PERTURB: A program for calculating vibrational energies by generalized algebraic quantization
NASA Astrophysics Data System (ADS)
Fried, Laurence E.; Ezra, Gregory S.
1988-09-01
We describe PERTURB, a special purpose algebraic manipulation program which calculates vibrational eigenvalues in coupled oscillator systems. PERTURB implements the method of generalized algebraic quantization (AQ), in which Van Vleck perturbation theory is formulated in a mock phase space. The phase space formulation enables quantum and classical perturbation theory to be treated on the same footing, and allows the systematic calculation of corrections to classical perturbation results in powers of h̷. Generalized AQ is a powerful and efficient technique for calculating semiclassical vibrational energy levels. In many cases, including just the first correction to classical perturbation theory yields highly accurate energies.
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.
Twisting algebraically special solutions in five dimensions
NASA Astrophysics Data System (ADS)
Bernardi de Freitas, Gabriel; Godazgar, Mahdi; Reall, Harvey S.
2016-05-01
We determine the general form of the solutions of the five-dimensional vacuum Einstein equations with cosmological constant for which (i) the Weyl tensor is everywhere type II or more special in the null alignment classification of Coley et al, and (ii) the 3 × 3 matrix encoding the expansion, shear and twist of the aligned null direction has rank 2. The dependence of the solution on two coordinates is determined explicitly, so the Einstein equation reduces to PDEs in the three remaining coordinates, just as for four-dimensional (4d) algebraically special solutions. The solutions fall into several families. One of these consists of warped products of 4d algebraically special solutions. The others are new.
Translating cosmological special relativity into geometric algebra
NASA Astrophysics Data System (ADS)
Horn, Martin Erik
2012-11-01
Geometric algebra and Clifford algebra are important tools to describe and analyze the physics of the world we live in. Although there is enormous empirical evidence that we are living in four dimensional spacetime, mathematical worlds of higher dimensions can be used to present the physical laws of our world in an aesthetical and didactical more appealing way. In physics and mathematics education we are therefore confronted with the question how these high dimensional spaces should be taught. But as an immediate confrontation of students with high dimensional compactified spacetimes would expect too much from them at the beginning of their university studies, it seems reasonable to approach the mathematics and physics of higher dimensions step by step. The first step naturally is the step from four dimensional spacetime of special relativity to a five dimensional spacetime world. As a toy model for this artificial world cosmological special relativity, invented by Moshe Carmeli, can be used. This five dimensional non-compactified approach describes a spacetime which consists not only of one time dimension and three space dimensions. In addition velocity is regarded as a fifth dimension. This model very probably will not represent physics correctly. But it can be used to discuss and analyze the consequences of an additional dimension in a clear and simple way. Unfortunately Carmeli has formulated cosmological special relativity in standard vector notation. Therefore a translation of cosmological special relativity into the mathematical language of Grassmann and Clifford (Geometric algebra) is given and the physics of cosmological special relativity is discussed.
Regular perturbation theory of relativistic corrections: II. Algebraic approximation
NASA Astrophysics Data System (ADS)
Rutkowski, A.; Kozłowski, R.; Rutkowska, D.
2001-01-01
A four-component equivalent of the Schrödinger equation, describing both the nonrelativistic electron and the nonrelativistic positron, is introduced. The difference between this equation and the Dirac equation is treated as a perturbation. The relevant perturbation equations and formulas for corrections to the energy are derived. Owing to the semibounded character of the Schrödinger Hamiltonian of the unperturbed equation the variational perturbation method is formulated. The Hylleraas functionals become then either upper or lower bounds to the respective exact corrections to the energy. In order to demonstrate the usefulness of this approach to the problem of the variational optimization of nonlinear parameters, the perturbation corrections to wave functions for the of hydrogenlike atoms have been approximated in terms of exponential basis functions. The Dirac equation in this algebraic approximation is solved iteratively starting with the solution of the Schrödinger equation.
Calculation of exchange energies using algebraic perturbation theory
Burrows, B. L.; Dalgarno, A.; Cohen, M.
2010-04-15
An algebraic perturbation theory is presented for efficient calculations of localized states and hence of exchange energies, which are the differences between low-lying states of the valence electron of a molecule, formed by the collision of an ion Y{sup +} with an atom X. For the case of a homonuclear molecule these are the gerade and ungerade states and the exchange energy is an exponentially decreasing function of the internuclear distance. For such homonuclear systems the theory is used in conjunction with the Herring-Holstein technique to give accurate exchange energies for a range of intermolecular separations R. Since the perturbation parameter is essentially 1/R, this method is suitable for large R. In particular, exchange energies are calculated for X{sub 2}{sup +} systems, where X is H, Li, Na, K, Rb, or Cs.
Investigating a hybrid perturbation-Galerkin technique using computer algebra
NASA Technical Reports Server (NTRS)
Andersen, Carl M.; Geer, James F.
1988-01-01
A two-step hybrid perturbation-Galerkin method is presented for the solution of a variety of differential equations type problems which involve a scalar parameter. The resulting (approximate) solution has the form of a sum where each term consists of the product of two functions. The first function is a function of the independent field variable(s) x, and the second is a function of the parameter lambda. In step one the functions of x are determined by forming a perturbation expansion in lambda. In step two the functions of lambda are determined through the use of the classical Bubnov-Gelerkin method. The resulting hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Bubnov-Galerkin methods applied separately, while combining some of the good features of each. In particular, the results can be useful well beyond the radius of convergence associated with the perturbation expansion. The hybrid method is applied with the aid of computer algebra to a simple two-point boundary value problem where the radius of convergence is finite and to a quantum eigenvalue problem where the radius of convergence is zero. For both problems the hybrid method apparently converges for an infinite range of the parameter lambda. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.
Algebraic special functions and SO(3,2)
Celeghini, E.; Olmo, M.A. del
2013-06-15
A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L{sup 2} functions defined on (−1,1)×Z and on the sphere S{sup 2}, respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining in this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L{sup 2} functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2). -- Highlights: •The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP). •ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation. •A ladder structure is the condition to get a Lie group representation defining “algebraic special functions”. •The “algebraic special functions” connect Lie algebras and L{sup 2} functions.
Perturbative quantization of Yang-Mills theory with classical double as gauge algebra
NASA Astrophysics Data System (ADS)
Ruiz Ruiz, F.
2016-02-01
Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov-Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary.
