Lie algebra of conformal Killing–Yano forms
NASA Astrophysics Data System (ADS)
Ertem, Ümit
2016-06-01
We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing–Yano forms. A new Lie bracket for conformal Killing–Yano forms that corresponds to slightly modified Schouten–Nijenhuis bracket of differential forms is proposed. We show that conformal Killing–Yano forms satisfy a graded Lie algebra in constant curvature manifolds. It is also proven that normal conformal Killing–Yano forms in Einstein manifolds also satisfy a graded Lie algebra. The constructed graded Lie algebras reduce to the graded Lie algebra of Killing–Yano forms and the Lie algebras of conformal Killing and Killing vector fields in special cases.
NASA Astrophysics Data System (ADS)
Kuipers, J.
2012-06-01
New features of the symbolic algebra package Form 4 are discussed. Most importantly, these features include polynomial factorization and polynomial gcd computation. Examples of their use are shown. One of them is an exact version of Mincer which gives answers in terms of rational polynomials and 5 master integrals.
A new algebra core for the minimal form' problem
Purtill, M.R. . Center for Communications Research); Oliveira, J.S.; Cook, G.O. Jr. )
1991-12-20
The demands of large-scale algebraic computation have led to the development of many new algorithms for manipulating algebraic objects in computer algebra systems. For instance, parallel versions of many important algorithms have been discovered. Simultaneously, more effective symbolic representations of algebraic objects have been sought. Also, while some clever techniques have been found for improving the speed of the algebraic simplification process, little attention has been given to the issue of restructuring expressions, or transforming them into minimal forms.'' By minimal form,'' we mean that form of an expression that involves a minimum number of operations. In a companion paper, we introduce some new algorithms that are very effective at finding minimal forms of expressions. These algorithms require algebraic and combinatorial machinery that is not readily available in most algebra systems. In this paper we describe a new algebra core that begins to provide the necessary capabilities.
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…
FormCalc 8: Better Algebra and Vectorization
NASA Astrophysics Data System (ADS)
Chokoufe Nejad, B.; Hahn, T.; Lang, J.-N.; Mirabella, E.
2014-06-01
We present Version 8 of the Feynman-diagram calculator FormCalc. New features include in particular significantly improved algebraic simplification as well as vectorization of the generated code. The Cuba Library, used in FormCalc, features checkpointing to disk for all integration algorithms.
The method of minimal normal forms
Mane, S.R.; Weng, W.T.
1992-01-01
Normal form methods for solving nonlinear differential equations are reviewed and the comparative merits of three methods are evaluated. The concept of the minimal normal form is explained and is shown to be superior to other choices. The method is then extended to apply to the evaluation of discrete maps of an accelerator or storage ring. Such an extension, as suggested in this paper, is more suited for accelerator-based applications than a formulation utilizing continuous differential equations. A computer code has been generated to systematically implement various normal form formulations for maps in two-dimensional phase space. Specific examples of quadratic and cubic nonlinear fields were used and solved by the method developed. The minimal normal form method shown here gives good results using relatively low order expansions.
The method of minimal normal forms
Mane, S.R.; Weng, W.T.
1992-12-31
Normal form methods for solving nonlinear differential equations are reviewed and the comparative merits of three methods are evaluated. The concept of the minimal normal form is explained and is shown to be superior to other choices. The method is then extended to apply to the evaluation of discrete maps of an accelerator or storage ring. Such an extension, as suggested in this paper, is more suited for accelerator-based applications than a formulation utilizing continuous differential equations. A computer code has been generated to systematically implement various normal form formulations for maps in two-dimensional phase space. Specific examples of quadratic and cubic nonlinear fields were used and solved by the method developed. The minimal normal form method shown here gives good results using relatively low order expansions.
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
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.
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.
A brief study of quasi-normal modes in relativistic stars using algebraic computation
Campos, M. de
2010-11-12
The damped oscillations in relativistic stars generate gravitational waves that in the literature appear under the general denomination of quasi-normal modes. In this brief note we want offer some information about the use of algebraic computation to obtain the field equations and the perturbed version of them, in the context of general relativity theory, that is the framework to study gravitational waves in this work.
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).
Resonant normal form and asymptotic normal form behaviour in magnetic bottle Hamiltonians
NASA Astrophysics Data System (ADS)
Efthymiopoulos, C.; Harsoula, M.; Contopoulos, G.
2015-04-01
We consider normal forms in ‘magnetic bottle’ type Hamiltonians of the form H=\\frac{1}{2}(ρ^2_ρ+ω^2_1ρ^2) +\\frac{1}{2}p^2_z+hot (second frequency ω2 equal to zero in the lowest order). Our main results are: (i) a novel method to construct the normal form in cases of resonance, and (ii) a study of the asymptotic behaviour of both the non-resonant and the resonant series. We find that, if we truncate the normal form series at order r, the series remainder in both constructions decreases with increasing r down to a minimum, and then it increases with r. The computed minimum remainder turns to be exponentially small in \\frac{1}{Δ E} , where ΔE is the mirror oscillation energy, while the optimal order scales as an inverse power of ΔE. We estimate numerically the exponents associated with the optimal order and the remainder's exponential asymptotic behaviour. In the resonant case, our novel method allows to compute a ‘quasi-integral’ (i.e. truncated formal integral) valid both for each particular resonance as well as away from all resonances. We applied these results to a specific magnetic bottle Hamiltonian. The non-resonant normal form yields theoretical invariant curves on a surface of section which fit well the empirical curves away from resonances. On the other hand the resonant normal form fits very well both the invariant curves inside the islands of a particular resonance as well as the non-resonant invariant curves. Finally, we discuss how normal forms allow to compute a critical threshold for the onset of global chaos in the magnetic bottle.
Algebras Generated by Geometric Scalar Forms and their Applications in Physics and Social Sciences
Keller, Jaime
2008-09-17
The present paper analyzes the consequences of defining that the geometric scalar form is not necessarily quadratic, but in general K-atic, that is obtained from the K{sup th} power of the linear form, requiring {l_brace}e{sub i};i = 1,...,N;(e{sub i}){sup K} = 1{r_brace} and d-vector {sigma}{sub i}x{sub i}e{sub i}. We consider the algebras which are thus generated, for positive integer K, a generalization of the geometric algebras we know under the names of Clifford or Grassmann algebras. We then obtain a set of geometric K-algebras. We also consider the generalization of special functions of geometry which corresponds to the K-order scalar forms (as trigonometric functions and other related geometric functions which are based on the use of quadratic forms). We present an overview of the use of quadratic forms in physics as in our general theory, we have called START. And, in order to give an introduction to the use of the more general K-algebras and to the possible application to sciences other than physics, the application to social sciences is considered.For the applications to physics we show that quadratic spaces are a fundamental clue to understand the structure of theoretical physics (see, for example, Keller in ICNAAM 2005 and 2006)
The Hilbert polynomial and linear forms in the logarithms of algebraic numbers
NASA Astrophysics Data System (ADS)
Aleksentsev, Yu M.
2008-12-01
We prove a new estimate for homogeneous linear forms with integer coefficients in the logarithms of algebraic numbers. We obtain a qualitative improvement of the estimate depending on the coefficients of the linear form and the best value of the constant in the estimate in the case when the number of logarithms is not too large.
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.
Real forms of very extended Kac-Moody algebras and theories with eight supersymmetries
NASA Astrophysics Data System (ADS)
Riccioni, Fabio; West, Peter; Van Proeyen, Antoine
2008-05-01
We consider all theories with eight supersymmetries whose reduction to three dimensions gives rise to scalars that parametrise symmetric manifolds. We conjecture that these theories are non-linear realisations of very-extended Kac-Moody algebras for suitable choices of real forms. We show for the most interesting cases that the bosonic sector of the supersymmetric theory is precisely reproduced by the corresponding non-linear realisation.
Gamow functionals on operator algebras
NASA Astrophysics Data System (ADS)
Castagnino, M.; Gadella, M.; Betán, R. Id; Laura, R.
2001-11-01
We obtain the precise form of two Gamow functionals representing the exponentially decaying part of a quantum resonance and its mirror image that grows exponentially, as a linear, positive and continuous functional on an algebra containing observables. These functionals do not admit normalization and, with an appropriate choice of the algebra, are time reversal of each other.
Form in Algebra: Reflecting, with Peacock, on Upper Secondary School Teaching.
ERIC Educational Resources Information Center
Menghini, Marta
1994-01-01
Discusses algebra teaching by looking back into the history of algebra and the work of George Peacock, who considered algebra from two points of view: symbolic and instrumental. Claims that, to be meaningful, algebra must be linked to real-world problems. (18 references) (MKR)
The use of normal forms for analysing nonlinear mechanical vibrations
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
Implementation of control point form of algebraic grid-generation technique
NASA Technical Reports Server (NTRS)
Choo, Yung K.; Miller, David P.; Reno, Charles J.
1991-01-01
The control point form (CPF) provides explicit control of physical grid shape and grid spacing through the movement of the control points. The control point array, called a control net, is a space grid type arrangement of locations in physical space with an index for each direction. As an algebraic method CPF is efficient and works well with interactive computer graphics. A family of menu-driven, interactive grid-generation computer codes (TURBO) is being developed by using CPF. Key features of TurboI (a TURBO member) are discussed and typical results are presented. TurboI runs on any IRIS 4D series workstation.
Motility in normal and filamentous forms of Rhodospirillum rubrum.
Lee, A G; Fitzsimons, J T
1976-04-01
By suitable choice of medium, Rhodospirillum rubrum has been grown both in normal (length 2 mum) and filamentous (length up to 60 mum) forms. Both forms were highly motile, and negatively-stained preparations showed bipolar flagellated cells, with an average of seven flagella at each pole. Motion consisted of a series of runs and tumbles, the ditribution of run time-lengths being Poissonian. Both forms tumbled in response to dark shock and showed negative chemotaxis to oxygen. The observation that the motility pattern was very similar in normal and filamentous forms makes chemical control of tumbling unlikely and favours a system involving membrane potentials. PMID:819618
Holomorphic normal form of nonlinear perturbations of nilpotent vector fields
NASA Astrophysics Data System (ADS)
Stolovitch, Laurent; Verstringe, Freek
2016-07-01
We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension n ≥ 3. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensures that the normalizing transformation is holomorphic at the fixed point.We shall show that this sufficient condition is a nilpotent version of Bruno's condition (A). In dimension 2, no condition is required since, according to Stróżyna-Żołladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and sl2(C)-representations.
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.
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.
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…
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…
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
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.
Efficient linear algebra routines for symmetric matrices stored in packed form.
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.
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.
Explicit solutions of normal form of driven oscillatory systems
NASA Astrophysics Data System (ADS)
Tsarouhas, George E.; Ross, John
1987-12-01
We consider an oscillatory dissipative system driven by external sinusoidal perturbations of given amplitude Q and frequency ω. The kinetic equations are transformed to normal form and solved for small Q, near a Hopf bifurcation to oscillations in the autonomous system, for ratios ωn to the autonomous frequency of irrational so that the response of the system is quasiperiodic. The system is assumed to have either two variables or is adequately described by two variables near the bifurcation, and we obtain explicit solutions for this general case. The equations show interesting effects of external perturbations on limit cycles, both stable and unstable. Next we treat a specific model (Brusselator) and show by comparison with results of numerical integration that the theory predicts well the shape of the perturbed limit cycle, its variation with changes in constraints and parameters, and the point of transition from quasiperiodic to periodic response.
Matrix Equations and Normal Forms for Context-Free Grammars
NASA Astrophysics Data System (ADS)
Rosenkrantz, Daniel J.
The relationship between the set of productions of a context-free grammar and the corresponding set of defining equations is first pointed out. The closure operation on a matrix of strings is defined and this concept is used to formalize the solution to a set of linear equations. A procedure is then given for rewriting a context-free grammar in Greibach normal form, where the replacement string of each production begins with a terminal symbol. An additional procedure is given for rewriting the grammar so that each replacement string both begins and ends with a terminal symbol. Neither procedure requires the evaluation of regular expressions over the total vocabulary of the grammar, as is required by Greibach’s procedure.
Genotype phenotype mapping in RNA viruses - disjunctive normal form learning.
Wu, Chuang; Walsh, Andrew S; Rosenfeld, Roni
2011-01-01
RNA virus phenotypic changes often result from multiple alternative molecular mechanisms, where each mechanism involves changes to a small number of key residues. Accordingly, we propose to learn genotype-phenotype functions, using Disjunctive Normal Form (DNF) as the assumed functional form. In this study we develop DNF learning algorithms that attempt to construct predictors as Boolean combinations of covariates. We demonstrate the learning algorithm's consistency and efficiency on simulated sequences, and establish their biological relevance using a variety of real RNA virus datasets representing different viral phenotypes, including drug resistance, antigenicity, and pathogenicity. We compare our algorithms with previously published machine learning algorithms in terms of prediction quality: leave-one-out performance shows superior accuracy to other machine learning algorithms on the HIV drug resistance dataset and the UCIs promoter gene dataset. The algorithms are powerful in inferring the genotype-phenotype mapping from a moderate number of labeled sequences, as are typically produced in mutagenesis experiments. They can also greedily learn DNFs from large datasets. The Java implementation of our algorithms will be made publicly available.
Programed First Course in Algebra, Revised Form H, Student's Text, Part II, Unit 61.
ERIC Educational Resources Information Center
Buck, R. Creighton; And Others
This is part two of a two-part SMSG Programed Algebra Text for high school students. The general plan of the course is to build upon the student's experience with arithmetic. This part begins with factorization of positive integers and then develops the manipulative skills of fractions, exponents, radicals, and polynomials. The text then moves to…
The maximal Abelian dimension of linear algebras formed by strictly upper triangular matrices
NASA Astrophysics Data System (ADS)
Benjumea, J. C.; Núñez, J.; Tenorio, Á. F.
2007-09-01
We compute the largest dimension of the Abelian Lie subalgebras contained in the Lie algebra mathfrak{g}_n of n×n strictly upper triangular matrices, where n ∈ ℕ {1}. We do this by proving a conjecture, which we previously advanced, about this dimension. We introduce an algorithm and use it first to study the two simplest particular cases and then to study the general case.
Algebraic Systems and Pushdown Automata
NASA Astrophysics Data System (ADS)
Petre, Ion; Salomaa, Arto
We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.
Carleman linearization and normal forms for differential systems with quasi-periodic coefficients.
Chermnykh, Sergey V
2016-01-01
We study the matrix representation of Poincaré normalization using the Carleman linearization technique for non-autonomous differential systems with quasi-periodic coefficients. We provide a rigorous proof of the validity of the matrix representation of the normalization and obtain a recursive algorithm for computing the normalizing transformation and the normal form of the differential systems. The algorithm provides explicit formulas for the coefficients of the normal form and the corresponding transformation. PMID:27588240
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…
NASA Astrophysics Data System (ADS)
Jiang, Heping; Jiang, Jiao; Song, Yongli
In this paper, we firstly employ the normal form theory of delayed differential equations according to Faria and Magalhães to derive the normal form of saddle-node-Hopf bifurcation for the general retarded functional differential equations. Then, the dynamical behaviors of a Leslie-Gower predator-prey model with time delay and nonmonotonic functional response are considered. Specially, the dynamical classification near the saddle-node-Hopf bifurcation point is investigated by using the normal form and the center manifold approaches. Finally, the numerical simulations are employed to support the theoretical results.
ERIC Educational Resources Information Center
Cuevas, Eduardo J.
1997-01-01
Discusses cornerstone of Montessori theory, normalization, which asserts that if a child is placed in an optimum prepared environment where inner impulses match external opportunities, the undeviated self emerges, a being totally in harmony with its surroundings. Makes distinctions regarding normalization, normalized, and normality, indicating how…
Normal form solutions of dynamical systems in the basin of attraction of their fixed points
NASA Astrophysics Data System (ADS)
Bountis, Tassos; Tsarouhas, George; Herman, Russell
1998-10-01
The normal form theory of Poincaré, Siegel and Arnol'd is applied to an analytically solvable Lotka-Volterra system in the plane, and a periodically forced, dissipative Duffing's equation with chaotic orbits in its 3-dimensional phase space. For the planar model, we determine exactly how the convergence region of normal forms about a nodal fixed point is limited by the presence of singularities of the solutions in the complex t-plane. Despite such limitations, however, we show, in the case of a periodically driven system, that normal forms can be used to obtain useful estimates of the basin of attraction of a stable fixed point of the Poincaré map, whose ``boundary'' is formed by the intersecting invariant manifolds of a second hyperbolic fixed point nearby.
ERIC Educational Resources Information Center
Pavelle, Richard; And Others
1981-01-01
Describes the nature and use of computer algebra and its applications to various physical sciences. Includes diagrams illustrating, among others, a computer algebra system and flow chart of operation of the Euclidean algorithm. (SK)
Design of a spatial data structure using the relational normal forms
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.
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.
New classification of lingual arch form in normal occlusion using three dimensional virtual models
Park, Kyung Hee; Bayome, Mohamed; Park, Jae Hyun; Lee, Jeong Woo; Baek, Seung-Hak
2015-01-01
Objective The purposes of this study were 1) to classify lingual dental arch form types based on the lingual bracket points and 2) to provide a new lingual arch form template based on this classification for clinical application through the analysis of three-dimensional virtual models of normal occlusion sample. Methods Maxillary and mandibular casts of 115 young adults with normal occlusion were scanned in their occluded positions and lingual bracket points were digitized on the virtual models by using Rapidform 2006 software. Sixty-eight cases (dataset 1) were used in K-means cluster analysis to classify arch forms with intercanine, interpremolar and intermolar widths and width/depth ratios as determinants. The best-fit curves of the mean arch forms were generated. The remaining cases (dataset 2) were mapped into the obtained clusters and a multivariate test was performed to assess the differences between the clusters. Results Four-cluster classification demonstrated maximum intercluster distance. Wide, narrow, tapering, and ovoid types were described according to the intercanine and intermolar widths and their best-fit curves were depicted. No significant differences in arch depths existed among the clusters. Strong to moderate correlations were found between maxillary and mandibular arch widths. Conclusions Lingual arch forms have been classified into 4 types based on their anterior and posterior dimensions. A template of the 4 arch forms has been depicted. Three-dimensional analysis of the lingual bracket points provides more accurate identification of arch form and, consequently, archwire selection. PMID:25798413
BRST charges for finite nonlinear algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2010-07-01
Some ingredients of the BRST construction for quantum Lie algebras are applied to a wider class of quadratic algebras of constraints. We build the BRST charge for a quantum Lie algebra with three generators and ghost-anti-ghosts commuting with constraints. We consider a one-parametric family of quadratic algebras with three generators and show that the BRST charge acquires the conventional form after a redefinition of ghosts. The modified ghosts form a quadratic algebra. The family possesses a nonlinear involution, which implies the existence of two independent BRST charges for each algebra in the family. These BRST charges anticommute and form a double BRST complex.
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.
Semiclassical spectrum of small Bose-Hubbard chains: A normal-form approach
Itin, A. P.
2011-12-15
We analyze the spectrum of the three-site Bose-Hubbard model with periodic boundary conditions using a semiclassical method. The Bohr-Sommerfeld quantization is applied to an effective classical Hamiltonian which we derive using resonance normal form theory. The derivation takes into account the 1:1 resonance between frequencies of a linearized classical system and brings nonlinear terms into a corresponding normal form. The obtained expressions reproduce the exact low-energy spectrum of the system remarkably well even for a small number of particles N corresponding to fillings of just two particles per site. Such small fillings are often used in current experiments, and it is inspiring to get insight into this quantum regime using essentially classical calculations.
NASA Astrophysics Data System (ADS)
DeVille, R. E. Lee; Harkin, Anthony; Holzer, Matt; Josić, Krešimir; Kaper, Tasso J.
2008-06-01
For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502-4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré-Birkhoff normal forms for these systems up to and including terms of O(ɛ2), where ɛ is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O(1/ɛ).
Normal and quasinormal forms for systems of difference and differential-difference equations
NASA Astrophysics Data System (ADS)
Kashchenko, Ilya; Kaschenko, Sergey
2016-09-01
The local dynamics of systems of difference and singularly perturbed differential-difference equations is studied in the neighborhood of a zero equilibrium state. Critical cases in the problem of stability of its state of equilibrium have infinite dimension. Special nonlinear evolution equations, which act as normal forms, are set up. It is shown that their dynamics defines the behavior of solutions to the initial system.
