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Sample records for algebraic normal form

  1. Novikov algebras with associative bilinear forms

    NASA Astrophysics Data System (ADS)

    Zhu, Fuhai; Chen, Zhiqi

    2007-11-01

    Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. The goal of this paper is to study Novikov algebras with non-degenerate associative symmetric bilinear forms, which we call quadratic Novikov algebras. Based on the classification of solvable quadratic Lie algebras of dimension not greater than 4 and Novikov algebras in dimension 3, we show that quadratic Novikov algebras up to dimension 4 are commutative. Furthermore, we obtain the classification of transitive quadratic Novikov algebras in dimension 4. But we find that not every quadratic Novikov algebra is commutative and give a non-commutative quadratic Novikov algebra in dimension 6.

  2. Bilinear forms on fermionic Novikov algebras

    NASA Astrophysics Data System (ADS)

    Chen, Zhiqi; Zhu, Fuhai

    2007-05-01

    Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian super-operator in a super-variable. In this paper, we show that there is a remarkable geometry on fermionic Novikov algebras with non-degenerate invariant symmetric bilinear forms, which we call pseudo-Riemannian fermionic Novikov algebras. They are related to pseudo-Riemannian Lie algebras. Furthermore, we obtain a procedure to classify pseudo-Riemannian fermionic Novikov algebras. As an application, we give the classification in dimension <=4. Motivated by the one in dimension 4, we construct some examples in high dimensions.

  3. A new algebra core for the minimal form' problem

    SciTech Connect

    Purtill, M.R. . Center for Communications Research); Oliveira, J.S.; Cook, G.O. Jr. )

    1991-12-20

    The demands of large-scale algebraic computation have led to the development of many new algorithms for manipulating algebraic objects in computer algebra systems. For instance, parallel versions of many important algorithms have been discovered. Simultaneously, more effective symbolic representations of algebraic objects have been sought. Also, while some clever techniques have been found for improving the speed of the algebraic simplification process, little attention has been given to the issue of restructuring expressions, or transforming them into minimal forms.'' By minimal form,'' we mean that form of an expression that involves a minimum number of operations. In a companion paper, we introduce some new algorithms that are very effective at finding minimal forms of expressions. These algorithms require algebraic and combinatorial machinery that is not readily available in most algebra systems. In this paper we describe a new algebra core that begins to provide the necessary capabilities.

  4. Constructing Involutive Tableaux with Guillemin Normal Form

    NASA Astrophysics Data System (ADS)

    Smith, Abraham D.

    2015-07-01

    Involutivity is the algebraic property that guarantees solutions to an analytic and torsion-free exterior differential system or partial differential equation via the Cartan-Kähler theorem. Guillemin normal form establishes that the prolonged symbol of an involutive system admits a commutativity property on certain subspaces of the prolonged tableau. This article examines Guillemin normal form in detail, aiming at a more systematic approach to classifying involutive systems. The main result is an explicit quadratic condition for involutivity of the type suggested but not completed in Chapter IV, § 5 of the book Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths. This condition enhances Guillemin normal form and characterizes involutive tableaux.

  5. Almost split real forms for hyperbolic Kac Moody Lie algebras

    NASA Astrophysics Data System (ADS)

    Ben Messaoud, Hechmi

    2006-11-01

    A Borel Tits theory was developed for almost split forms of symmetrizable Kac Moody Lie algebras. In this paper, we look to almost split real forms and their restricted root systems for symmetrizable hyperbolic Kac Moody Lie algebras. We establish a complete list of these forms, in terms of their Satake Tits index, for the strictly hyperbolic ones and for those which are obtained as (hyperbolic) canonical Lorentzian extensions of affine Lie algebras. These forms are of particular interest in theoretical physics because of their connection to supergravity theories.

  6. Diagonalization and Jordan Normal Form--Motivation through "Maple"[R

    ERIC Educational Resources Information Center

    Glaister, P.

    2009-01-01

    Following an introduction to the diagonalization of matrices, one of the more difficult topics for students to grasp in linear algebra is the concept of Jordan normal form. In this note, we show how the important notions of diagonalization and Jordan normal form can be introduced and developed through the use of the computer algebra package…

  7. Box products in nilpotent normal form theory: The factoring method

    NASA Astrophysics Data System (ADS)

    Murdock, James

    2016-01-01

    Let N be a nilpotent matrix and consider vector fields x ˙ = Nx + v (x) in normal form. Then v is equivariant under the flow eN*t for the inner product normal form or eMt for the sl2 normal form. These vector equivariants can be found by finding the scalar invariants for the Jordan blocks in N* or M; taking the box product of these to obtain the invariants for N* or M itself; and then boosting the invariants to equivariants by another box product. These methods, developed by Murdock and Sanders in 2007, are here given a self-contained exposition with new foundations and new algorithms yielding improved (simpler) Stanley decompositions for the invariants and equivariants. Ideas used include transvectants (from classical invariant theory), Stanley decompositions (from commutative algebra), and integer cones (from integer programming). This approach can be extended to covariants of sl2k for k > 1, known as SLOCC in quantum computing.

  8. Reusable Software Component Retrieval via Normalized Algebraic Specifications

    DTIC Science & Technology

    1991-12-01

    SOFTWARE COMPONENT RETRIEVAL VIA NORMALUZED ALGEBRAIC SPECIFICATIONS by Robert Allen Steigerwald Captain , United States Air Force B.S., United States...requirements. Aooession For NTIS GRA&I DTIC TAB 0 Unannounced fl Justifitatio By Avat lability Codes avail ead/or DLllt |Speblal iii TABLE OF CONTENTS

  9. Pancreastatin molecular forms in normal human plasma.

    PubMed

    Kitayama, N; Tateishi, K; Funakoshi, A; Miyasaka, K; Shimazoe, T; Kono, A; Iwamoto, N; Matsuoka, Y

    1994-01-01

    Circulating molecular forms with pancreastatin (PST)-like immunoreactivity in plasma from normal subjects were examined. An immunoreactive form corresponding to a human PST-like sequence [human chromogranin-A-(250-301)] (hPST-52) and a larger form (mol wt 15-21 kDa) were detected by gel filtration of plasma from normal subjects. On high performance liquid chromatography, predominant immunoreactive forms coeluted with the three larger forms which were purified from the xenograft of human pancreatic islet cell carcinoma cell line QGP-1N cells and with synthetic hPST-52. The fraction containing larger forms purified from xenograft of QGP-1N cells had biological activity equivalent to that of hPST-52 on the inhibition of pancreatic exocrine secretion. These results suggest that the larger molecular forms as well as hPST-52 may be physiologically important circulating forms of PST in human.

  10. Algebraic approach to form factors in the complex sinh-Gordon theory

    NASA Astrophysics Data System (ADS)

    Lashkevich, Michael; Pugai, Yaroslav

    2017-01-01

    We study form factors of the quantum complex sinh-Gordon theory in the algebraic approach. In the case of exponential fields the form factors can be obtained from the known form factors of the ZN-symmetric Ising model. The algebraic construction also provides an Ansatz for form factors of descendant operators. We obtain generating functions of such form factors and establish their main properties: the cluster factorization and reflection equations.

  11. A Cohomological Proof that Real Representations of Semisimple Lie Algebras Have Q-Forms

    NASA Astrophysics Data System (ADS)

    Morris, Dave Witte

    2015-04-01

    A Lie algebra g_Q over Q is said to be R-universal if every homomorphism from g_Q to gl(n,R) is conjugate to a homomorphism into gl(n,Q) (for every n). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an R-universal Q-form. We also provide a classification of the R-universal Lie algebras that are semisimple.

  12. Birkhoff Normal Form for Some Nonlinear PDEs

    NASA Astrophysics Data System (ADS)

    Bambusi, Dario

    We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear wave equation FORM="DISPLAY" DISC="MATH"> with Dirichlet boundary conditions on [0,π] g is an analytic skewsymmetric function which vanishes for u=0 and is periodic with period 2π in the x variable. We prove, under a nonresonance condition which is fulfilled for most g's, that for any integer M there exists a canonical transformation that puts the Hamiltonian in Birkhoff normal form up to a reminder of order M. The canonical transformation is well defined in a neighbourhood of the origin of a Sobolev type phase space of sufficiently high order. Some dynamical consequences are obtained. The technique of proof is applicable to quite general semilinear equations in one space dimension.

  13. Normal Forms for Nonautonomous Differential Equations

    NASA Astrophysics Data System (ADS)

    Siegmund, Stefan

    2002-01-01

    We extend Henry Poincarés normal form theory for autonomous differential equations x=f(x) to nonautonomous differential equations x=f(t, x). Poincarés nonresonance condition λj-∑ni=1 ℓiλi≠0 for eigenvalues is generalized to the new nonresonance condition λj∩∑ni=1 ℓiλi=∅ for spectral intervals.

  14. 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.

  15. ADDENDUM: The classification of Novikov algebras in low dimensions: invariant bilinear forms

    NASA Astrophysics Data System (ADS)

    Bai, Chengming; Meng, Daoji

    2001-10-01

    In this note, we give a complete classification of the (non-degenerate) symmetric invariant bilinear forms on Novikov algebras in dimension 2 and 3, which can be regarded as an addendum of the classification of Novikov algebras in low dimensions given in our previous work (Bai C M and Meng D J 2001 J. Phys. A: Math. Gen. 34 1581-94).

  16. An Error Analysis of Form 2 (Grade 7) Students in Simplifying Algebraic Expressions: A Descriptive Study

    ERIC Educational Resources Information Center

    Lim, Kok Seng

    2010-01-01

    Introduction: This study aimed to investigate the errors made by 265 Form 2 male students in simplifying algebraic expressions. Method: A total of 265 Form 2 (Grade 7) male students were selected for this study. 10 high, medium and low ability students in each group were selected for the interviews. 40 items were administered to the respondents to…

  17. The algebra of observables in Gaußian normal spacetime coordinates

    NASA Astrophysics Data System (ADS)

    Bodendorfer, Norbert; Duch, Paweł; Lewandowski, Jerzy; Świeżewski, Jędrzej

    2016-01-01

    We discuss the canonical structure of a spacetime version of the radial gauge, i.e. Gaußian normal spacetime coordinates. While it was found for the spatial version of the radial gauge that a "local" algebra of observables can be constructed, it turns out that this is not possible for the spacetime version. The technical reason for this observation is that the new gauge condition needed to upgrade the spatial to a spacetime radial gauge does not Poisson-commute with the previous gauge conditions. It follows that the involved Dirac bracket is inherently non-local in the sense that no complete set of observables can be found which is constructed locally and at the same time has local Dirac brackets. A locally constructed observable here is defined as a finite polynomial of the canonical variables at a given physical point specified by the Gaußian normal spacetime coordinates.

  18. Symbolic computation of the Birkhoff normal form in the problem of stability of the triangular libration points

    NASA Astrophysics Data System (ADS)

    Shevchenko, I. I.

    2008-05-01

    The problem of stability of the triangular libration points in the planar circular restricted three-body problem is considered. A software package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is designed so that normalization problems of high analytical complexity could be solved. It is used to obtain the Birkhoff normal form of the Hamiltonian in the given problem. The normalization is carried out up to the 6th order of expansion of the Hamiltonian in the coordinates and momenta. Analytical expressions for the coefficients of the normal form of the 6th order are derived. Though intermediary expressions occupy gigabytes of the computer memory, the obtained coefficients of the normal form are compact enough for presentation in typographic format. The analogue of the Deprit formula for the stability criterion is derived in the 6th order of normalization. The obtained floating-point numerical values for the normal form coefficients and the stability criterion confirm the results by Markeev (1969) and Coppola and Rand (1989), while the obtained analytical and exact numeric expressions confirm the results by Meyer and Schmidt (1986) and Schmidt (1989). The given computational problem is solved without constructing a specialized algebraic processor, i.e., the designed computer algebra package has a broad field of applicability.

  19. One Teacher's Dilemma in Mediating Translation from Written to Symbolic Form in a Multilingual Algebra Classroom

    ERIC Educational Resources Information Center

    Essien, Anthony A.

    2011-01-01

    This study investigated how a teacher in a multilingual classroom attempted to support learners who are struggling to translate written/verbal mathematics into a symbolic form. Thirty-six Grade ten learners in one multilingual classroom in South Africa were given a written test involving one algebraic question and then a discussion on the solution…

  20. Applications of the DA based normal form algorithm on parameter-dependent perturbations

    NASA Astrophysics Data System (ADS)

    Weisskopf, Adrian

    Many advanced models in physics use a simpler system as the foundation upon which problemspecific perturbation terms are added. There are many mathematical methods in perturbation theory which attempt to solve or at least approximate the solution for the advanced model based on the solution of the unperturbed system. The analytical approaches have the advantage that their approximation is an algebraic expression relating all involved quantities in the calculated solution up to a certain order. However, the complexity of the calculation often increases drastically with the number of iterations, variables, and parameters considered. On the other hand, the computer-based numerical approaches are fast once implemented, but their results are only numerical approximations without a symbolic form. A numerical integrator, for example, takes the initial values and integrates the ordinary differential equation up to the requested final state and yields the result as specific numbers. Therefore, no algebraic expression, much less a parameter dependence within the solution is given. The method presented in this work is based on the differential algebra (DA) framework, which was first developed to its current extent by Martin Berz et. al [3, 4, 5]. The used DA Normal Form Algorithm is an advancement by Martin Berz from the first arbitrary order algorithm by Forest, Berz, and Irwin [13], which was based on an DA-Lie approach. Both structures are already implemented in COSY INFINITY [18] documented in [7, 16, 17]. The result of the presented method is a numerically calculated algebraic expression of the solution up to an arbitrary truncation order. This method combines the effectiveness and automatic calculation of a computer-based numerical approximation and the algebraic relation between the involved quantities.

  1. Early universe cosmology, effective supergravity, and invariants of algebraic forms

    NASA Astrophysics Data System (ADS)

    Sinha, Kuver

    2015-09-01

    The presence of light scalars can have profound effects on early universe cosmology, influencing its thermal history as well as paradigms like inflation and baryogenesis. Effective supergravity provides a framework to make quantifiable, model-independent studies of these effects. The Riemannian curvature of the Kähler manifold spanned by scalars belonging to chiral superfields, evaluated along supersymmetry breaking directions, provides an order parameter (in the sense that it must necessarily take certain values) for phenomena as diverse as slow roll modular inflation, nonthermal cosmological histories, and the viability of Affleck-Dine baryogenesis. Within certain classes of UV completions, the order parameter for theories with n scalar moduli is conjectured to be related to invariants of n -ary cubic forms (for example, for models with three moduli, the order parameter is given by a function on the ring of invariants spanned by the Aronhold invariants). Within these completions, and under the caveats spelled out, this may provide an avenue to obtain necessary conditions for the above phenomena that are in principle calculable given nothing but the intersection numbers of a Calabi-Yau compactification geometry. As an additional result, abstract relations between holomorphic sectional and bisectional curvatures are utilized to constrain Affleck-Dine baryogenesis on a wide class of Kähler geometries.

  2. The most general form of deformation of the Heisenberg algebra from the generalized uncertainty principle

    NASA Astrophysics Data System (ADS)

    Masood, Syed; Faizal, Mir; Zaz, Zaid; Ali, Ahmed Farag; Raza, Jamil; Shah, Mushtaq B.

    2016-12-01

    In this paper, we will propose the most general form of the deformation of Heisenberg algebra motivated by the generalized uncertainty principle. This deformation of the Heisenberg algebra will deform all quantum mechanical systems. The form of the generalized uncertainty principle used to motivate these results will be motivated by the space fractional quantum mechanics, and non-locality in quantum mechanical systems. We also analyse a specific limit of this generalized deformation for one dimensional system, and in that limit, a nonlocal deformation of the momentum operator generates a local deformation of all one dimensional quantum mechanical systems. We analyse the low energy effects of this deformation on a harmonic oscillator, Landau levels, Lamb shift, and potential barrier. We also demonstrate that this deformation leads to a discretization of space.

  3. Electron spectral function and algebraic spin liquid for the normal state of underdoped high T(c) superconductors.

    PubMed

    Rantner, W; Wen, X G

    2001-04-23

    We propose to describe the spin fluctuations in the normal state (spin-pseudogap phase) of underdoped high T(c) cuprates as a manifestation of an algebraic spin liquid. Within the slave boson implementation of spin-charge separation, the normal state is described by massless Dirac fermions, charged bosons, and a gauge field. The gauge interaction, as an exact marginal perturbation, drives the mean-field free-spinon fixed point to a new spin-quantum fixed point-the algebraic spin liquid. Luttinger-liquid-like line shapes for the electron spectral function are obtained in the normal state, and we show how a coherent quasiparticle peak appears as spin and charge recombine.

  4. The use of normal forms for analysing nonlinear mechanical vibrations

    PubMed Central

    Neild, Simon A.; Champneys, Alan R.; Wagg, David J.; Hill, Thomas L.; Cammarano, Andrea

    2015-01-01

    A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations. PMID:26303917

  5. Trojan dynamics well approximated by a new Hamiltonian normal form

    NASA Astrophysics Data System (ADS)

    Páez, Rocío Isabel; Locatelli, Ugo

    2015-10-01

    We revisit a classical perturbative approach to the Hamiltonian related to the motions of Trojan bodies, in the framework of the planar circular restricted three-body problem, by introducing a number of key new ideas in the formulation. In some sense, we adapt the approach of Garfinkel to the context of the normal form theory and its modern techniques. First, we make use of Delaunay variables for a physically accurate representation of the system. Therefore, we introduce a novel manipulation of the variables so as to respect the natural behaviour of the model. We develop a normalization procedure over the fast angle which exploits the fact that singularities in this model are essentially related to the slow angle. Thus, we produce a new normal form, i.e. an integrable approximation to the Hamiltonian. We emphasize some practical examples of the applicability of our normalizing scheme, e.g. the estimation of the stable libration region. Finally, we compare the level curves produced by our normal form with surfaces of section provided by the integration of the non-normalized Hamiltonian, with very good agreement. Further precision tests are also provided. In addition, we give a step-by-step description of the algorithm, allowing for extensions to more complicated models.

  6. ηc elastic and transition form factors: Contact interaction and algebraic model

    NASA Astrophysics Data System (ADS)

    Bedolla, Marco A.; Raya, Khépani; Cobos-Martínez, J. J.; Bashir, Adnan

    2016-05-01

    For the flavor-singlet heavy-quark system of charmonia in the pseudoscalar [ηc(1 S ) ] channel, we calculate the elastic (EFF) and transition form factors (TFFs) [ηc(1 S )→γ γ* ] for a wide range of photon momentum transfer squared (Q2). The framework for this analysis is provided by a symmetry-preserving Schwinger-Dyson equation and Bethe-Salpeter equation treatment of a vector×vector contact interaction. We also employ an algebraic model, developed earlier to describe the light-quark systems. It correctly correlates infrared and ultraviolet dynamics of quantum chromodynamics (QCD). The contact interaction results agree with the lattice data for low Q2. For Q2≥Q02 , the results start deviating from the lattice results by more than 20%. Q02≈2.5 GeV2 for the EFF, and ≈25 GeV2 for the TFF. We also present the results for the EFF, TFF, and ηc(1 S ) parton distribution amplitude for the algebraic model. Wherever the comparison is possible, these results are in excellent agreement with the lattice, perturbative QCD, results obtained through a Schwinger-Dyson equation-Bethe-Salpeter equation study, employing refined truncations, and the experimental findings of the BABAR experiment.

  7. Cotangent bundle reduction and Poincaré-Birkhoff normal forms

    NASA Astrophysics Data System (ADS)

    Çiftçi, Ünver; Waalkens, Holger; Broer, Henk W.

    2014-02-01

    In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincaré-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincaré-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.

  8. Breather solutions for inhomogeneous FPU models using Birkhoff normal forms

    NASA Astrophysics Data System (ADS)

    Martínez-Farías, Francisco; Panayotaros, Panayotis

    2016-11-01

    We present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of Fermi-Pasta-Ulam (FPU) type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations. The main feature of the FPU lattices we consider is that the number of interacting neighbors varies from site to site, and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem. This property is seen in 1-D models, and in a 3-D model with a geometry obtained from protein data. The spectral analysis of these examples suggests some non-resonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic Birkhoff normal forms of the FPU systems. The invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way.

  9. Normal form analysis of a forced aeroelastic plate

    NASA Astrophysics Data System (ADS)

    Eugeni, Marco; Mastroddi, Franco; Dowell, Earl H.

    2017-03-01

    A nonlinear elastic plate in a supersonic unsteady flow forced by a dynamic excitation and a biaxial compressive load is studied. The physical behavior of the plate is modelized by the Von Kármán equations and the aerodynamic loads are modeled by using the piston theory including nonlinearities up to the third order. The space-continuum model is space-discretized by a Galerkin projection and then studied by a perturbation approach based on the Normal Form method in order to reduce the system to a simpler and essential form defined by its resonance conditions. A physical interpretation of the involved small divisors is given by analyzing how different equation parameters influence the reduced normal form model in the neighborhood of both static and dynamic bifurcation points.

  10. Automatic identification and normalization of dosage forms in drug monographs

    PubMed Central

    2012-01-01

    Background Each day, millions of health consumers seek drug-related information on the Web. Despite some efforts in linking related resources, drug information is largely scattered in a wide variety of websites of different quality and credibility. Methods As a step toward providing users with integrated access to multiple trustworthy drug resources, we aim to develop a method capable of identifying drug's dosage form information in addition to drug name recognition. We developed rules and patterns for identifying dosage forms from different sections of full-text drug monographs, and subsequently normalized them to standardized RxNorm dosage forms. Results Our method represents a significant improvement compared with a baseline lookup approach, achieving overall macro-averaged Precision of 80%, Recall of 98%, and F-Measure of 85%. Conclusions We successfully developed an automatic approach for drug dosage form identification, which is critical for building links between different drug-related resources. PMID:22336431

  11. Syntax and Meaning as Sensuous, Visual, Historical Forms of Algebraic Thinking

    ERIC Educational Resources Information Center

    Radford, Luis; Puig, Luis

    2007-01-01

    Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and…

  12. Using Graphing Software to Teach about Algebraic Forms: A Study of Technology-Supported Practice in Secondary-School Mathematics

    ERIC Educational Resources Information Center

    Ruthven, Kenneth; Deaney, Rosemary; Hennessy, Sara

    2009-01-01

    From preliminary analysis of teacher-nominated examples of successful technology-supported practice in secondary-school mathematics, the use of graphing software to teach about algebraic forms was identified as being an important archetype. Employing evidence from lesson observation and teacher interview, such practice was investigated in greater…

  13. Contributions of Domain-General Cognitive Resources and Different Forms of Arithmetic Development to Pre-Algebraic Knowledge

    ERIC Educational Resources Information Center

    Fuchs, Lynn S.; Compton, Donald L.; Fuchs, Douglas; Powell, Sarah R.; Schumacher, Robin F.; Hamlett, Carol L.; Vernier, Emily; Namkung, Jessica M.; Vukovic, Rose K.

    2012-01-01

    The purpose of this study was to investigate the contributions of domain-general cognitive resources and different forms of arithmetic development to individual differences in pre-algebraic knowledge. Children (n = 279, mean age = 7.59 years) were assessed on 7 domain-general cognitive resources as well as arithmetic calculations and word problems…

  14. Contributions of Domain-General Cognitive Resources and Different Forms of Arithmetic Development to Pre-Algebraic Knowledge

    PubMed Central

    Fuchs, Lynn S.; Compton, Donald L.; Fuchs, Douglas; Powell, Sarah R.; Schumacher, Robin F.; Hamlett, Carol L.; Vernier, Emily; Namkung, Jessica M.; Vukovic, Rose K.

    2012-01-01

    The purpose of this study was to investigate the contributions of domain-general cognitive resources and different forms of arithmetic development to individual differences in pre-algebraic knowledge. Children (n=279; mean age=7.59 yrs) were assessed on 7 domain-general cognitive resources as well as arithmetic calculations and word problems at start of 2nd grade and on calculations, word problems, and pre-algebraic knowledge at end of 3rd grade. Multilevel path analysis, controlling for instructional effects associated with the sequence of classrooms in which students were nested across grades 2–3, indicated arithmetic calculations and word problems are foundational to pre-algebraic knowledge. Also, results revealed direct contributions of nonverbal reasoning and oral language to pre-algebraic knowledge, beyond indirect effects that are mediated via arithmetic calculations and word problems. By contrast, attentive behavior, phonological processing, and processing speed contributed to pre-algebraic knowledge only indirectly via arithmetic calculations and word problems. PMID:22409764

  15. Shocks and metallicity gradients in normal star-forming galaxies

    NASA Astrophysics Data System (ADS)

    Ho, I.-Ting

    Gas flow is one of the most fundamental processes driving galaxy evolution. This thesis explores gas flows in local galaxies by studying metallicity gradients and galactic-scale outflows in normal star-forming galaxies. This is made possible by new integral field spectroscopy data that provide simultaneously spatial and spectral information of galaxies. First, I measure metallicity gradients in isolated disk galaxies and show that their metallicity gradients are remarkably simple and universal. When the metallicity gradients are normalized to galaxy sizes, all the 49 galaxies studied have virtually the same metallicity gradient. I model the common metallicity gradient using a simple chemical evolution model to understand its origin. The common metallicity gradient is a direct result of the coevolution of gas and stellar disk while galactic disks build up their masses from inside-out. Tight constraints on the mass outflow rates and inflow rates can be placed by the chemical evolution model. Second, I investigate galactic winds in normal star-forming galaxies using data from an integral field spectroscopy survey. I demonstrate how to search for galactic winds by probing emission line ratios, shocks, and gas kinematics. Galactic winds are found to be common even in normal star-forming galaxies that were not expected to host winds. By comparing galaxies with and without hosting winds, I show that galaxies with high star formation rate surface densities and bursty star formation histories are more likely to drive large-scale galactic winds. Finally, lzifu, a toolkit for fitting multiple emission lines simultaneously in integral field spectroscopy data, is developed in this thesis. I describe in detail the structure of the toolkit and demonstrate the capabilities of lzifu.

  16. A new quantum scheme for normal-form games

    NASA Astrophysics Data System (ADS)

    Fraçkiewicz, Piotr

    2015-06-01

    We give a strict mathematical description for a refinement of the Marinatto-Weber quantum game scheme. The model allows the players to choose projector operators that determine the state on which they perform their local operators. The game induced by the scheme generalizes finite strategic-form game. In particular, it covers normal representations of extensive games, i.e., strategic games generated by extensive ones. We illustrate our idea with an example of extensive game and prove that rational choices in the classical game and its quantum counterpart may lead to significantly different outcomes.

  17. Normal forms and gauge symmetry of local dynamics

    NASA Astrophysics Data System (ADS)

    Lyakhovich, S. L.; Sharapov, A. A.

    2009-08-01

    A systematic procedure is proposed for deriving all the gauge symmetries of the general, not necessarily variational, equations of motion. For the variational equations, this procedure reduces to the Dirac-Bergmann algorithm for the constrained Hamiltonian systems with certain extension: it remains applicable beyond the scope of Dirac's conjecture. Even though no pairing exists between the constraints and the gauge symmetry generators in general nonvariational dynamics, certain counterparts still can be identified of the first- and second-class constraints without appealing to any Poisson structure. It is shown that the general local gauge dynamics can be equivalently reformulated in an involutive normal form. The last form of dynamics always admits the BRST embedding, which does not require the classical equations to follow from any variational principle.

  18. Efficient linear algebra routines for symmetric matrices stored in packed form.

    PubMed

    Ahlrichs, Reinhart; Tsereteli, Kakha

    2002-01-30

    Quantum chemistry methods require various linear algebra routines for symmetric matrices, for example, diagonalization or Cholesky decomposition for positive matrices. We present a small set of these basic routines that are efficient and minimize memory requirements.

  19. A Deterministic Interfacial Cyclic Oxidation Spalling Model. Part 2; Algebraic Approximation, Descriptive Parameters, and Normalized Universal Curve

    NASA Technical Reports Server (NTRS)

    Smialek, James L.

    2002-01-01

    A cyclic oxidation interfacial spalling model has been developed in Part 1. The governing equations have been simplified here by substituting a new algebraic expression for the series (Good-Smialek approximation). This produced a direct relationship between cyclic oxidation weight change and model input parameters. It also allowed for the mathematical derivation of various descriptive parameters as a function of the inputs. It is shown that the maximum in weight change varies directly with the parabolic rate constant and cycle duration and inversely with the spall fraction, all to the 1/2 power. The number of cycles to reach maximum and zero weight change vary inversely with the spall fraction, and the ratio of these cycles is exactly 1:3 for most oxides. By suitably normalizing the weight change and cycle number, it is shown that all cyclic oxidation weight change model curves can be represented by one universal expression for a given oxide scale.

  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. Quasi-exact-solvability of the {{A}_{2}}/{{G}_{2}} elliptic model: algebraic forms, sl(3)/{{g}^{(2)}} hidden algebra, and polynomial eigenfunctions

    NASA Astrophysics Data System (ADS)

    Sokolov, Vladimir V.; Turbiner, Alexander V.

    2015-04-01

    The potential of the A2 quantum elliptic model (three-body Calogero-Moser elliptic model) is defined by the pairwise three-body interaction through the Weierstrass ℘-function and has a single coupling constant. A change of variables has been found, which are A2 elliptic invariants, such that the potential becomes a rational function, while the flat space metric, as well as its associated vector, are polynomials in two variables. It is shown that the model possesses the hidden sl(3) algebra—the Hamiltonian is an element of the universal enveloping algebra {{U}sl(3)} for the arbitrary coupling constant—thus, it is equivalent to the sl(3)-quantum Euler-Arnold top. The integral, in a form of the third order differential operator with polynomial coefficients, is constructed explicitly, being also an element of {{U}sl(3)}. It is shown that there exists a discrete sequence of the coupling constants for which a finite number of polynomial eigenfunctions, up to a (non-singular) gauge factor, occurs. For these values of the coupling constants there exists a particular integral: it commutes with the Hamiltonian in action on the space of polynomial eigenfunctions, and the Hamiltonian is invariant with respect to two-dimensional projective transformations. It is shown that the A2 model has another hidden algebra {{g}(2)} introduced in Rosenbaum et al (1998 Int. J. Mod. Phys. A 13 3885). The potential of the G2 quantum elliptic model (three-body Wolfes elliptic model) is defined by the pairwise and three-body interactions through the Weierstrass ℘-function and has two coupling constants. A change of variables has been found, which are G2 elliptic invariants, such that the potential becomes a rational function, while the flat space metric, as well as its associated vector, are polynomials in two variables. It is shown the model possesses the hidden {{g}(2)} algebra. It is shown that there exists a discrete family of the coupling constants for which a finite number of

  2. Simultaneous multiplicative column-normalized method (SMART) for 3-D ionosphere tomography in comparison to other algebraic methods

    NASA Astrophysics Data System (ADS)

    Gerzen, T.; Minkwitz, D.

    2016-01-01

    The accuracy and availability of satellite-based applications like GNSS positioning and remote sensing crucially depends on the knowledge of the ionospheric electron density distribution. The tomography of the ionosphere is one of the major tools to provide link specific ionospheric corrections as well as to study and monitor physical processes in the ionosphere. In this paper, we introduce a simultaneous multiplicative column-normalized method (SMART) for electron density reconstruction. Further, SMART+ is developed by combining SMART with a successive correction method. In this way, a balancing between the measurements of intersected and not intersected voxels is realised. The methods are compared with the well-known algebraic reconstruction techniques ART and SART. All the four methods are applied to reconstruct the 3-D electron density distribution by ingestion of ground-based GNSS TEC data into the NeQuick model. The comparative case study is implemented over Europe during two periods of the year 2011 covering quiet to disturbed ionospheric conditions. In particular, the performance of the methods is compared in terms of the convergence behaviour and the capability to reproduce sTEC and electron density profiles. For this purpose, independent sTEC data of four IGS stations and electron density profiles of four ionosonde stations are taken as reference. The results indicate that SMART significantly reduces the number of iterations necessary to achieve a predefined accuracy level. Further, SMART+ decreases the median of the absolute sTEC error up to 15, 22, 46 and 67 % compared to SMART, SART, ART and NeQuick respectively.