Special issue on cluster algebras in mathematical physics
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito
2014-02-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics. Editorial policy The Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki. The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutations cluster structure on Poisson varieties cluster algebras and soliton interactions cluster positivity conjecture Y-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjecture T-system of transfer matrices of integrable lattice models dilogarithm identities in conformal field theory wall crossing in 4d N = 2 supersymmetric gauge theories 4d N = 1 quiver gauge theories described by networks scattering amplitudes of 4d N = 4 theories 3d N = 2 gauge theories described by flat connections on 3-manifolds integrability of dimer/Ising models on graphs. All contributions will be refereed and processed according to the usual procedure of the journal. Guidelines for preparation of contributions The deadline for contributed papers is 31 March
Special issue on cluster algebras in mathematical physics
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito
2013-10-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics. Editorial policy The Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki. The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutations cluster structure on Poisson varieties cluster algebras and soliton interactions cluster positivity conjecture Y-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjecture T-system of transfer matrices of integrable lattice models dilogarithm identities in conformal field theory wall crossing in 4d N = 2 supersymmetric gauge theories 4d N = 1 quiver gauge theories described by networks scattering amplitudes of 4d N = 4 theories 3d N = 2 gauge theories described by flat connections on 3-manifolds integrability of dimer/Ising models on graphs. All contributions will be refereed and processed according to the usual procedure of the journal. Guidelines for preparation of contributions The deadline for contributed papers is 31 March
Special issue on cluster algebras in mathematical physics
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito
2013-12-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics. Editorial policy The Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki. The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutations cluster structure on Poisson varieties cluster algebras and soliton interactions cluster positivity conjecture Y-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjecture T-system of transfer matrices of integrable lattice models dilogarithm identities in conformal field theory wall crossing in 4d N = 2 supersymmetric gauge theories 4d N = 1 quiver gauge theories described by networks scattering amplitudes of 4d N = 4 theories 3d N = 2 gauge theories described by flat connections on 3-manifolds integrability of dimer/Ising models on graphs. All contributions will be refereed and processed according to the usual procedure of the journal. Guidelines for preparation of contributions The deadline for contributed papers is 31 March
Special issue on cluster algebras in mathematical physics
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito
2013-11-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics. Editorial policy The Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki. The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutations cluster structure on Poisson varieties cluster algebras and soliton interactions cluster positivity conjecture Y-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjecture T-system of transfer matrices of integrable lattice models dilogarithm identities in conformal field theory wall crossing in 4d N = 2 supersymmetric gauge theories 4d N = 1 quiver gauge theories described by networks scattering amplitudes of 4d N = 4 theories 3d N = 2 gauge theories described by flat connections on 3-manifolds integrability of dimer/Ising models on graphs. All contributions will be refereed and processed according to the usual procedure of the journal. Guidelines for preparation of contributions The deadline for contributed papers is 31 March
Revisiting Special Relativity: A Natural Algebraic Alternative to Minkowski Spacetime
Chappell, James M.; Iqbal, Azhar; Iannella, Nicolangelo; Abbott, Derek
2012-01-01
Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension , with the unit imaginary producing the correct spacetime distance , and the results of Einstein’s then recently developed theory of special relativity, thus providing an explanation for Einstein’s theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary , with the Clifford bivector for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis and . We find that with this model of planar spacetime, using a two-dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton’s scattering formula, and a simple formulation of Dirac’s and Maxwell’s equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane. PMID:23300566
Revisiting special relativity: a natural algebraic alternative to Minkowski spacetime.
Chappell, James M; Iqbal, Azhar; Iannella, Nicolangelo; Abbott, Derek
2012-01-01
Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension [Formula: see text], with the unit imaginary producing the correct spacetime distance [Formula: see text], and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary [Formula: see text], with the Clifford bivector [Formula: see text] for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis [Formula: see text] and [Formula: see text]. We find that with this model of planar spacetime, using a two-dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane. PMID:23300566
The perturbed universe in the deformed algebra approach of loop quantum cosmology
NASA Astrophysics Data System (ADS)
Grain, Julien
2016-06-01
Loop Quantum Cosmology (LQC) is a tentative approach to model the universe down to the Planck era where quantum gravity settings are needed. The quantization of the universe as a dynamical spacetime is inspired by Loop Quantum Gravity (LQG) ideas. In addition, LQC could bridge contact with astronomical observations, and thus potentially investigate quantum cosmology modelings in the light of observations. To do so however, modeling both the background evolution and its perturbations is needed. The latter described cosmic inhomogeneities that are the main cosmological observables. In this context, we present the so-called deformed algebra approach implementing the quantum corrections to the perturbed universe at an effective level by taking great care of gauge issues. We particularly highlight that in this framework, the algebra of hypersurface deformation receives quantum corrections, and we discuss their meaning. The primordial power spectra of scalar and tensor inhomogeneities are then presented, assuming initial conditions are set in the contracting phase preceding the quantum bounce and the well-known expanding phase of the cosmic history. These spectra are subsequently propagated to angular power spectra of the anisotropies of the cosmic microwave background. It is then shown that regardless of the choice for the initial conditions inside the effective approach for the background evolution (except that they are set in the contracting phase), the predicted angular power spectra of the polarized B-modes exceed the upper bound currently set by observations. The exclusion of this specific version of LQC establishes the falsifiability of the approach, though one shall not conclude here that either LQC or LQG excluded.
Specific optimization of genetic algorithm on special algebras
NASA Astrophysics Data System (ADS)
Habiballa, Hashim; Novak, Vilem; Dyba, Martin; Schenk, Jiri
2016-06-01
Searching for complex finite algebras can be succesfully done by the means of genetic algorithm as we showed in former works. This genetic algorithm needs specific optimization of crossover and mutation. We present details about these optimizations which are already implemented in software application for this task - EQCreator.
Analysis of Errors in a Special Perturbations Satellite Orbit Propagator
Beckerman, M.; Jones, J.P.
1999-02-01
We performed an analysis of error densities for the Special Perturbations orbit propagator using data for 29 satellites in orbits of interest to Space Shuttle and International Space Station collision avoidance. We find that the along-track errors predominate. These errors increase monotonically over each 36-hour prediction interval. The predicted positions in the along-track direction progressively either leap ahead of or lag behind the actual positions. Unlike the along-track errors the radial and cross-track errors oscillate about their nearly zero mean values. As the number of observations per fit interval decline the along-track prediction errors, and amplitudes of the radial and cross-track errors, increase.
ERIC Educational Resources Information Center
Bair, Sherry L.; Rich, Beverly S.
2011-01-01
This article characterizes the development of a deep and connected body of mathematical knowledge categorized by Ball and Bass' (2003b) model of Mathematical Knowledge for Teaching (MKT), as Specialized Content Knowledge for Teaching (SCK) in algebraic reasoning and number sense. The research employed multiple cases across three years from two…
Relations between elliptic multiple zeta values and a special derivation algebra
NASA Astrophysics Data System (ADS)
Broedel, Johannes; Matthes, Nils; Schlotterer, Oliver
2016-04-01
We investigate relations between elliptic multiple zeta values (eMZVs) and describe a method to derive the number of indecomposable elements of given weight and length. Our method is based on representing eMZVs as iterated integrals over Eisenstein series and exploiting the connection with a special derivation algebra. Its commutator relations give rise to constraints on the iterated integrals over Eisenstein series relevant for eMZVs and thereby allow to count the indecomposable representatives. Conversely, the above connection suggests apparently new relations in the derivation algebra. Under https://tools.aei.mpg.de/emzv we provide relations for eMZVs over a wide range of weights and lengths.