NASA Astrophysics Data System (ADS)
Heikkila, S.
2007-08-01
In this paper we apply generalized iteration methods to prove comparison results which show how fixed points of a multifunction can be bounded by least and greatest fixed points of single-valued functions. As an application we prove existence and comparison results for fixed points of multifunctions. These results are applied to normal-form games, by proving existence and comparison results for pure and mixed Nash equilibria and their utilities.
Algebraic integrability: a survey.
Vanhaecke, Pol
2008-03-28
We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems. PMID:17588863
Invertible linear transformations and the Lie algebras
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Tam, Honwah; Guo, Fukui
2008-07-01
With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.
ERIC Educational Resources Information Center
Schaufele, Christopher; Zumoff, Nancy
Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through…
ERIC Educational Resources Information Center
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
Hopf normal form with SN symmetry and reduction to systems of nonlinearly coupled phase oscillators
NASA Astrophysics Data System (ADS)
Ashwin, Peter; Rodrigues, Ana
2016-06-01
Coupled oscillator models where N oscillators are identical and symmetrically coupled to all others with full permutation symmetry SN are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive-and we characterize generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, ɛ (the strength of coupling) and λ (an unfolding parameter for the Hopf bifurcation). For small enough λ > 0 there is an attractor that is the product of N stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small ɛ > 0. Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with SN symmetry. For fixed N and taking the limit 0 < ɛ ≪ λ ≪ 1, we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the well-known Kuramoto-Sakaguchi system of coupled oscillators. The next order of approximation generically includes terms with up to four interacting phases, regardless of N. Using a normalization that maintains nontrivial interactions in the limit N → ∞, we show that the additional terms can lead to new phenomena in terms of coexistence of two-cluster states with the same phase difference but different cluster size.
High molecular gas fractions in normal massive star-forming galaxies in the young Universe.
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
Experimental Results on the Centipede Game in Normal Form: An Investigation on Learning.
Nagel; Tang
1998-06-01
We analyze behavior of an experiment on the centipede game played in the reduced normal form. In this game two players decide simultaneously when to split a pie which increases over time. The subjects repeat this game 100 times against randomly chosen opponents. We compare several static models and quantitative learning models, among them a quantal response, model reinforcement models and fictitious play. Furthermore, we structure behavior from period to period according to a simple cognitive process, called learning direction theory. We show that there is a significant difference in behavior from period to period whether a player has decided to split the pie before or after the opponent. Copyright 1998 Academic Press.
Normal form analysis of multiple bifurcations in incompletely mixed chemical reactors
NASA Astrophysics Data System (ADS)
Puhl, Andreas; Nicolis, Grégoire
1987-07-01
Using the theory of normal forms, we investigate the effects of mixing in a continuous flow stirred tank reactor (CSTR) for a reaction model exhibiting oscillatory behavior in the vicinity of a degenerated bifurcation point (here, a Takens-Bogdanov point). In addition we show without specification of a particular reaction system that, as long as reaction rates remain much slower than the inverse mixing time, incomplete mixing introduces a new bifurcation parameter for nonpremixed feeding conditions, whereas premixed feeding conditions merely lead to a renormalization of flow rate.
Implementation of Boolean functions with a bounded number of zeros by disjunctive normal forms
NASA Astrophysics Data System (ADS)
Maximov, Yu. V.
2013-09-01
The problem of constructing simple disjunctive normal forms (DNFs) of Boolean functions with a small number of zeros is considered. The problem is of interest in the complexity analysis of Boolean functions and in its applications to data analysis. The method used is a further development of the reduction approach to the construction of DNFs of Boolean functions. A key idea of the reduction method is that a Boolean function is represented as a disjunction of Boolean functions with fewer zeros. In a number of practically important cases, this technique makes it possible to considerably reduce the complexity of DNF implementations of Boolean functions.
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.
The Infrared Spectral Energy Distribution of Normal Star-forming Galaxies
NASA Astrophysics Data System (ADS)
Dale, Daniel A.; Helou, George; Contursi, Alessandra; Silbermann, Nancy A.; Kolhatkar, Sonali
2001-03-01
We present a new phenomenological model for the spectral energy distribution of normal star-forming galaxies between 3 and 1100 μm. A sequence of realistic galaxy spectra are constructed from a family of dust emission curves assuming a power-law distribution of dust mass over a wide range of interstellar radiation fields. For each interstellar radiation field heating intensity, we combine emission curves for large and very small grains and aromatic feature carriers. The model is constrained by IRAS and ISOCAM broadband photometric and ISOPHOT spectrophotometric observations for our sample of 69 normal galaxies; the model reproduces well the empirical spectra and infrared color trends. These model spectra allow us to determine the infrared energy budget for normal galaxies and in particular to translate far-infrared fluxes into total (bolometric) infrared fluxes. The 20-42 μm range appears to show the most significant growth in relative terms as the activity level increases, suggesting that the 20-42 μm continuum may be the best dust emission tracer of current star formation in galaxies. The redshift dependence of infrared color-color diagrams and the far-infrared-to-radio correlation for galaxies are also explored.
Unique form of rickets with low serum 25-hydroxyvitamin D in two normally nourished children.
Asami, T; Kawasaki, T; Uchiyama, M
1995-04-01
We present an unusual type of rickets involving two children: a 2 year old boy and a 15 month old boy, who presented with marked bowing of the lower extremities and bulging of costochondral junctions. Both children had normal growth, with their height and body weight greater than the 50th and 97th percentile for age. Roentgenograms of their extremities showed the typical changes of vitamin D refractory rickets. Serum alkaline phosphatase levels were elevated and serum levels of calcium and phosphate were both within the normal range. No primary cause for the rickets, including nutritional deficiencies, was found in the two patients. Characteristic findings were persistently low serum 25-hydroxyvitamin D (25-OH-D) and normal 1,25-dihydroxyvitamin D (1,25-(OH)2-D). Improvements in clinical and X-ray findings were observed after either oral administration of 1 alpha-(OH)-D3 (9-15 micrograms per day) or massive vitamin D2 therapy (600,000 IU single injection). The low serum levels of 25-OH-D did not increase unless massive vitamin D2 therapy was also given. These two cases represent a unique form of rickets that does not meet the criteria for any type of previously known rickets. PMID:7793252
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.
Theory and praxis pf map analsys in CHEF part 1: Linear normal form
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.
Theory and praxis of map analsys in CHEF part 2: Nonlinear normal form
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
Optimization of accelerator parameters using normal form methods on high-order transfer maps
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
NASA Astrophysics Data System (ADS)
Akiba, Sumitaka; Sato, Taisuke; Motoyoshi, Fumio
In this paper, we describe an improvement of a calculation procedure of logic programs. The procedure proposed before is the combination of a replacement procedure of logical formulae and a transformation procedure of equations to disjunctive normal form, and it can calculate logical consequences of the completion of any given first-order logic program (FLP), which is equivalent to the FLP in two-valued logic, soundly and completely in three-valued logic. The new procedure is also the combination of them, but the transformation procedure is improved to be able to calculate two-valued logical consequences of the FLP more than the old one. We prove that it can calculate logical consequences of a completed program, which is not equivalent to the completion of the FLP, soundly and completely in three-valued logic.
Interactive learning in 2×2 normal form games by neural network agents
NASA Astrophysics Data System (ADS)
Spiliopoulos, Leonidas
2012-11-01
This paper models the learning process of populations of randomly rematched tabula rasa neural network (NN) agents playing randomly generated 2×2 normal form games of all strategic classes. This approach has greater external validity than the existing models in the literature, each of which is usually applicable to narrow subsets of classes of games (often a single game) and/or to fixed matching protocols. The learning prowess of NNs with hidden layers was impressive as they learned to play unique pure strategy equilibria with near certainty, adhered to principles of dominance and iterated dominance, and exhibited a preference for risk-dominant equilibria. In contrast, perceptron NNs were found to perform significantly worse than hidden layer NN agents and human subjects in experimental studies.
Normal form and limit cycle bifurcation of piecewise smooth differential systems with a center
NASA Astrophysics Data System (ADS)
Wei, Lijun; Zhang, Xiang
2016-07-01
In this paper we prove that any Σ-center (either nondegenerate or degenerate) of a planar piecewise Cr smooth vector field Z is topologically equivalent to that of Z0: (x ˙ , y ˙) = (- 1 , 2 x) for y ≥ 0, (x ˙ , y ˙) = (1 , 2 x) for y ≤ 0, and that the homeomorphism between Z and Z0 is Cr smoothness when restricted to each side of the switching line except at the center p. We illustrate by examples that there are degenerate Σ-centers whose flows are conjugate to that of Z0, and also there exist nondegenerate Σ-centers whose flows cannot be conjugate to that of Z0. Finally applying the normal form Z0 together with the piecewise smooth equivalence, we study the number of limit cycles which can be bifurcated from the Σ-center of Z.
Formal scattering theory by an algebraic approach
NASA Astrophysics Data System (ADS)
Alhassid, Y.; Levine, R. D.
1985-02-01
Formal scattering theory is recast in a Lie-algebraic form. The central result is an algebraic Lippmann-Schwinger equation for the wave operator from which an algebraic form of the Born series (containing only linked terms) is obtained. When a finite Lie algebra is sufficient, The Mo/ller wave operator, on the energy shell, can be solved for explicitly as an element of the corresponding group. The method is illustrated for the separable potential whose relevant algebra is found to be U(1,1).
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)
Becchi-Rouet-Stora-Tyutin operators for W algebras
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2008-07-15
The study of quantum Lie algebras motivates a use of noncanonical ghosts and antighosts for nonlinear algebras, such as W-algebras. This leads, for the W{sub 3} and W{sub 3}{sup (2)} algebras, to the Becchi-Rouet-Stora-Tyutin operator having the conventional cubic form.
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.
Explicit solutions of normal form of driven oscillatory systems in entrainment bands
NASA Astrophysics Data System (ADS)
Tsarouhas, George E.; Ross, John
1988-11-01
As in a prior article (Ref. 1), we consider an oscillatory dissipative system driven by external sinusoidal perturbations of given amplitude Q and frequency ω. The kinetic equations are transformed to normal form and solved for small Q near a Hopf bifurcation to oscillations in the autonomous system. Whereas before we chose irrational ratios of the frequency of the autonomous system ωn to ω, with quasiperiodic response of the system to the perturbation, we now choose rational coprime ratios, with periodic response (entrainment). The dissipative system has either two variables or is adequately described by two variables near the bifurcation. We obtain explicit solutions and develop these in detail for ωn/ω=1; 1:2; 2:1; 1:3; 3:1. We choose a specific dissipative model (Brusselator) and test the theory by comparison with full numerical solutions. The analytic solutions of the theory give an excellent approximation for the autonomous system near the bifurcation. The theoretically predicted and calculated entrainment bands agree very well for small Q in the vicinity of the bifurcation (small μ); deviations increase with increasing Q and μ. The theory is applicable to one or two external periodic perturbations.
Walendziak, Andrzej
2015-01-01
The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained. PMID:26125050
NASA Technical Reports Server (NTRS)
Ruge, J. W.; Stueben, K.
1987-01-01
The state of the art in algebraic multgrid (AMG) methods is discussed. The interaction between the relaxation process and the coarse grid correction necessary for proper behavior of the solution probes is discussed in detail. Sufficient conditions on relaxation and interpolation for the convergence of the V-cycle are given. The relaxation used in AMG, what smoothing means in an algebraic setting, and how it relates to the existing theory are considered. Some properties of the coarse grid operator are discussed, and results on the convergence of two-level and multilevel convergence are given. Details of an algorithm particularly studied for problems obtained by discretizing a single elliptic, second order partial differential equation are given. Results of experiments with such problems using both finite difference and finite element discretizations are presented.
Sjaardema, G.; Gilkey, A.; Smith, M.; Forsythe, C.
2005-04-11
The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.
NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
NASA Astrophysics Data System (ADS)
Dankova, T. S.; Rosensteel, G.
1998-10-01
Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for an arbitrary Lie algebra L of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space L^* of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The mean field theories for the Elliott SU(3) and symplectic Sp(3,R) algebras are constructed explicitly in the coadjoint orbit framework.
A local construction of the Smith normal form of a matrix polynomial
Wilkening, Jon; Yu, Jia
2008-09-01
We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular operations to the original matrix row by row (or column by column). The performance of the algorithm in exact arithmetic is reported for several test cases.
Direct Observation of the Interconversion of Normal and Toxic Forms of α-Synuclein
Cremades, Nunilo; Cohen, Samuel I.A.; Deas, Emma; Abramov, Andrey Y.; Chen, Allen Y.; Orte, Angel; Sandal, Massimo; Clarke, Richard W.; Dunne, Paul; Aprile, Francesco A.; Bertoncini, Carlos W.; Wood, Nicholas W.; Knowles, Tuomas P.J.; Dobson, Christopher M.; Klenerman, David
2012-01-01
Summary Here, we use single-molecule techniques to study the aggregation of α-synuclein, the protein whose misfolding and deposition is associated with Parkinson's disease. We identify a conformational change from the initially formed oligomers to stable, more compact proteinase-K-resistant oligomers as the key step that leads ultimately to fibril formation. The oligomers formed as a result of the structural conversion generate much higher levels of oxidative stress in rat primary neurons than do the oligomers formed initially, showing that they are more damaging to cells. The structural conversion is remarkably slow, indicating a high kinetic barrier for the conversion and suggesting that there is a significant period of time for the cellular protective machinery to operate and potentially for therapeutic intervention, prior to the onset of cellular damage. In the absence of added soluble protein, the assembly process is reversed and fibrils disaggregate to form stable oligomers, hence acting as a source of cytotoxic species. PMID:22632969
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.
Gottwald, Georg A
2008-03-01
We present a bifurcation analysis of a normal form for traveling waves in one-dimensional excitable media. The normal form that has been recently proposed on phenomenological grounds is given in the form of a differential delay equation. The normal form exhibits a symmetry-preserving Hopf bifurcation that may coalesce with a saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans, which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.
Form of 15q proximal duplication appears to be a normal euchromatic variant
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.
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…
Solving Absolute Value Equations Algebraically and Geometrically
ERIC Educational Resources Information Center
Shiyuan, Wei
2005-01-01
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.
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.
NASA Astrophysics Data System (ADS)
Ellison, James A.; Heinemann, Klaus; Vogt, Mathias; Gooden, Matthew
2013-09-01
We present a mathematical analysis of planar motion of energetic electrons moving through a planar dipole undulator, excited by a fixed planar polarized plane wave Maxwell field in the x-ray free electron laser (FEL) regime. Our starting point is the 6D Lorentz system, which allows planar motions, and we examine this dynamical system as the wavelength λ of the traveling wave varies. By scalings and transformations the 6D system is reduced, without approximation, to a 2D system in a form for a rigorous asymptotic analysis using the method of averaging (MoA), a long-time perturbation theory. The two dependent variables are a scaled energy deviation and a generalization of the so-called ponderomotive phase. As λ varies the system passes through resonant and nonresonant (NonR) intervals and we develop NonR and near-to-resonant (NearR) MoA normal form approximations to the exact equations. The NearR normal forms contain a parameter which measures the distance from a resonance. For the planar motion, with the special initial condition that matches into the undulator design trajectory, and on resonance, the NearR normal form reduces to the well-known FEL pendulum system. We then state and prove NonR and NearR first-order averaging theorems which give explicit error bounds for the normal form approximations. We prove the theorems in great detail, giving the interested reader a tutorial on mathematically rigorous perturbation theory in a context where the proofs are easily understood. The proofs are novel in that they do not use a near-identity transformation and they use a system of differential inequalities. The NonR case is an example of quasiperiodic averaging where the small divisor problem enters in the simplest possible way. To our knowledge the planar problem has not been analyzed with the generality we aspire to here nor has the standard FEL pendulum system been derived with associated error bounds as we do here. We briefly discuss the low gain theory in light of
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…
Numerical algebraic geometry and algebraic kinematics
NASA Astrophysics Data System (ADS)
Wampler, Charles W.; Sommese, Andrew J.
In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.
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.
Trank, T V; Chen, C; Smith, J L
1996-10-01
1. Posture, hindlimb kinematics, and activity patterns of selected hindlimb muscles were compared for normal and crouched treadmill walking (0.5-0.6 m/s) for eight cats. To elicit crouched walking in which the trunk and head were lowered, cats were encouraged to walk under a light-weight Plexiglas ceiling suspended 17-20 cm above the treadmill belt. Kinematic data were obtained from high-speed ciné film, and electromyograms (EMGs)-synchronized with the kinematic records-were taken from 11 hindlimb muscles. 2. The postures for the two forms of walking were distinctly different. During crouched walking, each cat lowered its entire body keeping its trunk horizontal to the treadmill belt. Also the head was lowered, with the top of the head in line with the dorsal surface of the trunk. Hip height, used as a measure for hindlimb crouch, was reduced by 30%, from an average height of 23 cm to an average height of 16 cm above the belt during the entire step cycle. 3. Average cycle periods (766 +/- 30 ms, mean +/- SD) and percentage of time devoted to swing (30%) and stance (70%) were similar for normal and crouched walking. The profiles of the hindlimb kinematics were also similar for the hip, knee, ankle, and metatarsophalangeal (MTP) joints during the step cycle, but the timing of some of the motion reversal, as well as the ranges of motion during various phases, were different at some joints for the two forms of walking. 4. During the swing phase, the transition between the flexion and extension (F-E1 reversal) occurred later in the normalized swing phase at the hip, knee, and ankle joints, and the range of flexion was increased at each joint. With greater flexion at these joints, the anatomic axis of the hindlimb (measured from hip joint to toe) was decreased and the hind paw advanced in the narrow space between the abdomen and treadmill belt. At contact, the position of the paw was less anterior to the perpendicular reference line (hip joint marker to belt) and all
Lord, E M; Dutton, R W
1975-11-01
Treatment of mouse erythrocytes with the proteolytic enzymes, bromelain, reveals antigenic determinants not normally exposed on the erythrocyte surface. It was found that not only NZB mice, a known autoimmune strain, but also several normal strains of mice contain cells in small numbers in their spleens and in larger numbers in their peritoneal cavities which will form plaques against bromelain-treated MRBC. During in vitro culture the number of anti-BR-MRBC PFC increases slightly in the spleen cell populations whereas the number of these PFC in peritoneal cells increases dramatically to as many as 100,000 PFC/10(6) cells. The plaques detected in this assay contain a central lymphoid cell and their development, which requires the presence of complement and protein synthesis, is inhibited by anti-mouse immunoglobulin.
Math for All Learners: Algebra.
ERIC Educational Resources Information Center
Meader, Pam; Storer, Judy
This book consists of a series of activities aimed at providing a problem solving, hands-on approach so that students can experience concepts in algebra. Topics include ratio and proportion, patterns and formulas, integers, polynomials, linear equations, graphs, and probability. The activities come in the form of reproducible blackline masters…
Constructing a parasupersymmetric Virasoro algebra
NASA Astrophysics Data System (ADS)
Kuwata, S.
2011-03-01
We construct a para SUSY Virasoro algebra by generalizing the ordinary fermion in SUSY Virasoro algebra (Ramond or Neveu-Schwarz algebra) to the parafermion. First, we obtain a polynomial relation (PR) between different-mode parafermion fi's by generalizing the corresponding single-mode PR to such that is invariant under the unitary transformation of fi (Green's condition). Differently from a usual context, where the Green's condition is imposed only on the defining relation of fi (degree three with respect to fi and fi†), we impose it on any degree of PR. For the case of order-two parafermion (the simplest case of para SUSY), we calculate a PR between the parasupercharge G0, the bosonic hamiltonian LB0 and parafermionic one LF0, although it is difficult to obtain a PR between G0 and the total hamiltonian L0 (= LB0 + LF0). Finally, we construct a para SUSY Virasoro algebra by generalizing L0 to the Ln's such that form a Virasoro algebra.
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.
Bobodzhanov, A A; Safonov, V F
2013-07-31
The paper deals with extending the Lomov regularization method to classes of singularly perturbed Fredholm-type integro-differential systems, which have not so far been studied. In these the limiting operator is discretely noninvertible. Such systems are commonly known as problems with unstable spectrum. Separating out the essential singularities in the solutions to these problems presents great difficulties. The principal one is to give an adequate description of the singularities induced by 'instability points' of the spectrum. A methodology for separating singularities by using normal forms is developed. It is applied to the above type of systems and is substantiated in these systems. Bibliography: 10 titles.