  3. On Goursat Normal Forms, Prolongations, and Control Systems

    DTIC Science & Technology

    2007-11-02

    form, we also show how the exact linearization conditions for control systems can be restated in the language of Pfaffian systems. In addition, we give...area. We show that all of the main results in exact linearization of nonlinear systems can be restated in terms of exterior differential systems, and...to Goursat form can be specialized to give conditions for exact linearization . Theorem 6. Exact Linearization [5]. If a control system I defined

  4. The design of a parallel, dense linear algebra software library: Reduction to Hessenberg, tridiagonal, and bidiagonal form

    SciTech Connect

    Choi, J.; Dongarra, J.J. |; Walker, D.W.

    1994-09-01

    This paper discusses issues in the design of ScaLAPACK, a software library for performing dense linear algebra computations on distributed memory concurrent computers. These issues are illustrated using the ScaLAPACK routines for reducing matrices to Hessenberg, tridiagonal, and bidiagonal forms. These routines are important in the solution of eigenproblems. The paper focuses on how building blocks are used to create higher-level library routines. Results are presented that demonstrate the scalability of the reduction routines. The most commonly-used building blocks used in ScaLAPACK are the sequential BLAS, the Parallel Block BLAS (PB-BLAS) and the Basic Linear Algebra Communication Subprograms (BLACS). Each of the matrix reduction algorithms consists of a series of steps in each of which one block column (or panel), and/or block row, of the matrix is reduced, followed by an update of the portion of the matrix that has not been factorized so far. This latter phase is performed using distributed Level 3 BLAS routines, and contains the bulk of the computation. However, the panel reduction phase involves a significant amount of communication. And is important in determining the scalability of the algorithm. The simplest way to parallelize the panel reduction phase is to replace the appropriate Level 2 and Level 3 BLAS routines appearing in the LAPACK routine (mostly matrix-vector and matrix-matrix multiplications) with PB-BLAS routines.

  5. Teaching Algebra without Algebra

    ERIC Educational Resources Information Center

    Kalman, Richard S.

    2008-01-01

    Algebra is, among other things, a shorthand way to express quantitative reasoning. This article illustrates ways for the classroom teacher to convert algebraic solutions to verbal problems into conversational solutions that can be understood by students in the lower grades. Three reasonably typical verbal problems that either appeared as or…

  6. Pseudo-Riemannian Novikov algebras

    NASA Astrophysics Data System (ADS)

    Chen, Zhiqi; Zhu, Fuhai

    2008-08-01

    Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.

  7. The Normalized Reduced Form and Cell Mathematical Tools for Lattice Analysis-Symmetry and Similarity.

    PubMed

    Mighell, Alan D

    2003-01-01

    To intelligently and effectively use crystallographic databases, mathematical and computer tools are required that can elucidate diverse types of intra- and interlattice relationships. Two such tools are the normalized reduced form and normalized reduced cell. Practical experience has revealed that the first tool-the normalized reduced form-is very helpful in establishing lattice metric symmetry as it enables one to readily deduce significant relationships between the elements of the reduced form. Likewise research with crystallographic databases has demonstrated that the second tool-the normalized reduced cell-plays a vital role in determining metrically similar lattices. Knowledge of similar lattices has practical value in solving structures, in assignment of structure types, in materials design, and in nano-technology. In addition to using the reduced cell, it is recommended that lattice-matching strategies based on the normalized reduced cell be routinely carried out in database searching, in data evaluation, and in experimental work.

  8. The Normalized Reduced Form and Cell Mathematical Tools for Lattice Analysis—Symmetry and Similarity

    PubMed Central

    Mighell, Alan D.

    2003-01-01

    To intelligently and effectively use crystallographic databases, mathematical and computer tools are required that can elucidate diverse types of intra- and interlattice relationships. Two such tools are the normalized reduced form and normalized reduced cell. Practical experience has revealed that the first tool—the normalized reduced form—is very helpful in establishing lattice metric symmetry as it enables one to readily deduce significant relationships between the elements of the reduced form. Likewise research with crystallographic databases has demonstrated that the second tool—the normalized reduced cell—plays a vital role in determining metrically similar lattices. Knowledge of similar lattices has practical value in solving structures, in assignment of structure types, in materials design, and in nano-technology. In addition to using the reduced cell, it is recommended that lattice-matching strategies based on the normalized reduced cell be routinely carried out in database searching, in data evaluation, and in experimental work. PMID:27413622

  9. On weak Lie 2-algebras

    NASA Astrophysics Data System (ADS)

    Roytenberg, Dmitry

    2007-11-01

    A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and certain "up to homotopy" structures on the complex; for strictly skew-symmetric Lie 2-algebras these are L∞-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer-Cartan equation in some differential graded Lie algebras and L∞-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples.

  10. Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

    NASA Astrophysics Data System (ADS)

    Gouveia, Márcio R. A.; Llibre, Jaume; Novaes, Douglas D.; Pessoa, Claudio

    2016-04-01

    We consider an n-dimensional piecewise smooth vector field with two zones separated by a hyperplane Σ which admits an invariant hyperplane Ω transversal to Σ containing a period annulus A fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in A. When n = 3 we provide normal forms for the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the given normal forms.

  11. Design of a spatial data structure using the relational normal forms

    USGS Publications Warehouse

    van Roessel, Jan W.

    1987-01-01

    In previous work, a relational data structure aimed at the exchange of spatial data between systems was developed. As this data structure was relational it was of first normal form, but compliance with the higher normal forms was not investigated. Recently, a new procedural method for composing fully normalized data structures from the basic data fields has been developed by H. C. Smith, as an alternative to the process of non-loss decomposition which is difficult to understand. Smith's method has been applied to data fields required to store points, lines and polygons in a chain-node spatial data model. When geographic domain, coverage layer and map are also considered, the procedure naturally leads to a catalogue model, needed for the exchange of spatial data. Although the method produces a fully normalized data structure, it is not as easy to identify which normal forms are responsible for the ultimate arrangement of the data fields into relations, but the benefits of these criteria for data base development also apply to spatial data structures and related ancillary data.

  12. Integrating Boolean Queries in Conjunctive Normal Form with Probabilistic Retrieval Models.

    ERIC Educational Resources Information Center

    Losee, Robert M.; Bookstein, Abraham

    1988-01-01

    Presents a model that places Boolean database queries into conjunctive normal form, thereby allowing probabilistic ranking of documents and the incorporation of relevance feedback. Experimental results compare the performance of a sequential learning probabilistic retrieval model with the proposed integrated Boolean probabilistic model and a fuzzy…

  13. Normal forms for sub-Lorentzian metrics supported on Engel type distributions

    NASA Astrophysics Data System (ADS)

    Grochowski, Marek

    2014-06-01

    We construct normal forms for Lorentzian metrics on Engel distributions under the assumption that abnormal curves are timelike future directed Hamiltonian geodesics. Then we indicate some cases in which the abnormal timelike future directed curve initiating at the origin is geometrically optimal. We also give certain estimates for reachable sets from a point.

  14. A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

    NASA Technical Reports Server (NTRS)

    Freund, Roland W.; Huckle, Thomas

    1989-01-01

    In recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices appeared in the literature. We study restricted congruence transformation of Hermitian matrices M which, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain short proofs of most of the known inertia theorems and also derive some new results of this type. For some applications, a special class of almost unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form which, in particular, displays the eigenvalues of A. Finally, applications of this quasi-spectral decomposition to generalize inverses and Hermitian matrix pencils are discussed.

  15. Computing Matrix Representations of Filiform Lie Algebras

    NASA Astrophysics Data System (ADS)

    Ceballos, Manuel; Núñez, Juan; Tenorio, Ángel F.

    In this paper, we compute minimal faithful unitriangular matrix representations of filiform Lie algebras. To do it, we use the nilpotent Lie algebra, g_n, formed of n ×n strictly upper-triangular matrices. More concretely, we search the lowest natural number n such that the Lie algebra g_n contains a given filiform Lie algebra, also computing a representative of this algebra. All the computations in this paper have been done using MAPLE 9.5.

  16. Bicovariant quantum algebras and quantum Lie algebras

    NASA Astrophysics Data System (ADS)

    Schupp, Peter; Watts, Paul; Zumino, Bruno

    1993-10-01

    A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun(mathfrak{G}_q ) to U q g, given by elements of the pure braid group. These operators—the “reflection matrix” Y≡L + SL - being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for Y in SO q (N).

  17. 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.

  18. Kiddie Algebra

    ERIC Educational Resources Information Center

    Cavanagh, Sean

    2009-01-01

    As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…

  19. Applications of algebraic grid generation

    NASA Technical Reports Server (NTRS)

    Eiseman, Peter R.; Smith, Robert E.

    1990-01-01

    Techniques and applications of algebraic grid generation are described. The techniques are univariate interpolations and transfinite assemblies of univariate interpolations. Because algebraic grid generation is computationally efficient, the use of interactive graphics in conjunction with the techniques is advocated. A flexible approach, which works extremely well in an interactive environment, called the control point form of algebraic grid generation is described. The applications discussed are three-dimensional grids constructed about airplane and submarine configurations.

  20. Heterogeneity of normal prion protein in two- dimensional immunoblot: presence of various glycosylated and truncated forms.

    PubMed

    Pan, Tao; Li, Ruliang; Wong, Boon-Seng; Liu, Tong; Gambetti, Pierluigi; Sy, Man-Sun

    2002-06-01

    The common use of one-dimensional (1-D) immunoblot with a single monoclonal antibody (Mab) engenders the notion that the normal or cellular prion protein (PrP(C) ) comprises few and simple forms. In this study we used two-dimensional (2-D) immunoblot with a panel Mabs to various regions of the prion protein to demonstrate the complexity of the PrP(C) present in human brain. We distinguished over 50 immunoblot spots, each representing a distinct PrP(C) species based on combinations of different molecular weights and isoelectric points (pIs). The PrP(C) heterogeneity is due to the presence of a full-length and two major truncated forms as well as to the diversity of the glycans linked to most of these forms. The two major truncated forms result from distinct cleavage sites located at the N-terminus. In addition, enzymatic removal of sialic acid and lectin binding studies indicate that the glycans linked to the full-length and truncated PrP(C) forms differ in their structure and ratios of the glycoforms. The truncation of PrP(C) and the heterogeneity of the linked glycans may play a role in regulating PrP(C) function. Furthermore, the presence of relatively large quantities of different PrP(C) species may provide additional mechanisms by which the diversity of prion strains could be generated.

  1. High molecular gas fractions in normal massive star-forming galaxies in the young Universe.

    PubMed

    Tacconi, L J; Genzel, R; Neri, R; Cox, P; Cooper, M C; Shapiro, K; Bolatto, A; Bouché, N; Bournaud, F; Burkert, A; Combes, F; Comerford, J; Davis, M; Schreiber, N M Förster; Garcia-Burillo, S; Gracia-Carpio, J; Lutz, D; Naab, T; Omont, A; Shapley, A; Sternberg, A; Weiner, B

    2010-02-11

    Stars form from cold molecular interstellar gas. As this is relatively rare in the local Universe, galaxies like the Milky Way form only a few new stars per year. Typical massive galaxies in the distant Universe formed stars an order of magnitude more rapidly. Unless star formation was significantly more efficient, this difference suggests that young galaxies were much more molecular-gas rich. Molecular gas observations in the distant Universe have so far largely been restricted to very luminous, rare objects, including mergers and quasars, and accordingly we do not yet have a clear idea about the gas content of more normal (albeit massive) galaxies. Here we report the results of a survey of molecular gas in samples of typical massive-star-forming galaxies at mean redshifts of about 1.2 and 2.3, when the Universe was respectively 40% and 24% of its current age. Our measurements reveal that distant star forming galaxies were indeed gas rich, and that the star formation efficiency is not strongly dependent on cosmic epoch. The average fraction of cold gas relative to total galaxy baryonic mass at z = 2.3 and z = 1.2 is respectively about 44% and 34%, three to ten times higher than in today's massive spiral galaxies. The slow decrease between z approximately 2 and z approximately 1 probably requires a mechanism of semi-continuous replenishment of fresh gas to the young galaxies.

  2. Practical output tracking of switched nonlinear systems in p-normal form with unstable subsystems

    NASA Astrophysics Data System (ADS)

    Long, Lijun; Zhao, Jun

    2016-08-01

    This paper studies practical output tracking of switched nonlinear systems in p-normal form. No solvability of the practical output tracking problem for subsystems is required. A constructive scheme to solve the problem for a switched nonlinear system is set up by exploiting the single Lyapunov function method and the tool of adding a power integrator. Also, we design a proper switching law and construct state-feedback controllers of subsystems. A two inverted pendulums as a practical example, which cannot be handled by the existing approaches, illustrates our theoretical result.

  3. An asymptotic analysis of the 1:3:4 Hamiltonian normal form

    NASA Astrophysics Data System (ADS)

    Wang, L.; Bosley, D. L.; Kevorkian, J.

    The normal form of the Hamiltonian 1:3:4 resonance, which exhibits two simultaneous resonances of differing orders, is studied asymptotically. Since the two resonances have different strengths, the exact solution of the primary single resonance system may be used to construct an action-angle transformation. The resulting standard form system is solved asymptotically by canonical near-identity averaging transformations. In addition to the Hamiltonian itself and its unperturbed part, which are two exact constants of the motion, a third independent adiabatic invariant of the original Hamiltonian system is constructed. The results apply directly to the problem of a free electron laser with weak self-fields. A specific model problem is studied numerically to verify the asymptotic validity of the results over long times.

  4. Clifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras

    NASA Astrophysics Data System (ADS)

    Catto, Sultan; Gürcan, Yasemin; Khalfan, Amish; Kurt, Levent; Kato La, V.

    2016-10-01

    We discuss a construction scheme for Clifford numbers of arbitrary dimension. The scheme is based upon performing direct products of the Pauli spin and identity matrices. Conjugate fermionic algebras can then be formed by considering linear combinations of the Clifford numbers and the Hermitian conjugates of such combinations. Fermionic algebras are important in investigating systems that follow Fermi-Dirac statistics. We will further comment on the applications of Clifford algebras to Fueter analyticity, twistors, color algebras, M-theory and Leech lattice as well as unification of ancient and modern geometries through them.

  5. Invariant manifolds of an autonomous ordinary differential equation from its generalized normal forms.

    PubMed

    Palacián, Jesús

    2003-12-01

    A method to approximate some invariant sets of dynamical systems defined through an autonomous m-dimensional ordinary differential equation is presented. Our technique is based on the calculation of formal symmetries and generalized normal forms associated with the system of equations, making use of Lie transformations for smooth vector fields. Once a symmetry is determined up to a certain order, a reduction map allows us to pass from the equation in normal form to a related equation in a certain reduced space, the so-called reduced system of dimension s

  6. Theory and praxis pf map analsys in CHEF part 1: Linear normal form

    SciTech Connect

    Michelotti, Leo; /Fermilab

    2008-10-01

    This memo begins a series which, put together, could comprise the 'CHEF Documentation Project' if there were such a thing. The first--and perhaps only--three will telegraphically describe theory, algorithms, implementation and usage of the normal form map analysis procedures encoded in CHEF's collection of libraries. [1] This one will begin the sequence by explaining the linear manipulations that connect the Jacobian matrix of a symplectic mapping to its normal form. It is a 'Reader's Digest' version of material I wrote in Intermediate Classical Dynamics (ICD) [2] and randomly scattered across technical memos, seminar viewgraphs, and lecture notes for the past quarter century. Much of its content is old, well known, and in some places borders on the trivial.1 Nevertheless, completeness requires their inclusion. The primary objective is the 'fundamental theorem' on normalization written on page 8. I plan to describe the nonlinear procedures in a subsequent memo and devote a third to laying out algorithms and lines of code, connecting them with equations written in the first two. Originally this was to be done in one short paper, but I jettisoned that approach after its first section exceeded a dozen pages. The organization of this document is as follows. A brief description of notation is followed by a section containing a general treatment of the linear problem. After the 'fundamental theorem' is proved, two further subsections discuss the generation of equilibrium distributions and issue of 'phase'. The final major section reviews parameterizations--that is, lattice functions--in two and four dimensions with a passing glance at the six-dimensional version. Appearances to the contrary, for the most part I have tried to restrict consideration to matters needed to understand the code in CHEF's libraries.

  7. Quantum computation using geometric algebra

    NASA Astrophysics Data System (ADS)

    Matzke, Douglas James

    This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.

  8. Theory and praxis of map analsys in CHEF part 2: Nonlinear normal form

    SciTech Connect

    Michelotti, Leo; /FERMILAB

    2009-04-01

    This is the second of three memos describing how normal form map analysis is implemented in CHEF. The first [1] explained the manipulations required to assure that initial, linear transformations preserved Poincare invariants, thereby confirming correct normalization of action-angle coordinates. In this one, the transformation will be extended to nonlinear terms. The third, describing how the algorithms were implemented within the software of CHEF's libraries, most likely will never be written. The first section, Section 2, quickly lays out preliminary concepts and relationships. In Section 3, we shall review the perturbation theory - an iterative sequence of transformations that converts a nonlinear mapping into its normal form - and examine the equation which moves calculations from one step to the next. Following that is a section titled 'Interpretation', which identifies connections between the normalized mappings and idealized, integrable, fictitious Hamiltonian models. A final section contains closing comments, some of which may - but probably will not - preview work to be done later. My reasons for writing this memo and its predecessor have already been expressed. [1] To them can be added this: 'black box code' encourages users to proceed with little or no understanding of what it does or how it operates. So far, CHEF has avoided this trap admirably by failing to attract potential users. However, we reached a watershed last year: even I now have difficulty following the software through its maze of operations. Extensions to CHEF's physics functionalities, software upgrades, and even simple maintenance are becoming more difficult than they should. I hope these memos will mark parts of the maze for easier navigation in the future. Despite appearances to the contrary, I tried to include no (or very little) more than the minimum needed to understand what CHEF's nonlinear analysis modules do.1 As with the first memo, material has been lifted - and modified - from

  9. Accessing the Elastic Form-Factors of the $Delta(1232)$ Using the Beam-Normal Asymmetry

    SciTech Connect

    Dalton, Mark M.

    2016-08-01

    The beam-normal single-spin asymmetry, $B_n$, exists in the scattering of high energy electrons, polarized transverse to their direction of motion, from nuclear targets. To first order, this asymmetry is caused by the interference of the one-photon exchange amplitude with the imaginary part of the two-photon exchange amplitude. Measurements of $B_n$, for the production of a $\\Delta(1232)$ resonance from a proton target, will soon become available from the Qweak experiment at Jefferson Lab and the A4 experiment at Mainz. The imaginary part of two-photon exchange allows only intermediate states that are on-shell, including the $\\Delta$ itself. Therefore such data is sensitive to $\\gamma\\Delta\\Delta$, the elastic form-factors of the $\\Delta$. This article will introduce the form-factors of the $\\Delta$, discuss what might be learned about the elastic form-factors from these new data, describe ongoing efforts in calculation and measurement, and outline the possibility of future measurements.

  10. Optimization of accelerator parameters using normal form methods on high-order transfer maps

    SciTech Connect

    Snopok, Pavel

    2007-05-01

    Methods of analysis of the dynamics of ensembles of charged particles in collider rings are developed. The following problems are posed and solved using normal form transformations and other methods of perturbative nonlinear dynamics: (1) Optimization of the Tevatron dynamics: (a) Skew quadrupole correction of the dynamics of particles in the Tevatron in the presence of the systematic skew quadrupole errors in dipoles; (b) Calculation of the nonlinear tune shift with amplitude based on the results of measurements and the linear lattice information; (2) Optimization of the Muon Collider storage ring: (a) Computation and optimization of the dynamic aperture of the Muon Collider 50 x 50 GeV storage ring using higher order correctors; (b) 750 x 750 GeV Muon Collider storage ring lattice design matching the Tevatron footprint. The normal form coordinates have a very important advantage over the particle optical coordinates: if the transformation can be carried out successfully (general restrictions for that are not much stronger than the typical restrictions imposed on the behavior of the particles in the accelerator) then the motion in the new coordinates has a very clean representation allowing to extract more information about the dynamics of particles, and they are very convenient for the purposes of visualization. All the problem formulations include the derivation of the objective functions, which are later used in the optimization process using various optimization algorithms. Algorithms used to solve the problems are specific to collider rings, and applicable to similar problems arising on other machines of the same type. The details of the long-term behavior of the systems are studied to ensure the their stability for the desired number of turns. The algorithm of the normal form transformation is of great value for such problems as it gives much extra information about the disturbing factors. In addition to the fact that the dynamics of particles is represented

  11. 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.

  12. Exact traveling wave solutions of the van der Waals normal form for fluidized granular matter

    NASA Astrophysics Data System (ADS)

    Abourabia, A. M.; Morad, A. M.

    2015-11-01

    Analytical solutions of the van der Waals normal form for fluidized granular media have been done to study the phase separation phenomenon by using two different exact methods. The Painlevé analysis is discussed to illustrate the integrability of the model equation. An auto-Bäcklund transformation is presented via the truncated expansion and symbolic computation. The results show that the exact solutions of the model introduce solitary waves of different types. The solutions of the hydrodynamic model and the van der Waals equation exhibit a behavior similar to the one observed in molecular dynamic simulations such that two pairs of shock and rarefaction waves appear and move away, giving rise to the bubbles. The dispersion properties and the relation between group and phase velocities of the model equation are studied using the plane wave assumption. The diagrams are drawn to illustrate the physical properties of the exact solutions, and indicate their stability and bifurcation.

  13. Normal forms for linear mode conversion and Landau-Zener transitions in one dimension

    SciTech Connect

    Flynn, W.G.; Littlejohn, R.G.

    1994-09-01

    Standard eikonal methods for the asymptotic analysis of coupled linear wave equations may fail when two eigenvalues of a matrix (the dispersion matrix) associated with the wave operator are both small in the same region of wave phase space. In this region the two eikonal modes associated with the two small eigenvalues are coupled, leading to a process called linear mode conversion or Landau-Zener coupling. A theory of linear mode conversion is presented in which geometric structure is emphasized. This theory is then used to identify the most generic type of mode conversion which occurs in one dimension. Finally, a general solution for this generic mode conversion problem is derived by transforming an arbitrary equation exhibiting generic mode conversion into an easily solvable normal form. This solution is given as a connection rule, with which one may continue standard eikonal wave solutions through mode conversion regions. 51 refs., 13 figs.

  14. A Mathematical Framework for Critical Transitions: Normal Forms, Variance and Applications

    NASA Astrophysics Data System (ADS)

    Kuehn, Christian

    2013-06-01

    Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find early-warning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast-subsystem bifurcations up to codimension two. The theory is applied to several models: the Stommel-Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator-inhibitor switch from systems biology, a predator-prey system from ecology and to the Euler buckling problem from classical mechanics. For the Stommel-Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator-inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator-prey model explosive population growth near a codimension-two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.

  15. Nonlinear control design for stressed power systems using normal forms of vector fields

    NASA Astrophysics Data System (ADS)

    Jang, Gilsoo

    Large stressed interconnected power systems exhibit complicated dynamic behavior when subjected to disturbances. This nonlinear complex behavior is not well analyzed with present tools, and a complete theoretical analysis of this is not feasible in large systems. In stressed power systems, due to the presence of increased nonlinearity and the existence of nonlinear modal interactions, there exist some limitation to the use of conventional linear control design techniques. Therefore there is a need to understand the nature of nonlinear modal interactions and their influences on control performance for optimal controller setting. This work deals with control design in power systems using the method of normal forms. The objective of this work is to understand the effect of the nonlinear modal interaction on control performance and to develop a procedure to design controls incorporating the nonlinear information. For power systems equipped with fast exciters, the exciter gains have crucial influence on the system dynamic behavior. In order to be able to tune the exciter gains for optimal system performance, one has to understand, how the system response changes with different gain settings. In linear analysis, this consists of determining the eigenvalues for various gains, and computing the sensitivity of the eigenvalues under gain variations. If one takes into account the influence of the second order normal forms on the system response, then the corresponding interaction coefficients and their sensitivity with respect to gain variations has to be studied as well. This is the topic of the study presented here. The concept of nonlinear participation factors, and sensitivity of the normal forms coefficient, together with linear participation factors and eigenvalue sensitivity are used to vary control settings. The control settings are varied to obtain improved stability and to reduce the nonlinearity in the system. The proposed procedure was applied to the 50-generator

  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. 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.

  17. Algebraic mesh quality metrics

    SciTech Connect

    KNUPP,PATRICK

    2000-04-24

    Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally-invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. the singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. Condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Combined metrics for shape and volume, shape-volume-orientation are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to non-simplical elements. A series of numerical tests verify the theoretical properties of the metrics defined.

  18. Metacognition and Reading: Comparing Three Forms of Metacognition in Normally Developing Readers and Readers with Dyslexia

    PubMed Central

    Brunswick, Nicola

    2015-01-01

    Metacognition refers to ‘cognition about cognition’ and includes metacognitive knowledge, strategies and experiences (Efklides, 2008; Flavell, 1979). Research on reading has shown that better readers demonstrate more metacognitive knowledge than poor readers (Baker & Beall, 2009), and that reading ability improves through strategy instruction (Gersten, Fuchs, Williams, & Baker, 2001). The current study is the first to specifically compare the three forms of metacognition in dyslexic (N = 22) versus normally developing readers (N = 22). Participants read two factual texts, with learning outcome measured by a memory task. Metacognitive knowledge and skills were assessed by self-report. Metacognitive experiences were measured by predictions of performance and judgments of learning. Individuals with dyslexia showed insight into their reading problems, but less general knowledge of how to approach text reading. They more often reported lack of available reading strategies, but groups did not differ in the use of deep and surface strategies. Learning outcome and mean ratings of predictions of performance and judgments of learning were lower in dyslexic readers, but not the accuracy with which metacognitive experiences predicted learning. Overall, the results indicate that dyslexic reading and spelling problems are not generally associated with lower levels of metacognitive knowledge, metacognitive strategies or sensitivity to metacognitive experiences in reading situations. @ 2015 The Authors. Dyslexia published by John Wiley & Sons Ltd. PMID:26234622

  19. Metacognition and Reading: Comparing Three Forms of Metacognition in Normally Developing Readers and Readers with Dyslexia.

    PubMed

    Furnes, Bjarte; Norman, Elisabeth

    2015-08-01

    Metacognition refers to 'cognition about cognition' and includes metacognitive knowledge, strategies and experiences (Efklides, 2008; Flavell, 1979). Research on reading has shown that better readers demonstrate more metacognitive knowledge than poor readers (Baker & Beall, 2009), and that reading ability improves through strategy instruction (Gersten, Fuchs, Williams, & Baker, 2001). The current study is the first to specifically compare the three forms of metacognition in dyslexic (N = 22) versus normally developing readers (N = 22). Participants read two factual texts, with learning outcome measured by a memory task. Metacognitive knowledge and skills were assessed by self-report. Metacognitive experiences were measured by predictions of performance and judgments of learning. Individuals with dyslexia showed insight into their reading problems, but less general knowledge of how to approach text reading. They more often reported lack of available reading strategies, but groups did not differ in the use of deep and surface strategies. Learning outcome and mean ratings of predictions of performance and judgments of learning were lower in dyslexic readers, but not the accuracy with which metacognitive experiences predicted learning. Overall, the results indicate that dyslexic reading and spelling problems are not generally associated with lower levels of metacognitive knowledge, metacognitive strategies or sensitivity to metacognitive experiences in reading situations.

  20. GCD, LCM, and Boolean Algebra?

    ERIC Educational Resources Information Center

    Cohen, Martin P.; Juraschek, William A.

    1976-01-01

    This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)

  1. Steinberg conformal algebras

    NASA Astrophysics Data System (ADS)

    Mikhalev, A. V.; Pinchuk, I. A.

    2005-06-01

    The structure of Steinberg conformal algebras is studied; these are analogues of Steinberg groups (algebras, superalgebras).A Steinberg conformal algebra is defined as an abstract algebra by a system of generators and relations between the generators. It is proved that a Steinberg conformal algebra is the universal central extension of the corresponding conformal Lie algebra; the kernel of this extension is calculated.

  2. Becchi-Rouet-Stora-Tyutin operators for W algebras

    SciTech Connect

    Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.

    2008-07-15

    The study of quantum Lie algebras motivates a use of noncanonical ghosts and antighosts for nonlinear algebras, such as W-algebras. This leads, for the W{sub 3} and W{sub 3}{sup (2)} algebras, to the Becchi-Rouet-Stora-Tyutin operator having the conventional cubic form.

  3. Web Algebra.

    ERIC Educational Resources Information Center

    Capani, Antonio; De Dominicis, Gabriel

    This paper proposes a model for a general interface between people and Computer Algebra Systems (CAS). The main features in the CAS interface are data navigation and the possibility of accessing powerful remote machines. This model is based on the idea of session management, in which the main engine of the tool enables interactions with the…

  4. Current algebra and the nonlinear σ-model

    NASA Astrophysics Data System (ADS)

    Ghosh, S.

    2007-06-01

    We present the current algebra of a particular form in the nonlinear σ-model. The algebra has a non-Abelian form with field-dependent structure functions. We comment on the connection of the model with noncommutative space.

  5. Comparison between dental and basal arch forms in normal occlusion and Class III malocclusions utilizing cone-beam computed tomography

    PubMed Central

    Suk, Kyung Eun; Park, Jae Hyun; Bayome, Mohamed; Nam, Young-Ok; Sameshima, Glenn T.

    2013-01-01

    Objective The purpose of this study was to investigate the relationship between the mandibular dental and basal arch forms in subjects with normal occlusion and compare them with those of Class III malocclusion using cone-beam computed tomography (CBCT). Methods CBCT images of 32 normal occlusion (19 males, 13 females; 24.3 years) and 33 Class III malocclusion subjects (20 males, 13 females, 22.2 years) were selected. Facial axis and root center points were identified from the left to right mandibular first molars. Distances between the facial axis and root center points for each tooth were calculated, and 4 linear and 2 ratio variables were measured and calculated for each arch form. The variables were compared between groups by independent t-test. Pearson correlation coefficient was applied to assess the relationships between dental and basal variables within each group. Results The mandibular dental and basal intercanine widths were significantly greater in the Class III group than in normal occlusion subjects (p < 0.05). The dental and basal intercanine widths as well as the dental and basal intermolar widths were strongly correlated in normal occlusion and moderately correlated in Class III malocclusion. Conclusions The dental arch form demon strated a strong positive correlation with the basal arch form in the normal occlusion group and moderate correlation in the Class III malocclusion group. These results might be helpful for clinicians to have a better understanding of the importance of basal arch form in the alveolar bone. PMID:23504406

  6. Mössbauer spectroscopic study of the forms of iron in normal human liver and spleen tissue

    NASA Astrophysics Data System (ADS)

    Chua-Anusorn, W.; Pierre, T. G. St.; Webb, J.; Macey, D. J.; Yansukon, P.; Pootrakul, P.

    1994-12-01

    Mössbauer spectra of 12 normal human spleen and 12 normal human liver samples ( post mortem) from Australia and Thailand have been recorded at 78 K. The spectra show the presence of iron in the form of ferrihydrite, together with some deoxyhemoglobin and methemoglobin in some samples. The spectra were used in conjunction with elemental analysis to calculate the non-heme iron concentrations in the tissues. The mean non-heme iron concentration in the Thai livers was significantly less than that for the Australian samples. The goethite-like form of hemosiderin that has been observed in some pathological tissues was not detected.