NASA Astrophysics Data System (ADS)
Hack, Thomas-Paul
2014-11-01
We quantize the linearized Einstein-Klein-Gordon system on arbitrary on-shell backgrounds in a manifestly covariant and gauge-invariant manner. For the special case of perturbations in inflation, i.e. on-shell backgrounds of Friedmann-Lemaître-Robertson-Walker type, we compare our general quantization construction with the standard approach to the quantum theory of perturbations in inflation. We find that not all local quantum observables of the linearized Einstein-Klein-Gordon system can be split into local observables of scalar and tensor type as in the standard approach. However, we argue that this subclass of observables is sufficient for measuring perturbations that vanish at spatial infinity, which is in line with standard assumptions. Finally, we comment on a recent observation that, upon standard quantization, the quantum Bardeen potentials display a non-local behaviour and argue that a similar phenomenon occurs in any local quantum field theory. It is the hope of the author that the present work may constitute a bridge between the generally applicable and thus powerful framework of algebraic quantum field theory in curved spacetimes and the standard treatment of perturbations in inflation.
Liouvillian perturbations of black holes
NASA Astrophysics Data System (ADS)
Couch, W. E.; Holder, C. L.
2007-10-01
We apply the well-known Kovacic algorithm to find closed form, i.e., Liouvillian solutions, to the differential equations governing perturbations of black holes. Our analysis includes the full gravitational perturbations of Schwarzschild and Kerr, the full gravitational and electromagnetic perturbations of Reissner-Nordstrom, and specialized perturbations of the Kerr-Newman geometry. We also include the extreme geometries. We find all frequencies ω, in terms of black hole parameters and an integer n, which allow Liouvillian perturbations. We display many classes of black hole parameter values and their corresponding Liouvillian perturbations, including new closed-form perturbations of Kerr and Reissner-Nordstrom. We also prove that the only type 1 Liouvillian perturbations of Schwarzschild are the known algebraically special ones and that type 2 Liouvillian solutions do not exist for extreme geometries. In cases where we do not prove the existence or nonexistence of Liouvillian perturbations we obtain sequences of Diophantine equations on which decidability rests.
Special modulated beams for cylindrical coordinates in anisotropic media using computer algebra
NASA Astrophysics Data System (ADS)
Echeverri Chacón, Santiago
2010-04-01
An extension of the solution for the propagation of modulated beams through homogeneous media in cylindrical coordinates which results in a wave function described by Bessel Beams is the basis for this analysis of modulated beams through non homogeneous media in cylindrical coordinates]. By solving the wave equation analytically, including functions that describe the non-homogeneity, and using computer algebra software such as MAPLE©, we formulate new kinds of beams defined by special functions such as Airy, Kummer,and Hypergeometric functions. We also present convergence issues around the axis of propagation and possible applications for these new beams in telecommunication systems.
Twisted Quantum Toroidal Algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Liu, Rongjia
2014-09-01
We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.
Herbert, J.M.
1997-07-01
Rayleigh-Schroedinger perturbation theory is an effective and popular tool for describing low-lying vibrational and rotational states of molecules. This method, in conjunction with ab initio techniques for computation of electronic potential energy surfaces, can be used to calculate first-principles molecular vibrational-rotational energies to successive orders of approximation. Because of mathematical complexities, however, such perturbation calculations are rarely extended beyond the second order of approximation, although recent work by Herbert has provided a formula for the nth-order energy correction. This report extends that work and furnishes the remaining theoretical details (including a general formula for the Rayleigh-Schroedinger expansion coefficients) necessary for calculation of energy corrections to arbitrary order. The commercial computer algebra software Mathematica is employed to perform the prohibitively tedious symbolic manipulations necessary for derivation of generalized energy formulae in terms of universal constants, molecular constants, and quantum numbers. As a pedagogical example, a Hamiltonian operator tailored specifically to diatomic molecules is derived, and the perturbation formulae obtained from this Hamiltonian are evaluated for a number of such molecules. This work provides a foundation for future analyses of polyatomic molecules, since it demonstrates that arbitrary-order perturbation theory can successfully be applied with the aid of commercially available computer algebra software.
NASA Astrophysics Data System (ADS)
Ivanov, I. G.; Netov, N. C.; Bogdanova, B. C.
2015-10-01
This paper addresses the problem of solving a generalized algebraic Riccati equation with an indefinite sign of its quadratic term. We extend the approach introduced by Lanzon, Feng, Anderson and Rotkowitz (2008) for solving similar Riccati equations. We numerically investigate two types of iterative methods for computing the stabilizing solution. The first type of iterative methods constructs two matrix sequences, where the sum of them converges to the stabilizing solution. The second type of methods defines one matrix sequence which converges to the stabilizing solution. Computer realizations of the presented methods are numerically tested and compared on the test of family examples. Based on the experiments some conclusions are derived.
Non-singular orbital elements for special perturbations in the two-body problem
NASA Astrophysics Data System (ADS)
Baù, Giulio; Bombardelli, Claudio; Peláez, Jesús; Lorenzini, Enrico
2015-12-01
Seven spatial elements and a time element are proposed as the state variables of a new special perturbation method for the two-body problem. The new elements hold for zero eccentricity and inclination and for negative values of the total energy. They are developed by combining a spatial transformation into projective coordinates (as in the Burdet-Ferrándiz regularization) with a time transformation in which the exponent of the orbital radius is equal to one instead of two (as commonly done in the literature). By following this approach, we discover a new linearization of the two-body problem, from which the orbital elements can be generated by the variation of parameters method. The geometrical significance of the spatial quantities is revealed by a new intermediate frame which differs from a local vertical local horizontal frame by one rotation in the instantaneous orbital plane. Four elements parametrize the attitude in space of this frame, which in turn defines the orientation of the orbital plane and fixes the departure direction for the longitude of the propagated body. The remaining three elements determine the motion along the radial unit vector and the orbital longitude. The performance of the method, tested using a series of benchmark orbit propagation scenarios, is extremely good when compared to several regularized formulations, some of which have been modified and improved here for the first time.
Kappa Snyder deformations of Minkowski spacetime, realizations, and Hopf algebra
Meljanac, S.; Meljanac, D.; Samsarov, A.; Stojic, M.
2011-03-15
We present Lie-algebraic deformations of Minkowski space with undeformed Poincare algebra. These deformations interpolate between Snyder and {kappa}-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that, although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are isomorphic at the level of Hopf-algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from {kappa}-deformed Snyder to Snyder space is constructed. The deformed Leibniz rule, the Hopf structure, and the star product are found. Special cases, particularly Snyder and {kappa}-Minkowski in Maggiore-type realizations, are discussed. The same generalized Hopf-algebraic structures are considered as well in the case of an arbitrary allowable kind of realization, and results are given perturbatively up to second order in deformation parameters.
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…
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.
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.…
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.
Invariants of triangular Lie algebras
NASA Astrophysics Data System (ADS)
Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman
2007-07-01
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated.