Quantization of Algebraic Reduction
Sniatycki, Jeodrzej
2007-11-14
For a Poisson algebra obtained by algebraic reduction of symmetries of a quantizable system we develop an analogue of geometric quantization based on the quantization structure of the original system.
Learning Algebra in a Computer Algebra Environment
ERIC Educational Resources Information Center
Drijvers, Paul
2004-01-01
This article summarises a doctoral thesis entitled "Learning algebra in a computer algebra environment, design research on the understanding of the concept of parameter" (Drijvers, 2003). It describes the research questions, the theoretical framework, the methodology and the results of the study. The focus of the study is on the understanding of…
NASA Technical Reports Server (NTRS)
Iachello, Franco
1995-01-01
An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.
Profiles of Algebraic Competence
ERIC Educational Resources Information Center
Humberstone, J.; Reeve, R.A.
2008-01-01
The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…
Orientation in operator algebras
Alfsen, Erik M.; Shultz, Frederic W.
1998-01-01
A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. PMID:9618457
Developing Thinking in Algebra
ERIC Educational Resources Information Center
Mason, John; Graham, Alan; Johnson-Wilder, Sue
2005-01-01
This book is for people with an interest in algebra whether as a learner, or as a teacher, or perhaps as both. It is concerned with the "big ideas" of algebra and what it is to understand the process of thinking algebraically. The book has been structured according to a number of pedagogic principles that are exposed and discussed along the way,…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Structure of classical affine and classical affine fractional W-algebras
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.
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.
Gatti, G; Barzaghi, N; Attardo Parrinello, G; Vitiello, B; Perucca, E
1989-01-01
The pharmacokinetic profile of an innovative formulation of soluble aspirin (l-ornithine acetylsalicylate, ldB 1003) was compared with that of conventional tablets and two other soluble dosage forms (d, l-lysine acetylsalicylate and a buffered effervescent formulation of acetylsalicylic acid) after administration of single oral doses in six normal volunteers. All soluble forms showed a rapid absorption profile, peak plasma salicylic acid levels being attained after about 30 min on average and without statistically significant differences among the solutions tested. As compared to the soluble formulations, acetylsalicylic acid given as tablets resulted in slower absorption, with peak plasma salicylic acid levels being reached more than 1 h after dosing. Despite these differences in time course of plasma level profiles, the extent of absorption was similar for all formulations. Apart from the potential advantages in terms of improved gastric tolerability, the increased rate of absorption of aspirin solutions is therapeutically useful whenever a rapid onset of action is required. In this respect, the kinetic pattern of the innovative formulation compares favourably with that of other available soluble dosage forms.
Gatti, G; Barzaghi, N; Attardo Parrinello, G; Vitiello, B; Perucca, E
1989-01-01
The pharmacokinetic profile of an innovative formulation of soluble aspirin (l-ornithine acetylsalicylate, ldB 1003) was compared with that of conventional tablets and two other soluble dosage forms (d, l-lysine acetylsalicylate and a buffered effervescent formulation of acetylsalicylic acid) after administration of single oral doses in six normal volunteers. All soluble forms showed a rapid absorption profile, peak plasma salicylic acid levels being attained after about 30 min on average and without statistically significant differences among the solutions tested. As compared to the soluble formulations, acetylsalicylic acid given as tablets resulted in slower absorption, with peak plasma salicylic acid levels being reached more than 1 h after dosing. Despite these differences in time course of plasma level profiles, the extent of absorption was similar for all formulations. Apart from the potential advantages in terms of improved gastric tolerability, the increased rate of absorption of aspirin solutions is therapeutically useful whenever a rapid onset of action is required. In this respect, the kinetic pattern of the innovative formulation compares favourably with that of other available soluble dosage forms. PMID:2517497
NASA Astrophysics Data System (ADS)
Akiba, Sumitaka; Sato, Taisuke
In this paper, we describe the completeness of a calculation procedure of logic programs. The procedure is the combination of two procedures, a replacement procedure of atoms in the goal by the bodies or the negation of the bodies of rules in the program, and a transformation procedure of equations to disjunctive normal forms (DNF) equivalent under Clark's Equational Theory (CET). To combine replacement of atoms in the goal to logical formulae determined from the program and transformation of equations to DNF equivalent under CET is a method by which procedures with the capability of expressing answers in DNF can be build, so it is a leading method for expressing answers in a form including negation. Some procedures based on the method are devised, and their calculation capabilities are shown by applying the theory of completed programs. However, the procedure that uses the bodies or the negation of the bodies of rules for replacement has higher calculation capability, and is intuitively more natural than they. Therefore, to clarify the calculation capability of the procedure is considered an important subject for research into calculation procedures of logic programs with the capability for expressing answers in a form including negation. Moreover, since the completeness is realized by standing on the viewpoint of treating the implication symbol as a different implication symbol from usual, and interpreting logic programs in three-valued logic, examples which support the viewpoint are also described.
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.
Krukowska, Anna; Tarkowski, Andrzej K
2005-11-01
A mouse spermatozoon was injected into mouse secondary oocytes (ICSI) in the vicinity of the metaphase spindle. In 22% of oocytes injected successfully, the maternal chromatin (the haploid chromatids formed after the second meiotic division) and paternal chromatin (from the sperm nucleus) were surrounded by a common nuclear envelope to form one diploid bi-parental pronucleus. However, the use of spermatozoa in which BrdU had been incorporated into DNA during spermatogenesis revealed, that maternal and paternal chromatin occupied two separate compartments within the one pronucleus. In the living state, the diploid pronucleus could be distinguished from a haploid one by its distinctly larger size and by a greater number of "nucleolus-like bodies"-criteria confirmed karylogically at the 1st cleavage division. Such zygotes with one diploid pronucleus were able to develop in vitro into blastocysts as often as those with two haploid pronuclei [11/29 (38%) vs. 14/35 (40%)]. Seventy nine 2-cell embryos developing in vitro from zygotes with one diploid pronucleus were transplanted to the oviducts of pseudopregnant recipients: two females had six foetuses when killed on the 17th day, and two females gave birth to nine young, eight of which survived and developed into normal fertile animals. PMID:16047392
Constraint algebra in bigravity
Soloviev, V. O.
2015-07-15
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
Constraint algebra in bigravity
NASA Astrophysics Data System (ADS)
Soloviev, V. O.
2015-07-01
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
A Richer Understanding of Algebra
ERIC Educational Resources Information Center
Foy, Michelle
2008-01-01
Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number, pre-algebra, and algebra…
Algebraic operator approach to gas kinetic models
NASA Astrophysics Data System (ADS)
Il'ichov, L. V.
1997-02-01
Some general properties of the linear Boltzmann kinetic equation are used to present it in the form ∂ tϕ = - Â†Âϕ with the operators ÂandÂ† possessing some nontrivial algebraic properties. When applied to the Keilson-Storer kinetic model, this method gives an example of quantum ( q-deformed) Lie algebra. This approach provides also a natural generalization of the “kangaroo model”.
Algebraic sub-structuring for electromagnetic applications
Yang, Chao; Gao, Weiguo; Bai, Zhaojun; Li, Xiaoye; Lee, Lie-Quan; Husbands, Parry; Ng, Esmond G.
2004-09-14
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, we show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Algebraic Sub-Structuring for Electromagnetic Applications
Yang, C.; Gao, W.G.; Bai, Z.J.; Li, X.Y.S.; Lee, L.Q.; Husbands, P.; Ng, E.G.; /LBL, Berkeley /UC, Davis /SLAC
2006-06-30
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, they show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Georgiev, Georgi As; Yokoi, Norihiko; Ivanova, Slavyana; Tonchev, Vesselin; Nencheva, Yana; Krastev, Rumen
2014-08-14
The surface properties of human meibomian lipids (MGS), the major constituent of the tear film (TF) lipid layer, are of key importance for TF stability. The dynamic interfacial properties of films by MGS from normal eyes (nMGS) and eyes with meibomian gland dysfunction (dMGS) were studied using a Langmuir surface balance. The behavior of the samples during dynamic area changes was evaluated by surface pressure-area isotherms and isocycles. The surface dilatational rheology of the films was examined in the frequency range 10(-5) to 1 Hz by the stress-relaxation method. A significant difference was found, with dMGS showing slow viscosity-dominated relaxation at 10(-4) to 10(-3) Hz, whereas nMGS remained predominantly elastic over the whole range. A Cole-Cole plot revealed two characteristic processes contributing to the relaxation, fast (on the scale of characteristic time τ < 5 s) and slow (τ > 100 s), the latter prevailing in dMGS films. Brewster angle microscopy revealed better spreading of nMGS at the air-water interface, whereas dMGS layers were non-uniform and patchy. The distinctions in the interfacial properties of the films in vitro correlated with the accelerated degradation of meibum layer pattern at the air-tear interface and with the decreased stability of TF in vivo. These results, and also recent findings on the modest capability of meibum to suppress the evaporation of the aqueous subphase, suggest the need for a re-evaluation of the role of MGS. The probable key function of meibomian lipids might be to form viscoelastic films capable of opposing dilation of the air-tear interface. The impact of temperature on the meibum surface properties is discussed in terms of its possible effect on the normal structure of the film.
2003-06-03
The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran code is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.
2003-06-03
The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran codemore » is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.« less
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)
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.
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…
Polynomial Extensions of the Weyl C*-Algebra
NASA Astrophysics Data System (ADS)
Accardi, Luigi; Dhahri, Ameur
2015-09-01
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl C*-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.
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.
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
Kurita, Hiroshi; Koyama, Tomonori; Osada, Hirokazu
2005-08-01
A Japanese version of the Autism Spectrum Quotient (AQ), AQ-J was administered to 25 normally intelligent high-functioning pervasive developmental disorder (HPDD) patients (mean age, 24.2 years; 24 male, one female) and 215 controls (mean age, 30.4 years; 86 male, 129 female) randomly selected from the general population. The AQ-J had satisfactory internal consistency reliability (Cronbach's alpha > 0.70 in the two groups), test-retest reliability, and discriminant validity [i.e. the AQ-J score was significantly higher in the HPDD (mean, 29.6) than controls (mean, 22.2)]. At a cut-off of 26, the AQ-J had satisfactory sensitivity, specificity, and negative predictive value, but it had low positive predictive value (0.24) possibly due to the facts that the 25 mild HPDD patients scored lower and the controls scored higher on the AQ-J than British counterparts on the AQ. The AQ-J-21 (consisting of 21 items significantly associated with HPDD diagnosis) and the AQ-J-10 (consisting of 10 of the 21 items with an effect size > 0.17) had higher, although not satisfactory, positive predictive values of 0.35 and 0.46 at cut-offs of 12 and 7, respectively, than the AQ-J. The AQ-J and two short forms are useful not to predict but to rule out mild HPDD, the most difficult part of HPDD to be distinguished from non-PDD conditions, in persons scoring under the cut-offs and to consider professionals' examination of HPDD in persons scoring over them, because their negative predictive values were satisfactory.
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,…
Lie algebra extensions of current algebras on S3
NASA Astrophysics Data System (ADS)
Kori, Tosiaki; Imai, Yuto
2015-06-01
An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory.
Leibniz algebras associated with representations of filiform Lie algebras
NASA Astrophysics Data System (ADS)
Ayupov, Sh. A.; Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A.
2015-12-01
In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1. We introduce a Fock module for the algebra nn,1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra nn,1 with condition that the ideal I is a Fock nn,1-module, where I is the ideal generated by squares of elements from L. We also consider Leibniz algebras with corresponding Lie algebra nn,1 and such that the action I ×nn,1 → I gives rise to a minimal faithful representation of nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl( N;?)-case is discussed.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.
FAST TRACK COMMUNICATION: Kac Moody algebras and controlled chaos
NASA Astrophysics Data System (ADS)
Wesley, Daniel H.
2007-02-01
Compactification can control chaotic Mixmaster behaviour in gravitational systems with p-form matter: we consider this in light of the connection between supergravity models and Kac Moody algebras. We show that different compactifications define 'mutations' of the algebras associated with the noncompact theories. We list the algebras obtained in this way, and find novel examples of wall systems determined by Lorentzian (but not hyperbolic) algebras. Cosmological models with a smooth pre-big bang phase require that chaos is absent: we show that compactification alone cannot eliminate chaos in the simplest compactifications of the heterotic string on a Calabi Yau, or M theory on a manifold of G2 holonomy.
Gilkey, A.P.
1988-08-01
The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS database which can be read by plot programs. 8 refs.
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.
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.
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.…
Algebraic Semantics for Narrative
ERIC Educational Resources Information Center
Kahn, E.
1974-01-01
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
Geometric Algebra for Physicists
NASA Astrophysics Data System (ADS)
Doran, Chris; Lasenby, Anthony
2007-11-01
Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.
NASA Astrophysics Data System (ADS)
Hiley, B. J.
In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.
Dual number coefficient octonion algebra, field equations and conservation laws
NASA Astrophysics Data System (ADS)
Chanyal, B. C.; Chanyal, S. K.
2016-08-01
Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac-Maxwell's equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons.
Normal levels of the antiprion proteins Btn2 and Cur1 cure most newly formed [URE3] prion variants.
Wickner, Reed B; Bezsonov, Evgeny; Bateman, David A
2014-07-01
[URE3] is an amyloid prion of the Saccharomyces cerevisiae Ure2p, a regulator of nitrogen catabolism. Overproduction of Btn2p, involved in late endosome to Golgi protein transport, or its paralog Cur1p, cures [URE3]. Btn2p, in curing, is colocalized with Ure2p in a single locus, suggesting sequestration of Ure2p amyloid filaments. We find that most [URE3] variants generated in a btn2 cur1 double mutant are cured by restoring normal levels of Btn2p and Cur1p, with both proteins needed for efficient curing. The [URE3] variants cured by normal levels of Btn2p and Cur1p all have low seed number, again suggesting a seed sequestration mechanism. Hsp42 overproduction also cures [URE3], and Hsp42p aids Btn2 overproduction curing. Cur1p is needed for Hsp42 overproduction curing of [URE3], but neither Btn2p nor Cur1p is needed for overproduction curing by the other. Although hsp42Δ strains stably propagate [URE3-1], hsp26Δ destabilizes this prion. Thus, Btn2p and Cur1p are antiprion system components at their normal levels, acting with Hsp42. Btn2p is related in sequence to human Hook proteins, involved in aggresome formation and other transport activities.
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…
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…
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…
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…
SLAPP: A systolic linear algebra parallel processor
Drake, B.L.; Luk, F.T.; Speiser, J.M.; Symanski, J.J.
1987-07-01
Systolic array computer architectures provide a means for fast computation of the linear algebra algorithms that form the building blocks of many signal-processing algorithms, facilitating their real-time computation. For applications to signal processing, the systolic array operates on matrices, an inherently parallel view of the data, using numerical linear algebra algorithms that have been suitably parallelized to efficiently utilize the available hardware. This article describes work currently underway at the Naval Ocean Systems Center, San Diego, California, to build a two-dimensional systolic array, SLAPP, demonstrating efficient and modular parallelization of key matric computations for real-time signal- and image-processing problems.
Weak Lie symmetry and extended Lie algebra
Goenner, Hubert
2013-04-15
The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found ('extended Lie algebras') which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).
Devi, V. Kalpana; 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. PMID:22556626
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…
Adaptive Algebraic Multigrid Methods
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
On vertex algebra representations of the Schrödinger-Virasoro Lie algebra
NASA Astrophysics Data System (ADS)
Unterberger, Jérémie
2009-12-01
The Schrödinger-Virasoro Lie algebra sv is an extension of the Virasoro Lie algebra by a nilpotent Lie algebra formed with a bosonic current of weight 3/2 and a bosonic current of weight 1. It is also a natural infinite-dimensional extension of the Schrödinger Lie algebra, which — leaving aside the invariance under time-translation — has been proved to be a symmetry algebra for many statistical physics models undergoing a dynamics with dynamical exponent z=2. We define in this article general Schrödinger-Virasoro primary fields by analogy with conformal field theory, characterized by a 'spin' index and a (non-relativistic) mass, and construct vertex algebra representations of sv out of a charged symplectic boson and a free boson and its associated vertex operators. We also compute two- and three-point functions of still conjectural massive fields that are defined by an analytic continuation with respect to a formal parameter.
Computer Program For Linear Algebra
NASA Technical Reports Server (NTRS)
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
Algebra for Gifted Third Graders.
ERIC Educational Resources Information Center
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
Pseudo Algebraically Closed Extensions
NASA Astrophysics Data System (ADS)
Bary-Soroker, Lior
2009-07-01
This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that although there are many PAC extensions, the Galois closure of a proper PAC extension is separably closed. The dissertation also contains the following subjects. The group theoretical counterpart of pseudo algebraically closed extensions, the so-called projective pairs. Applications to seemingly unrelated subjects, e.g., an analog of Dirichlet's theorem about primes in arithmetic progression for polynomial rings in one variable over infinite fields.
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)
Assessing Elementary Algebra with STACK
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2007-01-01
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…
L∞-algebra models and higher Chern-Simons theories
NASA Astrophysics Data System (ADS)
Ritter, Patricia; Sämann, Christian
2016-10-01
We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In the first part, we review in detail how higher Chern-Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of L∞-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In the second part, we demonstrate that Nambu-Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie p-algebra extensions of 𝔰𝔬(p + 2). Finally, we study a number of L∞-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.
Hsa-miR-520d induces hepatoma cells to form normal liver tissues via a stemness-mediated process
Tsuno, Satoshi; Wang, Xinhui; Shomori, Kohei; Hasegawa, Junichi; Miura, Norimasa
2014-01-01
The human ncRNA gene RGM249 regulates the extent of differentiation of cancer cells and the conversion of 293FT cells to hiPSCs. To identify the factors underlying this process, we investigated the effects of lentivirally inducing miR-520d expression in 293FT and HLF cells in vitro. Subsequently, we evaluated tumor formation in a xenograft model. Transformed HLF cells were Oct4 and Nanog positive within 24 h, showed p53 upregulation and hTERT downregulation, and mostly lost their migration abilities. After lentiviral infection, the cells were intraperitoneally injected into mice, resulting in benign teratomas (6%), the absence of tumors (87%) or differentiation into benign liver tissues (7%) at the injection site after 1 month. We are the first to demonstrate the loss of malignant properties in cancer cells in vivo through the expression of a single microRNA (miRNA). This miRNA successfully converted 293FT and hepatoma cells to hiPSC-like cells. The regulation of malignancy by miR-520d appears to be through the conversion of cancer cells to normal stem cells, maintaining p53 upregulation. PMID:24458129
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.
Mid-J CO Emission from NGC 891: Microturbulent Molecular Shocks in Normal Star-forming Galaxies
NASA Astrophysics Data System (ADS)
Nikola, T.; Stacey, G. J.; Brisbin, D.; Ferkinhoff, C.; Hailey-Dunsheath, S.; Parshley, S.; Tucker, C.
2011-12-01
We have detected the CO(6-5), CO(7-6), and [C I] 370 μm lines from the nuclear region of NGC 891 with our submillimeter grating spectrometer ZEUS on the Caltech Submillimeter Observatory. These lines provide constraints on photodissociation region (PDR) and shock models that have been invoked to explain the H2 S(0), S(1), and S(2) lines observed with Spitzer. We analyze our data together with the H2 lines, CO(3-2), and infrared continuum from the literature using a combined PDR/shock model. We find that the mid-J CO originates almost entirely from shock-excited warm molecular gas; contributions from PDRs are negligible. Also, almost all of the H2 S(2) line and half of the S(1) line are predicted to emerge from shocks. Shocks with a pre-shock density of 2 × 104 cm-3 and velocities of 10 km s-1 and 20 km s-1 for C-shocks and J-shocks, respectively, provide the best fit. In contrast, the [C I] line emission arises exclusively from the PDR component, which is best parameterized by a density of 3.2 × 103 cm-3 and a far-ultraviolet field of Go = 100 for both PDR/shock-type combinations. Our mid-J CO observations show that turbulence is a very important heating source in molecular clouds, even in normal quiescent galaxies. The most likely energy sources for the shocks are supernovae or outflows from young stellar objects. The energetics of these shock sources favor C-shock excitation of the lines.