  7. Cartan calculus on quantum Lie algebras

    SciTech Connect

    Schupp, P.; Watts, P.; Zumino, B.

    1993-12-09

    A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions au into one big algebra, the ``Cartan Calculus.``

  8. MRI visualization of pathological forms by suppression of normal tissue signals

    NASA Astrophysics Data System (ADS)

    Pirogov, Yuri A.; Anisimov, Nikolai V.; Gubskiy, Leonid V.; Babich, Piotr I.

    2005-04-01

    To improve the visualization and 3D-reconstruction of some pathological formations of the brain, it is offered to use a new method of processing of MR images with suppression of signals from normal tissues. The special attention is offered to be given suppression of signals of fatty tissue, free water and partially bound water of mucous membranes. For such way realization, it is offered to lead two scans with simultaneous suppression of two normal components and to multiply the obtained images. Simultaneous suppression of signals from two normal tissues is realized with the help of pulse sequence twice using inversion-recovery effect. Delays in pulse sequence are selected in accordance with the times of longitudinal relaxation of fat, free water and partially bound water. In comparison with earlier described technique of simultaneous suppression of signals of water and fat, the new method is especially useful at research of pathological formations when the zone of defeat is placed in a zone of nose bosoms. Besides allocation of a zone of defeat, MIP reconstruction becomes simpler. The offered technique well proves at research of tumors and hemorrhages.

  9. Shear Stress-Normal Stress (Pressure) Ratio Decides Forming Callus in Patients with Diabetic Neuropathy.

    PubMed

    Amemiya, Ayumi; Noguchi, Hiroshi; Oe, Makoto; Takehara, Kimie; Ohashi, Yumiko; Suzuki, Ryo; Yamauchi, Toshimasa; Kadowaki, Takashi; Sanada, Hiromi; Mori, Taketoshi

    2016-01-01

    Aim. Callus is a risk factor, leading to severe diabetic foot ulcer; thus, prevention of callus formation is important. However, normal stress (pressure) and shear stress associated with callus have not been clarified. Additionally, as new valuables, a shear stress-normal stress (pressure) ratio (SPR) was examined. The purpose was to clarify the external force associated with callus formation in patients with diabetic neuropathy. Methods. The external force of the 1st, 2nd, and 5th metatarsal head (MTH) as callus predilection regions was measured. The SPR was calculated by dividing shear stress by normal stress (pressure), concretely, peak values (SPR-p) and time integral values (SPR-i). The optimal cut-off point was determined. Results. Callus formation region of the 1st and 2nd MTH had high SPR-i rather than noncallus formation region. The cut-off value of the 1st MTH was 0.60 and the 2nd MTH was 0.50. For the 5th MTH, variables pertaining to the external forces could not be determined to be indicators of callus formation because of low accuracy. Conclusions. The callus formation cut-off values of the 1st and 2nd MTH were clarified. In the future, it will be necessary to confirm the effect of using appropriate footwear and gait training on lowering SPR-i.

  10. Shear Stress-Normal Stress (Pressure) Ratio Decides Forming Callus in Patients with Diabetic Neuropathy

    PubMed Central

    Noguchi, Hiroshi; Takehara, Kimie; Ohashi, Yumiko; Suzuki, Ryo; Yamauchi, Toshimasa; Kadowaki, Takashi; Sanada, Hiromi

    2016-01-01

    Aim. Callus is a risk factor, leading to severe diabetic foot ulcer; thus, prevention of callus formation is important. However, normal stress (pressure) and shear stress associated with callus have not been clarified. Additionally, as new valuables, a shear stress-normal stress (pressure) ratio (SPR) was examined. The purpose was to clarify the external force associated with callus formation in patients with diabetic neuropathy. Methods. The external force of the 1st, 2nd, and 5th metatarsal head (MTH) as callus predilection regions was measured. The SPR was calculated by dividing shear stress by normal stress (pressure), concretely, peak values (SPR-p) and time integral values (SPR-i). The optimal cut-off point was determined. Results. Callus formation region of the 1st and 2nd MTH had high SPR-i rather than noncallus formation region. The cut-off value of the 1st MTH was 0.60 and the 2nd MTH was 0.50. For the 5th MTH, variables pertaining to the external forces could not be determined to be indicators of callus formation because of low accuracy. Conclusions. The callus formation cut-off values of the 1st and 2nd MTH were clarified. In the future, it will be necessary to confirm the effect of using appropriate footwear and gait training on lowering SPR-i. PMID:28050567

  11. Universal Algebraic Varieties and Ideals in Physics:. Field Theory on Algebraic Varieties

    NASA Astrophysics Data System (ADS)

    Iguchi, Kazumoto

    A class of universal algebraic varieties in physics is discussed herein using the concepts of determinant ideals in algebraic geometry. It is shown that these algebraic varieties arise with very different physical contexts in many branches of physics and mathematics from high energy physics theory to chaos theory. In these physical systems the models are constructed by using the fields on usual manifolds such as vector fields in a Euclidean space and a Minkowskian space. But there is a universal mathematical aspect of linear algebra for linear vector spaces, where the linear independency and dependency are described using the Gramians of the vectors. These Gramians form a class of hypersurfaces in a higher-dimensional mathematical space: If there exist g vectors vi in an n-dimensional Euclidean space, the Gramian Gg is given as a g × g determinant Gg=Det[xij] with the inner products xij=(vi,vj), and exists in a g(g-1)/2-[g(g+1)/2-] dimensional space if the vectors are (not) normalized, xii=1 (xii ≠ 1). It is also shown that the Gramians are invariant under automorphisms of the vectors. The mathematical structure of the Gramians is revealed to be equivalent to the concepts of determinant ideals Ig(v), each element of which is a g × g determinant constructed from components of an arbitrary N×N matrix with N>n and which have inclusion relation: R=I0(v)⊃ I1(v) ⊃···⊃ Ig(v) ⊃···, and Ig(v)=0 if g>n. In the various physical systems the ideals naturally emerge to give us dynamical flows on the hypersurfaces, and therefore, it is called the field theory on algebraic varieties. This viewpoint provides us a grand viewpoint in physics and mathematics.

  12. Algebraic trigonometry

    NASA Astrophysics Data System (ADS)

    Vaninsky, Alexander

    2011-04-01

    This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.

  13. Dynamical algebras for Poeschl-Teller Hamiltonian hierarchies

    SciTech Connect

    Kuru, S.; Negro, J.

    2009-12-15

    The dynamical algebras of the trigonometric and hyperbolic symmetric Poeschl-Teller Hamiltonian hierarchies are obtained. A kind of discrete-differential realizations of these algebras are found which are isomorphic to so(3, 2) Lie algebras. In order to get them, first the relation between ladder and factor operators is investigated. In particular, the action of the ladder operators on normalized eigenfunctions is found explicitly. Then, the whole dynamical algebras are generated in a straightforward way.

  14. Form of 15q proximal duplication appears to be a normal euchromatic variant

    SciTech Connect

    Jalal, S.M.; Persons, D.L.; DeWald, G.W.; Lindor, N.M.

    1994-10-01

    Deletions involving often leads to either Prader-Willi or Angelman syndrome, depending on the hereditary path of the deletion (paternal or maternal). A number of cases have been reported in which duplications involving 15q11.2-q13 have not been associated with any detectable phenotypic abnormalities. Ludowese et al. (1991) have summarized 25 such cases that include 10 of their own cases from 5 unrelated families. They conclude that duplication of 15q12-13 does not have an adverse phenotypic effect, though they do not completely rule out the possibility that, instead of 15q12-13 duplication, the extra material could be an insertion from another chromosome. Thus, the dilemma is when duplication of 15q11.2-q13 is clinically significant. We suggest that certain kinds of amplification or duplication involving distal 15q12 and 15q13 may represent a normal variant. 14 refs., 1 fig., 1 tab.

  15. 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,…

  16. Generation of Strategies for Environmental Deception in Two-Player Normal-Form Games

    DTIC Science & Technology

    2015-06-18

    two-player, strategic-form games. Environmental deception is defined as deception where one player has the ability to change the other’s perception ...of the state of the game through modification of their perception of the game’s payoff matrix, similar to the use of camouflage. The main...environmental deception as the deceiver changes the mark’s perception regarding the state of the world rather than his perception of the deceiver’s

  17. 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.

  18. Normal forms for germs of vector fields with quadratic leading part. The polynomial first integral case

    NASA Astrophysics Data System (ADS)

    Stróżyna, Ewa

    2015-12-01

    We study the problem of formal classification of the vector fields of the form x ˙ = ax2 + bxy + cy2 + … , y ˙ = dx2 + exy + fy2 + … using formal changes of the coordinates, but not using the changes of the time. We focus on one special case (which is the most complex one): when the quadratic homogeneous part has a polynomial first integral. In the proofs we avoid complicated calculations. The method we use is effective and it is based on the method introduced in our previous work concerning the Bogdanov-Takens singularity.

  19. Correlation between maxillary central incisor crown morphology and mandibular dental arch form in normal occlusion subjects.

    PubMed

    Paranhos, Luiz Renato; Lima, Carolina Souto; da Silva, Ricardo Henrique Alves; Daruge Júnior, Eduardo; Torres, Fernando Cesar

    2012-01-01

    The aim of this study was to evaluate the correlation between the morphology of the mandibular dental arch and the maxillary central incisor crown. Cast models from 51 Caucasian individuals, older than 15 years, with optimal occlusion, no previous orthodontic treatment, featuring 4 of the 6 keys to normal occlusion by Andrews (the first being mandatory) were observed. The models were digitalized using a 3D scanner, and images of the maxillary central incisor and mandibular dental arch were obtained. These were printed and placed in an album below pre-set models of arches and dental crowns, and distributed to 12 dental surgeons, who were asked to choose which shape was most in accordance with the models and crown presented. The Kappa test was performed to evaluate the concordance among evaluators while the chi-square test was used to verify the association between the dental arch and central incisor morphology, at a 5% significance level. The Kappa test showed moderate agreement among evaluators for both variables of this study, and the chi-square test showed no significant association between tooth shape and mandibular dental arch morphology. It may be concluded that the use of arch morphology as a diagnostic method to determine the shape of the maxillary central incisor is not appropriate. Further research is necessary to assess tooth shape using a stricter scientific basis.

  20. Analyzing dynamic performance of power systems over parameter space using the method of normal forms of vector fields

    NASA Astrophysics Data System (ADS)

    Zhu, Songzhe

    Today's power systems have become more and more stressed due to the high utilization of available facilities. The complex dynamic behavior of large stressed power systems following disturbances can not be fully explained with present tools, such as linear eigen-analysis tools and nonlinear time-domain simulation methods. This research work applies a nonlinear analytical tool, the method of normal forms of vector fields, to help understand the complex transient oscillations in stressed power systems. The method of normal forms is a well-known mathematical tool to study systems of differential equations. The basic idea is to simplify the dynamical system by a sequence of nonlinear coordinate transformations. If there is no resonance in the system, then the nonlinear vector field can be turned into a linear one by the transformations. Previous work applied the second-order normal form transformation under non-resonance condition to power system dynamical equations. The nonlinear interaction among the fundamental modes was investigated. Based on these efforts, this work extends the application of normal forms to evaluate the dynamic performance of power systems taking into account changing operation conditions. As the resonance and near-resonance could occur in parameter space, a new normal form transformation under second order resonance condition is derived. The analysis shows that the high nonlinearity resulting from the resonance and near-resonance among poorly damped oscillatory modes and control modes is detrimental to the system performance. An approach to determine the resonance and near-resonance regions in parameter space is developed. The modes contributing to the detrimental behavior associated with the near-resonance region are identified by a procedure based on certain modal interaction indices. The state variables showing detrimental behavior are then determined using nonlinear participation factors. The accuracy of the prediction is verified by

  1. 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…

  2. The SAMI Galaxy Survey: shocks and outflows in a normal star-forming galaxy

    NASA Astrophysics Data System (ADS)

    Ho, I.-Ting; Kewley, Lisa J.; Dopita, Michael A.; Medling, Anne M.; Allen, J. T.; Bland-Hawthorn, Joss; Bloom, Jessica V.; Bryant, Julia J.; Croom, Scott M.; Fogarty, L. M. R.; Goodwin, Michael; Green, Andy W.; Konstantopoulos, Iraklis S.; Lawrence, Jon S.; López-Sánchez, Á. R.; Owers, Matt S.; Richards, Samuel; Sharp, Rob

    2014-11-01

    We demonstrate the feasibility and potential of using large integral field spectroscopic surveys to investigate the prevalence of galactic-scale outflows in the local Universe. Using integral field data from the Sydney-AAO Multi-object Integral field spectrograph (SAMI) and the Wide Field Spectrograph, we study the nature of an isolated disc galaxy, SDSS J090005.05+000446.7 (z = 0.053 86). In the integral field data sets, the galaxy presents skewed line profiles changing with position in the galaxy. The skewed line profiles are caused by different kinematic components overlapping in the line-of-sight direction. We perform spectral decomposition to separate the line profiles in each spatial pixel as combinations of (1) a narrow kinematic component consistent with H II regions, (2) a broad kinematic component consistent with shock excitation, and (3) an intermediate component consistent with shock excitation and photoionization mixing. The three kinematic components have distinctly different velocity fields, velocity dispersions, line ratios, and electron densities. We model the line ratios, velocity dispersions, and electron densities with our MAPPINGS IV shock and photoionization models, and we reach remarkable agreement between the data and the models. The models demonstrate that the different emission line properties are caused by major galactic outflows that introduce shock excitation in addition to photoionization by star-forming activities. Interstellar shocks embedded in the outflows shock-excite and compress the gas, causing the elevated line ratios, velocity dispersions, and electron densities observed in the broad kinematic component. We argue from energy considerations that, with the lack of a powerful active galactic nucleus, the outflows are likely to be driven by starburst activities. Our results set a benchmark of the type of analysis that can be achieved by the SAMI Galaxy Survey on large numbers of galaxies.

  3. Effects of interchanging hyperopic defocus and form deprivation stimuli in normal and optic nerve-sectioned chicks.

    PubMed

    Choh, Vivian; Lew, MinJung Y; Nadel, Michel W; Wildsoet, Christine F

    2006-03-01

    To test the hypothesis that the same mechanisms mediate form deprivation and lens-induced myopia, the ocular growth responses of chicks alternately exposed to lenses and diffusers at regular intervals (3h) were compared to those of chicks exposed to either negative lenses or diffusers alone. In total, there were four experiments: (1) -15 D lenses and/or diffusers on normal birds, (2) -15 D lenses and/or diffusers on optic nerve-sectioned (ONS) birds, (3) -5/-10/-15 D lenses (sequentially applied) and/or diffusers on normal birds and (4) -5/-10/-15 D lenses and/or diffusers on ONS birds. All treatments were monocular. In all experiments, optical axial lengths (cornea-to-retina distances) in treated eyes were greater than in fellow eyes, irrespective of the optical device (diffuser, lens or switch), lens power (fixed or incremented) and optic nerve condition (intact or severed). In normal chicks, optical axial length responses in the switch group were significantly reduced relative to those of the diffuser but not to those of the -15 D lens group. For both groups of ONS birds, diffusers exaggerated the optical axial length changes. For all groups, the responses to the switch and lens groups were most similar. These results together suggest that the mechanisms mediating form deprivation- and lens-induced myopia are different.

  4. Post-UV colony-forming ability of normal fibroblast strains and of the xeroderma pigmentosum group G strain

    SciTech Connect

    Barrett, S.F.; Tarone, R.E.; Moshell, A.N.; Ganges, M.B.; Robbins, J.H.

    1981-01-01

    In xeroderma pigmentosum, an inherited disorder of defective DNA repair, post-uv colony-forming ability of fibroblasts from patients in complementation groups A through F correlates with the patients' neurological status. The first xeroderma pigmentosum patient assigned to the recently discovered group G had the neurological abnormalities of XP. Researchers have determined the post-uv colony-forming ability of cultured fibroblasts from this patient and from 5 more control donors. Log-phase fibroblasts were irradiated with 254 nm uv light from a germicidal lamp, trypsinized, and replated at known densities. After 2 to 4 weeks' incubation the cells were fixed, stained and scored for colony formation. The strains' post-uv colony-forming ability curves were obtained by plotting the log of the percent remaining post-uv colony-forming ability as a function of the uv dose. The post-uv colony-forming ability of 2 of the 5 new normal strains was in the previously defined control donor zone, but that of the other 3 extended down to the level of the most resistant xeroderma pigmentosum strain. The post-uv colony-forming ability curve of the group G fibroblasts was not significantly different from the curves of the group D fibroblast strains from patients with clinical histories similar to that of the group G patient.

  5. Rees algebras, Monomial Subrings and Linear Optimization Problems

    NASA Astrophysics Data System (ADS)

    Dupont, Luis A.

    2010-06-01

    In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.

  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. Close-form expression of one-tap normalized LMS carrier phase recovery in optical communication systems

    NASA Astrophysics Data System (ADS)

    Xu, Tianhua; Jacobsen, Gunnar; Popov, Sergei; Li, Jie; Liu, Tiegen; Zhang, Yimo

    2016-10-01

    The performance of long-haul high speed coherent optical fiber communication systems is significantly degraded by the laser phase noise and the equalization enhanced phase noise (EEPN). In this paper, the analysis of the one-tap normalized least-mean-square (LMS) carrier phase recovery (CPR) is carried out and the close-form expression is investigated for quadrature phase shift keying (QPSK) coherent optical fiber communication systems, in compensating both laser phase noise and equalization enhanced phase noise. Numerical simulations have also been implemented to verify the theoretical analysis. It is found that the one-tap normalized least-mean-square algorithm gives the same analytical expression for predicting CPR bit-error-rate (BER) floors as the traditional differential carrier phase recovery, when both the laser phase noise and the equalization enhanced phase noise are taken into account.

  8. Graphs and Networks for Years 7 to 10: Reasons for and Ways of Using Digital Technologies to teach Algebra and the Standard Normal Curve

    ERIC Educational Resources Information Center

    Padula, Janice

    2014-01-01

    If educators want to interest students in mathematics (and science), they must engage them in the lower forms of high school or even earlier (Fisher, 2012). So, teachers should always consider a topic's ability to interest students in the early years of instruction in high school and its topicality. Networks have come into prominence recently with…

  9. Algebraic grid generation with corner singularities

    NASA Technical Reports Server (NTRS)

    Vinokur, M.; Lombard, C. K.

    1983-01-01

    A simple noniterative algebraic procedure is presented for generating smooth computational meshes on a quadrilateral topology. Coordinate distribution and normal derivative are provided on all boundaries, one of which may include a slope discontinuity. The boundary conditions are sufficient to guarantee continuity of global meshes formed of joined patches generated by the procedure. The method extends to 3-D. The procedure involves a synthesis of prior techniques stretching functions, cubic blending functions, and transfinite interpolation - to which is added the functional form of the corner solution. The procedure introduces the concept of generalized blending, which is implemented as an automatic scaling of the boundary derivatives for effective interpolation. Some implications of the treatment at boundaries for techniques solving elliptic PDE's are discussed in an Appendix.

  10. 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.

  11. Algebraic Lattices in QFT Renormalization

    NASA Astrophysics Data System (ADS)

    Borinsky, Michael

    2016-07-01

    The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.

  12. Role of L-forms of Nocardia caviae in the development of chronic mycetomas in normal and immunodeficient murine models.

    PubMed Central

    Beaman, B L; Scates, S M

    1981-01-01

    Single-cell suspensions of Nocardia caviae 112 were injected into normal, athymic, and asplenic mice by several different routes. The 50% lethal dose values, kill curve characteristics, histological and electron microscopic properties, organ clearance patterns, and induction of L-forms during the acute and chronic phase of disease were determined in groups of mice for up to 2 years after infection. From these data we concluded the following. (i) Athymic and asplenic animals were significantly more susceptible to N. caviae than their littermate controls regardless of inoculation route. (ii) All mice were most susceptible to lethal infection after intranasal administration and least affected when the organisms were injected into the peritoneal cavity. (iii) Chronic, progressive disease leading to the formation of mycetomas occurred only in mice injected intravenously. (iv) T-cell-deficient animals were impaired in the development of typical mycetomas. (v) L-forms of N. caviae were induced within immunocompetent hosts, whereas the cell wall-less state of the bacteria was not observed in the immunodeficient animals. (vi) Two colony types of the cell wall-deficient state were isolated from infected animals. (vii) These cell wall-deficient organisms were intimately involved in the pathogenesis of disease and bacterial persistence within the host. Finally (viii), with this strain of Nocardia, cell wall-deficient organisms played a major role in the development of the characteristic bacterial granule formed within the mycetomatous lesions 6 months to 1 year after intravenous inoculation. Images PMID:7287189

  13. Profiles of Algebraic Competence

    ERIC Educational Resources Information Center

    Humberstone, J.; Reeve, R.A.

    2008-01-01

    The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…

  14. Writing to Learn Algebra.

    ERIC Educational Resources Information Center

    Miller, L. Diane; England, David A.

    1989-01-01

    Describes a study in a large metropolitan high school to ascertain what influence the use of regular writing in algebra classes would have on students' attitudes towards algebra and their skills in algebra. Reports the simpler and more direct the writing topics the better. (MVL)

  15. 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.

  16. Applied Algebra Curriculum Modules.

    ERIC Educational Resources Information Center

    Texas State Technical Coll., Marshall.

    This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students…

  17. Connecting Arithmetic to Algebra

    ERIC Educational Resources Information Center

    Darley, Joy W.; Leapard, Barbara B.

    2010-01-01

    Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…

  18. Ternary Virasoro - Witt algebra.

    SciTech Connect

    Zachos, C.; Curtright, T.; Fairlie, D.; High Energy Physics; Univ. of Miami; Univ. of Durham

    2008-01-01

    A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.

  19. Structure of classical affine and classical affine fractional W-algebras

    SciTech Connect

    Suh, Uhi Rinn

    2015-01-15

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

  20. Toxic and non-toxic aggregates from the SBMA and normal forms of androgen receptor have distinct oligomeric structures.

    PubMed

    Jochum, Tobias; Ritz, Manuela E; Schuster, Christoph; Funderburk, Sarah F; Jehle, Katja; Schmitz, Katja; Brinkmann, Falko; Hirtz, Michael; Moss, David; Cato, Andrew C B

    2012-06-01

    Hormone-dependent aggregation of the androgen receptor (AR) with a polyglutamine (polyQ) stretch amplification (>38) is considered to be the causative agent of the neurodegenerative disorder spinal and bulbar muscular atrophy (SBMA), consistent with related neurodegenerative diseases involving polyQ-extended proteins. In spite of the widespread acceptance of this common causal hypothesis, little attention has been paid to its apparent incompatibility with the observation of AR aggregation in healthy individuals with no polyQ stretch amplification. Here we used atomic force microscopy (AFM) to characterize sub-micrometer scale aggregates of the wild-type (22 glutamines) and the SBMA form (65 glutamines), as well as a polyQ deletion mutant (1 glutamine) and a variant with a normal length polyQ stretch but with a serine to alanine double mutation elsewhere in the protein. We used a baculovirus-insect cell expression system to produce full-length proteins for these structural analyses. We related the AFM findings to cytotoxicity as measured by expression of the receptors in Drosophila motoneurons or in neuronal cells in culture. We found that the pathogenic AR mutants formed oligomeric fibrils up to 300-600nm in length. These were clearly different from annular oligomers 120-180nm in diameter formed by the nonpathogenic receptors. We could also show that melatonin, which is known to ameliorate the pathological phenotype in the fly model, caused polyQ-extended AR to form annular oligomers. Further comparative investigation of these reproducibly distinct toxic and non-toxic oligomers could advance our understanding of the molecular basis of the polyQ pathologies.

  1. 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.

  2. Discrete Minimal Surface Algebras

    NASA Astrophysics Data System (ADS)

    Arnlind, Joakim; Hoppe, Jens

    2010-05-01

    We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.

  3. Gauged Ads-Maxwell Algebra and Gravity

    NASA Astrophysics Data System (ADS)

    Durka, R.; Kowalski-Glikman, J.; Szczachor, M.

    We deform the anti-de Sitter algebra by adding additional generators {Z}ab, forming in this way the negative cosmological constant counterpart of the Maxwell algebra. We gauge this algebra and construct a dynamical model with the help of a constrained BF theory. It turns out that the resulting theory is described by the Einstein-Cartan action with Holst term, and the gauge fields associated with the Maxwell generators {Z}ab appear only in topological terms that do not influence dynamical field equations. We briefly comment on the extension of this construction, which would lead to a nontrivial Maxwell fields dynamics.

  4. Cyclic Cocycles on Twisted Convolution Algebras

    NASA Astrophysics Data System (ADS)

    Angel, Eitan

    2013-01-01

    We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper étale groupoids, Tu and Xu (Adv Math 207(2):455-483, 2006) provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to the construction of Mathai and Stevenson (Adv Math 200(2):303-335, 2006). When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.

  5. Generalized conformal realizations of Kac-Moody algebras

    NASA Astrophysics Data System (ADS)

    Palmkvist, Jakob

    2009-01-01

    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 3×3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n =2. Moreover, we obtain their infinite-dimensional extensions for n ≥3. In the case of 2×2 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).

  6. Human dihydrofolate reductase and thymidylate synthase form a complex in vitro and co-localize in normal and cancer cells.

    PubMed

    Antosiewicz, Anna; Jarmuła, Adam; Przybylska, Dorota; Mosieniak, Grażyna; Szczepanowska, Joanna; Kowalkowska, Anna; Rode, Wojciech; Cieśla, Joanna

    2016-08-05

    Enzymes involved in thymidylate biosynthesis, thymidylate synthase (TS), and dihydrofolate reductase (DHFR) are well-known targets in cancer chemotherapy. In this study, we demonstrated for the first time, that human TS and DHFR form a strong complex in vitro and co-localize in human normal and colon cancer cell cytoplasm and nucleus. Treatment of cancer cells with methotrexate or 5-fluorouracil did not affect the distribution of either enzyme within the cells. However, 5-FU, but not MTX, lowered the presence of DHFR-TS complex in the nucleus by 2.5-fold. The results may suggest the sequestering of TS by FdUMP in the cytoplasm and thereby affecting the translocation of DHFR-TS complex to the nucleus. Providing a strong likelihood of DHFR-TS complex formation in vivo, the latter complex is a potential new drug target in cancer therapy. In this paper, known 3D structures of human TS and human DHFR, and some protozoan bifunctional DHFR-TS structures as templates, are used to build an in silico model of human DHFR-TS complex structure, consisting of one TS dimer and two DHFR monomers. This complex structure may serve as an initial 3D drug target model for prospective inhibitors targeting interfaces between the DHFR and TS enzymes.

  7. Dynamics, Bifurcations and Normal Forms in Arrays of Magnetostrictive Energy Harvesters with All-to-All Coupling

    NASA Astrophysics Data System (ADS)

    Matus-Vargas, Antonio; González-Hernandez, Hugo G.; Chan, Bernard S.; Palacios, Antonio; Buono, Pietro-Luciano; in, Visarath; Naik, Suketu; Phipps, Alex; Longhini, Patrick

    Modeling and bifurcation analysis of an energy harvesting system composed of coupled resonators using the Galfenol-based magnetostrictive material are presented. The analysis in this work should be broad enough to be applicable to a large class of vibratory-based energy harvesting systems since various types of vibratory harvesters share the same normal forms, e.g. magnetostrictive and piezoelectric materials. A combined model of the mechanical and electrical domains of a single energy harvester is discussed first. Building on this model, the governing equations of the coupled system are derived, leading to a system of differential equations with an all-to-all coupling between the resonators. A bifurcation analysis of the system equations reveals different patterns of collective oscillations. Among the many different patterns, a synchronous state exists and it is stable over a broad region of parameter space. This pattern has the potential to yield significant increases in power output and it will be used as a starting point to guide future experimental work. A Hamiltonian approach is employed to study analytically the nature of the bifurcations and to calculate an expression for the onset of synchronization valid for any number of harvesters.

  8. Algebraic operator approach to gas kinetic models

    NASA Astrophysics Data System (ADS)

    Il'ichov, L. V.

    1997-02-01

    Some general properties of the linear Boltzmann kinetic equation are used to present it in the form ∂ tϕ = - †Âϕ with the operators Âand† possessing some nontrivial algebraic properties. When applied to the Keilson-Storer kinetic model, this method gives an example of quantum ( q-deformed) Lie algebra. This approach provides also a natural generalization of the “kangaroo model”.

  9. Algebraic Sub-Structuring for Electromagnetic Applications

    SciTech Connect

    Yang, C.; Gao, W.G.; Bai, Z.J.; Li, X.Y.S.; Lee, L.Q.; Husbands, P.; Ng, E.G.; /LBL, Berkeley /UC, Davis /SLAC

    2006-06-30

    Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, they show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.

  10. Algebraic sub-structuring for electromagnetic applications

    SciTech Connect

    Yang, Chao; Gao, Weiguo; Bai, Zhaojun; Li, Xiaoye; Lee, Lie-Quan; Husbands, Parry; Ng, Esmond G.

    2004-09-14

    Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, we show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.

  11. Light polarization: A geometric-algebra approach

    NASA Astrophysics Data System (ADS)

    Baylis, W. E.; Bonenfant, J.; Derbyshire, J.; Huschilt, J.

    1993-06-01

    The geometric algebra of three-dimensional space (the ``Pauli algebra'') is known to provide an efficient geometric description of electromagnetic phenomena. Here, it is applied to the three-dimensional Stokes subspace to describe the polarization of an approximately monochromatic collimated beam of electromagnetic radiation. The coherency density ρ is a real element of the algebra whose components are the four Stokes parameters: a scalar representing the total photon flux density plus a three-dimensional vector whose direction and length in the Poincaré sphere give the type and degree of polarization. The detection of the radiation and the incoherent and coherent modification of the polarization by various optical elements are calculated by algebraic multiplication which has faithful representations in 2×2 matrices. One matrix representation of ρ is the coherency matrix with which Jones and Mueller matrices are related whereas another representation is the spin density matrix. However, the calculations are simplest to perform and interpret in the algebraic form independent of any particular matrix representation. It is shown that any possible change in the Stokes parameters can be treated algebraically by a combination of attenuation, depolarization, polarization, and rotation transformations of ρ. The geometric algebra thus unifies Stokes parameters, the Poincaré sphere, Jones and Mueller matrices, and the coherency and density matrices in a single, simple formalism.

  12. Conformal current algebra in two dimensions

    NASA Astrophysics Data System (ADS)

    Ashok, Sujay K.; Benichou, Raphael; Troost, Jan

    2009-06-01

    We construct a non-chiral current algebra in two dimensions consistent with conformal invariance. We show that the conformal current algebra is realized in non-linear sigma-models on supergroup manifolds with vanishing Killing form, with or without a Wess-Zumino term. The current algebra is computed using two distinct methods. First we exploit special algebraic properties of supergroups to compute the exact two- and three-point functions of the currents and from them we infer the current algebra. The algebra is also calculated by using conformal perturbation theory about the Wess-Zumino-Witten point and resumming the perturbation series. We also prove that these models realize a non-chiral Kac-Moody algebra and construct an infinite set of commuting operators that is closed under the action of the Kac-Moody generators. The supergroup models that we consider include models with applications to statistical mechanics, condensed matter and string theory. In particular, our results may help to systematically solve and clarify the quantum integrability of PSU(n|n) models and their cosets, which appear prominently in string worldsheet models on anti-deSitter spaces.

  13. Prediction of Algebraic Instabilities

    NASA Astrophysics Data System (ADS)

    Zaretzky, Paula; King, Kristina; Hill, Nicole; Keithley, Kimberlee; Barlow, Nathaniel; Weinstein, Steven; Cromer, Michael

    2016-11-01

    A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A new methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to the governing differential equation (or dispersion relation) describing wave propagation in dispersive media. Several types of algebraic instabilities are explored in the context of both linear and nonlinear waves.