Azai, Chihiro; Sano, Yuko; Kato, Yuki; Noguchi, Takumi; Oh-oka, Hirozo
2016-01-01
Homodimeric photosynthetic reaction centers (RCs) in green sulfur bacteria and heliobacteria are functional homologs of Photosystem (PS) I in oxygenic phototrophs. They show unique features in their electron transfer reactions; however, detailed structural information has not been available so far. We mutated PscA-Leu688 and PscA-Val689 to cysteine residues in the green sulfur bacterium Chlorobaculum tepidum; these residues were predicted to interact with the special pair P840, based on sequence comparison with PS I. Spectroelectrochemical measurements showed that the L688C and V689C mutations altered a near-infrared difference spectrum upon P840 oxidation, as well as the redox potential of P840. Light-induced Fourier transform infrared difference measurements showed that the L688C mutation induced a differential signal of the S-H stretching vibration in the P840+/P840 spectrum, as reported in P800+/P800 difference spectrum in a heliobacterial RC. Spectral changes in the 131-keto C=O region, caused by both mutations, revealed corresponding changes in the electronic structure of P840 and in the hydrogen-bonding interaction at the 131-keto C=O group. These results suggest that there is a common spatial configuration around the special pair sites among type 1 RCs. The data also provided evidence that P840 has a symmetric electronic structure, as expected from a homodimeric RC. PMID:26804137
Canonical density matrix perturbation theory.
Niklasson, Anders M N; Cawkwell, M J; Rubensson, Emanuel H; Rudberg, Elias
2015-12-01
Density matrix perturbation theory [Niklasson and Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] is generalized to canonical (NVT) free-energy ensembles in tight-binding, Hartree-Fock, or Kohn-Sham density-functional theory. The canonical density matrix perturbation theory can be used to calculate temperature-dependent response properties from the coupled perturbed self-consistent field equations as in density-functional perturbation theory. The method is well suited to take advantage of sparse matrix algebra to achieve linear scaling complexity in the computational cost as a function of system size for sufficiently large nonmetallic materials and metals at high temperatures. PMID:26764847
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-04-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-07-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
ERIC Educational Resources Information Center
Luna, Pat
Designed for middle school students, this award winning, six-day teaching unit helped students learn about the concepts of specialization, interdependence, efficiency, and profit. At the onset of the lesson the students were already familiar with the concepts of scarcity, goods, services, profits, supply, demand, and opportunity costs. The unit's…
A new model for algebraic Rossby solitary waves in rotation fluid and its solution
NASA Astrophysics Data System (ADS)
Chen, Yao-Deng; Yang, Hong-Wei; Gao, Yu-Fang; Yin, Bao-Shu; Feng, Xing-Ru
2015-09-01
A generalized Boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transformations of time and space. Using this equation, the conservation laws of algebraic Rossby solitary waves are discussed. It is found that the mass, the momentum, the energy, and the velocity of center of gravity of the algebraic solitary waves are conserved in the propagation process. Finally, the analytical solution of the equation is generated. Based on the analytical solution, the properties of the algebraic solitary waves and the dissipation effect are discussed. The results point out that, similar to classic solitary waves, the dissipation can cause the amplitude and the speed of solitary waves to decrease; however, unlike classic solitary waves, the algebraic solitary waves can split during propagation and the decrease of the detuning parameter can accelerate the occurrence of the solitary waves fission phenomenon. Project supported by the Shandong Provincial Key Laboratory of Marine Ecology and Environment and Disaster Prevention and Mitigation Project, China (Grant No. 2012010), the National Natural Science Foundation of China (Grant Nos. 41205082 and 41476019), the Special Funds for Theoretical Physics of the National Natural Science Foundation of China (Grant No. 11447205), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), China.
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.
Flatbands under Correlated Perturbations
NASA Astrophysics Data System (ADS)
Bodyfelt, Joshua D.; Leykam, Daniel; Danieli, Carlo; Yu, Xiaoquan; Flach, Sergej
2014-12-01
Flatband networks are characterized by the coexistence of dispersive and flatbands. Flatbands (FBs) are generated by compact localized eigenstates (CLSs) with local network symmetries, based on destructive interference. Correlated disorder and quasiperiodic potentials hybridize CLSs without additional renormalization, yet with surprising consequences: (i) states are expelled from the FB energy EFB, (ii) the localization length of eigenstates vanishes as ξ ˜1 /ln (E -EFB) , (iii) the density of states diverges logarithmically (particle-hole symmetry) and algebraically (no particle-hole symmetry), and (iv) mobility edge curves show algebraic singularities at EFB . Our analytical results are based on perturbative expansions of the CLSs and supported by numerical data in one and two lattice dimensions.
Flatbands under correlated perturbations.
Bodyfelt, Joshua D; Leykam, Daniel; Danieli, Carlo; Yu, Xiaoquan; Flach, Sergej
2014-12-01
Flatband networks are characterized by the coexistence of dispersive and flatbands. Flatbands (FBs) are generated by compact localized eigenstates (CLSs) with local network symmetries, based on destructive interference. Correlated disorder and quasiperiodic potentials hybridize CLSs without additional renormalization, yet with surprising consequences: (i) states are expelled from the FB energy E_{FB}, (ii) the localization length of eigenstates vanishes as ξ∼1/ln(E-E_{FB}), (iii) the density of states diverges logarithmically (particle-hole symmetry) and algebraically (no particle-hole symmetry), and (iv) mobility edge curves show algebraic singularities at E_{FB}. Our analytical results are based on perturbative expansions of the CLSs and supported by numerical data in one and two lattice dimensions. PMID:25526142
C∗-completions and the DFR-algebra
NASA Astrophysics Data System (ADS)
Forger, Michael; Paulino, Daniel V.
2016-02-01
The aim of this paper is to present the construction of a general family of C∗-algebras which includes, as a special case, the "quantum spacetime algebra" introduced by Doplicher, Fredenhagen, and Roberts. It is based on an extension of the notion of C∗-completion from algebras to bundles of algebras, compatible with the usual C∗-completion of the appropriate algebras of sections, combined with a novel definition for the algebra of the canonical commutation relations using Rieffel's theory of strict deformation quantization. Taking the C∗-algebra of continuous sections vanishing at infinity, we arrive at a functor associating a C∗-algebra to any Poisson vector bundle and recover the original DFR-algebra as a particular example.
Algebraic vs physical N = 6 3-algebras
Cantarini, Nicoletta; Kac, Victor G.
2014-01-15
In our previous paper, we classified linearly compact algebraic simple N = 6 3-algebras. In the present paper, we classify their “physical” counterparts, which actually appear in the N = 6 supersymmetric 3-dimensional Chern-Simons theories.
Renormalized Lie perturbation theory
Rosengaus, E.; Dewar, R.L.
1981-07-01
A Lie operator method for constructing action-angle transformations continuously connected to the identity is developed for area preserving mappings. By a simple change of variable from action to angular frequency a perturbation expansion is obtained in which the small denominators have been renormalized. The method is shown to lead to the same series as the Lagrangian perturbation method of Greene and Percival, which converges on KAM surfaces. The method is not superconvergent, but yields simple recursion relations which allow automatic algebraic manipulation techniques to be used to develop the series to high order. It is argued that the operator method can be justified by analytically continuing from the complex angular frequency plane onto the real line. The resulting picture is one where preserved primary KAM surfaces are continuously connected to one another.
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, Inc., Reston, VA.