ERIC Educational Resources Information Center
Benjamin, Carl; And Others
Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra II. Topics covered include: differencing and complements; real numbers; factoring; fractions; linear equations; exponents and radicals; complex numbers,…
Thinking Visually about Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2015-01-01
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
Computer Algebra versus Manipulation
ERIC Educational Resources Information Center
Zand, Hossein; Crowe, David
2004-01-01
In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…
ERIC Educational Resources Information Center
Glick, David
1995-01-01
Presents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)
ERIC Educational Resources Information Center
Nwabueze, Kenneth K.
2004-01-01
The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…
González-Salgado, Amaia; Steinmann, Michael; Major, Louise L; Sigel, Erwin; Reymond, Jean-Louis; Smith, Terry K; Bütikofer, Peter
2015-06-01
myo-Inositol is a building block for all inositol-containing phospholipids in eukaryotes. It can be synthesized de novo from glucose-6-phosphate in the cytosol and endoplasmic reticulum. Alternatively, it can be taken up from the environment via Na(+)- or H(+)-linked myo-inositol transporters. While Na(+)-coupled myo-inositol transporters are found exclusively in the plasma membrane, H(+)-linked myo-inositol transporters are detected in intracellular organelles. In Trypanosoma brucei, the causative agent of human African sleeping sickness, myo-inositol metabolism is compartmentalized. De novo-synthesized myo-inositol is used for glycosylphosphatidylinositol production in the endoplasmic reticulum, whereas the myo-inositol taken up from the environment is used for bulk phosphatidylinositol synthesis in the Golgi complex. We now provide evidence that the Golgi complex-localized T. brucei H(+)-linked myo-inositol transporter (TbHMIT) is essential in bloodstream-form T. brucei. Downregulation of TbHMIT expression by RNA interference blocked phosphatidylinositol production and inhibited growth of parasites in culture. Characterization of the transporter in a heterologous expression system demonstrated a remarkable selectivity of TbHMIT for myo-inositol. It tolerates only a single modification on the inositol ring, such as the removal of a hydroxyl group or the inversion of stereochemistry at a single hydroxyl group relative to myo-inositol.
González-Salgado, Amaia; Steinmann, Michael; Major, Louise L.; Sigel, Erwin; Reymond, Jean-Louis
2015-01-01
myo-Inositol is a building block for all inositol-containing phospholipids in eukaryotes. It can be synthesized de novo from glucose-6-phosphate in the cytosol and endoplasmic reticulum. Alternatively, it can be taken up from the environment via Na+- or H+-linked myo-inositol transporters. While Na+-coupled myo-inositol transporters are found exclusively in the plasma membrane, H+-linked myo-inositol transporters are detected in intracellular organelles. In Trypanosoma brucei, the causative agent of human African sleeping sickness, myo-inositol metabolism is compartmentalized. De novo-synthesized myo-inositol is used for glycosylphosphatidylinositol production in the endoplasmic reticulum, whereas the myo-inositol taken up from the environment is used for bulk phosphatidylinositol synthesis in the Golgi complex. We now provide evidence that the Golgi complex-localized T. brucei H+-linked myo-inositol transporter (TbHMIT) is essential in bloodstream-form T. brucei. Downregulation of TbHMIT expression by RNA interference blocked phosphatidylinositol production and inhibited growth of parasites in culture. Characterization of the transporter in a heterologous expression system demonstrated a remarkable selectivity of TbHMIT for myo-inositol. It tolerates only a single modification on the inositol ring, such as the removal of a hydroxyl group or the inversion of stereochemistry at a single hydroxyl group relative to myo-inositol. PMID:25888554
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
A Hard X-Ray Study of the Normal Star-forming Galaxy M83 with NuSTAR
NASA Astrophysics Data System (ADS)
Yukita, M.; Hornschemeier, A. E.; Lehmer, B. D.; Ptak, A.; Wik, D. R.; Zezas, A.; Antoniou, V.; Maccarone, T. J.; Replicon, V.; Tyler, J. B.; Venters, T.; Argo, M. K.; Bechtol, K.; Boggs, S.; Christensen, F. E.; Craig, W. W.; Hailey, C.; Harrison, F.; Krivonos, R.; Kuntz, K.; Stern, D.; Zhang, W. W.
2016-06-01
We present the results from sensitive, multi-epoch NuSTAR observations of the late-type star-forming galaxy M83 (d = 4.6 Mpc). This is the first investigation to spatially resolve the hard (E\\gt 10 keV) X-ray emission of this galaxy. The nuclear region and ˜20 off-nuclear point sources, including a previously discovered ultraluminous X-ray source, are detected in our NuSTAR observations. The X-ray hardnesses and luminosities of the majority of the point sources are consistent with hard X-ray sources resolved in the starburst galaxy NGC 253. We infer that the hard X-ray emission is most likely dominated by intermediate accretion state black hole binaries and neutron star low-mass X-ray binaries (Z-sources). We construct the X-ray binary luminosity function (XLF) in the NuSTAR band for an extragalactic environment for the first time. The M83 XLF has a steeper XLF than the X-ray binary XLF in NGC 253, which is consistent with previous measurements by Chandra at softer X-ray energies. The NuSTAR integrated galaxy spectrum of M83 drops quickly above 10 keV, which is also seen in the starburst galaxies NGC 253, NGC 3310, and NGC 3256. The NuSTAR observations constrain any active galactic nucleus (AGN) to be either highly obscured or to have an extremely low luminosity of ≲1038 erg s‑1 (10–30 keV), implying that it is emitting at a very low Eddington ratio. An X-ray point source that is consistent with the location of the nuclear star cluster with an X-ray luminosity of a few times 1038 erg s‑1 may be a low-luminosity AGN but is more consistent with being an X-ray binary.
Teaching normal birth, normally.
Hotelling, Barbara A
2009-01-01
Teaching normal-birth Lamaze classes normally involves considering the qualities that make birth normal and structuring classes to embrace those qualities. In this column, teaching strategies are suggested for classes that unfold naturally, free from unnecessary interventions. PMID:19436595
Le, Jamie A.; Wilson, Heather M.; Shehu, Aurora; Mao, Jifang; Devi, Y. Sangeeta; Halperin, Julia; Aguilar, Tetley; Seibold, Anita; Maizels, Evelyn; Gibori, Geula
2011-01-01
ABSTRACT Prolactin (PRL), a pleiotropic hormone essential for maintenance of corpus luteum (CL) function and pregnancy, transduces its signal through two types of receptors, a short form (PRLR-S) and a long form (PRLR-L). Both types of receptors are expressed in the CL, yet their individual roles are not well defined. We have shown previously that female transgenic mice expressing only PRLR-S display total infertility characterized by defective follicular development and early degeneration of CL, suggesting that expression of PRLR-L is a prerequisite for normal follicular development and maintenance of CL. To determine whether PRLR-L alone is the sole receptor required to maintain normal CL formation, differentiation, and progesterone secretion, we generated two transgenic mice which express only PRLR-L, either ubiquitously (Tg-RL) or in a CL-specific manner (CL-RL). To generate CL-specific expression, we used the HSD17B7 promoter. We found both transgenic mice models cycled normally, displayed no apparent defect in follicular development, and had normal ovulation rates. The STAT5 signaling pathway, considered essential for luteinization and progesterone production, was activated by PRL in both transgenic mice models. However, soon after mating, Tg-RL and CL-RL mice showed early regression of CL, lack of progesterone production, and implantation failure that rendered them totally infertile. Embryo transfer studies demonstrated no embryo abnormalities, and supplementation with progesterone rescued implantation failure in these mice. Close observation revealed lack of luteinization and reduced expression of proteins involved in progesterone biosynthesis despite normal levels of LHCGR (LH-R), ESR1 (ER-alpha), CEBPB (C/EBP-beta) and CDKN1B (p27), proteins essential for luteinization. However, we found VEGFA, a key regulator of angiogenesis and vascularization, to be dramatically reduced in both Tg-RL and CL-RL mice. We also found collagen IV, a marker for the basal
Chang, Boksoon; Han, Seo Goo; Kim, Wooyoul; Ko, Yousang; Song, Junwhi; Hong, Goohyeon; Eom, Jung Seop; Lee, Ji Hyun; Jhun, Byung Woo; Koh, Won-Jung
2013-07-01
Carbohydrate antigen 19-9 (CA 19-9) is a widely-used tumor marker in patients with pancreatic cancer. However, some patients with respiratory disease also exhibit elevated serum CA 19-9 levels. We report a case of normalization of elevated serum CA 19-9 levels after treatment of the nodular bronchiectatic form of Mycobacterium ab scessus lung disease. A 40-year-old man visited our hospital because of chronic cough and sputum. A computed tomography scan revealed severe bronchiectasis in the right upper and right middle lobes. Nontuberculous mycobacteria were repeatedly isolated and identified as M. abscessus. The serum CA 19-9 level was elevated to 142.35 U/mL (normal range, <37 U/mL). Surgical resection was performed because of failure of sputum conversion after antibiotic treatment. The serum CA 19-9 level returned to the normal range after surgery. This case suggested that serum CA 19-9 levels could be elevated in patients with the nodular bronchiectatic form of M. abscessus lung disease.
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.
ERIC Educational Resources Information Center
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.
Inequalities, assessment and computer algebra
NASA Astrophysics Data System (ADS)
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary curricula. We consider the formal mathematical processes by which such inequalities are solved, and we consider the notation and syntax through which solutions are expressed. We review the extent to which current CAS can accurately solve these inequalities, and the form given to the solutions by the designers of this software. Finally, we discuss the functionality needed to deal with students' answers, i.e. to establish equivalence (or otherwise) of expressions representing unions of intervals. We find that while contemporary CAS accurately solve inequalities there is a wide variety of notation used.
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.
D-algebra structure of topological insulators
NASA Astrophysics Data System (ADS)
Estienne, B.; Regnault, N.; Bernevig, B. A.
2012-12-01
In the quantum Hall effect, the density operators at different wave vectors generally do not commute and give rise to the Girvin-MacDonald-Plazmann (GMP) algebra, with important consequences such as ground-state center-of-mass degeneracy at fractional filling fraction, and W1+∞ symmetry of the filled Landau levels. We show that the natural generalization of the GMP algebra to higher-dimensional topological insulators involves the concept of a D commutator. For insulators in even-dimensional space, the D commutator is isotropic and closes, and its structure factors are proportional to the D/2 Chern number. In odd dimensions, the algebra is not isotropic, contains the weak topological insulator index (layers of the topological insulator in one fewer dimension), and does not contain the Chern-Simons θ form. This algebraic structure paves the way towards the identification of fractional topological insulators through the counting of their excitations. The possible relation to D-dimensional volume-preserving diffeomorphisms and parallel transport of extended objects is also discussed.
On the cohomology of Leibniz conformal algebras
NASA Astrophysics Data System (ADS)
Zhang, Jiao
2015-04-01
We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.
Assessing Algebraic Solving Ability: A Theoretical Framework
ERIC Educational Resources Information Center
Lian, Lim Hooi; Yew, Wun Thiam
2012-01-01
Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s. PMID:26806075
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.
Second-Order Algebraic Theories
NASA Astrophysics Data System (ADS)
Fiore, Marcelo; Mahmoud, Ola
Fiore and Hur [10] recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.
Pawlak Algebra and Approximate Structure on Fuzzy Lattice
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. PMID:25152922
Pawlak algebra and approximate structure on fuzzy lattice.
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.
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…
Algebraic independence of p-adic numbers
NASA Astrophysics Data System (ADS)
Nesterenko, Yu V.
2008-06-01
We prove lower bounds for the transcendence degree of fields generated by values of the p-adic exponential function. In particular, we estimate the transcendence degree of the field \\mathbb Q(e^{\\alpha_1},\\dots,e^{\\alpha_d}), where \\alpha_1,\\dots,\\alpha_d are algebraic (over the field of rational numbers) p-adic numbers that form a basis of a finite extension of \\mathbb Q.
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.
2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras
NASA Astrophysics Data System (ADS)
Ayupov, Shavkat; Kudaybergenov, Karimbergen
2016-03-01
The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2n (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation.
Plethystic algebras and vector symmetric functions.
Rota, G C; Stein, J A
1994-01-01
An isomorphism is established between the plethystic Hopf algebra Pleth(Super[L]) and the algebra of vector symmetric functions. The Hall inner product of symmetric function theory is extended to the Hopf algebra Pleth(Super[L]). PMID:11607504
Richgels, M A; Biffle, J H
1980-09-01
ALGEBRA is a program that allows the user to process output data from finite-element analysis codes before they are sent to plotting routines. These data take the form of variable values (stress, strain, and velocity components, etc.) on a tape that is both the output tape from the analyses code and the input tape to ALGEBRA. The ALGEBRA code evaluates functions of these data and writes the function values on an output tape that can be used as input to plotting routines. Convenient input format and error detection capabilities aid the user in providing ALGEBRA with the functions to be evaluated. 1 figure.
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Handheld Computer Algebra Systems in the Pre-Algebra Classroom
ERIC Educational Resources Information Center
Gantz, Linda Ann Galofaro
2010-01-01
This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…
Statecharts Via Process Algebra
NASA Technical Reports Server (NTRS)
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumpsmore » and an anisotropy in one part.« less
Asymptotic structure of free product von Neumann algebras
NASA Astrophysics Data System (ADS)
Houdayer, Cyril; Ueda, Yoshimichi
2016-11-01
Let $(M, \\varphi) = (M_1, \\varphi_1) \\ast (M_2, \\varphi_2)$ be the free product of any $\\sigma$-finite von Neumann algebras endowed with any faithful normal states. We show that whenever $Q \\subset M$ is a von Neumann subalgebra with separable predual such that both $Q$ and $Q \\cap M_1$ are the ranges of faithful normal conditional expectations and such that both the intersection $Q \\cap M_1$ and the central sequence algebra $Q' \\cap M^\\omega$ are diffuse (e.g. $Q$ is amenable), then $Q$ must sit inside $M_1$. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion $M_1 \\subset M$ in arbitrary free product von Neumann algebras.
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.
Open-closed homotopy algebra in mathematical physics
Kajiura, Hiroshige; Stasheff, Jim
2006-02-15
In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach's quantum open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich's deformation quantization and with the B-models of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A{sub {infinity}} algebras) by closed strings (L{sub {infinity}} algebras)
A natural history of mathematics: George Peacock and the making of English algebra.
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.
A natural history of mathematics: George Peacock and the making of English algebra.
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. PMID:23961689
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…
Linear Algebra and Image Processing
ERIC Educational Resources Information Center
Allali, Mohamed
2010-01-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
Linear algebra and image processing
NASA Astrophysics Data System (ADS)
Allali, Mohamed
2010-09-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty.
A Programmed Course in Algebra.
ERIC Educational Resources Information Center
Mewborn, Ancel C.; Hively, Wells II
This programed textbook consists of short sections of text interspersed with questions designed to aid the student in understanding the material. The course is designed to increase the student's understanding of some of the basic ideas of algebra. Some general experience and manipulative skill with respect to high school algebra is assumed.…
ERIC Educational Resources Information Center
1997
Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…
Online Algebraic Tools for Teaching
ERIC Educational Resources Information Center
Kurz, Terri L.
2011-01-01
Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
ERIC Educational Resources Information Center
Instructional Objectives Exchange, Los Angeles, CA.
A complete set of behavioral objectives for first-year algebra taught in any of grades 8 through 12 is presented. Three to six sample test items and answers are provided for each objective. Objectives were determined by surveying the most used secondary school algebra textbooks. Fourteen major categories are included: (1) whole numbers--operations…
Condensing Algebra for Technical Mathematics.
ERIC Educational Resources Information Center
Greenfield, Donald R.
Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-week…
ERIC Educational Resources Information Center
Buerman, Margaret
2007-01-01
Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…
Algebraic description of intrinsic modes in nuclei
Leviatan, A.
1989-01-01
We present a procedure for extracting normal modes in algebraic number-conserving systems of interacting bosons relevant for collective states in even-even nuclei. The Hamiltonian is resolved into intrinsic (bandhead related) and collective (in-band related) parts. Shape parameters are introduced through non-spherical boson bases. Intrinsic modes decoupled from the spurious modes are obtained from the intinsic part of the Hamiltonian in the limit of large number of bosons. Intrinsic states are constructed and serve to evaluate electromagnetic transition rates. The method is illustrated for systems with one type of boson as well as with proton-neutron bosons. 28 refs., 1 fig.
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)
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.
Algebraic grid generation with control points
NASA Technical Reports Server (NTRS)
Eiseman, Peter R.; Choo, Yung K.; Smith, Robert E.
1992-01-01
The control-point form (CPF) formulation is an algebraically defined class of coordinate transformations by means of which the interior form of the coordinates can be manipulated in the local fashion, and any boundary can be either specified or manipulated in a similar manner. Currently, the most intense activity involving CPF is with such graphic interactive codes as TurboI and TurboT, for which detailed illustrative examples are given; these have furnished experience on whose basis future interactive strategies can be developed.
Quantum algebra of N superspace
Hatcher, Nicolas; Restuccia, A.; Stephany, J.
2007-08-15
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.
Chen, J.; Safro, I.
2011-01-01
Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
SD-CAS: Spin Dynamics by Computer Algebra System.
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.
SD-CAS: Spin Dynamics by Computer Algebra System
NASA Astrophysics Data System (ADS)
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.
Readiness and Preparation for Beginning Algebra.
ERIC Educational Resources Information Center
Rotman, Jack W.
Drawing from experience at Lansing Community College (LCC), this paper discusses how to best prepare students for success in a beginning algebra course. First, an overview is presented of LCC's developmental math sequence, which includes Basic Arithmetic (MTH 008), Pre-Algebra (MTH 009), Beginning Algebra (MTH 012), and Intermediate Algebra (MTH…
Hopf algebras and Dyson-Schwinger equations
NASA Astrophysics Data System (ADS)
Weinzierl, Stefan
2016-06-01
In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
Two-parameter twisted quantum affine algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Zhang, Honglian
2016-09-01
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.
On the complexity of the binary expansions of algebraic irrational numbers (survey)
NASA Astrophysics Data System (ADS)
Kaneko, Hajime
2010-07-01
Borel conjectured that all irrational numbers are normal in any integral base α. For each positive number ξ and integer α greater than 1, ξ is normal in base α if and only if the sequence ξαn (n = 0,1,…) is uniformly distributed modulo 1. In this paper we survey not only the digit of algebraic irrational numbers in integral base but also the fractional parts of geometric progressions whose common ratios are algebraic numbers greater than 1. In our main results, we give new lower bounds for the number of digit changes in the binary expansions of algebraic irrational numbers.
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras
Yu, Zhang; Zhang, Yufeng
2009-01-01
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings. PMID:20084092
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.
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.
Description of DASSL: a differential/algebraic system solver
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.
Spontaneous Meta-Arithmetic as the First Step toward School Algebra
ERIC Educational Resources Information Center
Caspi, Shai; Sfard, Anna
2012-01-01
Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the "official" algebraic form of talk. In this paper we take a look at the very beginning of…
Cartooning in Algebra and Calculus
ERIC Educational Resources Information Center
Moseley, L. Jeneva
2014-01-01
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
NASA Technical Reports Server (NTRS)
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
Algebra and statistics of the solar wind
NASA Astrophysics Data System (ADS)
Veselovsky, I. S.; Dmitriev, A. V.; Suvorova, A. V.
2010-04-01
Statistical studies of properties of the solar wind and interplanetary magnetic field, based on an extended database for the period 1963-2007 including four solar cycles, show that the Gaussian approximation well suites for some parameters as the probability distribution of their numerical values, while for others the lognormal law is preferred. This paper gives an interpretation of these results as associated with predominance of linear or nonlinear processes in composition and interaction of various disturbances and irregularities propagating and originating in the interior of the Sun and its atmosphere, including the solar corona and the solar wind running away from it. Summation of independent random components of disturbances leads, according to the central limit theorem of the probability theory, to the normal (Gaussian) distributions of quantities proper, while their multiplication leads to the normal distributions of logarithms. Thus, one can discuss the algebra of events and associate observed statistical distinctions with one or another process of formation of irregularities in the solar wind. Among them there are impossible events (having null probability) and reliable events (occurring with 100% probability). For better understanding of the relationship between algebra and statistics of events in the solar wind further investigations are necessary.
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).
Wakimoto realizations of current algebras: an explicit construction
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.
Bases for representations of quantum algebras
NASA Astrophysics Data System (ADS)
Atakishiyev, N. M.; Winternitz, P.