  14. Connecting Algebra and Chemistry.

    ERIC Educational Resources Information Center

    O'Connor, Sean

    2003-01-01

    Correlates high school chemistry curriculum with high school algebra curriculum and makes the case for an integrated approach to mathematics and science instruction. Focuses on process integration. (DDR)

  15. The effects and interactions of GABAergic and dopaminergic agents in the prevention of form deprivation myopia by brief periods of normal vision.

    PubMed

    Schmid, Katrina L; Strasberg, Gal; Rayner, Cassie L; Hartfield, Perry J

    2013-05-01

    Intravitreal injections of GABA antagonists, dopamine agonists and brief periods of normal vision have been shown separately to inhibit form-deprivation myopia (FDM). Our study had three aims: (i) establish whether GABAergic agents modify the myopia protective effect of normal vision, (ii) investigate the receptor sub-type specificity of any observed effect, and (iii) consider an interaction with the dopamine (DA) system. Prior to the period of normal vision GABAergic agents were applied either (i) individually, (ii) in combination with other GABAergic agents (an agonist with an antagonist), or (iii) in combination with DA agonists and antagonists. Water injections were given to groups not receiving drug treatments so that all experimental eyes received intravitreal injections. As shown previously, constant form-deprivation resulted in high myopia and when diffusers were removed for 2 h per day the period of normal vision greatly reduced the FDM that developed. GABA agonists inhibited the protective effect of normal vision whereas antagonists had the opposite effect. GABAA/C agonists and D2 DA antagonists when used in combination were additive in suppressing the protective effect of normal vision. A D2 DA agonist restored some of the protective effect of normal vision that was inhibited by a GABA agonist (muscimol). The protective effect of normal vision against form-deprivation is modifiable by both the GABAergic and DAergic pathways.

  16. Comparing the Effectiveness of Collaborative Instructional Practices in Algebra

    ERIC Educational Resources Information Center

    Triaga, Russell D.

    2014-01-01

    The use of multiple forms of collaborative instruction to teach integrated algebra makes it difficult for teachers to determine which collaborative form is best suited for the curriculum. An inconsistent approach to integrated algebra instruction at the study school needed to be addressed for the benefit of teacher effectiveness and student…

  17. Twisted vertex algebras, bicharacter construction and boson-fermion correspondences

    NASA Astrophysics Data System (ADS)

    Anguelova, Iana I.

    2013-12-01

    The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence of type D-A. Further, we define a new concept of twisted vertex algebra of order N, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and D-A are isomorphisms of twisted vertex algebras.

  18. 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…

  19. Parastatistics Algebras and Combinatorics

    NASA Astrophysics Data System (ADS)

    Popov, T.

    2005-03-01

    We consider the algebras spanned by the creation parafermionic and parabosonic operators which give rise to generalized parastatistics Fock spaces. The basis of such a generalized Fock space can be labelled by Young tableaux which are combinatorial objects. By means of quantum deformations a nice combinatorial structure of the algebra of the plactic monoid that lies behind the parastatistics is revealed.

  20. Algebraic Reasoning through Patterns

    ERIC Educational Resources Information Center

    Rivera, F. D.; Becker, Joanne Rossi

    2009-01-01

    This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…

  1. 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,…

  2. 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.

  3. 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,…

  4. 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…

  5. Titration Calculations with Computer Algebra Software

    ERIC Educational Resources Information Center

    Lachance, Russ; Biaglow, Andrew

    2012-01-01

    This article examines the symbolic algebraic solution of the titration equations for a diprotic acid, as obtained using "Mathematica," "Maple," and "Mathcad." The equilibrium and conservation equations are solved symbolically by the programs to eliminate the approximations that normally would be performed by the student. Of the three programs,…

  6. Algebraic Nonlinear Collective Motion

    NASA Astrophysics Data System (ADS)

    Troupe, J.; Rosensteel, G.

    1998-11-01

    Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real numberΛ. TheΛ=0 solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear group action on Euclidean space transforms a certain family of deformed droplets among themselves. For positiveΛ, the droplets have a neck that becomes more pronounced asΛincreases; for negativeΛ, the droplets contain a spherical bubble of radius |Λ|1/3. The nonlinear vector field algebra is extended to the nonlinear general collective motion algebra gcm(3) which includes the inertia tensor. The quantum algebraic models of nonlinear nuclear collective motion are given by irreducible unitary representations of the nonlinear gcm(3) Lie algebra. These representations model fissioning isotopes (Λ>0) and bubble and two-fluid nuclei (Λ<0).

  7. Lie n-algebras of BPS charges

    NASA Astrophysics Data System (ADS)

    Sati, Hisham; Schreiber, Urs

    2017-03-01

    We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie ( p + 1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie ( p+1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.

  8. The structure of split regular BiHom-Lie algebras

    NASA Astrophysics Data System (ADS)

    Calderón, Antonio J.; Sánchez, José M.

    2016-12-01

    We introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. We show that an arbitrary split regular BiHom-Lie algebra L is of the form L = U +∑jIj with U a linear subspace of a fixed maximal abelian subalgebra H and any Ij a well described (split) ideal of L, satisfying [Ij ,Ik ] = 0 if j ≠ k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its simple ideals.

  9. Algebraic invariants for homotopy types

    NASA Astrophysics Data System (ADS)

    Blanc, David

    1999-11-01

    We define a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the [Pi]-algebra [pi][low asterisk]X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract [Pi]-algebra can be realized as the homotopy [Pi]-algebra of a space.

  10. 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,…

  11. A process algebra model of QED

    NASA Astrophysics Data System (ADS)

    Sulis, William

    2016-03-01

    The process algebra approach to quantum mechanics posits a finite, discrete, determinate ontology of primitive events which are generated by processes (in the sense of Whitehead). In this ontology, primitive events serve as elements of an emergent space-time and of emergent fundamental particles and fields. Each process generates a set of primitive elements, using only local information, causally propagated as a discrete wave, forming a causal space termed a causal tapestry. Each causal tapestry forms a discrete and finite sampling of an emergent causal manifold (space-time) M and emergent wave function. Interactions between processes are described by a process algebra which possesses 8 commutative operations (sums and products) together with a non-commutative concatenation operator (transitions). The process algebra possesses a representation via nondeterministic combinatorial games. The process algebra connects to quantum mechanics through the set valued process and configuration space covering maps, which associate each causal tapestry with sets of wave functions over M. Probabilities emerge from interactions between processes. The process algebra model has been shown to reproduce many features of the theory of non-relativistic scalar particles to a high degree of accuracy, without paradox or divergences. This paper extends the approach to a semi-classical form of quantum electrodynamics.

  12. Measuring the Readability of Elementary Algebra Using the Cloze Technique.

    ERIC Educational Resources Information Center

    Kulm, Gerald

    The relationship to readability of ten variables characterizing structural properties of mathematical prose was investigated in elementary algebra textbooks. Readability was measured by algebra student's responses to two forms of cloze tests. Linear and currilinear correlations were calculated between each structural variable and the cloze test.…

  13. Towards a cladistics of double Yangians and elliptic algebras*

    NASA Astrophysics Data System (ADS)

    Arnaudon, D.; Avan, J.; Frappat, L.; Ragoucy, E.; Rossi, M.

    2000-09-01

    A self-contained description of algebraic structures, obtained by combinations of various limit procedures applied to vertex and face sl(2) elliptic quantum affine algebras, is given. New double Yangian structures of dynamical type are defined. Connections between these structures are established. A number of them take the form of twist-like actions. These are conjectured to be evaluations of universal twists.

  14. Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations

    NASA Astrophysics Data System (ADS)

    Fortunati, Alessandro; Wiggins, Stephen

    2016-06-01

    The paper deals with the problem of the existence of a normal form for a nearly-integrable real-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly) in time. In particular, in the case of an isochronous integrable part, the system can be cast in an exact normal form, regardless of the properties of the frequency vector. The general case is treated by a suitable adaptation of the finite order normalization techniques usually used for Nekhoroshev arguments. The key point is that the so called "geometric part" is not necessary in this case. As a consequence, no hypotheses on the integrable part are required, apart from analyticity. The work, based on two different perturbative approaches developed by Giorgilli et al., is a generalisation of the techniques used by the same authors to treat more specific aperiodically time-dependent problems.

  15. Weyl n-Algebras

    NASA Astrophysics Data System (ADS)

    Markarian, Nikita

    2017-03-01

    We introduce Weyl n-algebras and show how their factorization complex may be used to define invariants of manifolds. In the appendix, we heuristically explain why these invariants must be perturbative Chern-Simons invariants.

  16. Developing Algebraic Thinking.

    ERIC Educational Resources Information Center

    Alejandre, Suzanne

    2002-01-01

    Presents a teaching experience that resulted in students getting to a point of full understanding of the kinesthetic activity and the algebra behind it. Includes a lesson plan for a traffic jam activity. (KHR)

  17. Jordan Algebraic Quantum Categories

    NASA Astrophysics Data System (ADS)

    Graydon, Matthew; Barnum, Howard; Ududec, Cozmin; Wilce, Alexander

    2015-03-01

    State cones in orthodox quantum theory over finite dimensional complex Hilbert spaces enjoy two particularly essential features: homogeneity and self-duality. Orthodox quantum theory is not, however, unique in that regard. Indeed, all finite dimensional formally real Jordan algebras -- arenas for generalized quantum theories with close algebraic kinship to the orthodox theory -- admit homogeneous self-dual positive cones. We construct categories wherein these theories are unified. The structure of composite systems is cast from universal tensor products of the universal C*-algebras enveloping ambient spaces for the constituent state cones. We develop, in particular, a notion of composition that preserves the local distinction of constituent systems in quaternionic quantum theory. More generally, we explicitly derive the structure of hybrid quantum composites with subsystems of arbitrary Jordan algebraic type.

  18. Accounting Equals Applied Algebra.

    ERIC Educational Resources Information Center

    Roberts, Sondra

    1997-01-01

    Argues that students should be given mathematics credits for completing accounting classes. Demonstrates that, although the terminology is different, the mathematical concepts are the same as those used in an introductory algebra class. (JOW)

  19. Aprepro - Algebraic Preprocessor

    SciTech Connect

    2005-08-01

    Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.

  20. ONEOptimal: A Maple Package for Generating One-Dimensional Optimal System of Finite Dimensional Lie Algebra

    NASA Astrophysics Data System (ADS)

    Miao, Qian; Hu, Xiao-Rui; Chen, Yong

    2014-02-01

    We present a Maple computer algebra package, ONEOptimal, which can calculate one-dimensional optimal system of finite dimensional Lie algebra for nonlinear equations automatically based on Olver's theory. The core of this theory is viewing the Killing form of the Lie algebra as an invariant for the adjoint representation. Some examples are given to demonstrate the validity and efficiency of the program.

  1. The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra

    NASA Astrophysics Data System (ADS)

    Hallowell, Karl; Waldron, Andrew

    2007-09-01

    Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R). These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra! was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.

  2. Biochemical effects in normal and stone forming rats treated with the ripe kernel juice of plantain (musa paradisiaca).

    PubMed

    Devi, V K; Baskar, R; Varalakshmi, P

    1993-01-01

    The effect of Musa paradisiaca stem kernel juice was investigated in experimental urolithiatic rats. Stone forming rats exhibited a significant elevation in the activities of two oxalate synthesizing enzymes - Glycollic acid oxidase and Lactate dehydrogenase. Deposition and excretion of stone forming constituents in kidney and urine were also increased in these rats. The enzyme activities and the level of crystalline components were lowered with the extract treatment. The extract also reduced the activities of urinary alkaline phosphatase, lactate dehydrogenase, r-glutamyl transferase, inorganic pyrophosphatase and β-glucuronidase in calculogenic rats. No appreciable changes were noticed with leucine amino peptidase activity in treated rats.

  3. Temsirolimus induced structural transition of cancerous renal cystatin to normal form in rats: In vitro mechanistic approach underlying renal cancer prevention.

    PubMed

    Shamsi, Anas; Ahmed, Azaj; Bano, Bilqees

    2017-03-01

    Globally, renal cell carcinomas (RCCs) represent a major portion of patients suffering from cancer. Temsirolimus is an anti-renal cancer drug that has already been approved in poor-risk metastatic RCC (mRCC) patients. In our present study, we have evaluated the in vitro effect of varying concentrations of temsirolimus on cancerous rat kidney cystatin; renal cancer was induced in rats making use of dimethylnitrosamine (DMN). It has already been reported that cancerous rat kidney cystatin performs its activity in an efficacious manner as compared to normal rat kidney cystatin, so here an attempt was made to see the effect of temsirolimus on this increased activity of cystatin in renal cancers. Anti-papain activity assay was utilized to see this effect and it was found that temsirolimus reduces the increased activity of cancerous rat kidney cystatin similar to that of normal rat kidney cystatin. Further, to have an insight into temsirolimus induced structural alterations in cancerous rat kidney cystatin; various spectroscopic assays viz. UV, Fluorescence, Circular dichroism (CD) and FTIR spectroscopy were employed. UV and Fluorescence spectroscopy shows cancerous rat kidney cystatin transformation to normal form in the presence of temsirolimus. FTIR and CD spectroscopy confirmed the complete structural reversion of cancerous rat kidney cystatin to normal form in the presence of 40μM temsirolimus. Thus, it can said that temsirolimus causes renal cystatin to revert to normal form; the increased activity of renal cystatin observed in incidences of renal cancer is restored back to normal thereby halting the progression of renal cancer.

  4. Cartan-Weyl 3-algebras and the BLG theory. I: classification of Cartan-Weyl 3-algebras

    NASA Astrophysics Data System (ADS)

    Chu, Chong-Sun

    2010-10-01

    As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H I and a number of step generators E α that are characterized by a root space of non-degenerate one-forms α. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.

  5. Generalizing: The Core of Algebraic Thinking

    ERIC Educational Resources Information Center

    Kinach, Barbara M.

    2014-01-01

    Generalizing--along with conjecturing, representing, justifying, and refuting--are forms of mathematical reasoning important in all branches of mathematics (Lannin, Ellis, and Elliott 2011). Increasingly, however, generalizing is recognized as the essence of thinking in algebra (Mason, Graham, and Johnston-Wilder 2010; Kaput, Carraher, and Blanton…

  6. On a Equation in Finite Algebraically Structures

    ERIC Educational Resources Information Center

    Valcan, Dumitru

    2013-01-01

    Solving equations in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial equation of the form (1). We specify that this equation has been…

  7. 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…

  8. Modules as Learning Tools in Linear Algebra

    ERIC Educational Resources Information Center

    Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff; Loch, Sergio

    2014-01-01

    This paper reports on the experience of STEM and mathematics faculty at four different institutions working collaboratively to integrate learning theory with curriculum development in a core undergraduate linear algebra context. The faculty formed a Professional Learning Community (PLC) with a focus on learning theories in mathematics and…

  9. Weak Lie symmetry and extended Lie algebra

    SciTech Connect

    Goenner, Hubert

    2013-04-15

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

  10. Riemannian manifolds as Lie-Rinehart algebras

    NASA Astrophysics Data System (ADS)

    Pessers, Victor; van der Veken, Joeri

    2016-07-01

    In this paper, we show how Lie-Rinehart algebras can be applied to unify and generalize the elementary theory of Riemannian geometry. We will first review some necessary theory on a.o. modules, bilinear forms and derivations. We will then translate some classical theory on Riemannian geometry to the setting of Rinehart spaces, a special kind of Lie-Rinehart algebras. Some generalized versions of classical results will be obtained, such as the existence of a unique Levi-Civita connection, inducing a Levi-Civita connection on a submanifold, and the construction of spaces with constant sectional curvature.

  11. Characterization of proliferating cell nuclear antigen (PCNA) isoforms in normal and cancer cells: there is no cancer-associated form of PCNA.

    PubMed

    Naryzhny, Stanislav N; Lee, Hoyun

    2007-10-16

    In order to clarify the status of PCNA in normal and transformed cells, we performed analysis of this protein by 2D-PAGE, Western blot and mass spectrometry. All the cell lines examined contained the major PCNA form (pI 4.57/30kDa), that is not post-translationally modified. In addition to the major form, two minor isoforms (pI 4.52/30kDa and pI 4.62/30kDa) were also detected in all the cell lines examined. However, the level of PCNA in cancer cells is 5-6 folds higher than those in primary and most of the immortalized cells. Taken together, the significant difference in PCNA status between cancer and normal cells is not at the post-translational modifications but in the overall levels of PCNA.

  12. Operator algebra in logarithmic conformal field theory

    SciTech Connect

    Nagi, Jasbir

    2005-10-15

    For some time now, conformal field theories in two dimensions have been studied as integrable systems. Much of the success of these studies is related to the existence of an operator algebra of the theory. In this paper, some of the extensions of this machinery to the logarithmic case are studied and used. More precisely, from Moebius symmetry constraints, the generic three- and four-point functions of logarithmic quasiprimary fields are calculated in closed form for arbitrary Jordan rank. As an example, c=0 disordered systems with nondegenerate vacua are studied. With the aid of two-, three-, and four-point functions, the operator algebra is obtained and associativity of the algebra studied.

  13. Graphical methods of psychometrical indicators in Georgian women (20-40 years old) of normal physical development according to forms of temperament and character.

    PubMed

    Nadashvili, L

    2009-04-01

    To establish temperament and forms of character and graphical image, we have studied 120 Georgian women of normal physical development, who were divided by 4 age groups with 5 years intervals. To establish temperament and forms of character we used Eysenck's questionnaire (57 questions) and Sheldon's scale of temperament. The material was worked out by the recognized methods of mathematical psychology. It was stated that Georgian women of young age (20-40 years old) are of sanguine temperament, by character extraverts, which means that they are strong, moving, balanced, stable.

  14. The temperature gradient-forming device, an accessory unit for normal light microscopes to study the biology of hyperthermophilic microorganisms.

    PubMed

    Mora, Maximilian; Bellack, Annett; Ugele, Matthias; Hopf, Johann; Wirth, Reinhard

    2014-08-01

    To date, the behavior of hyperthermophilic microorganisms in their biotope has been studied only to a limited degree; this is especially true for motility. One reason for this lack of knowledge is the requirement for high-temperature microscopy-combined, in most cases, with the need for observations under strictly anaerobic conditions-for such studies. We have developed a custom-made, low-budget device that, for the first time, allows analyses in temperature gradients up to 40°C over a distance of just 2 cm (a biotope-relevant distance) with heating rates up to ∼5°C/s. Our temperature gradient-forming device can convert any upright light microscope into one that works at temperatures as high as 110°C. Data obtained by use of this apparatus show how very well hyperthermophiles are adapted to their biotope: they can react within seconds to elevated temperatures by starting motility-even after 9 months of storage in the cold. Using the temperature gradient-forming device, we determined the temperature ranges for swimming, and the swimming speeds, of 15 selected species of the genus Thermococcus within a few months, related these findings to the presence of cell surface appendages, and obtained the first evidence for thermotaxis in Archaea.

  15. 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

  16. 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…

  17. Current algebra formulation of M-theory based on E11 Kac-Moody algebra

    NASA Astrophysics Data System (ADS)

    Sugawara, Hirotaka

    2017-02-01

    Quantum M-theory is formulated using the current algebra technique. The current algebra is based on a Kac-Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the E11 Kac-Moody algebra that was shown recently1‑5 to contain all the ingredients of M-theory. Both the internal symmetry and the external Lorentz symmetry can be realized inside E11, so that, by constructing the current algebra of E11, I obtain both internal gauge theory and gravity theory. The energy-momentum tensor is constructed as the bilinear form of the currents, yielding a system of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way. The so-called “field-current identity” is built in and, for example, the gravitino field is itself a conserved supercurrent. One unanticipated outcome is that the quantum gravity equation is not identical to the one obtained from the Einstein-Hilbert action.

  18. Adaptive Algebraic Multigrid Methods

    SciTech Connect

    Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J

    2004-04-09

    Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.

  19. Resonant algebras and gravity

    NASA Astrophysics Data System (ADS)

    Durka, R.

    2017-04-01

    The S-expansion framework is analyzed in the context of a freedom in closing the multiplication tables for the abelian semigroups. Including the possibility of the zero element in the resonant decomposition, and associating the Lorentz generator with the semigroup identity element, leads to a wide class of the expanded Lie algebras introducing interesting modifications to the gauge gravity theories. Among the results, we find all the Maxwell algebras of type {{B}m} , {{C}m} , and the recently introduced {{D}m} . The additional new examples complete the resulting generalization of the bosonic enlargements for an arbitrary number of the Lorentz-like and translational-like generators. Some further prospects concerning enlarging the algebras are discussed, along with providing all the necessary constituents for constructing the gravity actions based on the obtained results.

  20. Invariant differential operators for non-compact Lie algebras parabolically related to conformal Lie algebras

    NASA Astrophysics Data System (ADS)

    Dobrev, V. K.

    2013-02-01

    In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G ' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E 7(7) which is parabolically related to the CLA E 7(-25) , the parabolic subalgebras including E 6(6) and E 6(-26). Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so( n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so( n - 1, 1) and its analogs so( p - 1, q - 1). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14) , the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so( p, q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of shadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not.

  1. 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)

  2. A Holistic Approach to Algebra.

    ERIC Educational Resources Information Center

    Barbeau, Edward J.

    1991-01-01

    Described are two examples involving recursive mathematical sequences designed to integrate a holistic approach to learning algebra. These examples promote pattern recognition with algebraic justification, full class participation, and mathematical values that can be transferred to other situations. (MDH)

  3. 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.

  4. Paving the Way To Algebraic Thought Using Residue Designs.

    ERIC Educational Resources Information Center

    Johnson, Iris DeLoach

    1998-01-01

    Presents a brief definition and examples of residue designs while sharing some of the algebraic thought that a student used to form generalizations about the patterns discovered during the investigations of residue designs. (ASK)

  5. Combinatorial Hopf Algebras in Quantum Field Theory I

    NASA Astrophysics Data System (ADS)

    Figueroa, Héctor; Gracia-Bondía, José M.

    This paper stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Sec. 1.1 is the introduction, and contains an elementary invitation to the subject as well. The rest of Sec. 1 is devoted to the basics of Hopf algebra theory and examples in ascending level of complexity. Section 2 turns around the all-important Faà di Bruno Hopf algebra. Section 2.1 contains a first, direct approach to it. Section 2.2 gives applications of the Faà di Bruno algebra to quantum field theory and Lagrange reversion. Section 2.3 rederives the related Connes-Moscovici algebras. In Sec. 3, we turn to the Connes-Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Sec. 3.1, we describe the first. Then in Sec. 3.2, we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Sec. 3.3, general incidence algebras are introduced, and the Faà di Bruno bialgebras are described as incidence bialgebras. In Sec. 3.4, deeper lore on Rota's incidence algebras allows us to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained. The structure results for commutative Hopf algebras are found in Sec. 4. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota-Baxter map in renormalization.

  6. 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.

  7. Three-dimensional polarization algebra.

    PubMed

    R Sheppard, Colin J; Castello, Marco; Diaspro, Alberto

    2016-10-01

    If light is focused or collected with a high numerical aperture lens, as may occur in imaging and optical encryption applications, polarization should be considered in three dimensions (3D). The matrix algebra of polarization behavior in 3D is discussed. It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system, which are in the literature. Explicit transformation matrices for converting the column vector form of these different matrices are extended to the 3D case, where they are large (81×81) but can be generated using simple rules. It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell-Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily. Explicit expressions are given for 3D complex field components in terms of Chandrasekhar-Stokes parameters.

  8. Sources of variation in growth, form, and survival in dwarf and normal-stature pitch pines (Pinus rigida, Pinaceae) in long-term transplant experiments.

    PubMed

    Fang, Wei; Taub, Daniel R; Fox, Gordon A; Landis, R Matthew; Natali, Susan; Gurevitch, Jessica

    2006-08-01

    Determining the relative contributions of genetic and environmental factors to phenotypic variation is critical for understanding the evolutionary ecology of plant species, but few studies have examined the sources of phenotypic differentiation between nearby populations of woody plants. We conducted reciprocal transplant experiments to examine sources of variation in growth rate, form, survival, and maturation in a globally rare dwarf population of pitch pine (Pinus rigida) and in surrounding populations of normal-stature pitch pines on Long Island, New York. Transplants were monitored over a 6-yr period. The influence of seedling origin on height, growth rate, survival, and form (single-stemmed vs. multi-stemmed growth habit) was much smaller than the effect of transplanting location. Both planting site and seed origin were important factors in determining time to reproduction; seedlings originating from dwarf populations and seedlings planted at the normal-stature site reproduced earliest. These results suggest that many of the differences between dwarf and normal-stature pitch pines may be due more to plastic responses to environmental factors than to genetic differentiation among populations. Therefore, preservation of the dwarf pine habitat is essential for preserving dwarf pine communities; the dwarf pines cannot be preserved ex situ.

  9. From Arithmetic to Algebra

    ERIC Educational Resources Information Center

    Ketterlin-Geller, Leanne R.; Jungjohann, Kathleen; Chard, David J.; Baker, Scott

    2007-01-01

    Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…

  10. Computers in Abstract Algebra

    ERIC Educational Resources Information Center

    Nwabueze, Kenneth K.

    2004-01-01

    The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…

  11. 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)

  12. Computer Algebra versus Manipulation

    ERIC Educational Resources Information Center

    Zand, Hossein; Crowe, David

    2004-01-01

    In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…

  13. Regulation of cell growth by Notch signaling and its differential requirement in normal vs. tumor-forming stem cells in Drosophila.

    PubMed

    Song, Yan; Lu, Bingwei

    2011-12-15

    Cancer stem cells (CSCs) are postulated to be a small subset of tumor cells with tumor-initiating ability that shares features with normal tissue-specific stem cells. The origin of CSCs and the mechanisms underlying their genesis are poorly understood, and it is uncertain whether it is possible to obliterate CSCs without inadvertently damaging normal stem cells. Here we show that a functional reduction of eukaryotic translation initiation factor 4E (eIF4E) in Drosophila specifically eliminates CSC-like cells in the brain and ovary without having discernable effects on normal stem cells. Brain CSC-like cells can arise from dedifferentiation of transit-amplifying progenitors upon Notch hyperactivation. eIF4E is up-regulated in these dedifferentiating progenitors, where it forms a feedback regulatory loop with the growth regulator dMyc to promote cell growth, particularly nucleolar growth, and subsequent ectopic neural stem cell (NSC) formation. Cell growth regulation is also a critical component of the mechanism by which Notch signaling regulates the self-renewal of normal NSCs. Our findings highlight the importance of Notch-regulated cell growth in stem cell maintenance and reveal a stronger dependence on eIF4E function and cell growth by CSCs, which might be exploited therapeutically.

  14. Poisson and symplectic structures on Lie algebras. I

    NASA Astrophysics Data System (ADS)

    Alekseevsky, D. V.; Perelomov, A. M.

    1997-06-01

    The purpose of this paper is to describe a new class of Poisson and symplectic structures on Lie algebras. This gives a new class of solutions of the classical Yang-Baxter equation. The class of elementary Lie algebras is defined and the Poisson and symplectic structures for them are described. The algorithm is given for description of all closed 2-forms and of symplectic structures on any Lie algebra G, which is decomposed into semidirect sum of elementary subalgebras. Using these results we obtain the description of closed 2-forms and symplectic forms (if they exist) on the Borel subalgebra B(G) of semisimple Lie algebra G. As a byproduct, we get description of the second cohomology group H2( B( G)).

  15. LINPACK. Simultaneous Linear Algebraic Equations

    SciTech Connect

    Miller, M.A.

    1990-05-01

    LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic equations. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).

  16. LINPACK. Simultaneous Linear Algebraic Equations

    SciTech Connect

    Dongarra, J.J.

    1982-05-02

    LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic equations. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).

  17. The pathophysiology of the aqueduct stroke volume in normal pressure hydrocephalus: can co-morbidity with other forms of dementia be excluded?

    PubMed

    Bateman, Grant A; Levi, Christopher R; Schofield, Peter; Wang, Yang; Lovett, Elizabeth C

    2005-10-01

    Variable results are obtained from the treatment of normal pressure hydrocephalus (NPH) by shunt insertion. There is a high correlation between NPH and the pathology of Alzheimer's disease (AD) on brain biopsy. There is an overlap between AD and vascular dementia (VaD), suggesting that a correlation exists between NPH and other forms of dementia. This study seeks to (1) understand the physiological factors behind, and (2) define the ability of, the aqueduct stroke volume to exclude dementia co-morbidity. Twenty-four patients from a dementia clinic were classified as having either early AD or VaD on the basis of clinical features, Hachinski score and neuropsychological testing. They were compared with 16 subjects with classical clinical findings of NPH and 12 aged-matched non-cognitively impaired subjects. MRI flow quantification was used to measure aqueduct stroke volume and arterial pulse volume. An arterio-cerebral compliance ratio was calculated from the two volumes in each patient. The aqueduct stroke volume was elevated in all three forms of dementia, with no significant difference noted between the groups. The arterial pulse volume was elevated by 24% in VaD and reduced by 35% in NPH, compared to normal (P = 0.05 and P = 0.002, respectively), and was normal in AD. There was a spectrum of relative compliance with normal compliance in VaD and reduced compliance in AD and NPH. The aqueduct stroke volume depends on the arterial pulse volume and the relative compliance between the arterial tree and brain. The aqueduct stroke volume cannot exclude significant co-morbidity in NPH.

  18. Algebraic connectivity and graph robustness.

    SciTech Connect

    Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.

    2009-07-01

    Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.

  19. On Dunkl angular momenta algebra

    NASA Astrophysics Data System (ADS)

    Feigin, Misha; Hakobyan, Tigran

    2015-11-01

    We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.

  20. Double conformal space-time algebra

    NASA Astrophysics Data System (ADS)

    Easter, Robert Benjamin; Hitzer, Eckhard

    2017-01-01

    The Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra G 4,8that extends the concepts introduced with the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA) G 8,2 with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the Space-Time Algebra (STA) G 1,3. Two Conformal Space-Time subalgebras (CSTA) G 2,4 provide spacetime entities for points, flats (incl. worldlines), and hyperbolics, and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reflection, and (pseudo)spherical inversion in rounds or hyperbolics. The DCSTA G 4,8 is a doubling product of two G 2,4 CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new bivector entities for (pseudo)quadrics and Darboux (pseudo)cyclides in spacetime that are also transformed by the doubled versors. The "pseudo" surface entities are spacetime hyperbolics or other surface entities using the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in rounds or hyperbolics. An operation for the directed non-uniform scaling (anisotropic dilation) of the bivector general quadric entities is defined using the boost operator and a spatial projection. DCSTA allows general quadric surfaces to be transformed in spacetime by the same complete set of doubled CSTA versor (i.e., DCSTA versor) operations that are also valid on the doubled CSTA point entity (i.e., DCSTA point) and the other doubled CSTA entities. The new DCSTA bivector entities are formed by extracting values from the DCSTA point entity using specifically defined inner product extraction operators. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the length

  1. Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras

    SciTech Connect

    Marquette, Ian

    2013-07-15

    We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.

  2. The Propositional Logic Induced by Means of Basic Algebras

    NASA Astrophysics Data System (ADS)

    Chajda, I.

    2015-12-01

    A propositional logic induced by means of commutative basic algebras was already described by M. Botur and R. Halaš. It turns out that this is a kind of non-associative fuzzy logic which can be used e.g. in expert systems. Unfortunately, there are other important classes of basic algebras which are not commutative, e.g. orthomodular lattices which are used as an axiomatization of the logic of quantum mechanics. This motivated us to develop another axioms and derivation rules which form a propositional logic induced by basic algebras in general. We show that this logic is algebraizable in the sense of W. J. Blok and D. Pigozzi.