This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…
Complex Kumjian-Pask algebras of 2-graphs
NASA Astrophysics Data System (ADS)
Yusnitha, Isnie; Rosjanuardi, Rizky
2016-02-01
Let Λ be a row-finitek-graph without sources and R be any field. The Kumjian-Pask algebras KPR(Λ) is an algebraic analog of k-graph algebrasC*(Λ). When the field R is the complex field ℂ, there is a special relationship between the complex Kumjian-Pask algebras KP𝕔(Λ) and k-graph algebrasC*(Λ). We examine this relationship particularly to the case 2-graph 𝔽θ+, 2-graph on single vertex generated by m blue edges and n red edges with θ respect to some commutation relations, by analyzing the associated C*-algebras of 𝔽θ+ . As the presence of cycles on 2-graph 𝔽θ+, we can imply that 2-graph algebras C*(𝔽F+ ) is infinite-dimensional. Hence, the complex Kumjian-Pask algebras KP𝕔 (𝔽θ+ ) is also infinite dimensional.
Figueroa-O'Farrill, Jose Miguel
2009-11-15
We phrase deformations of n-Leibniz algebras in terms of the cohomology theory of the associated Leibniz algebra. We do the same for n-Lie algebras and for the metric versions of n-Leibniz and n-Lie algebras. We place particular emphasis on the case of n=3 and explore the deformations of 3-algebras of relevance to three-dimensional superconformal Chern-Simons theories with matter.
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
Algebras with convergent star products and their representations in Hilbert spaces
Soloviev, M. A.
2013-07-15
We study star product algebras of analytic functions for which the power series defining the products converge absolutely. Such algebras arise naturally in deformation quantization theory and in noncommutative quantum field theory. We consider different star products in a unifying way and present results on the structure and basic properties of these algebras, which are useful for applications. Special attention is given to the Hilbert space representation of the algebras and to the exact description of their corresponding operator algebras.
Deforming the Maxwell-Sim algebra
Gibbons, G. W.; Gomis, Joaquim; Pope, C. N.
2010-09-15
The Maxwell algebra is a noncentral extension of the Poincare algebra, in which the momentum generators no longer commute, but satisfy [P{sub {mu}},P{sub {nu}}]=Z{sub {mu}{nu}}. The charges Z{sub {mu}{nu}} commute with the momenta, and transform tensorially under the action of the angular momentum generators. If one constructs an action for a massive particle, invariant under these symmetries, one finds that it satisfies the equations of motion of a charged particle interacting with a constant electromagnetic field via the Lorentz force. In this paper, we explore the analogous constructions where one starts instead with the ISim subalgebra of Poincare, this being the symmetry algebra of very special relativity. It admits an analogous noncentral extension, and we find that a particle action invariant under this Maxwell-Sim algebra again describes a particle subject to the ordinary Lorentz force. One can also deform the ISim algebra to DISim{sub b}, where b is a nontrivial dimensionless parameter. We find that the motion described by an action invariant under the corresponding Maxwell-DISim algebra is that of a particle interacting via a Finslerian modification of the Lorentz force. In an appendix is it shown that the DISim{sub b} algebra is isomorphic to the extended Schroedinger algebra with its standard deformation parameter z, when b=(1/1-z).
Algorithmic Questions for Linear Algebraic Groups. Ii
NASA Astrophysics Data System (ADS)
Sarkisjan, R. A.
1982-04-01
It is proved that, given a linear algebraic group defined over an algebraic number field and satisfying certain conditions, there exists an algorithm which determines whether or not two double cosets of a special type coincide in its adele group, and which enumerates all such double cosets. This result is applied to the isomorphism problem for finitely generated nilpotent groups, and also to other problems.Bibliography: 18 titles.
Learning Algebra in a Computer Algebra Environment
ERIC Educational Resources Information Center
Drijvers, Paul
2004-01-01
This article summarises a doctoral thesis entitled "Learning algebra in a computer algebra environment, design research on the understanding of the concept of parameter" (Drijvers, 2003). It describes the research questions, the theoretical framework, the methodology and the results of the study. The focus of the study is on the understanding of…
Realizations of Galilei algebras
NASA Astrophysics Data System (ADS)
Nesterenko, Maryna; Pošta, Severin; Vaneeva, Olena
2016-03-01
All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations.
NASA Technical Reports Server (NTRS)
Iachello, Franco
1995-01-01
An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.
Orientation in operator algebras
Alfsen, Erik M.; Shultz, Frederic W.
1998-01-01
A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. PMID:9618457
Developing Thinking in Algebra
ERIC Educational Resources Information Center
Mason, John; Graham, Alan; Johnson-Wilder, Sue
2005-01-01
This book is for people with an interest in algebra whether as a learner, or as a teacher, or perhaps as both. It is concerned with the "big ideas" of algebra and what it is to understand the process of thinking algebraically. The book has been structured according to a number of pedagogic principles that are exposed and discussed along the way,…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Applied Algebra Curriculum Modules.
ERIC Educational Resources Information Center
Texas State Technical Coll., Marshall.
This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students…
Profiles of Algebraic Competence
ERIC Educational Resources Information Center
Humberstone, J.; Reeve, R.A.
2008-01-01
The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…
Ternary Virasoro - Witt algebra.
Zachos, C.; Curtright, T.; Fairlie, D.; High Energy Physics; Univ. of Miami; Univ. of Durham
2008-01-01
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
Cosmological perturbations in massive bigravity
Lagos, Macarena; Ferreira, Pedro G. E-mail: p.ferreira1@physics.ox.ac.uk
2014-12-01
We present a comprehensive analysis of classical scalar, vector and tensor cosmological perturbations in ghost-free massive bigravity. In particular, we find the full evolution equations and analytical solutions in a wide range of regimes. We show that there are viable cosmological backgrounds but, as has been found in the literature, these models generally have exponential instabilities in linear perturbation theory. However, it is possible to find stable scalar cosmological perturbations for a very particular choice of parameters. For this stable subclass of models we find that vector and tensor perturbations have growing solutions. We argue that special initial conditions are needed for tensor modes in order to have a viable model.
Quantum walks, deformed relativity and Hopf algebra symmetries.
Bisio, Alessandro; D'Ariano, Giacomo Mauro; Perinotti, Paolo
2016-05-28
We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014Phys. Rev. A90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras-the usual Poincaré and theκ-Poincaré algebras. PMID:27091171
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.
Aspects of perturbative unitarity
NASA Astrophysics Data System (ADS)
Anselmi, Damiano
2016-07-01
We reconsider perturbative unitarity in quantum field theory and upgrade several arguments and results. The minimum assumptions that lead to the largest time equation, the cutting equations and the unitarity equation are identified. Using this knowledge and a special gauge, we give a new, simpler proof of perturbative unitarity in gauge theories and generalize it to quantum gravity, in four and higher dimensions. The special gauge interpolates between the Feynman gauge and the Coulomb gauge without double poles. When the Coulomb limit is approached, the unphysical particles drop out of the cuts and the cutting equations are consistently projected onto the physical subspace. The proof does not extend to nonlocal quantum field theories of gauge fields and gravity, whose unitarity remains uncertain.