2000-08-01
We derive an explicit expression for the eigenfunctions and the corresponding eigenvalues of the operator [q1/4J+(q) + q-1/4J-(q)] qJ3(q)/2 in an arbitrary irreducible representation of the algebra suq(2). The general form of the intertwining operator AJ(q), which is a q-extension of the classical su(2)-operator aJ, J1aJ = aJJ3, is also found. The matrix elements of AJ(q) are expressed in terms of the dual q-Kravchuk polynomials.
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
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.
Some Remarks on Kite Pseudo Effect Algebras
NASA Astrophysics Data System (ADS)
Dvurečenskij, Anatolij; Holland, W. Charles
2014-05-01
Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group G, a set I and with two bijections λ, ρ: I→ I. Using a clever construction on the ordinal sum of ( G +) I and ( G -) I , we can define a pseudo effect algebra which can be non-commutative even if G is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop.
Operator product expansion algebra
Holland, Jan; Hollands, Stefan
2013-07-15
We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ{sup 4}-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hep-th/1105.3375, that the 3-point OPE,
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…
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'…
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.
Anisotropy without tensors: a novel approach using geometric algebra.
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.
Algebraic field descriptions in three-dimensional Euclidean space
NASA Astrophysics Data System (ADS)
Salingaros, Nikos; Ilamed, Yehiel
1984-08-01
In this paper, we use the differential forms of three-dimensional Euclidean space to realize a Clifford algebra which is isomorphic to the algebra of the Pauli matrices or the complex quaternions. This is an associative vector-antisymmetric tensor algebra with division: We provide the algebraic inverse of an eight-component spinor field which is the sum of a scalar + vector + pseudovector + pseudoscalar. A surface of singularities is defined naturally by the inverse of an eight-component spinor and corresponds to a generalized Minkowski “double” light cone in the parameter space. A general description of finite spatial rotations, which utilizes the Baker-Campbell-Hausdorff formula, generalizes the usual infinitesimal treatments of the rotation group. We derive an explicit expression for the angle corresponding to two successive finite rotations in any direction. We also discuss Lorentz transformations and duality rotations of the electromagnetic field and exhibit a relationship between the algebraic inverse and a duality rotated field. Using a combined transformation, one can always transform an arbitrary electromagnetic field ( E≠0) into a pure electric field, but never into a pure magnetic field.
Quadratic algebras for three-dimensional superintegrable systems
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.
A Metric Conceptual Space Algebra
NASA Astrophysics Data System (ADS)
Adams, Benjamin; Raubal, Martin
The modeling of concepts from a cognitive perspective is important for designing spatial information systems that interoperate with human users. Concept representations that are built using geometric and topological conceptual space structures are well suited for semantic similarity and concept combination operations. In addition, concepts that are more closely grounded in the physical world, such as many spatial concepts, have a natural fit with the geometric structure of conceptual spaces. Despite these apparent advantages, conceptual spaces are underutilized because existing formalizations of conceptual space theory have focused on individual aspects of the theory rather than the creation of a comprehensive algebra. In this paper we present a metric conceptual space algebra that is designed to facilitate the creation of conceptual space knowledge bases and inferencing systems. Conceptual regions are represented as convex polytopes and context is built in as a fundamental element. We demonstrate the applicability of the algebra to spatial information systems with a proof-of-concept application.
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.
Colored Quantum Algebra and Its Bethe State
NASA Astrophysics Data System (ADS)
Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun
2014-12-01
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.
Generalized Galilean algebras and Newtonian gravity
NASA Astrophysics Data System (ADS)
González, N.; Rubio, G.; Salgado, P.; Salgado, S.
2016-04-01
The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.
Motivating Activities that Lead to Algebra
ERIC Educational Resources Information Center
Menon, Ramakrishnan
2004-01-01
Four activities consisting of puzzles are introduced, which help students to recognize the strength of algebraic generalizations. They also assist them to comprehend algebraic concepts, and enable them to develop their individual puzzles and games.
Scalable Parallel Algebraic Multigrid Solvers
Bank, R; Lu, S; Tong, C; Vassilevski, P
2005-03-23
The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.
Discrimination in a General Algebraic Setting.
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.
Discrimination in a General Algebraic Setting
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras. PMID:26171421
Characteristic Numbers of Matrix Lie Algebras
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Fan, En-Gui
2008-04-01
A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.
Spatial-Operator Algebra For Robotic Manipulators
NASA Technical Reports Server (NTRS)
Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.
1991-01-01
Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
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)
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)
Bilinear covariants and spinor fields duality in quantum Clifford algebras
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.
New family of Maxwell like algebras
NASA Astrophysics Data System (ADS)
Concha, P. K.; Durka, R.; Merino, N.; Rodríguez, E. K.
2016-08-01
We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.
Unifying the Algebra for All Movement
ERIC Educational Resources Information Center
Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.
2015-01-01
There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…
Build an Early Foundation for Algebra Success
ERIC Educational Resources Information Center
Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela
2016-01-01
Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…
Difficulties in Initial Algebra Learning in Indonesia
ERIC Educational Resources Information Center
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-01-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…
A Balancing Act: Making Sense of Algebra
ERIC Educational Resources Information Center
Gavin, M. Katherine; Sheffield, Linda Jensen
2015-01-01
For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…
Algebra? A Gate! A Barrier! A Mystery!
ERIC Educational Resources Information Center
Mathematics Educatio Dialogues, 2000
2000-01-01
This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…
Computer Algebra Systems, Pedagogy, and Epistemology
ERIC Educational Resources Information Center
Bosse, Michael J.; Nandakumar, N. R.
2004-01-01
The advent of powerful Computer Algebra Systems (CAS) continues to dramatically affect curricula, pedagogy, and epistemology in secondary and college algebra classrooms. However, epistemological and pedagogical research regarding the role and effectiveness of CAS in the learning of algebra lags behind. This paper investigates concerns regarding…
Teaching Strategies to Improve Algebra Learning
ERIC Educational Resources Information Center
Zbiek, Rose Mary; Larson, Matthew R.
2015-01-01
Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…
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.
Entropy algebras and Birkhoff factorization
NASA Astrophysics Data System (ADS)
Marcolli, Matilde; Tedeschi, Nicolas
2015-11-01
We develop notions of Rota-Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.
Algebraic Activities Aid Discovery Lessons
ERIC Educational Resources Information Center
Wallace-Gomez, Patricia
2013-01-01
After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…
ERIC Educational Resources Information Center
Bosse, Michael J.; Ries, Heather; Chandler, Kayla
2012-01-01
Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…
Dimension independence in exterior algebra.
Hawrylycz, M
1995-01-01
The identities between homogeneous expressions in rank 1 vectors and rank n - 1 covectors in a Grassmann-Cayley algebra of rank n, in which one set occurs multilinearly, are shown to represent a set of dimension-independent identities. The theorem yields an infinite set of nontrivial geometric identities from a given identity. PMID:11607520
Exploring Algebraic Misconceptions with Technology
ERIC Educational Resources Information Center
Sakow, Matthew; Karaman, Ruveyda
2015-01-01
Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…
ERIC Educational Resources Information Center
Oishi, Lindsay
2011-01-01
"Solve for x." While many people first encountered this enigmatic instruction in high school, the last 20 years have seen a strong push to get students to take algebra in eighth grade or even before. Today, concerns about the economy highlight a familiar worry: American eighth-graders trailed their peers in five Asian countries on the 2007 TIMSS…
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…
Algebraic methods in system theory
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Willems, J. C.; Willsky, A. S.
1975-01-01
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
Algebra from Chips and Chopsticks
ERIC Educational Resources Information Center
Yun, Jeong Oak; Flores, Alfinio
2012-01-01
Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…
Celestial mechanics with geometric algebra
NASA Technical Reports Server (NTRS)
Hestenes, D.
1983-01-01
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
Algebra for All. Research Brief
ERIC Educational Resources Information Center
Bleyaert, Barbara
2009-01-01
The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless national…
Kinds of Knowledge in Algebra.
ERIC Educational Resources Information Center
Lewis, Clayton
Solving equations in elementary algebra requires knowledge of the permitted operations, and knowledge of what operation to use at a given point in the solution process. While just these kinds of knowledge would be adequate for an ideal solver, human solvers appear to need and use other kinds of knowledge. First, many errors seem to indicate that…
Adventures in Flipping College Algebra
ERIC Educational Resources Information Center
Van Sickle, Jenna
2015-01-01
This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…
ERIC Educational Resources Information Center
Deakin, Michael A. B.
1974-01-01
Euler's famous formula, e to the (i, pi) power equals -1, is developed by a purely algebraic method that avoids the use of both trigonometry and calculus. A heuristic outline is given followed by the rigorous theory. Pedagogical considerations for classroom presentation are suggested. (LS)
Elementary Algebra Connections to Precalculus
ERIC Educational Resources Information Center
Lopez-Boada, Roberto; Daire, Sandra Arguelles
2013-01-01
This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…
Inequalities, Assessment and Computer Algebra
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in…
Algebra, Home Mortgages, and Recessions
ERIC Educational Resources Information Center
Mariner, Jean A. Miller; Miller, Richard A.
2009-01-01
The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…
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…
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.
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.
A uniform algebraically-based approach to computational physics and efficient programming
NASA Astrophysics Data System (ADS)
Raynolds, James; Mullin, Lenore
2007-03-01
We present an approach to computational physics in which a common formalism is used both to express the physical problem as well as to describe the underlying details of how computation is realized on arbitrary multiprocessor/memory computer architectures. This formalism is the embodiment of a generalized algebra of multi-dimensional arrays (A Mathematics of Arrays) and an efficient computational implementation is obtained through the composition of of array indices (the psi-calculus) of algorithms defined using matrices, tensors, and arrays in general. The power of this approach arises from the fact that multiple computational steps (e.g. Fourier Transform followed by convolution, etc.) can be algebraically composed and reduced to an simplified expression (i.e. Operational Normal Form), that when directly translated into computer code, can be mathematically proven to be the most efficient implementation with the least number of temporary variables, etc. This approach will be illustrated in the context of a cache-optimized FFT that outperforms or is competitive with established library routines: ESSL, FFTW, IMSL, NAG.
Filiform Lie algebras of order 3
Navarro, R. M.
2014-04-15
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.
Atomic effect algebras with compression bases
Caragheorgheopol, Dan; Tkadlec, Josef
2011-01-15
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.
Linear algebra algorithms for divisors on an algebraic curve
NASA Astrophysics Data System (ADS)
Khuri-Makdisi, Kamal
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to work with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Hess' 1999 Ph.D. thesis, which works with function fields as extensions of $k[x]$. However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor), superelliptic curves (Galbraith, Paulus, and Smart), and $C_{ab}$ curves (Harasawa and Suzuki); in all those cases, one can attain a complexity of $O(g^2)$.
Block algebra in two-component BKP and D type Drinfeld-Sokolov hierarchies
Li, Chuanzhong He, Jingsong
2013-11-15
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.
Alternative algebraic approaches in quantum chemistry
Mezey, Paul G.
2015-01-22
Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.
The algebras of large N matrix mechanics
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
Ranganathan, Perungavar N; Whalen, Richard; Boyer, Thomas D
2005-03-15
GSTP1 (glutathione S-transferase pi) is involved in stress responses and in cellular proliferation pathways as an inhibitor of JNK (c-Jun N-terminal kinase). It has been proposed that monomeric GSTP1 functions as a JNK inhibitor. All of the studies to date have been performed using rodent cells, and it is unclear if monomeric GSTP1 exists in human cells. Monomeric GSTP1 was sought in human gastric cancer cells (Kato III) and in normal human erythrocytes using gel filtration, ELISA and Western blots. Monomeric GSTP1 was found in conditioned medium, in cytosol of Kato III cells and in cytosol of erythrocytes. GSTP1 subunits from Kato III cells and erythrocytes were heterogeneous when analysed by MALDI-TOF (matrix-assisted laser-desorption ionization-time-of-flight) MS, suggesting that there were post-translational modifications to GSTP1. One post-translational modification, phosphorylation of a serine residue in the C-terminal portion of GSTP1 where JNK binds, was identified in GSTP1 purified from Kato III cells, but not in GSTP1 purified from human erythrocytes. Therefore normal and malignant human cells contain GSTP1 monomers with post-translational modifications, and it is likely that GSTP1 monomers regulate JNK activity in human cells in the same manner as in rodent cells. PMID:15471539
2004-01-01
GSTP1 (glutathione S-transferase pi) is involved in stress responses and in cellular proliferation pathways as an inhibitor of JNK (c-Jun N-terminal kinase). It has been proposed that monomeric GSTP1 functions as a JNK inhibitor. All of the studies to date have been performed using rodent cells, and it is unclear if monomeric GSTP1 exists in human cells. Monomeric GSTP1 was sought in human gastric cancer cells (Kato III) and in normal human erythrocytes using gel filtration, ELISA and Western blots. Monomeric GSTP1 was found in conditioned medium, in cytosol of Kato III cells and in cytosol of erythrocytes. GSTP1 subunits from Kato III cells and erythrocytes were heterogeneous when analysed by MALDI–TOF (matrix-assisted laser-desorption ionization–time-of-flight) MS, suggesting that there were post-translational modifications to GSTP1. One post-translational modification, phosphorylation of a serine residue in the C-terminal portion of GSTP1 where JNK binds, was identified in GSTP1 purified from Kato III cells, but not in GSTP1 purified from human erythrocytes. Therefore normal and malignant human cells contain GSTP1 monomers with post-translational modifications, and it is likely that GSTP1 monomers regulate JNK activity in human cells in the same manner as in rodent cells. PMID:15471539
BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1994-01-01
The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.
Computer algebra and transport theory.
Warsa, J. S.
2004-01-01
Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.
Introduction to Image Algebra Ada
NASA Astrophysics Data System (ADS)
Wilson, Joseph N.
1991-07-01
Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.
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.…
Kumjian-Pask algebras of desourcification
NASA Astrophysics Data System (ADS)
Rosjanuardi, Rizky; Yusnitha, Isnie
2016-02-01
Kumjian-Pask algebra which was introduced by Pino, Clark, an Huef and Raeburn [1] in 2013, gives a purely algebraic version of a k-graph algebra. Rosjanuardi [2] gave necessary and sufficient condition of finitely dimensional complex Kumjian-Pask algebra of row-finite k-graph without sources. We will improve the previous results which allows us to deal with sources. We will consider Kumjian-Pask algebra for locally convex row-finite k-graph which was introduced by Clark, Flynn and an Huef [3], and use the desourcification of the graph to get conditions which characterise when the complex Kumjian-Pask algebra of locally convex row-finite k-graph is finite dimensional.
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
NASA Astrophysics Data System (ADS)
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
Classical Becchi-Rouet-Stora-Tyutin charge for nonlinear algebras
NASA Astrophysics Data System (ADS)
Buchbinder, I. L.; Lavrov, P. M.
2007-08-01
We study the construction of the classical nilpotent canonical Becchi-Rouet-Stora-Tyutin (BRST) charge for the nonlinear gauge algebras, where a commutator (in terms of Poisson brackets) of the constraints is a finite order polynomial of the constraints. Such a polynomial is characterized by the coefficients forming a set of higher order structure constraints. Assuming the set of constraints to be linearly independent, we find the restrictions on the structure constraints when the nilpotent BRST charge can be written in a simple and universal form. In the case of quadratically nonlinear algebras, we find the expression for third order contribution in the ghost fields to the BRST charge without the use of any additional restrictions on the structure constants.
Non-Abelian gerbes and enhanced Leibniz algebras
NASA Astrophysics Data System (ADS)
Strobl, Thomas
2016-07-01
We present the most general gauge-invariant action functional for coupled 1- and 2-form gauge fields with kinetic terms in generic dimensions, i.e., dropping eventual contributions that can be added in particular space-time dimensions only such as higher Chern-Simons terms. After appropriate field redefinitions it coincides with a truncation of the Samtleben-Szegin-Wimmer action. In the process one sees explicitly how the existence of a gauge-invariant functional enforces that the most general semistrict Lie 2-algebra describing the bundle of a non-Abelian gerbe gets reduced to a very particular structure, which, after the field redefinition, can be identified with the one of an enhanced Leibniz algebra. This is the first step towards a systematic construction of such functionals for higher gauge theories, with kinetic terms for a tower of gauge fields up to some highest form degree p , solved here for p =2 .
Algebraic structure of general electromagnetic fields and energy flow
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.
Coverings of topological semi-abelian algebras
NASA Astrophysics Data System (ADS)
Mucuk, Osman; Demir, Serap
2016-08-01
In this work, we study on a category of topological semi-abelian algebras which are topological models of given an algebraic theory T whose category of models is semi-abelian; and investigate some results on the coverings of topological models of such theories yielding semi-abelian categories. We also consider the internal groupoid structure in the semi-abelian category of T-algebras, and give a criteria for the lifting of internal groupoid structure to the covering groupoids.
New algorithms for the minimal form'' problem
Oliveira, J.S.; Cook, G.O. Jr. ); Purtill, M.R. . Center for Communications Research)
1991-12-20
It is widely appreciated that large-scale algebraic computation (performing computer algebra operations on large symbolic expressions) places very significant demands upon existing computer algebra systems. Because of this, parallel versions of many important algorithms have been successfully sought, and clever techniques have been found for improving the speed of the algebraic simplification process. In addition, some attention has been given to the issue of restructuring large 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 the sense that no simple transformation on the expression leads to a form involving fewer operations. Unfortunately, the progress that has been achieved to date on this very hard problem is not adequate for the very significant demands of large computer algebra problems. In response to this situation, we have developed some efficient algorithms for constructing minimal forms.'' In this paper, the multi-stage algorithm in which these new algorithms operate is defined and the features of these algorithms are developed. In a companion paper, we introduce the core algebra engine of a new tool that provides the algebraic framework required for the implementation of these new algorithms.
Multicloning and Multibroadcasting in Operator Algebras
NASA Astrophysics Data System (ADS)
Kaniowski, Krzysztof; Lubnauer, Katarzyna; Łuczak, Andrzej
2015-12-01
We investigate multicloning and multibroadcasting in the general operator algebra framework in arbitrary dimension, generalizing thus results obtained in this framework for simple cloning and broadcasting.
On Realization of Generalized Effect Algebras
NASA Astrophysics Data System (ADS)
Paseka, Jan
2012-12-01
A well-known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice L(H) of all closed subspaces of a separable complex Hilbert space. We show that a generalized effect algebra is representable in the operator generalized effect algebra G(H) of effects of a complex Hilbert space H iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Riečanová and Zajac. Further, any operator generalized effect algebra G(H) possesses an order determining set of generalized states.
NASA Astrophysics Data System (ADS)
Chajda, Ivan
2014-10-01
Commutative BCI-algebras can be considered as semilattices whose sections are equipped with certain involutions. A similar view can be applied to commutative BCK-algebras. However, for general BCK-algebras a certain construction was settled by the author and J. Kühr (Miskolc Math. Notes 8:11-21, 2007) showing that they can be considered as structures essentially weaker than semilattices but still with certain involutions in sections. The aim of this paper is to involve a similar approach for BCI-algebras.
Difficulties in initial algebra learning in Indonesia
NASA Astrophysics Data System (ADS)
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-12-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.
Literal algebra for satellite dynamics. [perturbation analysis
NASA Technical Reports Server (NTRS)
Gaposchkin, E. M.
1975-01-01
A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.
Entanglement and algebraic independence in fermion systems
NASA Astrophysics Data System (ADS)
Benatti, Fabio; Floreanini, Roberto
2014-04-01
In the case of systems composed of identical particles, a typical instance in quantum statistical mechanics, the standard approach to separability and entanglement ought to be reformulated and rephrased in terms of correlations between operators from subalgebras localized in spatially disjoint regions. While this algebraic approach is straightforward for bosons, in the case of fermions it is subtler since one has to distinguish between micro-causality, that is the anti-commutativity of the basic creation and annihilation operators, and algebraic independence that is the commutativity of local observables. We argue that a consistent algebraic formulation of separability and entanglement should be compatible with micro-causality rather than with algebraic independence.