  3. The soluble form of Alzheimer's amyloid beta protein is complexed to high density lipoprotein 3 and very high density lipoprotein in normal human plasma.

    PubMed

    Koudinov, A; Matsubara, E; Frangione, B; Ghiso, J

    1994-12-15

    The amyloid fibrils of Alzheimer's neuritic plaques and cerebral blood vessels are mainly composed of aggregated forms of a 39 to 44 amino acids peptide, named amyloid beta (A beta). A similar although soluble form of A beta (sA beta) has been identified in plasma, cerebrospinal fluid and cell culture supernatants, indicating that it is produced under physiologic conditions. We report here that sA beta in normal human plasma is associated with lipoprotein particles, in particular to the HDL3 and VHDL fractions where it is complexed to ApoJ and, to a lesser extent, to ApoAI. This was assessed by immunoprecipitation experiments of purified plasma lipoproteins and lipoprotein-depleted plasma and confirmed by means of amino acid sequence analysis. Moreover, biotinylated synthetic peptide A beta 1-40 was traced in normal human plasma in in vitro experiments. As in the case of sA beta, biotinylated A beta 1-40 was specifically recovered in the HDL3 and VHDL fractions. This data together with the previous demonstration that A beta 1-40 is taken up into the brain via a specific mechanism and possibly as an A beta 1-40-ApoJ complex indicate a role for HDL3- and VHDL-containing ApoJ in the transport of the peptide in circulation and suggest their involvement in the delivery of sA beta across the blood-brain barrier.

  4. Algebraic Multigrid Benchmark

    SciTech Connect

    2013-05-06

    AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.

  5. Algebra of Majorana doubling.

    PubMed

    Lee, Jaehoon; Wilczek, Frank

    2013-11-27

    Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.

  6. 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®,…

  7. Priority in Process Algebras

    NASA Technical Reports Server (NTRS)

    Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.

    1999-01-01

    This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.

  8. Algebraic and geometric structures of analytic partial differential equations

    NASA Astrophysics Data System (ADS)

    Kaptsov, O. V.

    2016-11-01

    We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.

  9. Pawlak algebra and approximate structure on fuzzy lattice.

    PubMed

    Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai

    2014-01-01

    The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties.

  10. 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.

  11. Beyond the Schwinger boson representation of the su(2)-algebra

    NASA Astrophysics Data System (ADS)

    Tsue, Yasuhiko; Providência, Constança; da Providência, João; Yamamura, Masatoshi

    2015-04-01

    With the use of two kinds of boson operators, a new boson representation of the su(2)-algebra is proposed. The basic idea comes from the pseudo su(1,1)-algebra recently given by the present authors [Y. Tsue et al., Prog. Theor. Exp. Phys. 2013, 103D04 (2013)]. It forms a striking contrast to the Schwinger boson representation of the su(2)-algebra, which is also based on two kinds of bosons. It is proved that this new boson representation obeys the su(2)-algebra in a certain subspace in the whole boson space constructed by the Schwinger boson representation of the su(1,1)-algebra. This representation may be suitable for describing the time dependence of the system interacting with the external environment in the framework of the thermo field dynamics formalism, i.e., phase space doubling. Further, several deformations related to the su(2)-algebra in this boson representation are discussed. On the basis of these deformed algebras, various types of time evolution of a simple boson system are investigated.

  12. How Structure Sense for Algebraic Expressions or Equations Is Related to Structure Sense for Abstract Algebra

    ERIC Educational Resources Information Center

    Novotna, Jarmila; Hoch, Maureen

    2008-01-01

    Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…

  13. Asymptotics of bivariate generating functions with algebraic singularities

    NASA Astrophysics Data System (ADS)

    Greenwood, Torin

    Flajolet and Odlyzko (1990) derived asymptotic formulae the coefficients of a class of uni- variate generating functions with algebraic singularities. Gao and Richmond (1992) and Hwang (1996, 1998) extended these results to classes of multivariate generating functions, in both cases by reducing to the univariate case. Pemantle and Wilson (2013) outlined new multivariate ana- lytic techniques and used them to analyze the coefficients of rational generating functions. After overviewing these methods, we use them to find asymptotic formulae for the coefficients of a broad class of bivariate generating functions with algebraic singularities. Beginning with the Cauchy integral formula, we explicity deform the contour of integration so that it hugs a set of critical points. The asymptotic contribution to the integral comes from analyzing the integrand near these points, leading to explicit asymptotic formulae. Next, we use this formula to analyze an example from current research. In the following chapter, we apply multivariate analytic techniques to quan- tum walks. Bressler and Pemantle (2007) found a (d + 1)-dimensional rational generating function whose coefficients described the amplitude of a particle at a position in the integer lattice after n steps. Here, the minimal critical points form a curve on the (d + 1)-dimensional unit torus. We find asymptotic formulae for the amplitude of a particle in a given position, normalized by the number of steps n, as n approaches infinity. Each critical point contributes to the asymptotics for a specific normalized position. Using Groebner bases in Maple again, we compute the explicit locations of peak amplitudes. In a scaling window of size the square root of n near the peaks, each amplitude is asymptotic to an Airy function.

  14. A Topos for Algebraic Quantum Theory

    NASA Astrophysics Data System (ADS)

    Heunen, Chris; Landsman, Nicolaas P.; Spitters, Bas

    2009-10-01

    The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos {mathcal{T}(A)} in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra {A} . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum {\\underline{Σ}(A)} in {mathcal{T}(A)} , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on {\\underline{Σ}} , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from {\\underline{Σ}} to Scott’s interval domain. Noting that open subsets of {\\underline{Σ}(A)} correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos {mathcal{T}(A)}. These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.

  15. 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…

  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. Statecharts Via Process Algebra

    NASA Technical Reports Server (NTRS)

    Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance

    1999-01-01

    Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics

  18. 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.

  19. 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.

  20. 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.

  1. 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…

  2. Patterns to Develop Algebraic Reasoning

    ERIC Educational Resources Information Center

    Stump, Sheryl L.

    2011-01-01

    What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…

  3. Viterbi/algebraic hybrid decoder

    NASA Technical Reports Server (NTRS)

    Boyd, R. W.; Ingels, F. M.; Mo, C.

    1980-01-01

    Decoder computer program is hybrid between optimal Viterbi and optimal algebraic decoders. Tests have shown that hybrid decoder outperforms any strictly Viterbi or strictly algebraic decoder and effectively handles compound channels. Algorithm developed uses syndrome-detecting logic to direct two decoders to assume decoding load alternately, depending on real-time channel characteristics.

  4. Online Algebraic Tools for Teaching

    ERIC Educational Resources Information Center

    Kurz, Terri L.

    2011-01-01

    Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)

  5. 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…

  6. Linear algebra and image processing

    NASA Astrophysics Data System (ADS)

    Allali, Mohamed

    2010-09-01

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

  7. 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.)

  8. 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…

  9. 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…

  10. 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…

  11. Tsunami Flow Speed Estimates Using Inverse Modeling Of Normally-Graded Sandy Deposits Formed During The 29 September 2009 Tsunami Near Satitoa, East Upolu, Samoa

    NASA Astrophysics Data System (ADS)

    Jaffe, B. E.; Buckley, M. L.; Richmond, B. M.; Strotz, L. C.; Etienne, S.; Clark, K.; Gelfenbaum, G. R.

    2010-12-01

    Sandy deposits formed by the 29 September 2009 tsunami on the southeast coast of Upolu, Samoa were investigated to document their sedimentary characteristics and learn what information about the tsunami could be extracted from them. Deposits formed from ~25 to ~250 m inland of the shoreline, landward of a zone of erosion and were from 6 to 15 cm thick. We interpret the deposits to be composed of 2 layers, formed by the uprush from 2 waves, based on vertical changes in the grain size distribution, contacts, and vertical variation in shell content. Deposits at 3 locations along a shore-normal trancect (100, 170, and 240 m inland from the shoreline) were predominately normally graded (~80%), but contained massive sections (~15%) and inversely graded sections (~5%). About 75% of the normally graded intervals exhibit a signature of sediment falling out of suspension, termed distribution grading by Middleton (1967). Distribution grading describes a shift in the distribution to finer sizes moving upward in the deposit as there is a loss of coarser sediment and a gain in finer sediment. This shift occurs because grains with higher settling velocities (larger for a given density and shape) deposit first and are absent in the water column during the later stages of deposition. The grains with lower settling velocities take longer to reach the bed and absent from bottom of deposit and present in the top of the deposit. The Jaffe and Gelfenbaum (2007) inverse sediment transport model was applied to deposit intervals with distribution grading to estimate tsunami flow speed. Using a Manning’s roughness coefficient, n, of 0.03 (z0 ~0.006 m) flow speeds for the top and bottom layers were from 3.6 to 3.8 m/s and from 4.1 to 4.4 m/s, respectively. Froude numbers calculated using these estimates and measured flow depths range from 0.69 to 1.01, which is consistent with Froude numbers from other studies of modern tsunamis. This research underscores the importance of modeling only

  12. Automorphic correction of the hyperbolic Kac-Moody algebra E10

    NASA Astrophysics Data System (ADS)

    Kim, Henry H.; Lee, Kyu-Hwan

    2013-09-01

    In this paper, we study automorphic correction of the hyperbolic Kac-Moody algebra E10, using the Borcherds product for O(10, 2) attached to a weakly holomorphic modular form of weight -4 for SL_2({Z}). We also clarify some aspects of automorphic correction for Lorentzian Kac-Moody algebras and give heuristic reasons for the expectation that every Lorentzian Kac-Moody algebra has an automorphic correction.

  13. Symplectic Clifford Algebraic Field Theory.

    NASA Astrophysics Data System (ADS)

    Dixon, Geoffrey Moore

    We develop a mathematical framework on which is built a theory of fermion, scalar, and gauge vector fields. This field theory is shown to be equivalent to the original Weinberg-Salam model of weak and electromagnetic interactions, but since the new framework is more rigid than that on which the original Weinberg-Salam model was built, a concomitant reduction in the number of assumptions lying outside of the framework has resulted. In particular, parity violation is actually hiding within our framework, and with little difficulty we are able to manifest it. The mathematical framework upon which we build our field theory is arrived at along two separate paths. The first is by the marriage of a Clifford algebra and a Lie superalgebra, the result being called a super Clifford algebra. The second is by providing a new characterization for a Clifford algebra employing its generators and a symmetric array of metric coefficients. Subsequently we generalize this characterization to the case of an antisymmetric array of metric coefficients, and we call the algebra which results a symplectic Clifford algebra. It is upon one of these that we build our field theory, and it is shown that this symplectic Clifford algebra is a particular subalgebra of a super Clifford algebra. The final ingredient is the operation of bracketing which involves treating the elements of our algebra as endomorphisms of a particular inner product space, and employing this space and its inner product to provide us with maps from our algebra to the reals. It is this operation which enables us to manifest the parity violation hiding in our algebra.

  14. 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.

  15. 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)

  16. Quantum algebra of N superspace

    SciTech Connect

    Hatcher, Nicolas; Restuccia, A.; Stephany, J.

    2007-08-15

    We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.

  17. 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.

  18. Algebraic distance on graphs.

    SciTech Connect

    Chen, J.; Safro, I.

    2011-01-01

    Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.

  19. ISM and dust properties of normal star-forming galaxies at z~2 derived by Herschel and ALMA with the help of gravitational lensing

    NASA Astrophysics Data System (ADS)

    Schaerer, Daniel; Boone, Frederic; Dessauges-Zavadsky, Miroslava; Sklias, Panos

    2015-08-01

    Using strong gravitational lensing provided by massive galaxy clusters we have studied a sample of normal star-forming galaxies at z~1.5-3 selected from the Herschel Lensing Survey (HLS). The observations include deep ground-based, HST, Spitzer, and Herschel imaging, plus LABOCA/SCUBA2 data, and IRAM CO observations.Targetted [CII] 158 micron observations of one z=2.013 galaxy from this sample were recently obtained with ALMA, resulting in the first detection of this important ISM cooling line in a faint LIRG (with LIR~1.e11 Lsun), which is magnified by a factor ~50.We discuss the behavior of [CII] and CO emission with other physical properties such as IR luminosity, dust temperature, galaxy metallicity, specific star formation rate, and many other quantities which are measured for our lensed galaxies. We also compare the z~2 data to nearby galaxies and to recent detections and upper limits of [CII] in z>6 Lyman break galaxies and Lyman alpha emitters.

  20. A validated stability-indicating normal phase LC method for clopidogrel bisulfate and its impurities in bulk drug and pharmaceutical dosage form.

    PubMed

    Durga Rao, Dantu; Kalyanaraman, L; Sait, Shakil S; Venkata Rao, P

    2010-05-01

    A novel stability-indicating normal phase liquid chromatographic (NP-LC) method was developed for the determination of purity of clopidogrel drug substance and drug products in bulk samples and pharmaceutical dosage forms in the presence of its impurities and degradation products. This method is capable of separating all the related substances of clopidogrel along with the chiral impurities. This method can be also be used for the estimation of assay of clopidogrel in drug substance as well as in drug product. The method was developed using Chiralcel OJ-H (250mmx4.6mm, 5microm) column. n-Hexane, ethanol and diethyl amine in 95:5:0.05 (v/v/v) ratio was used as a mobile phase. The eluted compounds were monitored at 240nm. Clopidogrel bisulfate was subjected to the stress conditions of oxidative, acid, base, hydrolytic, thermal and photolytic degradation. The degradation products were well resolved from main peak and its impurities, proving the stability-indicating power of the method. The developed method was validated as per International Conference on Harmonization (ICH) guidelines with respect to specificity, limit of detection, limit of quantification, precision, linearity, accuracy, robustness and system suitability.

  1. Suppression of the inducible form of nitric oxide synthase prior to traumatic brain injury improves cytochrome c oxidase activity and normalizes cellular energy levels.

    PubMed

    Hüttemann, M; Lee, I; Kreipke, C W; Petrov, T

    2008-01-02

    We have previously shown that the observed immediate increase in nitric oxide (NO) plays a significant role in the control of the cerebral microcirculation following traumatic brain injury (TBI). However, a second consequence of increased NO production after TBI may be impaired mitochondrial function, due to the fact that NO is a well-known inhibitor of cytochrome c oxidase (CcO). CcO is a key enzyme of the mitochondrial oxidative phosphorylation (OxPhos) machinery, which creates cellular energy in the form of ATP. NO competes with oxygen at the heme a(3)-Cu(B) reaction center of CcO. We thus hypothesized that TBI triggers inhibition of CcO, which would in turn lead to a decreased energy production by OxPhos at a time of an elevated energy demand for tissue remodeling. Here we show that TBI as induced by an acceleration weight drop model of diffuse brain injury in rats leads to CcO inhibition and dramatically decreased ATP levels in brain cortex. CcO inhibition can be partially restored by application of iNOS antisense oligonucleotides prior to TBI, which leads to a normalization of ATP levels similar to the controls. We propose that a lack of energy after TBI caused by inhibition of CcO is an important aspect of trauma pathology.

  2. 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.

  3. Investigating Teacher Noticing of Student Algebraic Thinking

    ERIC Educational Resources Information Center

    Walkoe, Janet Dawn Kim

    2013-01-01

    Learning algebra is critical for students in the U.S. today. Algebra concepts provide the foundation for much advanced mathematical content. In addition, algebra serves as a gatekeeper to opportunities such as admission to college. Yet many students in the U.S. struggle in algebra classes. Researchers claim that one reason for these difficulties…

  4. Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?

    NASA Astrophysics Data System (ADS)

    Guttmann, A. J.; Jensen, I.; Maillard, J.-M.; Pantone, J.

    2016-12-01

    possible differentially algebraic forms.

  5. Central extensions of Lax operator algebras

    NASA Astrophysics Data System (ADS)

    Schlichenmaier, M.; Sheinman, O. K.

    2008-08-01

    Lax operator algebras were introduced by Krichever and Sheinman as a further development of Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this paper local cocycles and associated almost-graded central extensions of Lax operator algebras are classified. It is shown that in the case when the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives.

  6. Kac-Moody Algebra for Two Dimensional Principal Chiral Models

    NASA Astrophysics Data System (ADS)

    Chou, Kuang-Chao; Song, Xing-Chang

    A Darboux transformation depending on single continuous parameter t is constructed for a principal chiral field. The transformation forms a nonlinear representation of the group for any fixed value of t. Part of the kernel in the Riemann-Hilbert transform is shown to be related to the Darboux transformation with its generators forming a Kac-Moody algebra. Conserved currents associated with the Kac-Moody algebra of the linearized equations and the Nöether current for the group transformations with fixed value of t are obtained.

  7. Description of DASSL: a differential/algebraic system solver

    SciTech Connect

    Petzold, L.R.

    1982-09-01

    This paper describes a new code DASSL, for the numerical solution of implicit systems of differential/algebraic equations. These equations are written in the form F(t,y,y') = 0, and they can include systems which are substantially more complex than standard form ODE systems y' = f(t,y). Differential/algebraic equations occur in several diverse applications in the physical world. We outline the algorithms and strategies used in DASSL, and explain some of the features of the code. In addition, we outline briefly what needs to be done to solve a problem using DASSL.

  8. Charge transfer in algebraic quantum field theory

    NASA Astrophysics Data System (ADS)

    Wright, Jill Dianne

    We discuss aspects of the algebraic structure of quantum field theory. We take the view that the superselection structure of a theory should be determinable from the vacuum representation of the observable algebra, and physical properties of the charge. Hence one determines the nature of the charge transfer operations: the automorphisms of the observable algebra corresponding to the movement of charge along space-time paths. New superselection sectors are obtained from the vacuum sector by an automorphism which is a limit of charge transfer operations along paths with an endpoint tending to spacelike infinity. Roberts has shown that for a gauge theory of the first kind, the charge transfer operations for a given charge form a certain kind of 1-cocycle over Minkowski space. The local 1-cohomology group of their equivalence classes corresponds to the superselection structure. The exact definition of the cohomology group depends on the properties of the charge. Using displaced Fock representations of free fields, we develop model field theories which illustrate this structure. The cohomological classification of displaced Fock representations has been elucidated by Araki. For more general representations, explicit determination of the cohomology group is a hard problem. Using our models, we can illustrate ways in which fields with reasonable physical properties depart fromthe abovementioned structure. In 1+1 dimensions, we use the Streater-Wilde model to illustrate explicitly the representation-dependence of the cohomology structure, and the direction-dependence of the limiting charge transfer operation. The cohomology structure may also be representation-dependent in higher-dimensional theories without strict localization of charge, for example the electromagnetic field. The algebraic structure of the electromagnetic field has many other special features, which we discuss in relation to the concept of charge transfer. We also give some indication of the modifications

  9. Connected Representations of Knowledge: Do Undergraduate Students Relate Algebraic Rational Expressions to Rational Numbers?

    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 students' postsecondary success as STEM majors. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. In the present study, the connections…

  10. 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.

  11. Cartooning in Algebra and Calculus

    ERIC Educational Resources Information Center

    Moseley, L. Jeneva

    2014-01-01

    This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.

  12. 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.

  13. Path integral quantization corresponding to the deformed Heisenberg algebra

    SciTech Connect

    Pramanik, Souvik; Moussa, Mohamed; Faizal, Mir; Ali, Ahmed Farag

    2015-11-15

    In this paper, the deformation of the Heisenberg algebra, consistent with both the generalized uncertainty principle and doubly special relativity, has been analyzed. It has been observed that, though this algebra can give rise to fractional derivative terms in the corresponding quantum mechanical Hamiltonian, a formal meaning can be given to them by using the theory of harmonic extensions of function. Depending on this argument, the expression of the propagator of the path integral corresponding to the deformed Heisenberg algebra, has been obtained. In particular, the consistent expression of the one dimensional free particle propagator has been evaluated explicitly. With this propagator in hand, it has been shown that, even in free particle case, normal generalized uncertainty principle and doubly special relativity show very much different result.

  14. ALMA Observation of 158 μm [C II] Line and Dust Continuum of a z = 7 Normally Star-forming Galaxy in the Epoch of Reionization

    NASA Astrophysics Data System (ADS)

    Ota, Kazuaki; Walter, Fabian; Ohta, Kouji; Hatsukade, Bunyo; Carilli, Chris L.; da Cunha, Elisabete; González-López, Jorge; Decarli, Roberto; Hodge, Jacqueline A.; Nagai, Hiroshi; Egami, Eiichi; Jiang, Linhua; Iye, Masanori; Kashikawa, Nobunari; Riechers, Dominik A.; Bertoldi, Frank; Cox, Pierre; Neri, Roberto; Weiss, Axel

    2014-09-01

    We present ALMA observations of the [C II] line and far-infrared (FIR) continuum of a normally star-forming galaxy in the reionization epoch, the z = 6.96 Lyα emitter (LAE) IOK-1. Probing to sensitivities of σline = 240 μJy beam-1 (40 km s-1 channel) and σcont = 21 μJy beam-1, we found the galaxy undetected in both [C II] and continuum. Comparison of ultraviolet (UV)-FIR spectral energy distribution (SED) of IOK-1, including our ALMA limit, with those of several types of local galaxies (including the effects of the cosmic microwave background, CMB, on the FIR continuum) suggests that IOK-1 is similar to local dwarf/irregular galaxies in SED shape rather than highly dusty/obscured galaxies. Moreover, our 3σ FIR continuum limit, corrected for CMB effects, implies intrinsic dust mass M dust < 6.4 × 107 M ⊙, FIR luminosity L FIR < 3.7 × 1010 L ⊙ (42.5-122.5 μm), total IR luminosity L IR < 5.7 × 1010 L ⊙ (8-1000 μm), and dust-obscured star formation rate (SFR) < 10 M ⊙ yr-1, if we assume that IOK-1 has a dust temperature and emissivity index typical of local dwarf galaxies. This SFR is 2.4 times lower than one estimated from the UV continuum, suggesting that <29% of the star formation is obscured by dust. Meanwhile, our 3σ [C II] flux limit translates into [C II] luminosity, L [C II] < 3.4 × 107 L ⊙. Locations of IOK-1 and previously observed LAEs on the L [C II] versus SFR and L [C II]/L FIR versus L FIR diagrams imply that LAEs in the reionization epoch have significantly lower gas and dust enrichment than AGN-powered systems and starbursts at similar/lower redshifts, as well as local star-forming galaxies. Based in part on data collected with the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan; observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc

  15. ALMA observation of 158 μm [C II] line and dust continuum of a z = 7 normally star-forming galaxy in the epoch of reionization

    SciTech Connect

    Ota, Kazuaki; Walter, Fabian; Da Cunha, Elisabete; González-López, Jorge; Decarli, Roberto; Hodge, Jacqueline A.; Ohta, Kouji; Hatsukade, Bunyo; Nagai, Hiroshi; Iye, Masanori; Kashikawa, Nobunari; Carilli, Chris L.; Egami, Eiichi; Jiang, Linhua; Riechers, Dominik A.; Bertoldi, Frank; Cox, Pierre; Neri, Roberto; Weiss, Axel

    2014-09-01

    We present ALMA observations of the [C II] line and far-infrared (FIR) continuum of a normally star-forming galaxy in the reionization epoch, the z = 6.96 Lyα emitter (LAE) IOK-1. Probing to sensitivities of σ{sub line} = 240 μJy beam{sup –1} (40 km s{sup –1} channel) and σ{sub cont} = 21 μJy beam{sup –1}, we found the galaxy undetected in both [C II] and continuum. Comparison of ultraviolet (UV)-FIR spectral energy distribution (SED) of IOK-1, including our ALMA limit, with those of several types of local galaxies (including the effects of the cosmic microwave background, CMB, on the FIR continuum) suggests that IOK-1 is similar to local dwarf/irregular galaxies in SED shape rather than highly dusty/obscured galaxies. Moreover, our 3σ FIR continuum limit, corrected for CMB effects, implies intrinsic dust mass M {sub dust} < 6.4 × 10{sup 7} M {sub ☉}, FIR luminosity L {sub FIR} < 3.7 × 10{sup 10} L {sub ☉} (42.5-122.5 μm), total IR luminosity L {sub IR} < 5.7 × 10{sup 10} L {sub ☉} (8-1000 μm), and dust-obscured star formation rate (SFR) < 10 M {sub ☉} yr{sup –1}, if we assume that IOK-1 has a dust temperature and emissivity index typical of local dwarf galaxies. This SFR is 2.4 times lower than one estimated from the UV continuum, suggesting that <29% of the star formation is obscured by dust. Meanwhile, our 3σ [C II] flux limit translates into [C II] luminosity, L {sub [C} {sub II]} < 3.4 × 10{sup 7} L {sub ☉}. Locations of IOK-1 and previously observed LAEs on the L {sub [C} {sub II]} versus SFR and L {sub [C} {sub II]}/L {sub FIR} versus L {sub FIR} diagrams imply that LAEs in the reionization epoch have significantly lower gas and dust enrichment than AGN-powered systems and starbursts at similar/lower redshifts, as well as local star-forming galaxies.

  16. Coherent States for Hopf Algebras

    NASA Astrophysics Data System (ADS)

    Škoda, Zoran

    2007-07-01

    Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. If, in addition, the Hopf algebra has a left Haar integral, then a formula for noncommutative resolution of identity in terms of the family of coherent states holds. Examples come from quantum groups.

  17. Multiplier operator algebras and applications

    PubMed Central

    Blecher, David P.; Zarikian, Vrej

    2004-01-01

    The one-sided multipliers of an operator space X are a key to “latent operator algebraic structure” in X. We begin with a survey of these multipliers, together with several of the applications that they have had to operator algebras. We then describe several new results on one-sided multipliers, and new applications, mostly to one-sided M-ideals. PMID:14711990

  18. Hopf algebras and topological recursion

    NASA Astrophysics Data System (ADS)

    Esteves, João N.

    2015-11-01

    We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).

  19. Wakimoto realizations of current algebras: an explicit construction

    SciTech Connect

    de Boer, Jan; Feher, Laszlo

    1996-11-12

    A generalized Wakimoto realization of $\\widehat\\cal G_K$ can be associated with each parabolic subalgebra $\\cal P=(\\cal G_0 +\\cal G_+)$ of a simple Lie algebra $\\cal G$ according to an earlier proposal by Feigin and Frenkel. In this paper the proposal is made explicit by developing the construction of Wakimoto realizations from a simple but unconventional viewpoint. An explicit formula is derived for the Wakimoto current first at the Poisson bracket level by Hamiltonian symmetry reduction of the WZNW model. The quantization is then performed by normal ordering the classical formula and determining the required quantum correction for it to generate $\\widehat\\cal G_K$ by means of commutators. The affine-Sugawara stress-energy tensor is verified to have the expected quadratic form in the constituents, which are symplectic bosons belonging to $\\cal G_+$ and a current belonging to $\\cal G_0$. The quantization requires a choice of special polynomial coordinates on the big cell of the flag manifold $P\\backslash G$. The effect of this choice is investigated in detail by constructing quantum coordinate transformations. Finally, the explicit form of the screening charges for each generalized Wakimoto realization is determined, and some applications are briefly discussed.

  20. Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra

    NASA Astrophysics Data System (ADS)

    Klishevich, Sergey M.; Plyushchay, Mikhail S.

    2002-04-01

    Recently, it was noticed by us that the nonlinear holomorphic supersymmetry of order n∈ N, n>1 ( n-HSUSY) has an algebraic origin. We show that the Onsager algebra underlies n-HSUSY and investigate the structure of the former in the context of the latter. A new infinite set of mutually commuting charges is found which, unlike those from the Dolan-Grady set, include the terms quadratic in the Onsager algebra generators. This allows us to find the general form of the superalgebra of n-HSUSY and fix it explicitly for the cases of n=2,3,4,5,6. The similar results are obtained for a new, contracted form of the Onsager algebra generated via the contracted Dolan-Grady relations. As an application, the algebraic structure of the known 1D and 2D systems with n-HSUSY is clarified and a generalization of the construction to the case of nonlinear pseudo-supersymmetry is proposed. Such a generalization is discussed in application to some integrable spin models and with its help we obtain a family of quasi-exactly solvable systems appearing in the PT-symmetric quantum mechanics.

  1. Quantum Q systems: from cluster algebras to quantum current algebras

    NASA Astrophysics Data System (ADS)

    Di Francesco, Philippe; Kedem, Rinat

    2017-02-01

    This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the A_r quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593-2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra U_{√{q}}({n}[u,u^{-1}])subset U_{√{q}}(widehat{{sl}}_2), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97-152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.

  2. Computer Algebra Systems in Education Newsletter[s].

    ERIC Educational Resources Information Center

    Computer Algebra Systems in Education Newsletter, 1990

    1990-01-01

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

  3. The Progressive Development of Early Embodied Algebraic Thinking

    ERIC Educational Resources Information Center

    Radford, Luis

    2014-01-01

    In this article I present some results from a 5-year longitudinal investigation with young students about the genesis of embodied, non-symbolic algebraic thinking and its progressive transition to culturally evolved forms of symbolic thinking. The investigation draws on a cultural-historical theory of teaching and learning--the theory of…

  4. Excel Spreadsheets for Algebra: Improving Mental Modeling for Problem Solving

    ERIC Educational Resources Information Center

    Engerman, Jason; Rusek, Matthew; Clariana, Roy

    2014-01-01

    This experiment investigates the effectiveness of Excel spreadsheets in a high school algebra class. Students in the experiment group convincingly outperformed the control group on a post lesson assessment. The student responses, teacher observations involving Excel spreadsheet revealed that it operated as a mindtool, which formed the users'…

  5. Linear algebraic methods applied to intensity modulated radiation therapy.

    PubMed

    Crooks, S M; Xing, L

    2001-10-01

    Methods of linear algebra are applied to the choice of beam weights for intensity modulated radiation therapy (IMRT). It is shown that the physical interpretation of the beam weights, target homogeneity and ratios of deposited energy can be given in terms of matrix equations and quadratic forms. The methodology of fitting using linear algebra as applied to IMRT is examined. Results are compared with IMRT plans that had been prepared using a commercially available IMRT treatment planning system and previously delivered to cancer patients.

  6. Anisotropy without tensors: a novel approach using geometric algebra.

    PubMed

    Matos, Sérgio A; Ribeiro, Marco A; Paiva, Carlos R

    2007-11-12

    The most widespread approach to anisotropic media is dyadic analysis. However, to get a geometrical picture of a dielectric tensor, one has to resort to a coordinate system for a matrix form in order to obtain, for example, the index-ellipsoid, thereby obnubilating the deeper coordinate-free meaning of anisotropy itself. To overcome these shortcomings we present a novel approach to anisotropy: using geometric algebra we introduce a direct geometrical interpretation without the intervention of any coordinate system. By applying this new approach to biaxial crystals we show the effectiveness and insight that geometric algebra can bring to the optics of anisotropic media.

  7. The coquaternion algebra and complex partial differential equations

    NASA Astrophysics Data System (ADS)

    Dimiev, Stancho; Konstantinov, Mihail; Todorov, Vladimir

    2009-11-01

    In this paper we consider the problem of differentiation of coquaternionic functions. Let us recall that coquaternions are elements of an associative non-commutative real algebra with zero divisor, introduced by James Cockle (1849) under the name of split-quaternions or coquaternions. Developing two type complex representations for Cockle algebra (complex and paracomplex ones) we present the problem in a non-commutative form of the δ¯-type holomorphy. We prove that corresponding differentiable coquaternionic functions, smooth and analytic, satisfy PDE of complex, and respectively of real variables. Applications for coquaternionic polynomials are sketched.