From geometry to algebra: the Euclidean way with technology
NASA Astrophysics Data System (ADS)
Ferrarello, Daniela; Flavia Mammana, Maria; Pennisi, Mario
2016-05-01
In this paper, we present the results of an experimental classroom activity, history-based with a phylogenetic approach, to achieve algebra properties through geometry. In particular, we used Euclidean propositions, processed them by a dynamic geometry system and translate them into algebraic special products.
NASA Astrophysics Data System (ADS)
Tiburzi, Brian C.
The era of high-precision lattice QCD has led to synergy between lattice computations and phenomenological input from chiral perturbation theory. We provide an introduction to chiral perturbation theory with a bent towards understanding properties of the nucleon and other low-lying baryons. Four main topics are the basis for this chapter. We begin with a discussion of broken symmetries and the procedure to construct the chiral Lagrangian. The second topic concerns specialized applications of chiral perturbation theory tailored to lattice QCD, such as partial quenching, lattice discretization, and finite-volume effects. We describe inclusion of the nucleon in chiral perturbation theory using a heavy-fermion Euclidean action. Issues of convergence are taken up as our final topic. We consider expansions in powers of the strange-quark mass, and the appearance of unphysical singularities in the heavy-particle formulation. Our aim is to guide lattice practitioners in understanding the predictions chiral perturbation theory makes for baryons, and show how the lattice will play a role in testing the rigor of the chiral expansion at physical values of the quark masses.
Technology Transfer Automated Retrieval System (TEKTRAN)
Perturbing lignification is possible in multiple and diverse ways. Without obvious growth/development phenotypes, transgenic angiosperms can have lignin levels reduced to half the normal level, can have compositions ranging from very high-guaiacyl/low-syringyl to almost totally syringyl, and can eve...
The kinematic algebra from the self-dual sector
NASA Astrophysics Data System (ADS)
Monteiro, Ricardo; O'Connell, Donal
2011-07-01
We identify a diffeomorphism Lie algebra in the self-dual sector of Yang-Mills theory, and show that it determines the kinematic numerators of tree-level MHV amplitudes in the full theory. These amplitudes can be computed off-shell from Feynman diagrams with only cubic vertices, which are dressed with the structure constants of both the Yang-Mills colour algebra and the diffeomorphism algebra. Therefore, the latter algebra is the dual of the colour algebra, in the sense suggested by the work of Bern, Carrasco and Johansson. We further study perturbative gravity, both in the self-dual and in the MHV sectors, finding that the kinematic numerators of the theory are the BCJ squares of the Yang-Mills numerators.
Perturbation theory in light-cone quantization
Langnau, A.
1992-01-01
A thorough investigation of light-cone properties which are characteristic for higher dimensions is very important. The easiest way of addressing these issues is by analyzing the perturbative structure of light-cone field theories first. Perturbative studies cannot be substituted for an analysis of problems related to a nonperturbative approach. However, in order to lay down groundwork for upcoming nonperturbative studies, it is indispensable to validate the renormalization methods at the perturbative level, i.e., to gain control over the perturbative treatment first. A clear understanding of divergences in perturbation theory, as well as their numerical treatment, is a necessary first step towards formulating such a program. The first objective of this dissertation is to clarify this issue, at least in second and fourth-order in perturbation theory. The work in this dissertation can provide guidance for the choice of counterterms in Discrete Light-Cone Quantization or the Tamm-Dancoff approach. A second objective of this work is the study of light-cone perturbation theory as a competitive tool for conducting perturbative Feynman diagram calculations. Feynman perturbation theory has become the most practical tool for computing cross sections in high energy physics and other physical properties of field theory. Although this standard covariant method has been applied to a great range of problems, computations beyond one-loop corrections are very difficult. Because of the algebraic complexity of the Feynman calculations in higher-order perturbation theory, it is desirable to automatize Feynman diagram calculations so that algebraic manipulation programs can carry out almost the entire calculation. This thesis presents a step in this direction. The technique we are elaborating on here is known as light-cone perturbation theory.
A Richer Understanding of Algebra
ERIC Educational Resources Information Center
Foy, Michelle
2008-01-01
Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number, pre-algebra, and algebra…
NASA Astrophysics Data System (ADS)
Lesgourges, J.
2013-08-01
We present a self-contained summary of the theory of linear cosmological perturbations. We emphasize the effect of the six parameters of the minimal cosmological model, first, on the spectrum of Cosmic Microwave Background temperature anisotropies, and second, on the linear matter power spectrum. We briefly review at the end the possible impact of a few non-minimal dark matter and dark energy models.
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)
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.
Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees
Agarwala, Susama; Delaney, Colleen
2015-04-15
This paper defines a generalization of the Connes-Moscovici Hopf algebra, H(1), that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in non-commutative geometry, and the latter, a much studied object in perturbative quantum field theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.
Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees
NASA Astrophysics Data System (ADS)
Agarwala, Susama; Delaney, Colleen
2015-04-01
This paper defines a generalization of the Connes-Moscovici Hopf algebra, H ( 1 ) , that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in non-commutative geometry, and the latter, a much studied object in perturbative quantum field theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…
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
Semigroups and computer algebra in algebraic structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
Lie algebra extensions of current algebras on S3
NASA Astrophysics Data System (ADS)
Kori, Tosiaki; Imai, Yuto
2015-06-01
An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory.
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.
Coreflections in Algebraic Quantum Logic
NASA Astrophysics Data System (ADS)
Jacobs, Bart; Mandemaker, Jorik
2012-07-01
Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.
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)
Developing Algebraic Thinking.
ERIC Educational Resources Information Center
Alejandre, Suzanne
2002-01-01
Presents a teaching experience that resulted in students getting to a point of full understanding of the kinesthetic activity and the algebra behind it. Includes a lesson plan for a traffic jam activity. (KHR)
Algebraic integrability: a survey.
Vanhaecke, Pol
2008-03-28
We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems. PMID:17588863
Algebraic Semantics for Narrative
ERIC Educational Resources Information Center
Kahn, E.
1974-01-01
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
NASA Astrophysics Data System (ADS)
Matthews, Devin A.; Gong, Justin Z.; Stanton, John F.
2014-06-01
The derivation of analytic expressions for vibrational and rovibrational constants, for example the anharmonicity constants χij and the vibration-rotation interaction constants α^B_r, from second-order vibrational perturbation theory (VPT2) can be accomplished with pen and paper and some practice. However, the corresponding quantities from fourth-order perturbation theory (VPT4) are considerably more complex, with the only known derivations by hand extensively using many layers of complicated intermediates and for rotational quantities requiring specialization to orthorhombic cases or the form of Watson's reduced Hamiltonian. We present an automatic computer program for generating these expressions with full generality based on the adaptation of an existing numerical program based on the sum-over-states representation of the energy to a computer algebra context. The measures taken to produce well-simplified and factored expressions in an efficient manner are discussed, as well as the framework for automatically checking the correctness of the generated equations.
Kopeliovich, B. Z.; Pirner, H.-J.; Potashnikova, I. K.; Schmidt, Ivan; Tarasov, A. V.