NASA Astrophysics Data System (ADS)
Riechers, Dominik A.; Carilli, Chris Luke; Capak, Peter L.; COSMOS, HerMES
2016-01-01
Cold molecular and atomic gas plays a central role in our understanding of early galaxy formation and evolution. It represents the material that stars form out of, and its mass, distribution, excitation, and dynamics provide crucial insight into the physical processes that support the ongoing star formation and stellar mass buildup. We present some of the most recent progress in studies of gas-rich galaxies out to the highest redshifts through detailed investigations of the cold gas and dust with the most powerful facilities, i.e., the Karl G. Jansky Very Large Array (VLA), the NOrthern Extended Millimeter Array (NOEMA) and the Atacama Large (sub-) Millimeter Array (ALMA). Facilitating the impressive sensitivity of ALMA, this investigation encompasses a systematic study of the star-forming interstellar medium, gas dynamics, and dust obscuration in massive dusty starbursts and (much less luminous and massive) "typical" galaxies at such early epochs. These new results show that "typical" z>5 galaxies are significantly metal-enriched, but not heavily dust-obscured, consistent with a decreasing contribution of dust-obscured star formation to the star formation history of the universe towards the earliest cosmic epochs.
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
Some C∗-algebras which are coronas of non-C∗-Banach algebras
NASA Astrophysics Data System (ADS)
Voiculescu, Dan-Virgil
2016-07-01
We present results and motivating problems in the study of commutants of hermitian n-tuples of Hilbert space operators modulo normed ideals. In particular, the C∗-algebras which arise in this context as coronas of non-C∗-Banach algebras, the connections with normed ideal perturbations of operators, the hyponormal operators and the bidual Banach algebras one encounters are discussed.
ERIC Educational Resources Information Center
Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos
2016-01-01
This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…
Leibniz algebras associated with some finite-dimensional representation of Diamond Lie algebra
NASA Astrophysics Data System (ADS)
Camacho, Luisa M.; Ladra, Manuel; Karimjanov, Iqboljon A.; Omirov, Bakhrom A.
2016-03-01
In this paper we classify Leibniz algebras whose associated Lie algebra is four-dimensional Diamond Lie algebra 𝕯 and the ideal generated by squares of elements is represented by one of the finite-dimensional indecomposable D-modules Un 1, Un 2 or Wn 1 or Wn 2.
ERIC Educational Resources Information Center
Ozgun-Koca, S. Ash
2010-01-01
Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This…
Commutative n-ary superalgebras with an invariant skew-symmetric form
NASA Astrophysics Data System (ADS)
Vishnyakova, E. G.
2015-12-01
We study n-ary commutative superalgebras and L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their n-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative m-dimensional (m - 3) -ary algebras with an invariant form, and a classification of real simple m-dimensional Lie (m - 3) -algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for L∞-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric n-ary algebras.
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.
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.
Grubb, A O; López, C; Tejler, L; Mendez, E
1983-12-10
Human complex-forming glycoprotein, heterogeneous in charge (protein HC) has previously been isolated from urine and immunochemically shown to be present in low and high molecular weight forms in blood plasma (Tejler, L., and Grubb, A. O. (1976) Biochim. Biophys. Acta 439, 82-94). In the present work, the major low and high molecular weight forms of the protein were isolated from plasma by immunosorption followed by gel chromatography. The plasma low molecular weight protein HC and the urinary protein had similar, if not identical, molecular weight, amino acid composition, NH2-terminal and carboxyl-terminal amino acid sequences and electrophoretic mobility. The low molecular weight plasma protein HC carried a yellow chromophore like the urinary protein, but its molar extinction coefficient at 280 nm was lower and its charge heterogeneity less pronounced than that of urinary protein HC. The plasma high molecular weight protein HC had a hydrodynamic volume which was greater than that of monomeric IgA but smaller than that of dimeric IgA. Sodium dodecyl sulfate-polyacrylamide gel electrophoresis of the isolated high molecular weight protein followed by electrophoretic blotting and immunochemical analysis demonstrated that the protein contained four polypeptide chains: two light immunoglobulin chains (Mr = 23,000), one IgA alpha-chain (Mr = 54,000), and one chain with Mr approximately 90,000 which carried both alpha-chain and protein HC antigenic determinants. Whether the protein HC X IgA complex is a functionally significant part of the humoral immune system cannot be decided without further experimentation, but the complex was found to be completely absent from the blood plasma of patients with a selective deficiency of IgA-secreting immunocytes. The isolated low and high molecular weight plasma protein HC components were used as standard proteins in the construction of a quantitative crossed immunoelectrophoretic assay for the simultaneous quantitation of the two
Al-Hassan, Khader A
2013-11-01
An evidence is introduced through the b- and the twisted intramolecular charge transfer (TICT) fluorescence of ethyl-4-(N,N-dimethylamino)benzoate (EDMAB) and ethyl-4-(N,N-diethylamino)benzoate (EDEAB), confirming the role of donor size on the formation and emission of various inclusion complexes formed between these probes and α-CD in aqueous solution. A large variation in the b-fluorescence band of EDEAB as compared to that of EDMAB and a large variation in the TICT-fluorescence band of EDMAB as compared to that EDEAB, as the concentration of α-CD is increased in their aqueous solutions are observed. These variations are supported by time resolved fluorescence (TRF) spectra, fluorescence decay lifetimes and red edge effect (REE) results.
Cluster automorphism groups of cluster algebras with coefficients
NASA Astrophysics Data System (ADS)
Chang, Wen; Zhu, Bin
2016-10-01
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e. the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geometric coefficients, and cluster algebras from surfaces (except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.
The Structure of Parafermion Vertex Operator Algebras: General Case
NASA Astrophysics Data System (ADS)
Dong, Chongying; Wang, Qing
2010-11-01
The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.
Gene algebra from a genetic code algebraic structure.
Sanchez, R; Morgado, E; Grau, R
2005-10-01
By considering two important factors involved in the codon-anticodon interactions, the hydrogen bond number and the chemical type of bases, a codon array of the genetic code table as an increasing code scale of interaction energies of amino acids in proteins was obtained. Next, in order to consecutively obtain all codons from the codon AAC, a sum operation has been introduced in the set of codons. The group obtained over the set of codons is isomorphic to the group (Z(64), +) of the integer module 64. On the Z(64)-algebra of the set of 64(N) codon sequences of length N, gene mutations are described by means of endomorphisms f:(Z(64))(N)-->(Z(64))(N). Endomorphisms and automorphisms helped us describe the gene mutation pathways. For instance, 77.7% mutations in 749 HIV protease gene sequences correspond to unique diagonal endomorphisms of the wild type strain HXB2. In particular, most of the reported mutations that confer drug resistance to the HIV protease gene correspond to diagonal automorphisms of the wild type. What is more, in the human beta-globin gene a similar situation appears where most of the single codon mutations correspond to automorphisms. Hence, in the analyses of molecular evolution process on the DNA sequence set of length N, the Z(64)-algebra will help us explain the quantitative relationships between genes.
Dirac matrices as elements of a superalgebraic matrix algebra
NASA Astrophysics Data System (ADS)
Monakhov, V. V.
2016-08-01
The paper considers a Clifford extension of the Grassmann algebra, in which operators are built from Grassmann variables and by the derivatives with respect to them. It is shown that a subalgebra which is isomorphic to the usual matrix algebra exists in this algebra, the Clifford exten-sion of the Grassmann algebra is a generalization of the matrix algebra and contains superalgebraic operators expanding matrix algebra and produces supersymmetric transformations.
Automated Angular Momentum Recoupling Algebra
NASA Astrophysics Data System (ADS)
Williams, H. T.; Silbar, Richard R.
1992-04-01
We present a set of heuristic rules for algebraic solution of angular momentum recoupling problems. The general problem reduces to that of finding an optimal path from one binary tree (representing the angular momentum coupling scheme for the reduced matrix element) to another (representing the sub-integrals and spin sums to be done). The method lends itself to implementation on a microcomputer, and we have developed such an implementation using a dialect of LISP. We describe both how our code, called RACAH, works and how it appears to the user. We illustrate the use of RACAH for several transition and scattering amplitude matrix elements occurring in atomic, nuclear, and particle physics.
Automorphisms and Derivations of the Insertion-Elimination Algebra and Related Graded Lie Algebras
NASA Astrophysics Data System (ADS)
Ondrus, Matthew; Wiesner, Emilie
2016-07-01
This paper addresses several structural aspects of the insertion-elimination algebra {mathfrak{g}}, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of {mathfrak{g}}, the automorphism group of {mathfrak{g}}, the derivation Lie algebra of {mathfrak{g}}, and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.
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.
Investigating Algebraic Procedures Using Discussion and Writing
ERIC Educational Resources Information Center
Harper, Jonathan; Ford, Jeffrey
2012-01-01
This study reports on the implementation of an intermediate algebra curriculum centered on a framework of student-centered questions designed to investigate algebraic procedures. Instructional activities were designed to build discourse in the small-group discussion meetings of the course. Students were assigned writing prompts to emphasize the…
Practicing Algebraic Skills: A Conceptual Approach
ERIC Educational Resources Information Center
Friedlander, Alex; Arcavi, Abraham
2012-01-01
Traditionally, a considerable part of teaching and learning algebra has focused on routine practice and the application of rules, procedures, and techniques. Although today's computerized environments may have decreased the need to master algebraic skills, procedural competence is still a central component in any mathematical activity. However,…
Using Students' Interests as Algebraic Models
ERIC Educational Resources Information Center
Whaley, Kenneth A.
2012-01-01
Fostering algebraic thinking is an important goal for middle-grades mathematics teachers. Developing mathematical reasoning requires that teachers cultivate students' habits of mind. Teachers develop students' understanding of algebra by engaging them in tasks that involve modeling and representation. This study was designed to investigate how…
THE RADICAL OF A JORDAN ALGEBRA
McCrimmon, Kevin
1969-01-01
In this paper we define a Jacobson radical for Jordan algebras analogous to that for associative algebras and show that it enjoys many of the properties of the associative radical. We then relate the corresponding notion of “semisimplicity” to the previously defined notion of “nondegeneracy” (Jacobson, N., these Proceedings, 55, 243-251 (1966)). PMID:16591736
The operator algebra approach to quantum groups
Kustermans, Johan; Vaes, Stefaan
2000-01-01
A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory. PMID:10639116
Situated Learning in an Abstract Algebra Classroom
ERIC Educational Resources Information Center
Ticknor, Cindy S.
2012-01-01
Advisory committees of mathematics consider abstract algebra as an essential component of the mathematical preparation of secondary teachers, yet preservice teachers find it challenging to connect the topics addressed in this advanced course with the high school algebra they must someday teach. This study analyzed the mathematical content…
Predicting Turkish Ninth Grade Students' Algebra Performance
ERIC Educational Resources Information Center
Erbas, Ayhan Kursat
2005-01-01
The prediction of students' achievement in algebra in eighth and ninth grades has become a research interest for practical issues of placement. A group of simple, easily accessible variables was used to predict student performance in algebra after completion of eighth grade. The three variables of school type, grade level, and previous year…
Success in Algebra among Community College Students
ERIC Educational Resources Information Center
Reyes, Czarina
2010-01-01
College algebra is a required course for most majors, but is viewed by many as a gatekeeper course for degree completion by students. With almost half a million students taking college algebra each year, faculty are experimenting with new course lengths of time that might result in higher success, completion, and retention rates for college…
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…
How To Prepare Students for Algebra.
ERIC Educational Resources Information Center
Wu, H.
2001-01-01
Suggests that no matter how much algebraic thinking is introduced in the early grades, and no matter how worthwhile this might be, the failure rate in algebra will continue unless the teaching of fractions and decimals is radically revamped. The proper study of fractions provides a ramp that leads students gently from whole number arithmetic up to…
Using the Internet To Investigate Algebra.
ERIC Educational Resources Information Center
Sherwood, Walter
The lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approach algebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…
New directions in algebraic dynamical systems
NASA Astrophysics Data System (ADS)
Schmidt, Klaus; Verbitskiy, Evgeny
2011-02-01
The logarithmic Mahler measure of certain multivariate polynomials occurs frequently as the entropy or the free energy of solvable lattice models (especially dimer models). It is also known that the entropy of an algebraic dynamical system is the logarithmic Mahler measure of the defining polynomial. The connection between the lattice models and the algebraic dynamical systems is still rather mysterious.
Classical and quantum Kummer shape algebras
NASA Astrophysics Data System (ADS)
Odzijewicz, A.; Wawreniuk, E.
2016-07-01
We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras, here called Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, are also presented.
Fourier theory and C∗-algebras
NASA Astrophysics Data System (ADS)
Bédos, Erik; Conti, Roberto
2016-07-01
We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included.
Teaching Algebra to Students with Learning Disabilities
ERIC Educational Resources Information Center
Impecoven-Lind, Linda S.; Foegen, Anne
2010-01-01
Algebra is a gateway to expanded opportunities, but it often poses difficulty for students with learning disabilities. Consequently, it is essential to identify evidence-based instructional strategies for these students. The authors begin by identifying three areas of algebra difficulty experienced by students with disabilities: cognitive…
Arithmetic and Cognitive Contributions to Algebra
ERIC Educational Resources Information Center
Cirino, Paul T.; Tolar, Tammy D.; Fuchs, Lynn S.
2013-01-01
Algebra is a prerequisite for access to STEM careers and occupational success (NMAP, 2008a), yet algebra is difficult for students through high school (US DOE, 2008). Growth in children's conceptual and procedural arithmetical knowledge is reciprocal, although conceptual knowledge has more impact on procedural knowledge than the reverse…
Just Say Yes to Early Algebra!
ERIC Educational Resources Information Center
Stephens, Ana; Blanton, Maria; Knuth, Eric; Isler, Isil; Gardiner, Angela Murphy
2015-01-01
Mathematics educators have argued for some time that elementary school students are capable of engaging in algebraic thinking and should be provided with rich opportunities to do so. Recent initiatives like the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) have taken up this call by reiterating the place of early algebra in…
An Inquiry-Based Linear Algebra Class
ERIC Educational Resources Information Center
Wang, Haohao; Posey, Lisa
2011-01-01
Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…
Parabolas: Connection between Algebraic and Geometrical Representations
ERIC Educational Resources Information Center
Shriki, Atara
2011-01-01
A parabola is an interesting curve. What makes it interesting at the secondary school level is the fact that this curve is presented in both its contexts: algebraic and geometric. Being one of Apollonius' conic sections, the parabola is basically a geometric entity. It is, however, typically known for its algebraic characteristics, in particular…
Algebraic Geodesics on Three-Dimensional Quadrics
NASA Astrophysics Data System (ADS)
Kai, Yue
2015-12-01
By Hamilton-Jacobi method, we study the problem of algebraic geodesics on the third-order surface. By the implicit function theorem, we proved the existences of the real geodesics which are the intersections of two algebraic surfaces, and we also give some numerical examples.
ERIC Educational Resources Information Center
Rickard, Caroline
2008-01-01
Shortly after starting work for the University of Chichester in the School of Teacher Education, the author was planning a session relating to algebra and found herself inspired by an article in MT182: "Algebraic Infants" by Andrews and Sayers (2003). Based on the making of families of "Multilink" animals, Andrews and Sayers (2003) claim that…
Focus on Fractions to Scaffold Algebra
ERIC Educational Resources Information Center
Ooten, Cheryl Thomas
2013-01-01
Beginning algebra is a gatekeeper course into the pipeline to higher mathematics courses required for respected professions in engineering, science, statistics, mathematics, education, and technology. Beginning algebra can also be a perfect storm if the necessary foundational skills are not within a student's grasp. What skills ensure beginning…
Some Applications of Algebraic System Solving
ERIC Educational Resources Information Center
Roanes-Lozano, Eugenio
2011-01-01
Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact solve"…
A Technology-Intensive Approach to Algebra.
ERIC Educational Resources Information Center
Heid, M. Kathleen; Zbiek, Rose Mary
1995-01-01
Computer-Intensive Algebra (CIA) focuses on the use of technology to help develop a rich understanding of fundamental algebraic concepts in real-world settings using computing tools for easy access to numerical, graphical, and symbolic representations of mathematical ideas. (MKR)
Modern Algebra, Mathematics: 5293.36.
ERIC Educational Resources Information Center
Edwards, Raymond J.
This guidebook covers Boolean algebra, matrices, linear transformations of the plane, characteristic values, vectors, and algebraic structures. Overall course goals and performance objectives for each unit are specified; sequencing of units and various time schedules are suggested. A sample pretest and posttest are given, and an annotated list of…
Teaching Modeling and Axiomatization with Boolean Algebra.
ERIC Educational Resources Information Center
De Villiers, Michael D.
1987-01-01
Presented is an alternative approach to the traditional teaching of Boolean algebra for secondary school mathematics. The main aim of the approach is to use Boolean algebra to teach pupils such mathematical processes as modeling and axiomatization. A course using the approach is described. (RH)
Spacetime algebra as a powerful tool for electromagnetism
NASA Astrophysics Data System (ADS)
Dressel, Justin; Bliokh, Konstantin Y.; Nori, Franco
2015-08-01
We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field.
MODEL IDENTIFICATION AND COMPUTER ALGEBRA
Bollen, Kenneth A.; Bauldry, Shawn
2011-01-01
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods. PMID:21769158
MODEL IDENTIFICATION AND COMPUTER ALGEBRA.
Bollen, Kenneth A; Bauldry, Shawn
2010-10-01
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods.
Approximating smooth functions using algebraic-trigonometric polynomials
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
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.
Maass Forms and Quantum Modular Forms
NASA Astrophysics Data System (ADS)
Rolen, Larry
This thesis describes several new results in the theory of harmonic Maass forms and related objects. Maass forms have recently led to a flood of applications throughout number theory and combinatorics in recent years, especially following their development by the work of Bruinier and Funke the modern understanding Ramanujan's mock theta functions due to Zwegers. The first of three main theorems discussed in this thesis concerns the integrality properties of singular moduli. These are well-known to be algebraic integers, and they play a beautiful role in complex multiplication and explicit class field theory for imaginary quadratic fields. One can also study "singular moduli" for special non-holomorphic functions, which are algebraic but are not necessarily algebraic integers. Here we will explain the phenomenon of integrality properties and provide a sharp bound on denominators of symmetric functions in singular moduli. The second main theme of the thesis concerns Zagier's recent definition of a quantum modular form. Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous different topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions. Motivated by Zagier's example of the quantum modularity of Kontsevich's "strange" function F(q), we revisit work of Andrews, Jimenez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components. The final chapter of this thesis is devoted to a study of asymptotics of mock theta functions near roots of unity. In his famous deathbed letter, Ramanujan introduced the notion of a mock theta function, and he offered some alleged examples. The theory of mock theta functions has been brought to fruition using the framework of harmonic Maass forms, thanks to Zwegers. Despite this understanding, little attention has been given to Ramanujan's original definition. Here we prove that Ramanujan's examples do indeed satisfy his
Generalization of n-ary Nambu algebras and beyond
Ataguema, H.; Makhlouf, A.; Silvestrov, S.
2009-08-15
The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.
NASA Astrophysics Data System (ADS)
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.
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.
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.
NASA Astrophysics Data System (ADS)
Gainutdinov, A. M.; Read, N.; Saleur, H.
2016-01-01
We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed {gl(1|1)} spin-chain and its continuum limit—the {c=-2} symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245-288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289-329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL N , goes over in the continuum limit to a bigger algebra than {V}, the product of the left and right Virasoro algebras. This algebra, {S}—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field {S(z,bar{z})≡ S_{αβ} ψ^α(z)bar{ψ}^β(bar{z})}, with a symmetric form {S_{αβ}} and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL N in the {gl(1|1)} spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of {sp_{N-2}}. The semi-simple part of JTL N is represented by {U sp_{N-2}}, providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL N image represented in the spin-chain. On the continuum side, simple modules over {S} are identified with "fundamental" representations of {sp_∞}.
Working memory, worry, and algebraic ability.
Trezise, Kelly; Reeve, Robert A
2014-05-01
Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.
Weak homological dimensions and biflat Koethe algebras
Pirkovskii, A Yu
2008-06-30
The homological properties of metrizable Koethe algebras {lambda}(P) are studied. A criterion for an algebra A={lambda}(P) to be biflat in terms of the Koethe set P is obtained, which implies, in particular, that for such algebras the properties of being biprojective, biflat, and flat on the left are equivalent to the surjectivity of the multiplication operator A otimes-hat A{yields}A. The weak homological dimensions (the weak global dimension w.dg and the weak bidimension w.db) of biflat Koethe algebras are calculated. Namely, it is shown that the conditions w.db {lambda}(P)<=1 and w.dg {lambda}(P)<=1 are equivalent to the nuclearity of {lambda}(P); and if {lambda}(P) is non-nuclear, then w.dg {lambda}(P)=w.db {lambda}(P)=2. It is established that the nuclearity of a biflat Koethe algebra {lambda}(P), under certain additional conditions on the Koethe set P, implies the stronger estimate db {lambda}(P), where db is the (projective) bidimension. On the other hand, an example is constructed of a nuclear biflat Koethe algebra {lambda}(P) such that db {lambda}(P)=2 (while w.db {lambda}(P)=1). Finally, it is shown that many biflat Koethe algebras, while not being amenable, have trivial Hochschild homology groups in positive degrees (with arbitrary coefficients). Bibliography: 37 titles.