  8. Calculus and design of discrete velocity models using computer algebra

    NASA Astrophysics Data System (ADS)

    Babovsky, Hans; Grabmeier, Johannes

    2016-11-01

    In [2, 3], a framework for a calculus with Discrete Velocity Models (DVM) has been derived. The rotatonal symmetry of the discrete velocities can be modelled algebraically by the action of the cyclic group C4 - or including reflections of the dihedral group D4. Taking this point of view, the linearized collision operator can be represented in a compact form as a matrix of elements in the group algebra. Or in other words, by choosing a special numbering it exhibits a certain block structure which lets it appear as a matrix with entries in a certain polynomial ring. A convenient way for approaching such a structure is the use of a computer algebra system able to treat these (predefined) algebraic structures. We used the computer algebra system FriCAS/AXIOM [4, 5] for the generation of the velocity and the collision sets and for the analysis of the structure of the collision operator. Concerning the fluid dynamic limit, the system provides the characterization of sets of collisions and their contribution to the flow parameters. It allows the design of rotationally invariant symmetric models for prescribed Prandtl numbers. The implementation in FriCAS/AXIOM is explained and its results for a 25-velocity model are presented.

  9. Quadratic algebras for three-dimensional superintegrable systems

    SciTech Connect

    Daskaloyannis, C. Tanoudis, Y.

    2010-02-15

    The three-dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress, and Miller have proved that in the case of nondegenerate potentials with quadratic integrals of motion there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral implies that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. In this contribution we investigate the structure of this algebra. We show that in all the nondegenerate cases there is at least one subalgebra of three integrals having a Poisson quadratic algebra structure, which is similar to the two-dimensional case.

  10. Super-BMS3 algebras from {N}=2 flat supergravities

    NASA Astrophysics Data System (ADS)

    Lodato, Ivano; Merbis, Wout

    2016-11-01

    We consider two possible flat space limits of three dimensional {N}=(1, 1) AdS supergravity. They differ by how the supercharges are scaled with the AdS radius ℓ: the first limit (democratic) leads to the usual super-Poincaré theory, while a novel `twisted' theory of supergravity stems from the second (despotic) limit. We then propose boundary conditions such that the asymptotic symmetry algebras at null infinity correspond to supersymmetric extensions of the BMS algebras previously derived in connection to non- and ultra-relativistic limits of the {N}=(1, 1) Virasoro algebra in two dimensions. Finally, we study the supersymmetric energy bounds and find the explicit form of the asymptotic and global Killing spinors of supersymmetric solutions in both flat space supergravity theories.

  11. Automorphisms of Hilbert space effect algebras

    NASA Astrophysics Data System (ADS)

    Šemrl, Peter

    2015-05-01

    Let H be a Hilbert space and E (H) the effect algebra on H. A bijective map φ :E(H)\\to E(H) is called an ortho-order automorphism of E (H) if for every A,B\\in E(H) we have A≤slant B \\Longleftrightarrow φ (A)≤slant φ (B) and φ ({{A}\\bot })=φ {{(A)}\\bot }. The classical theorem of Ludwig states that every such ϕ is of the form φ (A)=UA{{U}*}, A\\in E(H), for some unitary or antiunitary operator U. It is also known that each bijective map on E (H) preserving order and coexistency in both directions is of the same form. Can we improve these two theorems by relaxing the bijectivity assumption and/or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion? For both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results. This research was supported by a grant from ARRS, Slovenia.

  12. Linearized gravity in terms of differential forms

    NASA Astrophysics Data System (ADS)

    Baykal, Ahmet; Dereli, Tekin

    2017-01-01

    A technique to linearize gravitational field equations is developed in which the perturbation metric coefficients are treated as second rank, symmetric, 1-form fields belonging to the Minkowski background spacetime by using the exterior algebra of differential forms.

  13. Colored Quantum Algebra and Its Bethe State

    NASA Astrophysics Data System (ADS)

    Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun

    2014-12-01

    We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.

  14. Using Number Theory to Reinforce Elementary Algebra.

    ERIC Educational Resources Information Center

    Covillion, Jane D.

    1995-01-01

    Demonstrates that using the elementary number theory in algebra classes helps students to use acquired algebraic skills as well as helping them to more clearly understand concepts that are presented. Discusses factoring, divisibility rules, and number patterns. (AIM)

  15. The Role of Proof in Comprehending and Teaching Elementary Linear Algebra.

    ERIC Educational Resources Information Center

    Uhlig, Frank

    2002-01-01

    Describes how elementary linear algebra can be taught successfully while introducing students to the concept and practice of mathematical proof. Suggests exploring the concept of solvability of linear systems first via the row echelon form (REF). (Author/KHR)

  16. Is Calculus Really That Different from Algebra? A More Logical Way To Understand and Teach Calculus.

    ERIC Educational Resources Information Center

    Elk, Seymour B.

    1998-01-01

    Discards the blinders that have hampered the traditional teaching of calculus and reexamines some of the intuitive ideas that underlie this subject matter. Analyzes the various indeterminate forms that arise through the blind application of algebraic operations. (Author/ASK)

  17. Bilinear covariants and spinor fields duality in quantum Clifford algebras

    SciTech Connect

    Abłamowicz, Rafał; Gonçalves, Icaro; Rocha, Roldão da

    2014-10-15

    Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields can be discussed. Thus, by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are thus compared to the classical (non quantum) ones. Classes of quantum spinor fields classes are introduced and compared with Lounesto's spinor field classification. A physical interpretation of the deformed parts and the underlying Z-grading is proposed. The existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity [S. W. Hawking, “The unpredictability of quantum gravity,” Commun. Math. Phys. 87, 395 (1982)]. Here, it is shown further to play a prominent role in the structure of Dirac, Weyl, and Majorana spinor fields, besides the most general flagpoles and flag-dipoles. We introduce a new duality between the standard and the quantum spinor fields, by showing that when Clifford algebras over vector spaces endowed with an arbitrary bilinear form are taken into account, a mixture among the classes does occur. Consequently, novel features regarding the spinor fields can be derived.

  18. A new algebraic transition model based on stress length function

    NASA Astrophysics Data System (ADS)

    Xiao, Meng-Juan; She, Zhen-Su

    2016-11-01

    Transition, as one of the two biggest challenges in turbulence research, is of critical importance for engineering application. For decades, the fundamental research seems to be unable to capture the quantitative details in real transition process. On the other hand, numerous empirical parameters in engineering transition models provide no unified description of the transition under varying physical conditions. Recently, we proposed a symmetry-based approach to canonical wall turbulence based on stress length function, which is here extended to describe the transition via a new algebraic transition model. With a multi-layer analytic form of the stress length function in both the streamwise and wall normal directions, the new model gives rise to accurate description of the mean field and friction coefficient, comparing with both the experimental and DNS results at different inlet conditions. Different types of transition process, such as the transition with varying incoming turbulence intensities or that with blow and suck disturbance, are described by only two or three model parameters, each of which has their own specific physical interpretation. Thus, the model enables one to extract physical information from both experimental and DNS data to reproduce the transition process, which may prelude to a new class of generalized transition model for engineering applications.

  19. Algebraic orbifold conformal field theories

    PubMed Central

    Xu, Feng

    2000-01-01

    The unitary rational orbifold conformal field theories in the algebraic quantum field theory and subfactor theory framework are formulated. Under general conditions, it is shown that the orbifold of a given unitary rational conformal field theory generates a unitary modular category. Many new unitary modular categories are obtained. It is also shown that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time. PMID:11106383

  20. Scalable Parallel Algebraic Multigrid Solvers

    SciTech Connect

    Bank, R; Lu, S; Tong, C; Vassilevski, P

    2005-03-23

    The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.

  1. Development of the Parent Form of the Preschool Children's Communication Skills Scale and Comparison of the Communication Skills of Children with Normal Development and with Autism Spectrum Disorder

    ERIC Educational Resources Information Center

    Aydin, Aydan

    2016-01-01

    This study aims at developing an assessment scale for identifying preschool children's communication skills, at distinguishing children with communication deficiencies and at comparing the communication skills of children with normal development (ND) and those with autism spectrum disorder (ASD). Participants were 427 children of up to 6 years of…

  2. Symmetry algebras of linear differential equations

    NASA Astrophysics Data System (ADS)

    Shapovalov, A. V.; Shirokov, I. V.

    1992-07-01

    The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.

  3. 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.

  4. Twining characters and orbit Lie algebras

    SciTech Connect

    Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph

    1996-12-05

    We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.

  5. Applications of Algebraic Logic and Universal Algebra to Computer Science

    DTIC Science & Technology

    1989-06-21

    conference, with roughly equal representation from Mathematics and Computer Science . The conference consisted of eight invited lectures (60 minutes...each) and 26 contributed talks (20-40 minutes each). There was also a round-table discussion on the role of algebra and logic in computer science . Keywords

  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. 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…

  8. 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…

  9. 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…

  10. 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…

  11. Embedding Algebraic Thinking throughout the Mathematics Curriculum

    ERIC Educational Resources Information Center

    Vennebush, G. Patrick; Marquez, Elizabeth; Larsen, Joseph

    2005-01-01

    This article explores the algebra that can be uncovered in many middle-grades mathematics tasks that, on first inspection, do not appear to be algebraic. It shows connections to the other four Standards that occur in traditional algebra problems, and it offers strategies for modifying activities so that they can be used to foster algebraic…

  12. 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"…

  13. 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…

  14. 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…

  15. 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…

  16. Cyclic homology for Hom-associative algebras

    NASA Astrophysics Data System (ADS)

    Hassanzadeh, Mohammad; Shapiro, Ilya; Sütlü, Serkan

    2015-12-01

    In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.

  17. A New Reynolds Stress Algebraic Equation Model

    NASA Technical Reports Server (NTRS)

    Shih, Tsan-Hsing; Zhu, Jiang; Lumley, John L.

    1994-01-01

    A general turbulent constitutive relation is directly applied to propose a new Reynolds stress algebraic equation model. In the development of this model, the constraints based on rapid distortion theory and realizability (i.e. the positivity of the normal Reynolds stresses and the Schwarz' inequality between turbulent velocity correlations) are imposed. Model coefficients are calibrated using well-studied basic flows such as homogeneous shear flow and the surface flow in the inertial sublayer. The performance of this model is then tested in complex turbulent flows including the separated flow over a backward-facing step and the flow in a confined jet. The calculation results are encouraging and point to the success of the present model in modeling turbulent flows with complex geometries.

  18. Visualization of stochastic Ca2+ signals in the formed somites during the early segmentation period in intact, normally developing zebrafish embryos.

    PubMed

    Leung, Christina F; Miller, Andrew L; Korzh, Vladimir; Chong, Shang-Wei; Sleptsova-Freidrich, Inna; Webb, Sarah E

    2009-09-01

    Localized Ca(2+) signals were consistently visualized in the formed somites of intact zebrafish embryos during the early segmentation period. Unlike the regular process of somitogenesis, these signals were stochastic in nature with respect to time and location. They did, however, occur predominantly at the medial and lateral boundaries within the formed somites. Embryos were treated with modulators of [Ca(2+)](i) to explore the signal generation mechanism and possible developmental function of the stochastic transients. Blocking elements in the phosphoinositol pathway eliminated the stochastic signals but had no obvious effect, stochastic or otherwise, on the formed somites. Such treatments did, however, result in the subsequently formed somites being longer in the mediolateral dimension. Targeted uncaging of buffer (diazo-2) or Ca(2+) (NP-ethyleneglycoltetraacetic acid [EGTA]) in the presomitic mesoderm, resulted in a regular mediolateral lengthening and shortening, respectively, of subsequently formed somites. These data suggest a requirement for IP(3) receptor-mediated Ca(2+) release during convergence cell movements in the presomitic mesoderm, which appears to have a distinct function from that of the IP(3) receptor-mediated stochastic Ca(2+) signaling in the formed somites.

  19. Carry Groups: Abstract Algebra Projects

    ERIC Educational Resources Information Center

    Miller, Cheryl Chute; Madore, Blair F.

    2004-01-01

    Carry Groups are a wonderful collection of groups to introduce in an undergraduate Abstract Algebra course. These groups are straightforward to define but have interesting structures for students to discover. We describe these groups and give examples of in-class group projects that were developed and used by Miller.

  20. Algebra, Home Mortgages, and Recessions

    ERIC Educational Resources Information Center

    Mariner, Jean A. Miller; Miller, Richard A.

    2009-01-01

    The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…

  1. Exploring Algebraic Misconceptions with Technology

    ERIC Educational Resources Information Center

    Sakow, Matthew; Karaman, Ruveyda

    2015-01-01

    Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…

  2. Easing Students' Transition to Algebra

    ERIC Educational Resources Information Center

    Baroudi, Ziad

    2006-01-01

    Traditionally, students learn arithmetic throughout their primary schooling, and this is seen as the ideal preparation for the learning of algebra in the junior secondary school. The four operations are taught and rehearsed in the early years and from this, it is assumed, "children will induce the fundamental structure of arithmetic" (Warren &…

  3. Algebra for All. Research Brief

    ERIC Educational Resources Information Center

    Bleyaert, Barbara

    2009-01-01

    The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless…

  4. 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.

  5. Inequalities, Assessment and Computer Algebra

    ERIC Educational Resources Information Center

    Sangwin, Christopher J.

    2015-01-01

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

  6. Adventures in Flipping College Algebra

    ERIC Educational Resources Information Center

    Van Sickle, Jenna

    2015-01-01

    This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…

  7. Elementary Algebra Connections to Precalculus

    ERIC Educational Resources Information Center

    Lopez-Boada, Roberto; Daire, Sandra Arguelles

    2013-01-01

    This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…

  8. Celestial mechanics with geometric algebra

    NASA Technical Reports Server (NTRS)

    Hestenes, D.

    1983-01-01

    Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.

  9. Math Sense: Algebra and Geometry.

    ERIC Educational Resources Information Center

    Howett, Jerry

    This book is designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four strands: (1) skill lessons…

  10. Weaving Geometry and Algebra Together

    ERIC Educational Resources Information Center

    Cetner, Michelle

    2015-01-01

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

  11. Algebraic Activities Aid Discovery Lessons

    ERIC Educational Resources Information Center

    Wallace-Gomez, Patricia

    2013-01-01

    After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…

  12. Algebraic Davis Decomposition and Asymmetric Doob Inequalities

    NASA Astrophysics Data System (ADS)

    Hong, Guixiang; Junge, Marius; Parcet, Javier

    2016-09-01

    In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let {({M}, τ)} be a noncommutative probability space equipped with a filtration of von Neumann subalgebras {({M}_n)_{n ≥ 1}}, whose union {bigcup_{n≥1}{M}_n} is weak-* dense in {{M}}. Let {{E}_n} denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for {1 < p < 2} and {x in L_p({M},τ)} one can find {a, b in L_p({M},τ)} and contractions {u_n, v_n in {M}} such that {E}_n(x) = a u_n + v_n b quad and quad max big{ |a|_p,|b|_p big} ≤ c_p |x|_p. Moreover, it turns out that {a u_n} and {v_n b} converge in the row/column Hardy spaces {{H}_p^r({M})} and {{H}_p^c({M})} respectively. In particular, this solves a problem posed by the Defant and Junge in 2004. In the case p = 1, our results establish a noncommutative form of the Davis celebrated theorem on the relation betwe en martingale maximal and square functions in L 1, whose noncommutative form has remained open for quite some time. Given {1 ≤ p ≤ 2}, we also provide new weak type maximal estimates, which imply in turn left/right almost uniform convergence of {{E}_n(x)} in row/column Hardy spaces. This improves the bilateral convergence known so far. Our approach is based on new forms of Davis martingale decomposition which are of independent interest, and an algebraic atomic description for the involved Hardy spaces. The latter results are new even for commutative von Neumann algebras.

  13. 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

  14. Explicit field realizations of W algebras

    NASA Astrophysics Data System (ADS)

    Wei, Shao-Wen; Liu, Yu-Xiao; Zhang, Li-Jie; Ren, Ji-Rong

    2009-06-01

    The fact that certain nonlinear W2,s algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize W2,s algebras from linear W1,2,s algebras. In this paper, we first construct the explicit field realizations of linear W1,2,s algebras with double scalar and double spinor, respectively. Then, after a change of basis, the realizations of W2,s algebras are presented. The results show that all these realizations are Romans-type realizations.

  15. Array algebra estimation in signal processing

    NASA Astrophysics Data System (ADS)

    Rauhala, U. A.

    A general theory of linear estimators called array algebra estimation is interpreted in some terms of multidimensional digital signal processing, mathematical statistics, and numerical analysis. The theory has emerged during the past decade from the new field of a unified vector, matrix and tensor algebra called array algebra. The broad concepts of array algebra and its estimation theory cover several modern computerized sciences and technologies converting their established notations and terminology into one common language. Some concepts of digital signal processing are adopted into this language after a review of the principles of array algebra estimation and its predecessors in mathematical surveying sciences.

  16. On special classes of n-algebras

    NASA Astrophysics Data System (ADS)

    Vainerman, L.; Kerner, R.

    1996-05-01

    We define n-algebras as linear spaces on which the internal composition law involves n elements: m:V⊗n■V. It is known that such algebraic structures are interesting for their applications to problems of modern mathematical physics. Using the notion of a commutant of two subalgebras of an n-algebra, we distinguish certain classes of n-algebras with reasonable properties: semisimple, Abelian, nilpotent, solvable. We also consider a few examples of n-algebras of different types, and show their properties.

  17. Recursion and feedback in image algebra

    NASA Astrophysics Data System (ADS)

    Ritter, Gerhard X.; Davidson, Jennifer L.

    1991-04-01

    Recursion and feedback are two important processes in image processing. Image algebra, a unified algebraic structure developed for use in image processing and image analysis, provides a common mathematical environment for expressing image processing transforms. It is only recently that image algebra has been extended to include recursive operations [1]. Recently image algebra was shown to incorporate neural nets [2], including a new type of neural net, the morphological neural net [3]. This paper presents the relationship of the recursive image algebra to the field of fractions of the ring of matrices, and gives the two dimensional moving average filter as an example. Also, the popular multilayer perceptron with back propagation and a morphology neural network with learning rule are presented in image algebra notation. These examples show that image algebra can express these important feedback concepts in a succinct way.

  18. Deformed Kac Moody and Virasoro algebras

    NASA Astrophysics Data System (ADS)

    Balachandran, A. P.; Queiroz, A. R.; Marques, A. M.; Teotonio-Sobrinho, P.

    2007-07-01

    Whenever the group {\\bb R}^n acts on an algebra {\\cal A} , there is a method to twist \\cal A to a new algebra {\\cal A}_\\theta which depends on an antisymmetric matrix θ (θμν = -θνμ = constant). The Groenewold-Moyal plane {\\cal A}_\\theta({\\bb R}^{d+1}) is an example of such a twisted algebra. We give a general construction to realize this twist in terms of {\\cal A} itself and certain 'charge' operators Qμ. For {\\cal A}_\\theta({\\bb R}^{d+1}), Q_\\mu are translation generators. This construction is then applied to twist the oscillators realizing the Kac-Moody (KM) algebra as well as the KM currents. They give different deformations of the KM algebra. From one of the deformations of the KM algebra, we construct, via the Sugawara construction, the Virasoro algebra. These deformations have an implication for statistics as well.

  19. Algebraic complexities and algebraic curves over finite fields

    PubMed Central

    Chudnovsky, D. V.; Chudnovsky, G. V.

    1987-01-01

    We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields. PMID:16593816

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

    ERIC Educational Resources Information Center

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

    2016-01-01

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

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

    NASA Astrophysics Data System (ADS)

    Li, Chuanzhong; He, Jingsong

    2013-11-01

    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.

  2. Alternative algebraic approaches in quantum chemistry

    SciTech Connect

    Mezey, Paul G.

    2015-01-22

    Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.

  3. Algebraic Methods to Design Signals

    DTIC Science & Technology

    2015-08-27

    group theory are employed to investigate the theory of their construction methods leading to new families of these arrays and some generalizations...sequences and arrays with desirable correlation properties. The methods used are very algebraic and number theoretic. Many new families of sequences...context of optical quantum computing, we prove that infinite families of anticirculant block weighing matrices can be obtained from generic weighing

  4. Functionally reduced sensorimotor connections form with normal specificity despite abnormal muscle spindle development: the role of spindle-derived neurotrophin 3.

    PubMed

    Shneider, Neil A; Mentis, George Z; Schustak, Joshua; O'Donovan, Michael J

    2009-04-15

    The mechanisms controlling the formation of synaptic connections between muscle spindle afferents and spinal motor neurons are believed to be regulated by factors originating from muscle spindles. Here, we find that the connections form with appropriate specificity in mice with abnormal spindle development caused by the conditional elimination of the neuregulin 1 receptor ErbB2 from muscle precursors. However, despite a modest ( approximately 30%) decrease in the number of afferent terminals on motor neuron somata, the amplitude of afferent-evoked synaptic potentials recorded in motor neurons was reduced by approximately 80%, suggesting that many of the connections that form are functionally silent. The selective elimination of neurotrophin 3 (NT3) from muscle spindles had no effect on the amplitude of afferent-evoked ventral root potentials until the second postnatal week, revealing a late role for spindle-derived NT3 in the functional maintenance of the connections. These findings indicate that spindle-derived factors regulate the strength of the connections but not their initial formation or their specificity.

  5. Functionally reduced sensorimotor connections form with normal specificity despite abnormal muscle spindle development: the role of spindle-derived NT3

    PubMed Central

    Shneider, Neil A.; Mentis, George Z.; Schustak, Joshua; O’Donovan, Michael J.

    2009-01-01

    Summary The mechanisms controlling the formation of synaptic connections between muscle spindle afferents and spinal motor neurons are believed to be regulated by factors originating from muscle spindles. Here, we find that the connections form with appropriate specificity in mice with abnormal spindle development caused by the conditional elimination of the neuregulin1 receptor ErbB2 from muscle precursors. However, despite a modest (~30%) decrease in the number of afferent terminals on motor neuron somata, the amplitude of afferent-evoked synaptic potentials recorded in motor neurons was reduced by ~80%, suggesting that many of the connections that form are functionally silent. The selective elimination of neurotrophin 3 (NT3) from muscle spindles had no effect on the amplitude of afferent-evoked ventral root potentials until the second postnatal week, revealing a late role for spindle-derived NT3 in the functional maintenance of the connections. These findings indicate that spindle-derived factors regulate the strength of the connections, but not their initial formation or their specificity. PMID:19369542

  6. Computer algebra and transport theory.

    SciTech Connect

    Warsa, J. S.

    2004-01-01

    Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.

  7. 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.

  8. Introduction to Image Algebra Ada

    NASA Astrophysics Data System (ADS)

    Wilson, Joseph N.

    1991-07-01

    Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.

  9. 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.…

  10. Algebraic structure of general electromagnetic fields and energy flow

    SciTech Connect

    Hacyan, Shahen

    2011-08-15

    Highlights: > Algebraic structure of general electromagnetic fields in stationary spacetime. > Eigenvalues and eigenvectors of the electomagnetic field tensor. > Energy-momentum in terms of eigenvectors and Killing vector. > Explicit form of reference frame with vanishing Poynting vector. > Application of formalism to Bessel beams. - Abstract: The algebraic structures of a general electromagnetic field and its energy-momentum tensor in a stationary space-time are analyzed. The explicit form of the reference frame in which the energy of the field appears at rest is obtained in terms of the eigenvectors of the electromagnetic tensor and the existing Killing vector. The case of a stationary electromagnetic field is also studied and a comparison is made with the standard short-wave approximation. The results can be applied to the general case of a structured light beams, in flat or curved spaces. Bessel beams are worked out as example.

  11. 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.

  12. A few Lie algebras and their applications for generating integrable hierarchies of evolution types

    NASA Astrophysics Data System (ADS)

    Zhang, Yufeng; Feng, Binlu

    2011-08-01

    A Lie algebra consisting of 3 × 3 matrices is introduced, whose induced Lie algebra by using an inverted linear transformation is obtained as well. As for application examples, we obtain a unified integrable model of the integrable couplings of the AKNS hierarchy, the D-AKNS hierarchy and the TD hierarchy as well as their induced integrable hierarchies. These integrable couplings are different from those results obtained before. However, the Hamiltonian structures of the integrable couplings cannot be obtained by using the quadratic-form identity or the variational identity. For solving the problem, we construct a higher-dimensional subalgebra R and its reduced algebra Q of the Lie algebra A2 by decomposing the induced Lie algebra and then again making some linear combinations. The subalgebras of the Lie algebras R and Q do not satisfy the relation ( G=G1⊕G2,[G1,G2]⊂G2), but we can deduce integrable couplings, which indicates that the above condition is not necessary to generate integrable couplings. As for application example, an expanding integrable model of the AKNS hierarchy is obtained whose Hamiltonian structure is generated by the trace identity. Finally, we give another Lie algebras which can be decomposed into two simple Lie subalgebras for which a nonlinear integrable coupling of the classical Boussinesq-Burgers (CBB) hierarchy is obtained.

  13. Numerical linear algebra algorithms and software

    NASA Astrophysics Data System (ADS)

    Dongarra, Jack J.; Eijkhout, Victor

    2000-11-01

    The increasing availability of advanced-architecture computers has a significant effect on all spheres of scientific computation, including algorithm research and software development in numerical linear algebra. Linear algebra - in particular, the solution of linear systems of equations - lies at the heart of most calculations in scientific computing. This paper discusses some of the recent developments in linear algebra designed to exploit these advanced-architecture computers. We discuss two broad classes of algorithms: those for dense, and those for sparse matrices.

  14. Symbolic Lie algebras manipulations using COMMON LISP

    NASA Astrophysics Data System (ADS)

    Cecchini, R.; Tarlini, M.

    1989-01-01

    We present a description and an implementation of a program in COMMON LISP to perform symbolic computations in a given Lie algebra. Using the general definitions of vector space Lie algebra and enveloping algebra, the program is able to compute commutators, to evaluate similarity transformations and the general Baker-Campbell-Hausdorff formula. All the computations are exact, including numerical coefficients. For the interactive user an optional menu facility and online help are available. LISP knowledge is unnecessary.

  15. Lie algebras of classical and stochastic electrodynamics

    NASA Astrophysics Data System (ADS)

    Neto, J. J. Soares; Vianna, J. D. M.

    1994-03-01

    The Lie algebras associated with infinitesimal symmetry transformations of third-order differential equations of interest to classical electrodynamics and stochastic electrodynamics have been obtained. The structure constants for a general case are presented and the Lie algebra for each particular application is easily achieved. By the method used here it is not necessary to know the explicit expressions of the infinitesimal generators in order to determine the structure constants of the Lie algebra.

  16. Algebraic logic of concepts and its machine implementation in the algebras of deontic and axiological notions

    NASA Astrophysics Data System (ADS)

    Manerowska, Anna; Nieznański, Edward; Mulawka, Jan

    2013-10-01

    Our aim is to present the algebra of concepts in two formal languages. First, after introducing a primary relation between concepts, which is subsumption, we shall specify in a language that uses quantifiers, the Boolean algebra of general concepts. Next, we shall note down the same algebra in simplified non-quantifying language, in order to use it as basis for two specific implementations, i.e. to create the Boolean algebras of deontic concepts and axiological concepts.

  17. Hidden symmetries and Lie algebra structures from geometric and supergravity Killing spinors

    NASA Astrophysics Data System (ADS)

    Açık, Özgür; Ertem, Ümit

    2016-08-01

    We consider geometric and supergravity Killing spinors and the spinor bilinears constructed out of them. The spinor bilinears of geometric Killing spinors correspond to the antisymmetric generalizations of Killing vector fields which are called Killing-Yano forms. They constitute a Lie superalgebra structure in constant curvature spacetimes. We show that the Dirac currents of geometric Killing spinors satisfy a Lie algebra structure up to a condition on 2-form spinor bilinears. We propose that the spinor bilinears of supergravity Killing spinors give way to different generalizations of Killing vector fields to higher degree forms. It is also shown that those supergravity Killing forms constitute a Lie algebra structure in six- and ten-dimensional cases. For five- and eleven-dimensional cases, the Lie algebra structure depends on an extra condition on supergravity Killing forms.

  18. Dispersion Operators Algebra and Linear Canonical Transformations

    NASA Astrophysics Data System (ADS)

    Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo

    2017-04-01

    This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.

  19. Dispersion Operators Algebra and Linear Canonical Transformations

    NASA Astrophysics Data System (ADS)

    Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo

    2017-02-01

    This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.

  20. Banach Algebras Associated to Lax Pairs

    NASA Astrophysics Data System (ADS)

    Glazebrook, James F.

    2015-04-01

    Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall- Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose-Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand-Naimark-Segal construction. Instrumental is the Plücker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan-Lie-Banach algebras structures can be compared.

  1. 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.

  2. Multicloning and Multibroadcasting in Operator Algebras

    NASA Astrophysics Data System (ADS)

    Kaniowski, Krzysztof; Lubnauer, Katarzyna; Łuczak, Andrzej

    2015-12-01

    We investigate multicloning and multibroadcasting in the general operator algebra framework in arbitrary dimension, generalizing thus results obtained in this framework for simple cloning and broadcasting.

  3. 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…

  4. 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…

  5. An algebraic approach to modeling in software engineering

    SciTech Connect

    Loegel, G.J. |; Ravishankar, C.V.

    1993-09-01

    Our work couples the formalism of universal algebras with the engineering techniques of mathematical modeling to develop a new approach to the software engineering process. Our purpose in using this combination is twofold. First, abstract data types and their specification using universal algebras can be considered a common point between the practical requirements of software engineering and the formal specification of software systems. Second, mathematical modeling principles provide us with a means for effectively analyzing real-world systems. We first use modeling techniques to analyze a system and then represent the analysis using universal algebras. The rest of the software engineering process exploits properties of universal algebras that preserve the structure of our original model. This paper describes our software engineering process and our experience using it on both research and commercial systems. We need a new approach because current software engineering practices often deliver software that is difficult to develop and maintain. Formal software engineering approaches use universal algebras to describe ``computer science`` objects like abstract data types, but in practice software errors are often caused because ``real-world`` objects are improperly modeled. There is a large semantic gap between the customer`s objects and abstract data types. In contrast, mathematical modeling uses engineering techniques to construct valid models for real-world systems, but these models are often implemented in an ad hoc manner. A combination of the best features of both approaches would enable software engineering to formally specify and develop software systems that better model real systems. Software engineering, like mathematical modeling, should concern itself first and foremost with understanding a real system and its behavior under given circumstances, and then with expressing this knowledge in an executable form.

  6. Disjointness preserving operators between little Lipschitz algebras

    NASA Astrophysics Data System (ADS)

    Jiménez-Vargas, A.

    2008-01-01

    Given a real number [alpha][set membership, variant](0,1) and a metric space (X,d), let Lip[alpha](X) be the algebra of all scalar-valued bounded functions f on X such that endowed with any one of the norms ||f||=max{p[alpha](f),||f||[infinity]} or ||f||=p[alpha](f)+||f||[infinity]. The little Lipschitz algebra lip[alpha](X) is the closed subalgebra of Lip[alpha](X) formed by all those functions f such that f(x)-f(y)/d(x,y)[alpha]->0 as d(x,y)->0. A linear mapping is called disjointness preserving if f[dot operator]g=0 in lip[alpha](X) implies (Tf)[dot operator](Tg)=0 in lip[alpha](Y). In this paper we study the representation and the automatic continuity of such maps T in the case in which X and Y are compact. We prove that T is essentially a weighted composition operator Tf=h[dot operator](f[circle, open][phi]) for some nonvanishing little Lipschitz function h and some continuous map [phi]. If, in addition, T is bijective, we deduce that h is a nonvanishing function in lip[alpha](Y) and [phi] is a Lipschitz homeomorphism from Y onto X and, in particular, we obtain that T is automatically continuous and T-1 is disjointness preserving. Moreover we show that there exists always a discontinuous disjointness preserving linear functional on lip[alpha](X), provided X is an infinite compact metric space.