2008-03-01
The Berger model of perturbative fragmentation of quarks to pions is improved by providing an absolute normalization and keeping all terms in a (1-z) expansion, which makes the calculation valid at all values of fractional pion momentum z. We also replace the nonrelativistic wave function of a loosely bound pion by the more realistic procedure of projecting to the light-cone pion wave function, which in turn is taken from well known models. The full calculation does not confirm the (1-z){sup 2} behavior of the fragmentation function (FF) predicted in [E. L. Berger, Z. Phys. C 4, 289 (1980); Phys. Lett. 89B, 241 (1980] for z>0.5, and only works at very large z>0.95, where it is in reasonable agreement with phenomenological FFs. Otherwise, we observe quite a different z-dependence which grossly underestimates data at smaller z. The disagreement is reduced after the addition of pions from decays of light vector mesons, but still remains considerable. The process dependent higher twist terms are also calculated exactly and found to be important at large z and/or p{sub T}.
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.
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
NASA Astrophysics Data System (ADS)
Dobrev, V. K.
2013-02-01
In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G ' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E 7(7) which is parabolically related to the CLA E 7(-25) , the parabolic subalgebras including E 6(6) and E 6(-26). Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so( n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so( n - 1, 1) and its analogs so( p - 1, q - 1). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14) , the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so( p, q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of shadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not.
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.
DG Poisson algebra and its universal enveloping algebra
NASA Astrophysics Data System (ADS)
Lü, JiaFeng; Wang, XingTing; Zhuang, GuangBin
2016-05-01
In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.
An algebraic approach to the Hubbard model
NASA Astrophysics Data System (ADS)
de Leeuw, Marius; Regelskis, Vidas
2016-02-01
We study the algebraic structure of an integrable Hubbard-Shastry type lattice model associated with the centrally extended su (2 | 2) superalgebra. This superalgebra underlies Beisert's AdS/CFT worldsheet R-matrix and Shastry's R-matrix. The considered model specializes to the one-dimensional Hubbard model in a certain limit. We demonstrate that Yangian symmetries of the R-matrix specialize to the Yangian symmetry of the Hubbard model found by Korepin and Uglov. Moreover, we show that the Hubbard model Hamiltonian has an algebraic interpretation as the so-called secret symmetry. We also discuss Yangian symmetries of the A and B models introduced by Frolov and Quinn.
Dually-Perturbed Matsumoto-Imai Signature (DPMS) Scheme
NASA Astrophysics Data System (ADS)
Gotaishi, Masahito; Tadaki, Kohtaro; Fujita, Ryo; Tsujii, Shigeo
A new signature scheme of MPKC is proposed. It is created by perturbing a traditional encryption scheme in two ways. The proposed perturbation polynomials successfully reinforce the Matsumoto-Imai cryptosystem This new signature scheme has a structure very difficult to cryptanalyze. Along with the security against algebraic attacks, its security against existing attacks is discussed. The experimental data imply that the scheme can create a both lightweight and secure signature system.
NASA Astrophysics Data System (ADS)
Roitman, Michael
2008-08-01
In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
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.
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
ERIC Educational Resources Information Center
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
Computer Program For Linear Algebra
NASA Technical Reports Server (NTRS)
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
NASA Technical Reports Server (NTRS)
Shahshahani, M.
1991-01-01
The performance characteristics are discussed of certain algebraic geometric codes. Algebraic geometric codes have good minimum distance properties. On many channels they outperform other comparable block codes; therefore, one would expect them eventually to replace some of the block codes used in communications systems. It is suggested that it is unlikely that they will become useful substitutes for the Reed-Solomon codes used by the Deep Space Network in the near future. However, they may be applicable to systems where the signal to noise ratio is sufficiently high so that block codes would be more suitable than convolutional or concatenated codes.
NASA Astrophysics Data System (ADS)
Bouwknegt, Peter
1988-06-01
We investigate extensions of the Virasoro algebra by a single primary field of integer or halfinteger conformal dimension Δ. We argue that for vanishing structure constant CΔΔΔ, the extended conformal algebra can only be associative for a generic c-value if Δ=1/2, 1, 3/2, 2 or 3. For the other Δ<=5 we compute the finite set of allowed c-values and identify the rational solutions. The case CΔΔΔ≠0 is also briefly discussed. I would like to thank Kareljan Schoutens for discussions and Sander Bais for a careful reading of the manuscript.
Teaching Arithmetic and Algebraic Expressions
ERIC Educational Resources Information Center
Subramaniam, K.; Banerjee, Rakhi
2004-01-01
A teaching intervention study was conducted with sixth grade students to explore the interconnections between students' growing understanding of arithmetic expressions and beginning algebra. Three groups of students were chosen, with two groups receiving instruction in arithmetic and algebra, and one group in algebra without arithmetic. Students…
Assessing Elementary Algebra with STACK
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2007-01-01
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…
Spinors in the hyperbolic algebra
NASA Astrophysics Data System (ADS)
Ulrych, S.
2006-01-01
The three-dimensional universal complex Clifford algebra Cbar3,0 is used to represent relativistic vectors in terms of paravectors. In analogy to the Hestenes spacetime approach spinors are introduced in an algebraic form. This removes the dependance on an explicit matrix representation of the algebra.
Topological basis realization for BMW algebra and Heisenberg XXZ spin chain model
NASA Astrophysics Data System (ADS)
Liu, Bo; Xue, Kang; Wang, Gangcheng; Liu, Ying; Sun, Chunfang
2015-04-01
In this paper, we study three-dimensional (3D) reduced Birman-Murakami-Wenzl (BMW) algebra based on topological basis theory. Several examples of BMW algebra representations are reviewed. We also discuss a special solution of BMW algebra, which can be used to construct Heisenberg XXZ model. The theory of topological basis provides a useful method to solve quantum spin chain models. It is also shown that the ground state of XXZ spin chain is superposition state of topological basis.
ERIC Educational Resources Information Center
Glick, David
1995-01-01
Presents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)
ERIC Educational Resources Information Center
Ketterlin-Geller, Leanne R.; Jungjohann, Kathleen; Chard, David J.; Baker, Scott
2007-01-01
Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…
Computer Algebra versus Manipulation
ERIC Educational Resources Information Center
Zand, Hossein; Crowe, David
2004-01-01
In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…
ERIC Educational Resources Information Center
Boiteau, Denise; Stansfield, David
This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative Numbers"; and (5) "Using Positive…
Thinking Visually about Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2015-01-01
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
ERIC Educational Resources Information Center
Kennedy, John
This text provides information and exercises on arithmetic topics which should be mastered before a student enrolls in an Elementary Algebra course. Section I describes the fundamental properties and relationships of whole numbers, focusing on basic operations, divisibility tests, exponents, order of operations, prime numbers, greatest common…
ERIC Educational Resources Information Center
Nwabueze, Kenneth K.
2004-01-01
The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…
Zeta functional equation on Jordan algebras of type II
NASA Astrophysics Data System (ADS)
Kayoya, J. B.
2005-02-01
Using the Jordan algebras methods, specially the properties of Peirce decomposition and the Frobenius transformation, we compute the coefficients of the zeta functional equation, in the case of Jordan algebras of type II. As particular cases of our result, we can cite the case of studied by Gelbart [Mem. Amer. Math. Soc. 108 (1971)] and Godement and Jacquet [Zeta functions of simple algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972], and the case of studied by Muro [Adv. Stud. Pure Math. 15 (1989) 429]. Let us also mention, that recently, Bopp and Rubenthaler have obtained a more general result on the zeta functional equation by using methods based on the algebraic properties of regular graded algebras which are in one-to-one correspondence with simple Jordan algebras [Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces, IRMA, Strasbourg, 2003]. The method used in this paper is a direct application of specific properties of Jordan algebras of type II.