Algebraic curves of maximal cyclicity
NASA Astrophysics Data System (ADS)
Caubergh, Magdalena; Dumortier, Freddy
2006-01-01
The paper deals with analytic families of planar vector fields, studying methods to detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit cycles that can locally bifurcate from it. It is known that this multi-parameter problem can be reduced to a single-parameter one, in the sense that there exist analytic curves in parameter space along which the maximal cyclicity can be attained. In that case one speaks about a maximal cyclicity curve (mcc) in case only the number is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is also taken into account. In view of obtaining efficient algorithms for detecting the cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear depending on certain general properties of the families or of their associated Bautin ideal. In any case by well chosen examples we show that prudence is appropriate.
Local Algebras of Differential Operators
NASA Astrophysics Data System (ADS)
Church, P. T.; Timourian, J. G.
2002-05-01
There is an increasing literature devoted to the study of boundary value problems using singularity theory. The resulting differential operators are typically Fredholm with index 0, defined on infinite-dimensional spaces, and they have often led to folds, cusps, and even higher-order Morin singularities. In this paper we develop some of the local algebras of germs of such differential Fredholm operators, extending the theory of the finite-dimensional case. We apply this work to nonlinear elliptic boundary value problems: in particular, we make further progress on a question proposed and initially studied by Ruf [1999, J. Differential Equations 151, 111-133]. We also make comments on several problems raised by others.
PC Basic Linear Algebra Subroutines
1992-03-09
PC-BLAS is a highly optimized version of the Basic Linear Algebra Subprograms (BLAS), a standardized set of thirty-eight routines that perform low-level operations on vectors of numbers in single and double-precision real and complex arithmetic. Routines are included to find the index of the largest component of a vector, apply a Givens or modified Givens rotation, multiply a vector by a constant, determine the Euclidean length, perform a dot product, swap and copy vectors, andmore » find the norm of a vector. The BLAS have been carefully written to minimize numerical problems such as loss of precision and underflow and are designed so that the computation is independent of the interface with the calling program. This independence is achieved through judicious use of Assembly language macros. Interfaces are provided for Lahey Fortran 77, Microsoft Fortran 77, and Ryan-McFarland IBM Professional Fortran.« less
Jucys-Murphy elements for Birman-Murakami-Wenzl algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Ogievetsky, O. V.
2011-05-01
The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.
Kinematical superalgebras and Lie algebras of order 3
Campoamor-Stursberg, R.; Rausch de Traubenberg, M.
2008-06-15
We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonue-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order 3.
Imperfect Cloning Operations in Algebraic Quantum Theory
NASA Astrophysics Data System (ADS)
Kitajima, Yuichiro
2015-01-01
No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal -imperfect cloning operation which tolerates a finite loss of fidelity in the cloned state, and show that an individual system's algebra of observables is abelian if and only if there is a universal -imperfect cloning operation in the case where the loss of fidelity is less than . Therefore in this case no universal -imperfect cloning operation is possible in algebraic quantum theory.
The algebra of two dimensional generalized Chebyshev-Koornwinder oscillator
Borzov, V. V.; Damaskinsky, E. V.
2014-10-15
In the previous works of Borzov and Damaskinsky [“Chebyshev-Koornwinder oscillator,” Theor. Math. Phys. 175(3), 765–772 (2013)] and [“Ladder operators for Chebyshev-Koornwinder oscillator,” in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillator-like system that is associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev-Koornwinder oscillator. In this paper, we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev-Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space H of functions that are defined on a region which is bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev-Koornwinder polynomials and these polynomials form an orthonormalized basis in the space H. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.
Intermediate grouping on remotely sensed data using Gestalt algebra
NASA Astrophysics Data System (ADS)
Michaelsen, Eckart
2014-10-01
Human observers often achieve striking recognition performance on remotely sensed data unmatched by machine vision algorithms. This holds even for thermal images (IR) or synthetic aperture radar (SAR). Psychologists refer to these capabilities as Gestalt perceptive skills. Gestalt Algebra is a mathematical structure recently proposed for such laws of perceptual grouping. It gives operations for mirror symmetry, continuation in rows and rotational symmetric patterns. Each of these operations forms an aggregate-Gestalt of a tuple of part-Gestalten. Each Gestalt is attributed with a position, an orientation, a rotational frequency, a scale, and an assessment respectively. Any Gestalt can be combined with any other Gestalt using any of the three operations. Most often the assessment of the new aggregate-Gestalt will be close to zero. Only if the part-Gestalten perfectly fit into the desired pattern the new aggregate-Gestalt will be assessed with value one. The algebra is suitable in both directions: It may render an organized symmetric mandala using random numbers. Or it may recognize deep hidden visual relationships between meaningful parts of a picture. For the latter primitives must be obtained from the image by some key-point detector and a threshold. Intelligent search strategies are required for this search in the combinatorial space of possible Gestalt Algebra terms. Exemplarily, maximal assessed Gestalten found in selected aerial images as well as in IR and SAR images are presented.
A note on derivations of Murray–von Neumann algebras
Kadison, Richard V.; Liu, Zhe
2014-01-01
A Murray–von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray–von Neumann algebras. We show that the “extended derivations” of a Murray–von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray–von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer’s seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements. PMID:24469831
I CAN Learn[R] Pre-Algebra and Algebra. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2009
2009-01-01
The I CAN Learn[R] Education System is an interactive, self-paced, mastery-based software system that includes the I CAN Learn[R] Fundamentals of Math (5th-6th grade math) curriculum, the I CAN Learn[R] Pre-Algebra curriculum, and the I CAN Learn[R] Algebra curriculum. College algebra credit is also available to students in participating schools…
Finding the Axis of Revolution of an Algebraic Surface of Revolution.
Alcazar, Juan G; Goldman, Ron
2016-09-01
We present an algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of the highest order form, and using univariate resultants and gcds. We compare and contrast the advantages and disadvantages of each of these three techniques and we derive conditions under which each technique is most appropriate. In addition, we provide several necessary conditions for an implicit algebraic equation to represent a surface of revolution.
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.
Highest-weight representations of Brocherd`s algebras
Slansky, R.
1997-01-01
General features of highest-weight representations of Borcherd`s algebras are described. to show their typical features, several representations of Borcherd`s extensions of finite-dimensional algebras are analyzed. Then the example of the extension of affine- su(2) to a Borcherd`s algebra is examined. These algebras provide a natural way to extend a Kac-Moody algebra to include the hamiltonian and number-changing operators in a generalized symmetry structure.
On \\delta-derivations of n-ary algebras
NASA Astrophysics Data System (ADS)
Kaygorodov, Ivan B.
2012-12-01
We give a description of \\delta-derivations of (n+1)-dimensional n-ary Filippov algebras and, as a consequence, of simple finite-dimensional Filippov algebras over an algebraically closed field of characteristic zero. We also give new examples of non-trivial \\delta-derivations of Filippov algebras and show that there are no non-trivial \\delta-derivations of the simple ternary Mal'tsev algebra M_8.
Supersymmetric extension of Galilean conformal algebras
Bagchi, Arjun; Mandal, Ipsita
2009-10-15
The Galilean conformal algebra has recently been realized in the study of the nonrelativistic limit of the AdS/CFT conjecture. This was obtained by a systematic parametric group contraction of the parent relativistic conformal field theory. In this paper, we extend the analysis to include supersymmetry. We work at the level of the coordinates in superspace to construct the N=1 super-Galilean conformal algebra. One of the interesting outcomes of the analysis is that one is able to naturally extend the finite algebra to an infinite one. This looks structurally similar to the N=1 superconformal algebra in two dimensions, but is different. We also comment on the extension of our construction to cases of higher N.
Algebraic structures of sequences of numbers
NASA Astrophysics Data System (ADS)
Huang, I.-Chiau
2012-09-01
For certain sequences of numbers, commutative rings with a module structure over a non-commutative ring are constructed. Identities of these numbers are considered as realizations of algebraic relations.
Representations of filtered solvable Lie algebras
Panov, Alexander N
2012-01-31
The representation theory of filtered solvable Lie algebras is constructed. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found. Bibliography: 9 titles.
Structure of The Planar Galilean Conformal Algebra
NASA Astrophysics Data System (ADS)
Gao, Shoulan; Liu, Dong; Pei, Yufeng
2016-08-01
In this paper, we compute the low-dimensional cohomology groups of the planar Galilean conformal algebra introduced by Bagchi and Goparkumar. Consequently we determine its derivations, central extensions, and automorphisms.
Applications: Using Algebra in an Accounting Practice.
ERIC Educational Resources Information Center
Eisner, Gail A.
1994-01-01
Presents examples of algebra from the field of accounting including proportional ownership of stock, separation of a loan payment into principal and interest portions, depreciation methods, and salary withholdings computations. (MKR)
Geometric and algebraic properties of minimal bases of singular systems
NASA Astrophysics Data System (ADS)
Karcanias, Nicos
2013-11-01
For a general singular system ? with an associated pencil T(S), a complete classification of the right polynomial vector pairs ?, connected with the ? rational vector space, is given according to the proper-nonproper property, characterising the relationship of the degrees of those two vectors. An integral part of the classification of right pairs is the development of the notions of canonical and normal minimal bases for ? and ? rational vector spaces, where R(s) is the state restriction pencil of ?. It is shown that the notions of canonical and normal minimal bases are equivalent; the first notion characterises the pure algebraic aspect of the classification, whereas the second is intimately connected to the real geometry properties and the underlying generation mechanism of the proper and nonproper state vectors ?. The results describe the algebraic and geometric dimensions of the invariant partitioning of the set of reachability indices of singular systems. The classification of all proper and nonproper polynomial vectors ? induces a corresponding classification for the reachability spaces to proper-nonproper and results related to the possible dimensions feedback-spectra assignment properties of them are also given. The classification of minimal bases introduces new feedback invariants for singular systems, based on the real geometry of polynomial minimal bases, and provides an extension of the standard theory for proper systems (Warren, M.E., & Eckenberg, A.E. (1975).
Using computer algebra and SMT solvers in algebraic biology
NASA Astrophysics Data System (ADS)
Pineda Osorio, Mateo
2014-05-01
Biologic processes are represented as Boolean networks, in a discrete time. The dynamics within these networks are approached with the help of SMT Solvers and the use of computer algebra. Software such as Maple and Z3 was used in this case. The number of stationary states for each network was calculated. The network studied here corresponds to the immune system under the effects of drastic mood changes. Mood is considered as a Boolean variable that affects the entire dynamics of the immune system, changing the Boolean satisfiability and the number of stationary states of the immune network. Results obtained show Z3's great potential as a SMT Solver. Some of these results were verified in Maple, even though it showed not to be as suitable for the problem approach. The solving code was constructed using Z3-Python and Z3-SMT-LiB. Results obtained are important in biology systems and are expected to help in the design of immune therapies. As a future line of research, more complex Boolean network representations of the immune system as well as the whole psychological apparatus are suggested.
Numerical linear algebra in data mining
NASA Astrophysics Data System (ADS)
Eldén, Lars
Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.
Vague Congruences and Quotient Lattice Implication Algebras
Qin, Xiaoyan; Xu, Yang
2014-01-01
The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given. PMID:25133207
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.
From Atiyah Classes to Homotopy Leibniz Algebras
NASA Astrophysics Data System (ADS)
Chen, Zhuo; Stiénon, Mathieu; Xu, Ping
2016-01-01
A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes T X [-1] into a Lie algebra object in D + ( X), the bounded below derived category of coherent sheaves on X. Furthermore, Kapranov proved that, for a Kähler manifold X, the Dolbeault resolution {Ω^{bullet-1}(T_X^{1, 0})} of T X [-1] is an L ∞ algebra. In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair ( L, A), i.e. a Lie algebroid L together with a Lie subalgebroid A, we define the Atiyah class α E of an A-module E as the obstruction to the existence of an A- compatible L-connection on E. We prove that the Atiyah classes α L/ A and α E respectively make L/ A[-1] and E[-1] into a Lie algebra and a Lie algebra module in the bounded below derived category {D^+(A)} , where {A} is the abelian category of left {U(A)} -modules and {U(A)} is the universal enveloping algebra of A. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of L/ A and E, and inducing the aforesaid Lie structures in {D^+(A)}.
An algebra of dimerization and its implications for G-protein coupled receptor signaling.
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.
A Z{sub 3} generalization of Pauli's principle, quark algebra and the Lorentz invariance
Kerner, Richard
2012-09-24
The fundamental difference between bosons and fermions is that they obey two alternative representations of the Z{sub 2} group, resulting in symmetric or anti-symmetric binary commutation relations. Our aim is to explore possibilities offered by ternary Z{sub 3} generalization commutation relations. This leads to cubic and ternary algebras which are a direct generalization of usual commutation relations, with Z{sub 3}-grading replacing the usual Z{sub 2}-grading. Properties and structure of such algebras are discussed, with special interest in a low-dimensional one, with two generators. Invariant cubic forms on such algebras are introduced, and it is shown how the SL(2,C) group arises naturally as the symmetry group preserving these forms. In the case of lowest dimension, with only two generators, it is shown how the cubic combinations of elements of the same Z{sub 3} grade behave like Lorentz spinors, while binary products of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors. The wave equation generalizing the Dirac operator to the Z{sub 3}-graded case is introduced, whose diagonalization leads to a third-order equation. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry obeyed by quark states.
Vortex lattice theory: A linear algebra approach
NASA Astrophysics Data System (ADS)
Chamoun, George C.
Vortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics-based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we require nullity( A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A 'Brownian ratchet' construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value of A. Distances between configurations and equilibria were measured using the Frobenius norm ||·||F and 2-norm ||·||2, and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves.
Synthetic normal-mode spectra: a full-coupling perspective
NASA Astrophysics Data System (ADS)
Yang, H. Y.; Tromp, J.
2014-12-01
Normal-mode spectra may be used to investigate the large-scale anelastic structure of the entire earth. The relevant theory was developed a few decades ago, however, mainly due to computational limitations, several approximations are commonly employed, and thus far the full merits of the complete theory have not been taken advantage of. In this study, we present an exact algebraic form of the theory for an aspherical, anelastic and rotating earth model in which either complex or real spherical harmonic bases are used. Physical dispersion is incorporated into the quadratic eigenvalue problem by expanding the logarithmic frequency term to 2nd order. In addition, we carry out numerical experiments up to 3 mHz to quantitatively evaluate the accuracy of commonly used approximate mode synthetics. We find that (1) approximating mode frequencies for realistic earth models with an average over degenerate frequencies of two coupled modes for physical dispersion, Coriolis effects and perturbed kinematic energy terms gives rise to subtle differences in mode spectra; (2) taking into account the exact normalization of modes instead of the one for a spherical, non-rotation model improves mode spectra by ~2%; (3) consideration of mode coupling in a narrow frequency band yields up to 10% discrepancies in mode spectra compared with wide-band coupling, indicating that the popular splitting function approach may introduce slight biases in normal-mode tomography.
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
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.
Construction of N = 2 superconformal algebra from affine algebras with extended symmetry: I
NASA Astrophysics Data System (ADS)
Cheng, Shun-Jen
1995-01-01
The purpose of this Letter is to use the idea of the Sugawara-Kač-Todorov construction of the N = 0 and N = 1 superconformal algebras to construct a very simple free-field realization of the N = 2 superconformal algebra.
ERIC Educational Resources Information Center
Okpube, Nnaemeka Michael; Anugwo, M. N.
2016-01-01
This study investigated the Card Games and Algebra tic-Tacmatics on Junior Secondary II Students' Achievement in Algebraic Expressions. Three research questions and three null hypotheses guided the study. The study adopted the pre-test, post-test control group design. A total of two hundred and forty (240) Junior Secondary School II students were…
ERIC Educational Resources Information Center
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…
Slower Algebra Students Meet Faster Tools: Solving Algebra Word Problems with Graphing Software
ERIC Educational Resources Information Center
Yerushalmy, Michal
2006-01-01
The article discusses the ways that less successful mathematics students used graphing software with capabilities similar to a basic graphing calculator to solve algebra problems in context. The study is based on interviewing students who learned algebra for 3 years in an environment where software tools were always present. We found differences…
Generalization of Patterns: The Tension between Algebraic Thinking and Algebraic Notation.
ERIC Educational Resources Information Center
Zazkis, Rina; Liljedahl, Peter
2002-01-01
Explores the attempts of a group of preservice elementary school teachers to generalize a repeating visual number pattern. Discusses students' emergent algebraic thinking. Indicates that students' ability to express generalities verbally was not accompanied by algebraic notation, but participants often perceived complete and accurate solutions…
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…
Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*
Bank, R.; Falgout, R. D.; Jones, T.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. W.
2015-10-29
In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods inmore » Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.« less
Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*
Bank, R.; Falgout, R. D.; Jones, T.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. W.
2015-10-29
In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.
Conn, Vicki S; Zerwic, Julie; Jefferson, Urmeka; Anderson, Cindy M; Killion, Cheryl M; Smith, Carol E; Cohen, Marlene Z; Fahrenwald, Nancy L; Herrick, Linda; Topp, Robert; Benefield, Lazelle E; Loya, Julio
2016-02-01
Getting turned down for grant funding or having a manuscript rejected is an uncomfortable but not unusual occurrence during the course of a nurse researcher's professional life. Rejection can evoke an emotional response akin to the grieving process that can slow or even undermine productivity. Only by "normalizing" rejection, that is, by accepting it as an integral part of the scientific process, can researchers more quickly overcome negative emotions and instead use rejection to refine and advance their scientific programs. This article provides practical advice for coming to emotional terms with rejection and delineates methods for working constructively to address reviewer comments. PMID:26041785
Classification of central extensions of Lax operator algebras
Schlichenmaier, Martin
2008-11-18
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
Capelli bitableaux and Z-forms of general linear Lie superalgebras.
Brini, A; Teolis, A G
1990-01-01
The combinatorics of the enveloping algebra UQ(pl(L)) of the general linear Lie superalgebra of a finite dimensional Z2-graded Q-vector space is studied. Three non-equivalent Z-forms of UQ(pl(L)) are introduced: one of these Z-forms is a version of the Kostant Z-form and the others are Lie algebra analogs of Rota and Stein's straightening formulae for the supersymmetric algebra Super[L P] and for its dual Super[L* P*]. The method is based on an extension of Capelli's technique of variabili ausiliarie to algebras containing positively and negatively signed elements. PMID:11607048
TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION
NASA Technical Reports Server (NTRS)
Smith, R. E.
1994-01-01
TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.
NASA Astrophysics Data System (ADS)
Shcheglova, A. A.
2009-09-01
Linear control differential algebraic equations are considered. The issue of minimum dimension of the control vector necessitated for complete controllability of the system on any closed interval from the domain of definition is investigated. The problem is analyzed in connection with the time invariant systems having regular matrix pencils and also systems with real-analytic or smooth coefficients, which possess some structural forms.
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…
Short Round Sub-Linear Zero-Knowledge Argument for Linear Algebraic Relations
NASA Astrophysics Data System (ADS)
Seo, Jae Hong
Zero-knowledge arguments allows one party to prove that a statement is true, without leaking any other information than the truth of the statement. In many applications such as verifiable shuffle (as a practical application) and circuit satisfiability (as a theoretical application), zero-knowledge arguments for mathematical statements related to linear algebra are essentially used. Groth proposed (at CRYPTO 2009) an elegant methodology for zero-knowledge arguments for linear algebraic relations over finite fields. He obtained zero-knowledge arguments of the sub-linear size for linear algebra using reductions from linear algebraic relations to equations of the form z = x *' y, where x, y ∈ Fnp are committed vectors, z ∈ Fp is a committed element, and *' : Fnp × Fnp → Fp is a bilinear map. These reductions impose additional rounds on zero-knowledge arguments of the sub-linear size. The round complexity of interactive zero-knowledge arguments is an important measure along with communication and computational complexities. We focus on minimizing the round complexity of sub-linear zero-knowledge arguments for linear algebra. To reduce round complexity, we propose a general transformation from a t-round zero-knowledge argument, satisfying mild conditions, to a (t - 2)-round zero-knowledge argument; this transformation is of independent interest.
