollahzadeh, Mahmoud; Sajadi, S Mohammad; Rostami-Vartooni, Akbar; Bagherzadeh, Mojtaba

    2015-06-15

    We report the green synthesis of palladium/CuO nanoparticles (Pd/CuO NPs) using Theobroma cacao L. seeds extract and their catalytic activity for the reduction of 4-nitrophenol and Heck coupling reaction under aerobic conditions. The catalyst was characterized using the powder XRD, TEM, EDS, UV-vis and FT-IR. This method has the advantages of high yields, elimination of surfactant, ligand and homogeneous catalysts, simple methodology and easy work up. The catalyst can be recovered from the reaction mixture and reused several times without any significant loss of catalytic activity. Copyright © 2015 Elsevier Inc. All rights reserved.

  9. ADAM: analysis of discrete models of biological systems using computer algebra.

    PubMed

    Hinkelmann, Franziska; Brandon, Madison; Guang, Bonny; McNeill, Rustin; Blekherman, Grigoriy; Veliz-Cuba, Alan; Laubenbacher, Reinhard

    2011-07-20

    Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, to gain a better understanding of them. The computational complexity to analyze the complete dynamics of these models grows exponentially in the number of variables, which impedes working with complex models. There exist software tools to analyze discrete models, but they either lack the algorithmic functionality to analyze complex models deterministically or they are inaccessible to many users as they require understanding the underlying algorithm and implementation, do not have a graphical user interface, or are hard to install. Efficient analysis methods that are accessible to modelers and easy to use are needed. We propose a method for efficiently identifying attractors and introduce the web-based tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other analysis methods for discrete models. ADAM converts several discrete model types automatically into polynomial dynamical systems and analyzes their dynamics using tools from computer algebra. Specifically, we propose a method to identify attractors of a discrete model that is equivalent to solving a system of polynomial equations, a long-studied problem in computer algebra. Based on extensive experimentation with both discrete models arising in systems biology and randomly generated networks, we found that the algebraic algorithms presented in this manuscript are fast for systems with the structure maintained by most biological systems, namely sparseness and robustness. For a large set of published complex discrete models, ADAM identified the attractors in less than one second. Discrete modeling techniques are a useful tool for analyzing complex biological systems and there is a need in the biological community for accessible efficient analysis tools. ADAM provides analysis methods based on mathematical algorithms as a web

  10. ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra

    PubMed Central

    2011-01-01

    Background Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, to gain a better understanding of them. The computational complexity to analyze the complete dynamics of these models grows exponentially in the number of variables, which impedes working with complex models. There exist software tools to analyze discrete models, but they either lack the algorithmic functionality to analyze complex models deterministically or they are inaccessible to many users as they require understanding the underlying algorithm and implementation, do not have a graphical user interface, or are hard to install. Efficient analysis methods that are accessible to modelers and easy to use are needed. Results We propose a method for efficiently identifying attractors and introduce the web-based tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other analysis methods for discrete models. ADAM converts several discrete model types automatically into polynomial dynamical systems and analyzes their dynamics using tools from computer algebra. Specifically, we propose a method to identify attractors of a discrete model that is equivalent to solving a system of polynomial equations, a long-studied problem in computer algebra. Based on extensive experimentation with both discrete models arising in systems biology and randomly generated networks, we found that the algebraic algorithms presented in this manuscript are fast for systems with the structure maintained by most biological systems, namely sparseness and robustness. For a large set of published complex discrete models, ADAM identified the attractors in less than one second. Conclusions Discrete modeling techniques are a useful tool for analyzing complex biological systems and there is a need in the biological community for accessible efficient analysis tools. ADAM provides analysis methods based on

  11. The noncommutative Poisson bracket and the deformation of the family algebras

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

    Wei, Zhaoting, E-mail: zhaotwei@indiana.edu

    The family algebras are introduced by Kirillov in 2000. In this paper, we study the noncommutative Poisson bracket P on the classical family algebra C{sub τ}(g). We show that P controls the first-order 1-parameter formal deformation from C{sub τ}(g) to Q{sub τ}(g) where the latter is the quantum family algebra. Moreover, we will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary, and therefore, the deformation is infinitesimally trivial. In the last part of this paper, we discuss the relation between Mackey’s analogue and the quantization problem of the family algebras.

  12. Chiral algebras in Landau-Ginzburg models

    NASA Astrophysics Data System (ADS)

    Dedushenko, Mykola

    2018-03-01

    Chiral algebras in the cohomology of the {\\overline{Q}}+ supercharge of two-dimensional N=(0,2) theories on flat spacetime are discussed. Using the supercurrent multiplet, we show that the answer is renormalization group invariant for theories with an R-symmetry. For N=(0,2) Landau-Ginzburg models, the chiral algebra is determined by the operator equations of motion, which preserve their classical form, and quantum renormalization of composite operators. We study these theories and then specialize to the N=(2,2) models and consider some examples.

  13. Optical systolic solutions of linear algebraic equations

    NASA Technical Reports Server (NTRS)

    Neuman, C. P.; Casasent, D.

    1984-01-01

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

  14. Algebraic Concepts: What's Really New in New Curricula?

    ERIC Educational Resources Information Center

    Star, Jon R.; Herbel-Eisenmann, Beth A.; Smith, John P., III

    2000-01-01

    Examines 8th grade units from the Connected Mathematics Project (CMP). Identifies differences in older and newer conceptions, fundamental objects of study, typical problems, and typical solution methods in algebra. Also discusses where the issue of what is new in algebra is relevant to many other innovative middle school curricula. (KHR)

  15. Mastering algebra retrains the visual system to perceive hierarchical structure in equations.

    PubMed

    Marghetis, Tyler; Landy, David; Goldstone, Robert L

    2016-01-01

    Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual system-in particular, object-based attention-is retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressions-but only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception.

  16. A new application of algebraic geometry to systems theory

    NASA Technical Reports Server (NTRS)

    Martin, C. F.; Hermann, R.

    1976-01-01

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

  17. College Algebra Students' Attitudes toward Mathematics in Their Careers

    ERIC Educational Resources Information Center

    Champion, Joe; Parker, Frieda; Mendoza-Spencer, Bernadette; Wheeler, Ann

    2011-01-01

    The purpose of this study was to identify the degree to which college algebra students' value mathematical skills in their prospective careers. A survey was administered to N = 144 students in 6 college algebra classes at a mid-sized doctoral granting university. Students in half the classes completed a data analysis project, and half of the…

  18. Ada Linear-Algebra Program

    NASA Technical Reports Server (NTRS)

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

    1988-01-01

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

  19. 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.

  20. Quantum walks, deformed relativity and Hopf algebra symmetries

    PubMed Central

    2016-01-01

    We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014 Phys. Rev. A 90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras–the usual Poincaré and the κ-Poincaré algebras. PMID:27091171