  7. Classification of filiform Lie algebras of order 3

    NASA Astrophysics Data System (ADS)

    Navarro, Rosa María

    2016-12-01

    Lie algebras of order 3 constitute a generalization of Lie algebras and superalgebras. Throughout this paper the classification problem of filiform Lie algebras of order 3 is considered and therefore this work is a continuation papers seen in the literature. We approach this classification by extending Vergne's result for filiform Lie algebras and by considering algebras of order 3 of high nilindex. We find the expression of the law to which any elementary filiform Lie algebra of order 3 is isomorphic.

  8. The Algebra of Lexical Semantics

    NASA Astrophysics Data System (ADS)

    Kornai, András

    The current generative theory of the lexicon relies primarily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the generative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of describing discrepancies between syntactic type (lexical category) and semantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR.

  9. Strengthening Effect Algebras in a Logical Perspective: Heyting-Wajsberg Algebras

    NASA Astrophysics Data System (ADS)

    Konig, Martinvaldo

    2014-10-01

    Heyting effect algebras are lattice-ordered pseudoboolean effect algebras endowed with a pseudocomplementation that maps on the center (i.e. Boolean elements). They are the algebraic counterpart of an extension of both Łukasiewicz many-valued logic and intuitionistic logic. We show that Heyting effect algebras are termwise equivalent to Heyting-Wajsberg algebras where the two different logical implications are defined as primitive operators. We prove this logic to be decidable, to be strongly complete and to have the deduction-detachment theorem.

  10. An algebraic criterion for the onset of chaos in nonlinear dynamic systems

    NASA Technical Reports Server (NTRS)

    Unal, A.; Tobak, M.

    1987-01-01

    The correspondence between iterated integrals and a noncommutative algebra is used to recast the given dynamical system from the time domain to the Laplace-Borel transform domain. It is then shown that the following algebraic criterion has to be satisfied for the outset of chaos: the limit (as tau approaches infinity and x sub 0 approaches infinity) of ((sigma(k=0) (tau sup k) / (k* x sub 0 sup k)) G II G = 0, where G is the generating power series of the trajectories, the symbol II is the shuffle product (le melange) of the noncommutative algebra, x sub 0 is a noncommutative variable, and tau is the correlation parameter. In the given equation, symbolic forms for both G and II can be obtained by use of one of the currently available symbolic languages such as PLI, REDUCE, and MACSYMA. Hence, the criterion is a computer-algebraic one.

  11. Algebraic Thinking: A Problem Solving Approach

    ERIC Educational Resources Information Center

    Windsor, Will

    2010-01-01

    Algebraic thinking is a crucial and fundamental element of mathematical thinking and reasoning. It initially involves recognising patterns and general mathematical relationships among numbers, objects and geometric shapes. This paper will highlight how the ability to think algebraically might support a deeper and more useful knowledge, not only of…

  12. Learning from Student Approaches to Algebraic Proofs

    ERIC Educational Resources Information Center

    D'Ambrosio, Beatriz S.; Kastberg, Signe E.; Viola dos Santos, Joao Ricardo

    2010-01-01

    Many mathematics teachers struggle to support their students' developing understanding of proof as an essential element in investigations of mathematics. The area of mathematics where the development of an understanding of proof is most challenging is algebra. In the case of algebraic proof, analysis of student written work on tasks that demand…

  13. From operator algebras to superconformal field theory

    SciTech Connect

    Kawahigashi, Yasuyuki

    2010-01-15

    We survey operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory of Jones, and certain aspects of noncommutative geometry of Connes.

  14. A Technology-Intensive Approach to Algebra.

    ERIC Educational Resources Information Center

    Heid, M. Kathleen; Zbiek, Rose Mary

    1995-01-01

    Computer-Intensive Algebra (CIA) focuses on the use of technology to help develop a rich understanding of fundamental algebraic concepts in real-world settings using computing tools for easy access to numerical, graphical, and symbolic representations of mathematical ideas. (MKR)

  15. 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…

  16. Post-Lie Algebras and Isospectral Flows

    NASA Astrophysics Data System (ADS)

    Ebrahimi-Fard, Kurusch; Lundervold, Alexander; Mencattini, Igor; Munthe-Kaas, Hans Z.

    2015-11-01

    In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical R-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation.

  17. Teaching Modeling and Axiomatization with Boolean Algebra.

    ERIC Educational Resources Information Center

    De Villiers, Michael D.

    1987-01-01

    Presented is an alternative approach to the traditional teaching of Boolean algebra for secondary school mathematics. The main aim of the approach is to use Boolean algebra to teach pupils such mathematical processes as modeling and axiomatization. A course using the approach is described. (RH)

  18. Arithmetic and Cognitive Contributions to Algebra

    ERIC Educational Resources Information Center

    Cirino, Paul T.; Tolar, Tammy D.; Fuchs, Lynn S.

    2013-01-01

    Algebra is a prerequisite for access to STEM careers and occupational success (NMAP, 2008a), yet algebra is difficult for students through high school (US DOE, 2008). Growth in children's conceptual and procedural arithmetical knowledge is reciprocal, although conceptual knowledge has more impact on procedural knowledge than the reverse…

  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. 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…

  1. An algebraic approach to the scattering equations

    NASA Astrophysics Data System (ADS)

    Huang, Rijun; Rao, Junjie; Feng, Bo; He, Yang-Hui

    2015-12-01

    We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

  2. 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

  3. 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

  4. 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…

  5. Teaching Algebra to Students with Learning Disabilities

    ERIC Educational Resources Information Center

    Impecoven-Lind, Linda S.; Foegen, Anne

    2010-01-01

    Algebra is a gateway to expanded opportunities, but it often poses difficulty for students with learning disabilities. Consequently, it is essential to identify evidence-based instructional strategies for these students. The authors begin by identifying three areas of algebra difficulty experienced by students with disabilities: cognitive…

  6. Gary M. Klingler Algebra Teacher Assistance Packages

    ERIC Educational Resources Information Center

    Klingler, Gary

    2005-01-01

    Several packages designed by Elizabeth Marquez for mathematics teachers of grades 6-12, officially entitled the Teacher Assistance Package in Support of Better Algebra Assessment, is a series of resources developed to accompany ET's End-of-Course Algebra Assessment. It is designed to enhance teachers classroom assessment by providing examples of…

  7. 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…

  8. 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,…

  9. SAYD Modules over Lie-Hopf Algebras

    NASA Astrophysics Data System (ADS)

    Rangipour, Bahram; Sütlü, Serkan

    2012-11-01

    In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.

  10. Algebraic Formulas for Areas between Curves.

    ERIC Educational Resources Information Center

    Gabai, Hyman

    1982-01-01

    Korean secondary school students preparing for college learn about a simple algebraic formula for area bounded by a parabola and line. The approach does not seem well-known among American students. It is noted that, while the formula derivations rely on integration, algebra students could use the formulas without proofs. (MP)

  11. 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…

  12. 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…

  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. Success in Algebra among Community College Students

    ERIC Educational Resources Information Center

    Reyes, Czarina

    2010-01-01

    College algebra is a required course for most majors, but is viewed by many as a gatekeeper course for degree completion by students. With almost half a million students taking college algebra each year, faculty are experimenting with new course lengths of time that might result in higher success, completion, and retention rates for college…

  15. Is Algebra Really Difficult for All Students?

    ERIC Educational Resources Information Center

    Egodawatte, Gunawardena

    2009-01-01

    Research studies have shown that students encounter difficulties in transitioning from arithmetic to algebra. Errors made by high school students were analyzed for patterns and their causes. The origins of errors were: intuitive assumptions, failure to understand the syntax of algebra, analogies with other familiar symbol systems such as the…

  16. Analysis of the SDS-PAGE patterns of outer membrane proteins from Escherichia coli strains that have lost the ability to form K1 antigen and varied in the susceptibility to normal human serum.

    PubMed

    Cisowska, Agnieszka; Bugla-Płoskońska, Gabriela

    2014-01-01

    We used SDS-polyacrylamide gel electrophoresis to investigate the outer membrane proteins (OMPs) band composition of 19 Escherichia coli K1 strains that have spontaneously lost the ability to form K1 polysaccharide capsule (E. coli K1-) and demonstrated different degrees of susceptibility to the bactericidal action of normal human serum. Presented results showed that there were differences between E. coli K1- strains in OMPs expressing capacity. The analysis performed on OMPs has not revealed a direct association between the different OMPs band composition and the susceptibility of these strains to the serum.

  17. MODEL IDENTIFICATION AND COMPUTER ALGEBRA.

    PubMed

    Bollen, Kenneth A; Bauldry, Shawn

    2010-10-07

    Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods.

  18. Hexagonal tessellations in image algebra

    NASA Astrophysics Data System (ADS)

    Eberly, David H.; Wenzel, Dennis J.; Longbotham, Harold G.

    1990-11-01

    In image algebra '' the concept of a coordinate set X is general in that such a set is simply a subset of ndimensional Euclidean space . The standard applications in 2-dimensional image processing use coordinate sets which are rectangular arrays X 72 x ZZm. However some applications may require other geometries for the coordinate set. We look at three such related applications in the context of image algebra. The first application is the modeling of photoreceptors in primate retinas. These receptors are inhomogeneously distributed on the retina. The largest receptor density occurs in the center of the fovea and decreases radially outwards. One can construct a hexagonal tessellation of the retina such that each hexagon contains approximately the same number of receptors. The resulting tessellation called a sunflower heart2 consists of concentric rings of hexagons whose sizes increase as the radius of the ring increases. The second application is the modeling of the primary visual . The neurons are assumed to be uniformly distributed as a regular hexagonal lattice. Cortical neural image coding is modeled by a recursive convolution of the retinal neural image using a special set of filters. The third application involves analysis of a hexagonally-tessellated image where the pixel resolution is variable .

  19. Approximating smooth functions using algebraic-trigonometric polynomials

    SciTech Connect

    Sharapudinov, Idris I

    2011-01-14

    The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form p{sub n}(t)+{tau}{sub m}(t), where p{sub n}(t) is an algebraic polynomial of degree n and {tau}{sub m}(t)=a{sub 0}+{Sigma}{sub k=1}{sup m}a{sub k} cos k{pi}t + b{sub k} sin k{pi}t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W{sup r}{sub {infinity}(}M) and an upper bound for similar approximations in the class W{sup r}{sub p}(M) with 4/3

  20. Approximating smooth functions using algebraic-trigonometric polynomials

    NASA Astrophysics Data System (ADS)

    Sharapudinov, Idris I.

    2011-01-01

    The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form p_n(t)+\\tau_m(t), where p_n(t) is an algebraic polynomial of degree n and \\tau_m(t)=a_0+\\sum_{k=1}^ma_k\\cos k\\pi t+b_k\\sin k\\pi t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W^r_\\infty(M) and an upper bound for similar approximations in the class W^r_p(M) with \\frac43 are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously. Bibliography: 13 titles.

  1. Gup-Based and Snyder Noncommutative Algebras, Relativistic Particle Models, Deformed Symmetries and Interaction: a Unified Approach

    NASA Astrophysics Data System (ADS)

    Pramanik, Souvik; Ghosh, Subir

    2013-08-01

    We have developed a unified scheme for studying noncommutative algebras based on generalized uncertainty principle (GUP) and Snyder form in a relativistically covariant point particle Lagrangian (or symplectic) framework. Even though the GUP-based algebra and Snyder algebra are very distinct, the more involved latter algebra emerges from an approximation of the Lagrangian model of the former algebra. Deformed Poincaré generators for the systems that keep space-time symmetries of the relativistic particle models have been studied thoroughly. From a purely constrained dynamical analysis perspective the models studied here are very rich and provide insights on how to consistently construct approximate models from the exact ones when nonlinear constraints are present in the system. We also study dynamics of the GUP particle in presence of external electromagnetic field.

  2. Gup-Based and Snyder Noncommutative Algebras, Relativistic Particle Models, Deformed Symmetries and Interaction: a Unified Approach

    NASA Astrophysics Data System (ADS)

    Pramanik, Souvik; Ghosh, Subir

    2013-10-01

    We have developed a unified scheme for studying noncommutative algebras based on generalized uncertainty principle (GUP) and Snyder form in a relativistically covariant point particle Lagrangian (or symplectic) framework. Even though the GUP-based algebra and Snyder algebra are very distinct, the more involved latter algebra emerges from an approximation of the Lagrangian model of the former algebra. Deformed Poincaré generators for the systems that keep space-time symmetries of the relativistic particle models have been studied thoroughly. From a purely constrained dynamical analysis perspective the models studied here are very rich and provide insights on how to consistently construct approximate models from the exact ones when nonlinear constraints are present in the system. We also study dynamics of the GUP particle in presence of external electromagnetic field.

  3. Generalization of n-ary Nambu algebras and beyond

    SciTech Connect

    Ataguema, H.; Makhlouf, A.; Silvestrov, S.

    2009-08-15

    The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.

  4. 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.

  5. The Impact of Handheld Graphing Calculator Use on Student Achievement in Algebra 1

    ERIC Educational Resources Information Center

    Heller, Joan I.; Curtis, Deborah A.; Jaffe, Rebecca; Verboncoeur, Carol J.

    2005-01-01

    This study investigated the relationship between instructional use of handheld graphing calculators and student achievement in Algebra 1. Three end-of-course test forms were administered (without calculators) using matrix sampling to 458 high-school students in two suburban school districts in Oregon and Kansas. Test questions on two forms were…

  6. 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.

  7. Jucys-Murphy elements for Birman-Murakami-Wenzl algebras

    NASA Astrophysics Data System (ADS)

    Isaev, A. P.; Ogievetsky, O. V.

    2011-05-01

    The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.

  8. Kinematical superalgebras and Lie algebras of order 3

    SciTech Connect

    Campoamor-Stursberg, R.; Rausch de Traubenberg, M.

    2008-06-15

    We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonue-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order 3.

  9. On computational complexity of Clifford algebra

    NASA Astrophysics Data System (ADS)

    Budinich, Marco

    2009-05-01

    After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced by Budinich and Budinich ["A spinorial formulation of the maximum clique problem of a graph," J. Math. Phys. 47, 043502 (2006)].

  10. The algebra of two dimensional generalized Chebyshev-Koornwinder oscillator

    NASA Astrophysics Data System (ADS)

    Borzov, V. V.; Damaskinsky, E. V.

    2014-10-01

    In the previous works of Borzov and Damaskinsky ["Chebyshev-Koornwinder oscillator," Theor. Math. Phys. 175(3), 765-772 (2013)] and ["Ladder operators for Chebyshev-Koornwinder oscillator," in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillator-like system that is associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev-Koornwinder oscillator. In this paper, we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev-Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space H of functions that are defined on a region which is bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev-Koornwinder polynomials and these polynomials form an orthonormalized basis in the space H. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.

  11. Rational solutions of CYBE for simple compact real Lie algebras

    NASA Astrophysics Data System (ADS)

    Pop, Iulia; Stolin, Alexander

    2007-04-01

    In [A.A. Stolin, On rational solutions of Yang-Baxter equation for sl(n), Math. Scand. 69 (1991) 57-80; A.A. Stolin, On rational solutions of Yang-Baxter equation. Maximal orders in loop algebra, Comm. Math. Phys. 141 (1991) 533-548; A. Stolin, A geometrical approach to rational solutions of the classical Yang-Baxter equation. Part I, in: Walter de Gruyter & Co. (Ed.), Symposia Gaussiana, Conf. Alg., Berlin, New York, 1995, pp. 347-357] a theory of rational solutions of the classical Yang-Baxter equation for a simple complex Lie algebra g was presented. We discuss this theory for simple compact real Lie algebras g. We prove that up to gauge equivalence all rational solutions have the form X(u,v)={Ω}/{u-v}+t1∧t2+⋯+t∧t2n, where Ω denotes the quadratic Casimir element of g and {ti} are linearly independent elements in a maximal torus t of g. The quantization of these solutions is also emphasized.

  12. Homomorphisms between C*-algebras and linear derivations on C*-algebras

    NASA Astrophysics Data System (ADS)

    Park, Choonkil; Boo, Deok-Hoon; An, Jong Su

    2008-01-01

    It is shown that every almost unital almost linear mapping of a unital C*-algebra to a unital C*-algebra is a homomorphism when h(3nuy)=h(3nu)h(y) holds for all unitaries , all , and all , and that every almost unital almost linear continuous mapping of a unital C*-algebra of real rank zero to a unital C*-algebra is a homomorphism when h(3nuy)=h(3nu)h(y) holds for all , and v is invertible}, all , and all . Furthermore, we prove the Hyers-Ulam-Rassias stability of *-homomorphisms between unital C*-algebras, and -linear *-derivations on unital C*-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300.

  13. Lie algebra type noncommutative phase spaces are Hopf algebroids

    NASA Astrophysics Data System (ADS)

    Meljanac, Stjepan; Škoda, Zoran; Stojić, Martina

    2016-11-01

    For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.

  14. Bias in parameter estimation of form errors

    NASA Astrophysics Data System (ADS)

    Zhang, Xiangchao; Zhang, Hao; He, Xiaoying; Xu, Min

    2014-09-01

    The surface form qualities of precision components are critical to their functionalities. In precision instruments algebraic fitting is usually adopted and the form deviations are assessed in the z direction only, in which case the deviations at steep regions of curved surfaces will be over-weighted, making the fitted results biased and unstable. In this paper the orthogonal distance fitting is performed for curved surfaces and the form errors are measured along the normal vectors of the fitted ideal surfaces. The relative bias of the form error parameters between the vertical assessment and orthogonal assessment are analytically calculated and it is represented as functions of the surface slopes. The parameter bias caused by the non-uniformity of data points can be corrected by weighting, i.e. each data is weighted by the 3D area of the Voronoi cell around the projection point on the fitted surface. Finally numerical experiments are given to compare different fitting methods and definitions of the form error parameters. The proposed definition is demonstrated to show great superiority in terms of stability and unbiasedness.

  15. Gradings on the real form 𝔢6,-26

    NASA Astrophysics Data System (ADS)

    Draper, Cristina; Guido, Valerio

    2016-10-01

    We describe four fine gradings on the real form 𝔢6,-26 of the complex Lie algebra 𝔢6. They are precisely the gradings whose complexifications are fine gradings on the complex algebra. The universal grading groups are Z2 6 , Z × Z2 4 , Z 2 × Z2 3 , and Z 4 × Z2 4 .

  16. 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

  17. 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

  18. The Progressive Development of Early Embodied Algebraic Thinking

    NASA Astrophysics Data System (ADS)

    Radford, Luis

    2014-06-01

    In this article I present some results from a 5-year longitudinal investigation with young students about the genesis of embodied, non-symbolic algebraic thinking and its progressive transition to culturally evolved forms of symbolic thinking. The investigation draws on a cultural-historical theory of teaching and learning—the theory of objectification. Within this theory, thinking is conceived of as a form of reflection and action that is simultaneously material and ideal: It includes inner and outer speech, sensuous forms of imagination and visualisation, gestures, rhythm, and their intertwinement with material culture (symbols, artifacts, etc.). The theory articulates a cultural view of development as an unfolding dialectic process between culturally and historically constituted forms of mathematical knowing and semiotically mediated classroom activity. Looking at the experimental data through these theoretical lenses reveals a developmental path where embodied forms of thinking are sublated or subsumed into more sophisticated ones through the mediation of properly designed classroom activity.

  19. Highest-weight representations of Brocherd`s algebras

    SciTech Connect

    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.

  20. BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras

    NASA Astrophysics Data System (ADS)

    Graziani, Giacomo; Makhlouf, Abdenacer; Menini, Claudia; Panaite, Florin

    2015-10-01

    A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α,β\\colon A→ A such that α (a)(bc)=(ab)β (c), for all a, b, cin A. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).

  1. Quantum integrable systems related to lie algebras

    NASA Astrophysics Data System (ADS)

    Olshanetsky, M. A.; Perelomov, A. M.

    1983-03-01

    Some quantum integrable finite-dimensional systems related to Lie algebras are considered. This review continues the previous review of the same authors [83] devoted to the classical aspects of these systems. The dynamics of some of these systems is closely related to free motion in symmetric spaces. Using this connection with the theory of symmetric spaces some results such as the forms of spectra, wave functions, S-matrices, quantum integrals of motion are derived. In specific cases the considered systems describe the one-dimensional n-body systems interacting pairwise via potentials g2v( q) of the following 5 types: vI( q) = q-2, vII( q) = sinh-2q, vIII( q) = sin-2q, v IV(q) = P(q) , vV( q) = q-2 + ω2q2. Here P(q) is the Weierstrass function, so that the first three cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbour potential exp( qjqj+ 1 ) is moreover considered. This review presents from a general and universal point of view results obtained mainly over the past fifteen years. Besides, it contains some new results both of physical and mathematical interest.

  2. Algebraic construction of the Darboux matrix revisited

    NASA Astrophysics Data System (ADS)

    Cieśliński, Jan L.

    2009-10-01

    We present algebraic construction of Darboux matrices for 1+1-dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux-Bäcklund transformation, based on different λ-dependences of the Darboux matrix: polynomial, sum of partial fractions or the transfer matrix form. We derive symmetric N-soliton formulae in the general case. The matrix spectral parameter and dressing actions in loop groups are also discussed. We describe reductions to twisted loop groups, unitary reductions, the matrix Lax pair for the KdV equation and reductions of chiral models (harmonic maps) to SU(n) and to Grassmann spaces. We show that in the KdV case the nilpotent Darboux matrix generates the binary Darboux transformation. The paper is intended as a review of known results (usually presented in a novel context) but some new results are included as well, e.g., general compact formulae for N-soliton surfaces and linear and bilinear constraints on the nonisospectral Lax pair matrices which are preserved by Darboux transformations.

  3. Multivariate normality

    NASA Technical Reports Server (NTRS)

    Crutcher, H. L.; Falls, L. W.

    1976-01-01

    Sets of experimentally determined or routinely observed data provide information about the past, present and, hopefully, future sets of similarly produced data. An infinite set of statistical models exists which may be used to describe the data sets. The normal distribution is one model. If it serves at all, it serves well. If a data set, or a transformation of the set, representative of a larger population can be described by the normal distribution, then valid statistical inferences can be drawn. There are several tests which may be applied to a data set to determine whether the univariate normal model adequately describes the set. The chi-square test based on Pearson's work in the late nineteenth and early twentieth centuries is often used. Like all tests, it has some weaknesses which are discussed in elementary texts. Extension of the chi-square test to the multivariate normal model is provided. Tables and graphs permit easier application of the test in the higher dimensions. Several examples, using recorded data, illustrate the procedures. Tests of maximum absolute differences, mean sum of squares of residuals, runs and changes of sign are included in these tests. Dimensions one through five with selected sample sizes 11 to 101 are used to illustrate the statistical tests developed.

  4. Using Schemas to Develop Algebraic Thinking

    ERIC Educational Resources Information Center

    Steele, Diana F.

    2005-01-01

    This article describes ways in which students develop schemas as they generalize and formalize patterns when solving related algebraic problems that involve size, shape, growth, and change. (Contains 7 figures and 3 tables.)

  5. Cohomological invariants of central simple algebras

    NASA Astrophysics Data System (ADS)

    Merkurjev, A. S.

    2016-10-01

    We determine the indecomposable degree 3 cohomological invariants of tuples of central simple algebras with linear relations. Equivalently, we determine the degree 3 reductive cohomological invariants of all split semisimple groups of type A.

  6. ALGEBRAIC DEPENDENCE THEOREMS ON COMPLEX PSEUDOCONCAVE SPACES

    DTIC Science & Technology

    The notion of pseudoconcave space is introduced and classical theorems on algebraic dependence of meromorphic functions are extended for this new class of spaces and for sections in a coherent sheaf. (Author)

  7. Applications: Using Algebra in an Accounting Practice.

    ERIC Educational Resources Information Center

    Eisner, Gail A.

    1994-01-01

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

  8. Lisa's Lemonade Stand: Exploring Algebraic Ideas.

    ERIC Educational Resources Information Center

    Billings, Esther M. H.; Lakatos, Tracy

    2003-01-01

    Presents an activity, "Lisa's Lemonade Stand," that actively engages students in algebraic thinking as they analyze change by investigating relationships between variables and gain experience describing and representing these relationships graphically. (YDS)

  9. Characterization of the clotting activities of structurally different forms of activated factor IX. Enzymatic properties of normal human factor IXa alpha, factor IXa beta, and activated factor IX Chapel Hill.

    PubMed Central

    Griffith, M J; Breitkreutz, L; Trapp, H; Briet, E; Noyes, C M; Lundblad, R L; Roberts, H R

    1985-01-01

    Two structurally different forms of activated human Factor IX (Factor IXa alpha and IXa beta) have been previously reported to have essentially identical clotting activity in vitro. Although it has been shown that activated Factor IX Chapel Hill, an abnormal Factor IX isolated from the plasma of a patient with mild hemophilia B, and normal Factor IXa alpha are structurally very similar, the clotting activity of activated Factor IX Chapel Hill is much lower (approximately fivefold) than that of normal Factor IXa beta. In the present study we have prepared activated Factor IX by incubating human Factor IX with calcium and Russell's viper venom covalently bound to agarose. Fractionation of the activated Factor IX by high-performance liquid chromatography demonstrated the presence of both Factors IXa alpha and IXa beta. On the basis of active site concentration, determined by titration with antithrombin III, the clotting activities of activated Factor IX Chapel Hill and IXa alpha were similar, but both activities were less than 20% of the clotting activity of Factor IXa beta. Activated Factor IX activity was also measured in the absence of calcium, phospholipid, and Factor VIII, by determination of the rate of Factor X activation in the presence of polylysine. In the presence of polylysine, the rates of Factor X activation by activated Factor IX Chapel Hill, Factor IXa alpha, and Factor IXa beta were essentially identical. We conclude that the clotting activity of activated Factor IX Chapel Hill is reduced when compared with that of Factor IXa beta but essentially normal when compared with that of Factor IXa alpha. PMID:3871202

  10. Dynamical systems and quantum bicrossproduct algebras

    NASA Astrophysics Data System (ADS)

    Arratia, Oscar; del Olmo, Mariano A.

    2002-06-01

    We present a unified study of some aspects of quantum bicrossproduct algebras of inhomogeneous Lie algebras, such as Poincaré, Galilei and Euclidean in N dimensions. The action associated with the bicrossproduct structure allows us to obtain a nonlinear action over a new group linked to the translations. This new nonlinear action associates a dynamical system with each generator which is the object of our study.

  11. 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

  12. Edge covers and independence: Algebraic approach

    NASA Astrophysics Data System (ADS)

    Kalinina, E. A.; Khitrov, G. M.; Pogozhev, S. V.

    2016-06-01

    In this paper, linear algebra methods are applied to solve some problems of graph theory. For ordinary connected graphs, edge coverings and independent sets are considered. Some results concerning minimum edge covers and maximum matchings are proved with the help of linear algebraic approach. The problem of finding a maximum matching of a graph is fundamental both practically and theoretically, and has numerous applications, e.g., in computational chemistry and mathematical chemistry.

  13. An algebra of dimerization and its implications for G-protein coupled receptor signaling.

    PubMed

    Woolf, Peter J; Linderman, Jennifer J

    2004-07-21

    Many species of receptors form dimers, but how can we use this information to make predictions about signal transduction? This problem is particularly difficult when receptors dimerize with many different species, leading to a combinatoric increase in the possible number of dimer pairs. As an example system, we focus on receptors in the G-protein coupled receptor (GPCR) family. GPCRs have been shown to reversibly form dimers, but this dimerization does not directly affect signal transduction. Here we present a new theoretical framework called a dimerization algebra. This algebra provides a systematic and rational way to represent, manipulate, and in some cases simplify large and often complicated networks of dimerization interactions. To compliment this algebra, Monte Carlo simulations are used to predict dimerization's effect on receptor organization on the membrane, signal transduction, and internalization. These simulation results are directly comparable to various experimental measures such as fluorescence resonance energy transfer (FRET), and as such provide a link between the dimerization algebra and experimental data. As an example, we show how the algebra and computational results can be used to predict the effects of dimerization on the dopamine D2 and somatastatin SSTR1 receptors. When these predictions were compared to experimental findings from the literature, good agreement was found, demonstrating the utility of our approach. Applications of this work to the development of a novel class of dimerization-modulating drugs are also discussed.

  14. The algebraic cluster model: Structure of 16O

    NASA Astrophysics Data System (ADS)

    Bijker, R.; Iachello, F.

    2017-01-01

    We discuss an algebraic treatment of four-body clusters which includes both continuous and discrete symmetries. In particular, tetrahedral configurations with Td symmetry are analyzed with respect to the energy spectrum, transition form factors and B (EL) values. It is concluded that the low-lying spectrum of 16O can be described by four α particles at the vertices of a regular tetrahedron, not as a rigid structure but rather a more floppy structure with relatively large rotation-vibration interactions and Coriolis forces.

  15. The algebraic theory of latent projectors in lambda matrices

    NASA Technical Reports Server (NTRS)

    Denman, E. D.; Leyva-Ramos, J.; Jeon, G. J.

    1981-01-01

    Multivariable systems such as a finite-element model of vibrating structures, control systems, and large-scale systems are often formulated in terms of differential equations which give rise to lambda matrices. The present investigation is concerned with the formulation of the algebraic theory of lambda matrices and the relationship of latent roots, latent vectors, and latent projectors to the eigenvalues, eigenvectors, and eigenprojectors of the companion form. The chain rule for latent projectors and eigenprojectors for the repeated latent root or eigenvalues is given.

  16. Analytical solutions for systems of partial differential-algebraic equations.

    PubMed

    Benhammouda, Brahim; Vazquez-Leal, Hector

    2014-01-01

    This work presents the application of the power series method (PSM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-one and index-three are solved to show that PSM can provide analytical solutions of PDAEs in convergent series form. What is more, we present the post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a useful strategy to find exact solutions. The main advantage of the proposed methodology is that the procedure is based on a few straightforward steps and it does not generate secular terms or depends of a perturbation parameter.

  17. Negative base encoding in optical linear algebra processors

    NASA Technical Reports Server (NTRS)

    Perlee, C.; Casasent, D.

    1986-01-01

    In the digital multiplication by analog convolution algorithm, the bits of two encoded numbers are convolved to form the product of the two numbers in mixed binary representation; this output can be easily converted to binary. Attention is presently given to negative base encoding, treating base -2 initially, and then showing that the negative base system can be readily extended to any radix. In general, negative base encoding in optical linear algebra processors represents a more efficient technique than either sign magnitude or 2's complement encoding, when the additions of digitally encoded products are performed in parallel.

  18. AMG (Algebraic Multigrid): Basic Development, Applications and Theory.

    DTIC Science & Technology

    1987-01-07

    NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE NUMBER 22c OFFICE SYMBOL I n iude .4 re4 Code Captain Thomas (202) 767-5025 NM DO FORM 1473.83 APR...31 (1977), 333-390, ICASE Report 76-27. (B2) A. Brandt; "Algebraic multigrid: theory", Proc. Int’l M3onf., Copper 1.buntain., C), Aprol, 1983. (B3) A... Copper Mtn., OD, April 1983. (Dl) J.E. Dendy, Jr.; "Black box multigrid," LA-UR-Sl-2337 Los Alamos National Laboratory, Los Alamos, New Mexico, J. Ccn

  19. I CAN Learn[R] Pre-Algebra and Algebra. What Works Clearinghouse Intervention Report

    ERIC Educational Resources Information Center

    What Works Clearinghouse, 2007

    2007-01-01

    The "I CAN Learn[R] Pre-Algebra" and "Algebra" computerized curricula are designed to cover mathematics and problem-solving skills for ethnically diverse, inner-city students in grades 6-12. The curricula are designed to equip students with the skills they need to meet district, state, and national math objectives through an…

  20. 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…

  1. 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…

  2. The Application of a Computer Algebra System as a Tool in College Algebra.

    ERIC Educational Resources Information Center

    Mayes, Robert L.