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.
Girard, Nadine; Koob, Meriam; Brunel, Herv
2016-01-01
Numerous events are involved in brain development, some of which are detected by neuroimaging. Major changes in brain morphology are depicted by brain imaging during the fetal period while changes in brain composition can be demonstrated in both pre- and postnatal periods. Although ultrasonography and computed tomography can show changes in brain morphology, these techniques are insensitive to myelination that is one of the most important events occurring during brain maturation. Magnetic resonance imaging (MRI) is therefore the method of choice to evaluate brain maturation. MRI also gives insight into the microstructure of brain tissue through diffusion-weighted imaging and diffusion tensor imaging. Metabolic changes are also part of brain maturation and are assessed by proton magnetic resonance spectroscopy. Understanding and knowledge of the different steps in brain development are required to be able to detect morphologic and structural changes on neuroimaging. Consequently alterations in normal development can be depicted. PMID:27430460
Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers
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.
Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers
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.
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
NASA Astrophysics Data System (ADS)
Mason, Geoffrey; Tuite, Michael P.
2010-12-01
We define the partition and n-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector n-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.
The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces
NASA Astrophysics Data System (ADS)
Alekseevskii, A. V.; Natanzon, S. M.
2008-08-01
We extend the definition of Hurwitz numbers to the case of seamed surfaces, which arise in new models of mathematical physics, and prove that they form a system of correlators for a Klein topological field theory in the sense defined in [1]. We find the corresponding Cardy-Frobenius algebras, which yield a method for calculating the Hurwitz numbers. As a by-product, we prove that the vector space generated by the bipartite graphs with n edges possesses a natural binary operation that makes this space into a non-commutative Frobenius algebra isomorphic to the algebra of intertwining operators for a representation of the symmetric group S_n on the space generated by the set of all partitions of a set of n elements.
Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces
NASA Astrophysics Data System (ADS)
Polakovič, Marcel; Riečanová, Zdenka
2011-04-01
Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.
Schwinger's Measurement Algebra, Preons and the Lepton Masses
NASA Astrophysics Data System (ADS)
Brannen, Carl
2006-04-01
In the 1950s and 1960s, Julian Schwinger developed an elegant general scheme for quantum kinematics and dynamics appropriate to systems with a finite number of dynamical variables, now knowns as ``Schwinger's Measurement Algebra'' (SMA). The SMA has seen little use, largely because it is non relativistic in that it does not allow for particle creation. In this paper, we apply the SMA to the problem of modeling tightly bound subparticles (preons) of the leptons and quarks. We discuss the structure of the ideals of Clifford algebras and, applying this to the elementary fermions, derive a preon substructure for the quarks and leptons. We show that matrices of SMA type elements can be used to model the quarks and leptons under the assumption that the preons are of such high energy that they cannot be created in normal interactions. This gives a definition of the SMA for the composite particle in terms of the SMA of its constituents. We solve the resulting matrix equation for the quarks and leptons. We show that the mass operator for the charged leptons is related to the democratic mass matrix used in the Koide mass formula.
1996-01-01
Cells of the mucosal mast cell line, RBL-2H3, are normally stimulated to degranulate after aggregation of high affinity receptors for IgE (Fc epsilon RI) by soluble cross-linking ligands. This cellular degranulation process requires sustained elevation of cytoplasmic Ca2+. In this study, we investigated the response of RBL-2H3 cells to 6- micron beads coated with IgE-specific ligands. These ligand-coated beads cause only small, transient Ca2+ responses, even though the same ligands added in soluble form cause larger, more sustained Ca2+ responses. The ligand-coated 6-micron beads also fail to stimulate significant degranulation of RBL-2H3 cells, whereas much larger ligand- coated Sepharose beads stimulate ample degranulation. Confocal fluorescence microscopy shows that the 6-micron beads (but not the Sepharose beads) are phagocytosed by RBL-2H3 cells and that, beginning with the initial stages of bead engulfment, there is exclusion of many plasma membrane components from the 6-micron bead/cell interface, including p53/56lyn and several other markers for detergent-resistant membrane domains, as well as an integrin and unliganded IgE-Fc epsilon RI. The fluorescent lipid probe DiIC16 is a marker for the membrane domains that is excluded from the cell/bead interface, whereas a structural analogue, fast DiI, which differs from DiIC16 by the presence of unsaturated acyl chains, is not substantially excluded from the interface. None of these components are excluded from the interface of RBL-2H3 cells and the large Sepharose beads. Additional confocal microscopy analysis indicates that microfilaments are involved in the exclusion of plasma membrane components from the cell/bead interface. These results suggest that initiation of phagocytosis diverts normal signaling pathways in a cytoskeleton-driven membrane clearance process that alters the physiological response of the cells. PMID:8830772
On Fusion Algebras and Modular Matrices
NASA Astrophysics Data System (ADS)
Gannon, T.; Walton, M. A.
We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal field theories, affine Kac-Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix S, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the Ar fusion algebra at level k. We prove that for many choices of rank r and level k, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most r and k. We also find new, systematic sources of zeros in the modular matrix S. In addition, we obtain a formula relating the entries of S at fixed points, to entries of S at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of S, and by the fusion (Verlinde) eigenvalues.
Optical systolic solutions of linear algebraic equations
NASA Technical Reports Server (NTRS)
Neuman, C. P.; Casasent, D.
1984-01-01
The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.
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.
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.
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.
Remedial Math: Its Effect on the Final Grade in Algebra.
ERIC Educational Resources Information Center
Head, L. Quinn; Lindsey, Jimmy D.
1984-01-01
The effectiveness of one remedial mathematics technique is examined. Results indicated that students who passed remedial math and then took college algebra had significantly higher final algebra grades than did undergraduates who failed remedial math. (MLW)
Geometric Algebra Software for Teaching Complex Numbers, Vectors and Spinors.
ERIC Educational Resources Information Center
Lounesto, Pertti; And Others
1990-01-01
Presents a calculator-type computer program, CLICAL, in conjunction with complex number, vector, and other geometric algebra computations. Compares the CLICAL with other symbolic programs for algebra. (Author/YP)
Rota-Baxter operators on Witt and Virasoro algebras
NASA Astrophysics Data System (ADS)
Gao, Xu; Liu, Ming; Bai, Chengming; Jing, Naihuan
2016-10-01
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.
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.
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.
Nijenhuis Operators on n-Lie Algebras
NASA Astrophysics Data System (ADS)
Liu, Jie-Feng; Sheng, Yun-He; Zhou, Yan-Qiu; Bai, Cheng-Ming
2016-06-01
In this paper, we study (n - 1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples. Supported by National Natural Science Foundation of China under Grant Nos. 11471139, 11271202, 11221091, 11425104, Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20120031110022, and National Natural Science Foundation of Jilin Province under Grant No. 20140520054JH
Bohr model as an algebraic collective model
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.
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.
Nijenhuis Operators on n-Lie Algebras
NASA Astrophysics Data System (ADS)
Liu, Jie-Feng; Sheng, Yun-He; Zhou, Yan-Qiu; Bai, Cheng-Ming
2016-06-01
In this paper, we study (n ‑ 1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples. Supported by National Natural Science Foundation of China under Grant Nos. 11471139, 11271202, 11221091, 11425104, Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20120031110022, and National Natural Science Foundation of Jilin Province under Grant No. 20140520054JH
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…
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,…
The Algebra Initiative Colloquium. Volume 2: Working Group Papers.
ERIC Educational Resources Information Center
Lacampagne, Carole B., Ed.; And Others
This volume presents recommendations from four working groups at a conference on reform in algebra held in Leesburg, Virginia, December 9-12, 1993. Working Group 1: Creating an Appropriate Algebra Experience for All Grades K-12 Students produced the following papers: (1) "Report" (A. H. Schoenfeld); (2) "Five Questions About Algebra Reform (and a…
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…
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.
Supersymmetry algebra cohomology. I. Definition and general structure
Brandt, Friedemann
2010-12-15
This paper concerns standard supersymmetry algebras in diverse dimensions, involving bosonic translational generators and fermionic supersymmetry generators. A cohomology related to these supersymmetry algebras, termed supersymmetry algebra cohomology, and corresponding 'primitive elements' are defined by means of a BRST (Becchi-Rouet-Stora-Tyutin)-type coboundary operator. A method to systematically compute this cohomology is outlined and illustrated by simple examples.
Supersymmetry algebra cohomology. I. Definition and general structure
NASA Astrophysics Data System (ADS)
Brandt, Friedemann
2010-12-01
This paper concerns standard supersymmetry algebras in diverse dimensions, involving bosonic translational generators and fermionic supersymmetry generators. A cohomology related to these supersymmetry algebras, termed supersymmetry algebra cohomology, and corresponding "primitive elements" are defined by means of a BRST (Becchi-Rouet-Stora-Tyutin)-type coboundary operator. A method to systematically compute this cohomology is outlined and illustrated by simple examples.
Placement Tools for Developmental Mathematics and Intermediate Algebra
ERIC Educational Resources Information Center
Donovan, William J.; Wheland, Ethel R.
2008-01-01
This paper investigates the placement of students at an urban Ohio college campus in developmental mathematics and Intermediate Algebra courses. We have found that the ACT Mathematics and COMPASS Domain I (Algebra) Placement scores both correlate well with success in the Intermediate Algebra course and that, although females have lower placement…
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…
The Algebra Initiative Colloquium. Volume 1: Plenary and Reactor Papers.
ERIC Educational Resources Information Center
Lacampagne, Carole B., Ed.; And Others
This volume contains the plenary or reactor papers presented at a conference on reform in algebra held in Leesburg, Virginia, December 9-12, 1993. Papers included are: (1) "Introduction" (C. B. Lacampagne); (2) "Summary" (C. B. Lacampagne); (3) "Recommendations" (C. B. Lacampagne); (4) "The Development of Algebra and Algebra Education" (V. J.…
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…
Assessing Mathematics Automatically Using Computer Algebra and the Internet
ERIC Educational Resources Information Center
Sangwin, Chris
2004-01-01
This paper reports some recent developments in mathematical computer-aided assessment which employs computer algebra to evaluate students' work using the Internet. Technical and educational issues raised by this use of computer algebra are addressed. Working examples from core calculus and algebra which have been used with first year university…
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…
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…
ERIC Educational Resources Information Center
Falcon, Raymond
2009-01-01
Teachers use action research in order to improve their teaching and student learning. This action research will analyze students' algebraic reasoning in finding values of variables in systems of equations pictorially and algebraically. This research will look at students solving linear systems of equations without knowing the algebraic algorithms.…
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"…
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…
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.
Noise limitations in optical linear algebra processors.
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.
A Microcomputer Lab for Algebra & Calculus.
ERIC Educational Resources Information Center
Avery, Chris; And Others
An overview is provided of De Anza College's use of computerized instruction in its mathematics courses. After reviewing the ways in which computer technology is changing math instruction, the paper looks at the use of computers in several course sequences. The instructional model for the algebra sequence is based on a large group format of…
Representable states on quasilocal quasi *-algebras
Bagarello, F.; Trapani, C.; Triolo, S.
2011-01-15
Continuing a previous analysis originally motivated by physics, we consider representable states on quasilocal quasi *-algebras, starting with examining the possibility for a compatible family of local states to give rise to a global state. Some properties of local modifications of representable states and some aspects of their asymptotic behavior are also considered.
Applications of Maple To Algebraic Cryptography.
ERIC Educational Resources Information Center
Sigmon, Neil P.
1997-01-01
Demonstrates the use of technology to enhance the appreciation of applications involving abstract algebra. The symbolic manipulator Maple can perform computations required for a linear cryptosystem. One major benefit of this process is that students can encipher and decipher messages using a linear cryptosystem without becoming confused and…
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…
Remedial Math and College Algebra Grades.
ERIC Educational Resources Information Center
Head, L. Quinn
This investigation tried to determine if a statistically significant relationship exists between different sequences of enrollment in remedial mathematics and grades obtained in college algebra classes at Jacksonville State University. Groups consisting of five different enrollment sequences in mathematics were studied. The data collected supports…
Hypercontractivity in finite-dimensional matrix algebras
Junge, Marius; Palazuelos, Carlos
2015-02-15
We obtain hypercontractivity estimates for a large class of semigroups defined on finite-dimensional matrix algebras M{sub n}. These semigroups arise from Poisson-like length functions ψ on ℤ{sub n} × ℤ{sub n} and provide new hypercontractive families of quantum channels when ψ is conditionally negative. We also study the optimality of our estimates.
A Linear Algebra Measure of Cluster Quality.
ERIC Educational Resources Information Center
Mather, Laura A.
2000-01-01
Discussion of models for information retrieval focuses on an application of linear algebra to text clustering, namely, a metric for measuring cluster quality based on the theory that cluster quality is proportional to the number of terms that are disjoint across the clusters. Explains term-document matrices and clustering algorithms. (Author/LRW)
The geometric semantics of algebraic quantum mechanics.
Cruz Morales, John Alexander; Zilber, Boris
2015-08-01
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.
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…
Proof in Algebra: Reasoning beyond Examples
ERIC Educational Resources Information Center
Otten, Samuel; Herbel-Eisenmann, Beth A.; Males, Lorraine M.
2010-01-01
The purpose of this article is to provide an image of what proof could look like in beginning algebra, a course that nearly every secondary school student encounters. The authors present an actual classroom vignette in which a rich opportunity for student reasoning arose. After analyzing the proof schemes at play, the authors provide a…
Window of Opportunity? Adolescence, Music, and Algebra
ERIC Educational Resources Information Center
Helmrich, Barbara H.
2010-01-01
Research has suggested that musicians process music in the same cortical regions that adolescents process algebra. An early adolescence synaptogenesis might present a window of opportunity during middle school for music to create and strengthen enduring neural connections in those regions. Six school districts across Maryland provided scores from…
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.
Expansion of real numbers by algebraic numbers
NASA Astrophysics Data System (ADS)
Hajime, Kaneko
2008-01-01
In this paper we represent the fractional part of ξαn, where ξ is a nonzero real number and α is an algebraic number. By using this representation, we give new lower bounds for the distance from ξαn to the nearest integer.
Mathematics: Algebra and Geometry. GED Scoreboost.
ERIC Educational Resources Information Center
Hoyt, Cathy
GED "Scoreboost" materials target exactly the skills one needs to pass the General Educational Development (GED) tests. This book focuses on the GED Mathematics test. To prepare for the test, the test taker needs to learn skills in number and operation sense, data and statistics, geometry and measurement, and algebra. To pass the test, the test…
Invariant algebraic surfaces for a virus dynamics
NASA Astrophysics Data System (ADS)
Valls, Claudia
2015-08-01
In this paper, we provide a complete classification of the invariant algebraic surfaces and of the rational first integrals for a well-known virus system. In the proofs, we use the weight-homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.
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.)
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…
Connecting Functions in Geometry and Algebra
ERIC Educational Resources Information Center
Steketee, Scott; Scher, Daniel
2016-01-01
One goal of a mathematics education is that students make significant connections among different branches of mathematics. Connections--such as those between arithmetic and algebra, between two-dimensional and three-dimensional geometry, between compass-and-straight-edge constructions and transformations, and between calculus and analytic…
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.
Modern Geometric Algebra: A (Very Incomplete!) Survey
ERIC Educational Resources Information Center
Suzuki, Jeff
2009-01-01
Geometric algebra is based on two simple ideas. First, the area of a rectangle is equal to the product of the lengths of its sides. Second, if a figure is broken apart into several pieces, the sum of the areas of the pieces equals the area of the original figure. Remarkably, these two ideas provide an elegant way to introduce, connect, and…
Euler and the Fundamental Theorem of Algebra.
ERIC Educational Resources Information Center
Duham, William
1991-01-01
The complexity of the proof of the Fundamental Theorem of Algebra makes it inaccessible to lower level students. Described are more understandable attempts of proving the theorem and a historical account of Euler's efforts that relates the progression of the mathematical process used and indicates some of the pitfalls encountered. (MDH)
Algebra II. Mathematics Curriculum Guide (Career Oriented).
ERIC Educational Resources Information Center
Ohmer, Merlin M.; And Others
The curriculum guide for Albegra 2 correlates algebraic concepts with career-oriented concepts and activities. The curriculum outline format gives the concepts to be taught, matched with related career-oriented performance objectives, concepts, and suggested instructional activities in facing page layouts. The suggested curriculum outline is…
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.…
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…
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.…
Parallel Algebraic Multigrids for Structural mechanics
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.
Some Unexpected Results Using Computer Algebra Systems.
ERIC Educational Resources Information Center
Alonso, Felix; Garcia, Alfonsa; Garcia, Francisco; Hoya, Sara; Rodriguez, Gerardo; de la Villa, Agustin
2001-01-01
Shows how teachers can often use unexpected outputs from Computer Algebra Systems (CAS) to reinforce concepts and to show students the importance of thinking about how they use the software and reflecting on their results. Presents different examples where DERIVE, MAPLE, or Mathematica does not work as expected and suggests how to use them as a…
Digital Maps, Matrices and Computer Algebra
ERIC Educational Resources Information Center
Knight, D. G.
2005-01-01
The way in which computer algebra systems, such as Maple, have made the study of complex problems accessible to undergraduate mathematicians with modest computational skills is illustrated by some large matrix calculations, which arise from representing the Earth's surface by digital elevation models. Such problems are often considered to lie in…
A Visual Approach to Algebra Concepts.
ERIC Educational Resources Information Center
Morelli, Lynn
1992-01-01
Presents activities to visually explore the algebraic concepts of variable, constant, the distributive property, and combining like terms. Presents four transparencies that use visual models to understand exercises in students perform the same mental calculations on a number of their choice and obtain the same result. (MDH)
The Krichever map, vector bundles over algebraic curves, and Heisenberg algebras
NASA Astrophysics Data System (ADS)
Adams, M. R.; Bergvelt, M. J.
1993-06-01
We study the Grassmannian Gr {/x n } consisting of equivalence classes of rank n algebraic vector bundles over a Riemann surface X with an holomorphic trivialization at a fixed point p. Commutative subalgebras of gl(n, H λ), H λ being the ring of functions holomorphic on a punctured disc about p, define flows on the Grassmannian, giving rise to classes of solutions to multi-component KP hierarchies. These commutative subalgebras correspond to Heisenberg algebras in the Kac-Moody algebra associated to gl(n, H λ). One can obtain, by the Krichever map, points of Gr {/x n } (and solutions of mcKP) from coverings f: Y→X and other geometric data. Conversely for every point of Gr {/x n } and for every choice of Heisenberg algebra we construct, using the cotangent bundle of Gr {/x n }, an algebraic curve covering X and other data, thus inverting the Krichever map. We show the explicit relation between the choice of Heisenberg algebra and the geometry of the covering space.
A note on probabilistic models over strings: the linear algebra approach.
Bouchard-Côté, Alexandre
2013-12-01
Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the "TKF91" model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems. PMID:24135792
A note on probabilistic models over strings: the linear algebra approach.
Bouchard-Côté, Alexandre
2013-12-01
Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the "TKF91" model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems.
Relation of deformed nonlinear algebras with linear ones
NASA Astrophysics Data System (ADS)
Nowicki, A.; Tkachuk, V. M.
2014-01-01
The relation between nonlinear algebras and linear ones is established. For a one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to a linear one with three operators. We also establish the relation between the Lie algebra of total angular momentum and corresponding nonlinear one. This relation gives a possibility to simplify and to solve the eigenvalue problem for the Hamiltonian in a nonlinear case using the reduction of this problem to the case of linear algebra. It is demonstrated in an example of a harmonic oscillator.
Differential geometry on Hopf algebras and quantum groups
Watts, P.
1994-12-15
The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined.
Extending Fourier transformations to Hamilton's quaternions and Clifford's geometric algebras
NASA Astrophysics Data System (ADS)
Hitzer, Eckhard
2013-10-01
We show how Fourier transformations can be extended to Hamilton's algebra of quaternions. This was initially motivated by applications in nuclear magnetic resonance and electric engineering. Followed by an ever wider range of applications in color image and signal processing. Hamilton's algebra of quaternions is only one example of the larger class of Clifford's geometric algebras, complete algebras encoding a vector space and all its subspace elements. We introduce how Fourier transformations are extended to Clifford algebras and applied in electromagnetism, and in the processing of images, color images, vector field and climate data.