    1995-01-01

    Students (n=61) in an experimental course stressing active student involvement and the use of a computer algebra system scored higher than students (n=76) in a traditional college algebra course on final measures of inductive reasoning, visualization, and problem solving while maintaining equivalent manipulation and computation skills. (Author/MLB)

  3. Algebra Is a Civil Right: Increasing Achievement for African American Males in Algebra through Collaborative Inquiry

    ERIC Educational Resources Information Center

    Davies Gomez, Lisa

    2012-01-01

    Algebra is the gatekeeper of access to higher-level math and science courses, higher education and future earning opportunities. Unequal numbers of African-American males drop out of Algebra and mathematics courses and underperform on tests of mathematical competency and are thus denied both essential skills and a particularly important pathway to…

  4. Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras

    NASA Astrophysics Data System (ADS)

    Arnlind, Joakim; Makhlouf, Abdenacer; Silvestrov, Sergei

    2010-04-01

    The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom-Nambu-Lie algebras obtained using this construction.

  5. Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*

    SciTech Connect

    Bank, R.; Falgout, R. D.; Jones, T.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. W.

    2015-10-29

    In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.

  6. 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

  7. Classification of central extensions of Lax operator algebras

    SciTech Connect

    Schlichenmaier, Martin

    2008-11-18

    Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.

  8. Linearizing W2,4 and WB2 algebras

    NASA Astrophysics Data System (ADS)

    Bellucci, S.; Krivonos, S.; Sorin, A.

    1995-02-01

    It has recently been shown that the W3 and W3(2) algebras can be considered as subalgebras in some linear conformal algebras. In this paper we show that the nonlinear algebras W2,4 and WB2 as well as Zamolodchikov's spin {5}/{2} superalgebra also can be embedded as subalgebras into some linear conformal algebras with a finite set of currents. These linear algebras give rise to new realizations of the nonlinear algebras which could be suitable in the construction of W-string theories.

  9. Classification of central extensions of Lax operator algebras

    NASA Astrophysics Data System (ADS)

    Schlichenmaier, Martin

    2008-11-01

    Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.

  10. Normals to a Parabola

    ERIC Educational Resources Information Center

    Srinivasan, V. K.

    2013-01-01

    Given a parabola in the standard form y[superscript 2] = 4ax, corresponding to three points on the parabola, such that the normals at these three points P, Q, R concur at a point M = (h, k), the equation of the circumscribing circle through the three points P, Q, and R provides a tremendous opportunity to illustrate "The Art of Algebraic…

  11. 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…

  12. Visual, Algebraic and Mixed Strategies in Visually Presented Linear Programming Problems.

    ERIC Educational Resources Information Center

    Shama, Gilli; Dreyfus, Tommy

    1994-01-01

    Identified and classified solution strategies of (n=49) 10th-grade students who were presented with linear programming problems in a predominantly visual setting in the form of a computerized game. Visual strategies were developed more frequently than either algebraic or mixed strategies. Appendix includes questionnaires. (Contains 11 references.)…

  13. Algebraic construction of a Nambu bracket for the two-dimensional vorticity equation.

    PubMed

    Sommer, M; Brazda, K; Hantel, M

    2011-08-29

    So far fluid mechanical Nambu brackets have mainly been given on an intuitive basis. Alternatively an algorithmic construction of such a bracket for the two-dimensional vorticity equation is presented here. Starting from the Lie-Poisson form and its algebraic properties it is shown how the Nambu representation can be explicitly constructed as the continuum limit from the structure preserving Zeitlin discretization.

  14. Explicit generators in rectangular affine W-algebras of type A

    NASA Astrophysics Data System (ADS)

    Arakawa, Tomoyuki; Molev, Alexander

    2016-10-01

    We produce in an explicit form free generators of the affine W-algebra of type A associated with a nilpotent matrix whose Jordan blocks are of the same size. This includes the principal nilpotent case and we thus recover the quantum Miura transformation of Fateev and Lukyanov.

  15. Explicit generators in rectangular affine W-algebras of type A

    NASA Astrophysics Data System (ADS)

    Arakawa, Tomoyuki; Molev, Alexander

    2017-01-01

    We produce in an explicit form free generators of the affine W-algebra of type A associated with a nilpotent matrix whose Jordan blocks are of the same size. This includes the principal nilpotent case and we thus recover the quantum Miura transformation of Fateev and Lukyanov.

  16. TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION

    NASA Technical Reports Server (NTRS)

    Smith, R. E.

    1994-01-01

    TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.

  17. Spinor representations of affine Lie algebras

    PubMed Central

    Frenkel, I. B.

    1980-01-01

    Let [unk] be an infinite-dimensional Kac-Moody Lie algebra of one of the types Dl+1(2), Bl(1), or Dl(1). These algebras are characterized by the property that an elimination of any endpoint of their Dynkin diagrams gives diagrams of types Bl or Dl of classical orthogonal Lie algebras. We construct two representations of a Lie algebra [unk], which we call spinor representations, following the analogy with the classical case. We obtain that every spinor representation is either irreducible or has two irreducible components. This provides us with an explicit construction of fundamental representations of [unk], two for the type Dl+1(2), three for Bl(1), and four for Dl(1). We note the profound connection of our construction with quantum field theory—in particular, with fermion fields. Comparing the character formulas of our representations with another construction of the fundamental representations of Kac-Moody Lie algebras of types Al(1), Dl(1), El(1), we obtain classical Jacobi identities and addition formulas for elliptic θ-functions. PMID:16592912

  18. ADA interpretative system for image algebra

    NASA Astrophysics Data System (ADS)

    Murillo, Juan J.; Wilson, Joseph N.

    1992-06-01

    An important research problem in image processing is to find appropriate tools to support algorithm development. There have been efforts to build algorithm development support systems for image algebra in several languages, but these systems still have the disadvantage of the time consuming algorithm development style associated with compilation-oriented programming. This paper starts with a description of the Run-Time Support Library (RTSL), which serves as the base for executing programs on both the Image Algebra Ada Translator (IAAT) and Image Algebra Ada Interpreter (IAAI). A presentation on the current status of IAAT and its capabilities is followed by a brief introduction to the utilization of the Image Display Manager (IDM) for image manipulation and analysis. We then discuss in detail the current development stage of IAAI and its relation with RTSL and IDM. The last section describes the design of a syntax-directed graphical user interface for IAAI. We close with an analysis of the current performance of IAAI, and future trends are discussed. Appendix A gives a brief introduction to Image Algebra (IA), and in Appendix B the reader is presented to the Image Algebra Ada (IAA) grammar.

  19. Permutation centralizer algebras and multimatrix invariants

    NASA Astrophysics Data System (ADS)

    Mattioli, Paolo; Ramgoolam, Sanjaye

    2016-03-01

    We introduce a class of permutation centralizer algebras which underly the combinatorics of multimatrix gauge-invariant observables. One family of such noncommutative algebras is parametrized by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of two-matrix models. The structure of the algebra, notably its dimension, its center and its maximally commuting subalgebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The center of the algebra allows efficient computation of a sector of multimatrix correlators. These generate the counting of a certain class of bicoloured ribbon graphs with arbitrary genus.

  20. TRACER version 1.1 A mathematica package for γ-algebra in arbitrary dimensions

    NASA Astrophysics Data System (ADS)

    Jamin, Matthias; Lautenbacher, Markus E.

    1993-02-01

    This paper describes the first MATHEMATICA implementation of γ-algebra in arbitrary space-time dimensions according to the 't Hooft-Veltman scheme. It is the only system based on a general purpose computer algebra system treating the γ 5-problem mathematically consistently in arbitrary dimensions. The TRACER package is capable of doing just purely algebraic manipulations as well as trace operations on strings of γ-algebra objects. In addition, it provides a set of utility functions for reordering, simplifying and improving the readability of the output. Optionally the output can be obtained in a form suitable to be fed into a T EX system for high quality text processing. As a whole, the TRACER package is intended as a computerized aid to a researcher working on higher order corrections in Relativistic Quantum Field Theories. A short comparison of procedural versus rule-based programming approaches is given and the discussion is supplemented by a toy implementation of the γ-algebra in rule-based style. The paper describes in detail the usage of the TRACER package and for further illustration a correlation function of two weak currents is calculated with the aid of TRACER. Finally, data on the performance of TRACER on some common platforms are given.

  1. From matrix models' topological expansion to topological string theories: counting surfaces with algebraic geometry

    NASA Astrophysics Data System (ADS)

    Orantin, N.

    2007-09-01

    The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links and extend them beyond the matrix models, following my work's evolution. First, I take care to define properly the hermitian 2 matrix model which gives rise to generating functions of discrete surfaces equipped with a spin structure. Then, I show how to compute all the terms in the topological expansion of any observable by using algebraic geometry tools. They are obtained as differential forms on an algebraic curve associated to the model: the spectral curve. In a second part, I show how to define such differentials on any algebraic curve even if it does not come from a matrix model. I then study their numerous symmetry properties under deformations of the algebraic curve. In particular, I show that these objects coincide with the topological expansion of the observable of a matrix model if the algebraic curve is the spectral curve of this model. Finally, I show that fine tuning the parameters ensure that these objects can be promoted to modular invariants and satisfy the holomorphic anomaly equation of the Kodaira-Spencer theory. This gives a new hint that the Dijkgraaf-Vafa conjecture is correct.

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

    SciTech Connect

    Nataf, J.M.; Winkelmann, F.

    1992-09-01

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

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

    SciTech Connect

    Nataf, J.M.; Winkelmann, F.

    1992-09-01

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

  4. IIA/IIB supergravity and ten-forms

    NASA Astrophysics Data System (ADS)

    Bergshoeff, E. A.; Hartong, J.; Howe, P. S.; Ortín, T.; Riccioni, F.

    2010-05-01

    We perform a careful investigation of which p-form fields can be introduced consistently with the supersymmetry algebra of IIA and/or IIB ten-dimensional supergravity. In particular the ten-forms, also known as “top-forms”, require a careful analysis since in this case, as we will show, closure of the supersymmetry algebra at the linear level does not imply closure at the non-linear level. Consequently, some of the (IIA and IIB) ten-form potentials introduced in earlier work of some of us are discarded. At the same time we show that new ten-form potentials, consistent with the full non-linear supersymmetry algebra can be introduced. We give a superspace explanation of our work. All of our results are precisely in line with the predictions of the E 11 algebra.

  5. 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.

  6. Algebraic quantum gravity (AQG): II. Semiclassical analysis

    NASA Astrophysics Data System (ADS)

    Giesel, K.; Thiemann, T.

    2007-05-01

    In the previous paper (Giesel and Thiemann 2006 Conceptual setup Preprint gr-qc/0607099) a new combinatorial and thus purely algebraical approach to quantum gravity, called algebraic quantum gravity (AQG), was introduced. In the framework of AQG, existing semiclassical tools can be applied to operators that encode the dynamics of AQG such as the master constraint operator. In this paper, we will analyse the semiclassical limit of the (extended) algebraic master constraint operator and show that it reproduces the correct infinitesimal generators of general relativity. Therefore, the question of whether general relativity is included in the semiclassical sector of the theory, which is still an open problem in LQG, can be significantly improved in the framework of AQG. For the calculations, we will substitute SU(2) with U(1)3. That this substitution is justified will be demonstrated in the third paper (Giesel and Thiemann 2006 Semiclassical perturbation theory Preprint gr-qc/0607101) of this series.

  7. 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.

  8. Sound Off! A Dialogue between Calculator and Algebra

    ERIC Educational Resources Information Center

    Wade, William R.

    2006-01-01

    This article illustrates the fact that unless tempered by algebraic reasoning, a graphing calculator can lead one to erroneous conclusions. It also demonstrates that some problems can be solved by combining technology with algebra.

  9. Infinitesimal deformations of naturally graded filiform Leibniz algebras

    NASA Astrophysics Data System (ADS)

    Khudoyberdiyev, A. Kh.; Omirov, B. A.

    2014-12-01

    In the present paper we describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any n-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra Fn3(0) . We establish that in the same way any n-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras Fn1,Fn2and Fn3(α) . Moreover, we describe the linear integrable deformations of the above-mentioned algebras with a fixed basis of HL2 in the set of all n-dimensional Leibniz algebras. Among these deformations one new rigid algebra has been found.

  10. Kac-Moody algebra and nonlinear sigma model

    NASA Astrophysics Data System (ADS)

    Ogura, Waichi; Hosoya, Akio

    1985-12-01

    We investigate the nonlinear sigma model over an arbitrary homogeneous space. Then it is shown that the sigma model realizes the Kac-Moody algebra as current algebra only if the homogeneous space is restricted to the group manifold.

  11. Upper bound for the length of commutative algebras

    NASA Astrophysics Data System (ADS)

    Markova, Ol'ga V.

    2009-12-01

    By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field one means the least positive integer k such that the words of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, an upper bound for the length of a commutative algebra in terms of a function of two invariants of the algebra, the dimension and the maximal degree of the minimal polynomial for the elements of the algebra, is obtained. As a corollary, a formula for the length of the algebra of diagonal matrices over an arbitrary field is obtained. Bibliography: 8 titles.

  12. Supersymmetry in physics: an algebraic overview

    SciTech Connect

    Ramond, P.

    1983-01-01

    In 1970, while attempting to generalize the Veneziano model (string model) to include fermions, I introduced a new algebraic structure which turned out to be a graded Lie algebra; it was used as a spectrum-generating algebra. This approach was soon after generalized to include interactions, yielding a complete model of fermions and boson (RNS model). In an unrelated work in the Soviet Union, it was shown how to generalize the Poincare group to include fermionic charges. However it was not until 1974 that an interacting field theory invariant under the Graded Poincare group in 3 + 1 dimensions was built (WZ model). Supersymmetric field theories turned out to have less divergent ultraviolet behavior than non-supersymmetric field theories. Gravity was generalized to include supersymmetry, to a theory called supergravity. By now many interacting local field theories exhibiting supersymmetry have been built and studied from 1 + 1 to 10 + 1 dimensions. Supersymmetric local field theories in less than 9 + 1 dimensions, can be understood as limits of multilocal (string) supersymmetric theories, in 9 + 1 dimensions. On the other hand, graded Lie algebras have been used in non-relativistic physics as approximate symmetries of Hamiltonians. The most striking such use so far helps comparing even and odd nuclei energy levels. It is believed that graded Lie algebras can be used whenever paired and unpaired fermions excitations can coexist. In this overview of a tremendously large field, I will only survey finite graded Lie algebras and their representations. For non-relativistic applications, all of GLA are potentially useful, while for relativistic applications, only these which include the Poincare group are to be considered.

  13. Constraint algebra for interacting quantum systems

    NASA Astrophysics Data System (ADS)

    Fubini, S.; Roncadelli, M.

    1988-04-01

    We consider relativistic constrained systems interacting with external fields. We provide physical arguments to support the idea that the quantum constraint algebra should be the same as in the free quantum case. For systems with ordering ambiguities this principle is essential to obtain a unique quantization. This is shown explicitly in the case of a relativistic spinning particle, where our assumption about the constraint algebra plus invariance under general coordinate transformations leads to a unique S-matrix. On leave from Dipartimento di Fisica Nucleare e Teorica, Università di Pavia and INFN, I-27100 Pavia, Italy.

  14. Algebraic surface design and finite element meshes

    NASA Technical Reports Server (NTRS)

    Bajaj, Chandrajit L.

    1992-01-01

    Some of the techniques are summarized which are used in constructing C sup 0 and C sup 1 continuous meshes of low degree, implicitly defined, algebraic surface patches in three dimensional space. These meshes of low degree algebraic surface patches are used to construct accurate computer models of physical objects. These meshes are also used in the finite element simulation of physical phenomena (e.g., heat dissipation, stress/strain distributions, fluid flow characteristics) required in the computer prototyping of both the manufacturability and functionality of the geometric design.

  15. Fréchet-algebraic deformation quantizations

    NASA Astrophysics Data System (ADS)

    Waldmann, S.

    2014-09-01

    In this review I present some recent results on the convergence properties of formal star products. Based on a general construction of a Fréchet topology for an algebra with countable vector space basis I discuss several examples from deformation quantization: the Wick star product on the flat phase space m2n gives a first example of a Fréchet algebraic framework for the canonical commutation relations. More interesting, the star product on the Poincare disk can be treated along the same lines, leading to a non-trivial example of a convergent star product on a curved Kahler manifold.

  16. Shapes and stability of algebraic nuclear models

    NASA Technical Reports Server (NTRS)

    Lopez-Moreno, Enrique; Castanos, Octavio

    1995-01-01

    A generalization of the procedure to study shapes and stability of algebraic nuclear models introduced by Gilmore is presented. One calculates the expectation value of the Hamiltonian with respect to the coherent states of the algebraic structure of the system. Then equilibrium configurations of the resulting energy surface, which depends in general on state variables and a set of parameters, are classified through the Catastrophe theory. For one- and two-body interactions in the Hamiltonian of the interacting Boson model-1, the critical points are organized through the Cusp catastrophe. As an example, we apply this Separatrix to describe the energy surfaces associated to the Rutenium and Samarium isotopes.

  17. Bohr model as an algebraic collective model

    SciTech Connect

    Rowe, D. J.; Welsh, T. A.; Caprio, M. A.

    2009-05-15

    Developments and applications are presented of an algebraic version of Bohr's collective model. Illustrative examples show that fully converged calculations can be performed quickly and easily for a large range of Hamiltonians. As a result, the Bohr model becomes an effective tool in the analysis of experimental data. The examples are chosen both to confirm the reliability of the algebraic collective model and to show the diversity of results that can be obtained by its use. The focus of the paper is to facilitate identification of the limitations of the Bohr model with a view to developing more realistic, computationally tractable models.

  18. 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.

  19. Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups

    NASA Astrophysics Data System (ADS)

    Batat, W.; Onda, K.

    2017-04-01

    We study algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are solvsolitons. In particular, we obtain new solitons on G2, G5, and G6, and we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not be algebraic Ricci solitons.

  20. Capability and Schur multiplier of a pair of Lie algebras

    NASA Astrophysics Data System (ADS)

    Johari, Farangis; Parvizi, Mohsen; Niroomand, Peyman

    2017-04-01

    The aim of this work is to find some criteria for detecting the capability of a pair of Lie algebras. We characterize the exact structure of all pairs of capable Lie algebras in the class of abelian and Heisenberg ones. Among the other results, we also give some exact sequences on the Schur multiplier and exterior product of Lie algebras.

  1. 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…

  2. The Impact of Early Algebra: Results from a Longitudinal Intervention

    ERIC Educational Resources Information Center

    Brizuela, Bárbara M.; Martinez, Mara V.; Cayton-Hodges, Gabrielle A.

    2013-01-01

    In this paper, we provide evidence of the impact of early algebra (EA) over time. We document this impact in the following ways: (a) by showing the performance over time of an experimental group of 15 children on an algebra assessment, from 3rd to 5th grade; and (b) by showing how the performance on an algebra assessment of children from an…

  3. Changing Pre-Service Elementary Teachers' Attitudes to Algebra.

    ERIC Educational Resources Information Center

    McGowen, Mercedes A.; Davis, Gary E.

    This article addresses the question: "What are the implications for the preparation of prospective elementary teachers of 'early algebra' in the elementary grades curriculum?" Part of the answer involves language aspects of algebra: in particular, how a change in pre-service teachers' attitudes to algebra, from instrumental to relational, is…

  4. A Research Base Supporting Long Term Algebra Reform?

    ERIC Educational Resources Information Center

    Kaput, James J.

    This paper discusses three dimensions of algebra reform: breadth, integration, and pedagogy. Breadth of algebra includes algebra as: generalizing and formalizing patterns and constraints; syntactically-guided manipulation of formalisms; study of structures abstracted from computations and relations; study of functions, relations, and joint…

  5. 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…

  6. Remarks on Virasoro and Kac-Moody Algebras

    NASA Astrophysics Data System (ADS)

    Grabowski, J.; Marmo, G.; Perelomov, A.; Simoni, A.

    Parametric realizations of Virasoro or Kac-Moody algebras are constructed on a generic manifold carrying an appropriate vector field. It is shown that the centrally extended algebras cannot be realized as algebras of vector fields on finite-dimensional manifolds.

  7. Processes Used by College Students in Understanding Basic Algebra.

    ERIC Educational Resources Information Center

    Rachlin, Sidney Lee

    The purpose of this study was to uncover information about and gain a greater insight into the extent to which students who are successful in a basic algebra course: l) demonstrate a reversibility of reasoning processes when solving algebraic problems; 2) demonstrate a flexibility of reasoning processes when solving algebraic problems; 3)…

  8. 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…

  9. 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,…

  10. Static friction, differential algebraic systems and numerical stability

    NASA Astrophysics Data System (ADS)

    Chen, Jian; Schinner, Alexander; Matuttis, Hans-Georg

    We show how Differential Algebraic Systems (Ordinary Differential Equations with algebraic constraints) in mechanics are affected by stability issues and we implement Lubich's projection method to reduce the error to practically zero. Then, we explain how the "numerically exact" implementation for static friction by Differential Algebraic Systems can be stabilized. We conclude by comparing the corresponding steps in the "Contact mechanics" introduced by Moreau.

  11. Pilot Study on Algebra Learning among Junior Secondary Students

    ERIC Educational Resources Information Center

    Poon, Kin-Keung; Leung, Chi-Keung

    2010-01-01

    The purpose of the study reported herein was to identify the common mistakes made by junior secondary students in Hong Kong when learning algebra and to compare teachers' perceptions of students' ability with the results of an algebra test. An algebra test was developed and administered to a sample of students (aged between 13 and 14 years). From…

  12. Evolution of a Teaching Approach for Beginning Algebra

    ERIC Educational Resources Information Center

    Banerjee, Rakhi; Subramaniam, K.

    2012-01-01

    The article reports aspects of the evolution of a teaching approach over repeated trials for beginning symbolic algebra. The teaching approach emphasized the structural similarity between arithmetic and algebraic expressions and aimed at supporting students in making a transition from arithmetic to beginning algebra. The study was conducted with…

  13. Should College Algebra be a Prerequisite for Taking Psychology Statistics?

    ERIC Educational Resources Information Center

    Sibulkin, Amy E.; Butler, J. S.

    2008-01-01

    In order to consider whether a course in college algebra should be a prerequisite for taking psychology statistics, we recorded students' grades in elementary psychology statistics and in college algebra at a 4-year university. Students who earned credit in algebra prior to enrolling in statistics for the first time had a significantly higher mean…

  14. 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…

  15. Algebraic direct methods for few-atoms structure models.

    PubMed

    Hauptman, Herbert A; Guo, D Y; Xu, Hongliang; Blessing, Robert H

    2002-07-01

    As a basis for direct-methods phasing at very low resolution for macromolecular crystal structures, normalized structure-factor algebra is presented for few-atoms structure models with N = 1, 2, 3, em leader equal atoms or polyatomic globs per unit cell. Main results include: [see text]. Triplet discriminant Delta(hk) and triplet weight W(hk) parameters, a approximately 4.0 and b approximately 3.0, respectively, were determined empirically in numerical error analyses. Tests with phases calculated for few-atoms 'super-glob' models of the protein apo-D-glyceraldehyde-3-phosphate dehydrogenase (approximately 10000 non-H atoms) showed that low-resolution phases from the new few-atoms tangent formula were much better than conventional tangent formula phases for N = 2 and 3; phases from the two formulae were essentially the same for N > or = 4.

  16. 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…

  17. Algebra Reform, Resarch, and the Classroom: A Reaction to a Research Base Supporting Long Term Algebra Reform.

    ERIC Educational Resources Information Center

    Burrill, Gail

    This paper is a reaction to a plenary address, "A Research Base Supporting Long Term Algebra Reform?" by James Kaput (SE 057 182). Three dimensions of algebra reform identified by Kaput (breadth, integration, and pedagogy) are discussed and contrasted with the draft version of the Algebra Document from the National Council of Teachers of…

  18. 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…

  19. Algebraic Foundations of Stability Theory: A Computerized Linear Algebra Bibliography

    DTIC Science & Technology

    1976-09-30

    permanent, an ele - mentary symmetric function of singular values squared, the property of being unitary, or the property of being of rank 1. The set of...slightly different form to the eigenvalues of products or minors. (ii) A family of inequalities known as the Amir- Moez inequalities, which were believed for

  20. Regular Gleason Measures and Generalized Effect Algebras

    NASA Astrophysics Data System (ADS)

    Dvurečenskij, Anatolij; Janda, Jiří

    2015-12-01

    We study measures, finitely additive measures, regular measures, and σ-additive measures that can attain even infinite values on the quantum logic of a Hilbert space. We show when particular classes of non-negative measures can be studied in the frame of generalized effect algebras.

  1. A Photographic Assignment for Abstract Algebra

    ERIC Educational Resources Information Center

    Warrington, Gregory S.

    2009-01-01

    We describe a simple photographic assignment appropriate for an abstract algebra (or other) course. Students take digital pictures around campus of various examples of symmetry. They then classify these pictures according to which of the 17 plane symmetry groups they belong. (Contains 2 figures.)

  2. Using Group Explorer in Teaching Abstract Algebra

    ERIC Educational Resources Information Center

    Schubert, Claus; Gfeller, Mary; Donohue, Christopher

    2013-01-01

    This study explores the use of Group Explorer in an undergraduate mathematics course in abstract algebra. The visual nature of Group Explorer in representing concepts in group theory is an attractive incentive to use this software in the classroom. However, little is known about students' perceptions on this technology in learning concepts in…

  3. I Teach Economics, Not Algebra and Calculus

    ERIC Educational Resources Information Center

    Hey, John D.

    2005-01-01

    Most people learn to drive without knowing how the engine works. In a similar vein, the author believes that students can learn economics without knowing the algebra and calculus underlying the results. If instructors follow the philosophy of other economics courses in using graphs to illustrate the results, and draw the graphs accurately, then…

  4. Hungry for Early Spatial and Algebraic Reasoning

    ERIC Educational Resources Information Center

    Cross, Dionne I.; Adefope, Olufunke; Lee, Mi Yeon; Perez, Arnulfo

    2012-01-01

    Tasks that develop spatial and algebraic reasoning are crucial for learning and applying advanced mathematical ideas. In this article, the authors describe how two early childhood teachers used stories as the basis for a unit that supports spatial reasoning in kindergartners and first graders. Having mathematical experiences that go beyond…

  5. Journal Writing: Enlivening Elementary Linear Algebra.

    ERIC Educational Resources Information Center

    Meel, David E.

    1999-01-01

    Examines the various issues surrounding the implementation of journal writing in an undergraduate linear algebra course. Identifies the benefits of incorporating journal writing into an undergraduate mathematics course, which are supported with students' comments from their journals and their reflections on the process. Contains 14 references.…

  6. The geometric semantics of algebraic quantum mechanics.

    PubMed

    Cruz Morales, John Alexander; Zilber, Boris

    2015-08-06

    In this paper, we will present an ongoing project that aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We argue that this approach provides a geometric semantics for such a formalism by means of establishing a (non-commutative) duality between certain algebraic and geometric objects.

  7. An Evaluation of Saxon's Algebra Test.

    ERIC Educational Resources Information Center

    Johnson, Dale M.; Smith, Blaine

    1987-01-01

    John Saxon's incremental development model has been proclaimed as a superior teaching strategy for mathematics. This study evaluated the Saxon approach and textbook using 276 Algebra I students in experimental and control groups. The groups were compared in cognitive and affective areas. Results are presented. (Author/MT)

  8. Programed Instruction in Elementary Algebra: An Experiment

    ERIC Educational Resources Information Center

    Lial, Margaret L.

    1970-01-01

    Report of an experiment which investigated the use of a programed elementary algebra text as a teaching method. The method was evaluated on the basis of student evaluation of the course and the percentage of students achieving a grade of C or better. Results indicated that the use of programed texts was superior to the traditional approach using…

  9. Private quantum subsystems and quasiorthogonal operator algebras

    NASA Astrophysics Data System (ADS)

    Levick, Jeremy; Jochym-O'Connor, Tomas; Kribs, David W.; Laflamme, Raymond; Pereira, Rajesh

    2016-03-01

    We generalize a recently discovered example of a private quantum subsystem to find private subsystems for Abelian subgroups of the n-qubit Pauli group, which exist in the absence of private subspaces. In doing so, we also connect these quantum privacy investigations with the theory of quasiorthogonal operator algebras through the use of tools from group theory and operator theory.

  10. Pre-Algebra Groups. Concepts & Applications.

    ERIC Educational Resources Information Center

    Montgomery County Public Schools, Rockville, MD.

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

  11. Parallel Algebraic Multigrids for Structural mechanics

    SciTech Connect

    Brezina, M; Tong, C; Becker, R

    2004-05-11

    This paper presents the results of a comparison of three parallel algebraic multigrid (AMG) preconditioners for structural mechanics applications. In particular, they are interested in investigating both the scalability and robustness of the preconditioners. Numerical results are given for a range of structural mechanics problems with various degrees of difficulty.

  12. Thinking Algebraically across the Elementary School Curriculum

    ERIC Educational Resources Information Center

    Soares, June; Blanton, Maria L.; Kaput, James J.

    2006-01-01

    With testing and accountability on everyone's mind, teachers are looking for creative ways to teach "all" subjects. Literacy is on the top of the list for testing, so it seems to get top priority. But how can teachers make sure that mathematics, especially a crucial area such as algebraic thinking, is a priority as well? Integrating subject matter…

  13. 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.…

  14. Using Technology to Balance Algebraic Explorations

    ERIC Educational Resources Information Center

    Kurz, Terri L.

    2013-01-01

    In 2000, the "National Council of Teachers of Mathematics" recommended that Algebra Standards, "instructional programs from prekindergarten through grade 12 should enable all students to use mathematical models to represent and understand quantitative relationships." In this article, the authors suggest the "Balance"…

  15. Noise limitations in optical linear algebra processors.

    PubMed

    Batsell, S G; Jong, T L; Walkup, J F; Krile, T F

    1990-05-10

    A general statistical noise model is presented for optical linear algebra processors. A statistical analysis which includes device noise, the multiplication process, and the addition operation is undertaken. We focus on those processes which are architecturally independent. Finally, experimental results which verify the analytical predictions are also presented.

  16. Lie algebras and linear differential equations.

    NASA Technical Reports Server (NTRS)

    Brockett, R. W.; Rahimi, A.

    1972-01-01

    Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

  17. Constructive Learning in Undergraduate Linear Algebra

    ERIC Educational Resources Information Center

    Chandler, Farrah Jackson; Taylor, Dewey T.

    2008-01-01

    In this article we describe a project that we used in our undergraduate linear algebra courses to help our students successfully master fundamental concepts and definitions and generate interest in the course. We describe our philosophy and discuss the projects overall success.

  18. Stability of Linear Equations--Algebraic Approach

    ERIC Educational Resources Information Center

    Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.

    2012-01-01

    This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…

  19. Fundamental Theorems of Algebra for the Perplexes

    ERIC Educational Resources Information Center

    Poodiak, Robert; LeClair, Kevin

    2009-01-01

    The fundamental theorem of algebra for the complex numbers states that a polynomial of degree n has n roots, counting multiplicity. This paper explores the "perplex number system" (also called the "hyperbolic number system" and the "spacetime number system") In this system (which has extra roots of +1 besides the usual [plus or minus]1 of the…

  20. Using geometric algebra to study optical aberrations

    SciTech Connect

    Hanlon, J.; Ziock, H.

    1997-05-01

    This paper uses Geometric Algebra (GA) to study vector aberrations in optical systems with square and round pupils. GA is a new way to produce the classical optical aberration spot diagrams on the Gaussian image plane and surfaces near the Gaussian image plane. Spot diagrams of the third, fifth and seventh order aberrations for square and round pupils are developed to illustrate the theory.