On the Generalized Unitary Parasupersymmetry Algebra of Beckers-Debergh
NASA Astrophysics Data System (ADS)
Chenaghlou, A.; Fakhri, H.
An appropriate generalization of the unitary parasupersymmetry algebra of Beckers-Debergh to arbitrary order is presented in this paper. A special representation for realizing the even arbitrary order unitary parasupersymmetry algebra of Beckers-Debergh is analyzed by one-dimensional shape invariance solvable models, 2D and 3D quantum solvable models obtained from the shape invariance theory as well. In particular, in the special representation, it is shown that the isospectrum Hamiltonians consist of the two partner Hamiltonians of the shape invariance theory.
NASA Astrophysics Data System (ADS)
Zhang, Ming; Yao, JingTao
2004-04-01
The XML is a new standard for data representation and exchange on the Internet. There are studies on XML query languages as well as XML algebras in literature. However, attention has not been paid to research on XML algebras for data mining due to partially the fact that there is no widely accepted definition of XML mining tasks. This paper tries to examine the XML mining tasks and provide guidelines to design XML algebras for data mining. Some summarization and comparison have been done to existing XML algebras. We argue that by adding additional operators for mining tasks, XML algebras may work well for data mining with XML documents.
Maximizing algebraic connectivity in air transportation networks
NASA Astrophysics Data System (ADS)
Wei, Peng
In air transportation networks the robustness of a network regarding node and link failures is a key factor for its design. An experiment based on the real air transportation network is performed to show that the algebraic connectivity is a good measure for network robustness. Three optimization problems of algebraic connectivity maximization are then formulated in order to find the most robust network design under different constraints. The algebraic connectivity maximization problem with flight routes addition or deletion is first formulated. Three methods to optimize and analyze the network algebraic connectivity are proposed. The Modified Greedy Perturbation Algorithm (MGP) provides a sub-optimal solution in a fast iterative manner. The Weighted Tabu Search (WTS) is designed to offer a near optimal solution with longer running time. The relaxed semi-definite programming (SDP) is used to set a performance upper bound and three rounding techniques are discussed to find the feasible solution. The simulation results present the trade-off among the three methods. The case study on two air transportation networks of Virgin America and Southwest Airlines show that the developed methods can be applied in real world large scale networks. The algebraic connectivity maximization problem is extended by adding the leg number constraint, which considers the traveler's tolerance for the total connecting stops. The Binary Semi-Definite Programming (BSDP) with cutting plane method provides the optimal solution. The tabu search and 2-opt search heuristics can find the optimal solution in small scale networks and the near optimal solution in large scale networks. The third algebraic connectivity maximization problem with operating cost constraint is formulated. When the total operating cost budget is given, the number of the edges to be added is not fixed. Each edge weight needs to be calculated instead of being pre-determined. It is illustrated that the edge addition and the
On Dunkl angular momenta algebra
NASA Astrophysics Data System (ADS)
Feigin, Misha; Hakobyan, Tigran
2015-11-01
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
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.
Primary School Students' Strategies in Early Algebra Problem Solving Supported by an Online Game
ERIC Educational Resources Information Center
van den Heuvel-Panhuizen, Marja; Kolovou, Angeliki; Robitzsch, Alexander
2013-01-01
In this study we investigated the role of a dynamic online game on students' early algebra problem solving. In total 253 students from grades 4, 5, and 6 (10-12 years old) used the game at home to solve a sequence of early algebra problems consisting of contextual problems addressing covarying quantities. Special software monitored the…
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…
Marquette, Ian
2013-07-15
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.
NASA Astrophysics Data System (ADS)
Dankova, T. S.; Rosensteel, G.
1998-10-01
Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for an arbitrary Lie algebra L of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space L^* of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The mean field theories for the Elliott SU(3) and symplectic Sp(3,R) algebras are constructed explicitly in the coadjoint orbit framework.
ERIC Educational Resources Information Center
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.
Vertex Algebras, Kac-Moody Algebras, and the Monster
NASA Astrophysics Data System (ADS)
Borcherds, Richard E.
1986-05-01
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
Invariants of triangular Lie algebras with one nil-independent diagonal element
NASA Astrophysics Data System (ADS)
Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman
2007-08-01
The invariants of solvable triangular Lie algebras with one nil-independent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartan's method of moving frames and the special technique developed for triangular and closed algebras in Boyko et al (J. Phys. A: Math. Theor. 2007 40 7557). The conjecture of Tremblay and Winternitz (J. Phys. A: Math. Gen. 2001 34 9085) on the number and form of elements in the bases is completed and proved.
NASA Astrophysics Data System (ADS)
Palmkvist, Jakob
2014-01-01
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D - 2 - p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
NASA Technical Reports Server (NTRS)
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
Palmkvist, Jakob
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras
NASA Astrophysics Data System (ADS)
Paseka, Jan; Riečanová, Zdenka
2010-12-01
We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.
Locally finite dimensional Lie algebras
NASA Astrophysics Data System (ADS)
Hennig, Johanna
We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L. Then we prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic p which give sufficient conditions for the algebras to be of the form [K(R, *), K( R, *)] / (Z(R) ∩ [ K(R, *), K(R, *)]) for a simple, locally finite dimensional associative algebra R with involution *. Lastly, we explore the noncommutative geometry of locally simple representations of the diagonal locally finite Lie algebras sl(ninfinity), o( ninfinity), and sp(n infinity).
Quantum computation using geometric algebra
NASA Astrophysics Data System (ADS)
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
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
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…
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. PMID:20843716
ERIC Educational Resources Information Center
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
Higher level twisted Zhu algebras
Ekeren, Jethro van
2011-05-15
The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra V, graded by {Gamma}/Z for some subgroup {Gamma} of R containing Z, and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras Zhu{sub p,{Gamma}}(V), and we prove the following theorems: For each p, there is a bijection between the irreducible Zhu{sub p,{Gamma}}(V)-modules and the irreducible {Gamma}-twisted positive energy V-modules, and V is ({Gamma}, H)-rational if and only if all its Zhu algebras Zhu{sub p,{Gamma}}(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra Vir{sup c} and the universal affine Kac-Moody vertex algebra V{sup k}(g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.
Feynman graph generation and calculations in the Hopf algebra of Feynman graphs
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2014-12-01
Two programs for the computation of perturbative expansions of quantum field theory amplitudes are provided. feyngen can be used to generate Feynman graphs for Yang-Mills, QED and φk theories. Using dedicated graph theoretic tools feyngen can generate graphs of comparatively high loop orders. feyncop implements the Hopf algebra of those Feynman graphs which incorporates the renormalization procedure necessary to calculate finite results in perturbation theory of the underlying quantum field theory. feyngen is validated by comparison to explicit calculations of zero dimensional quantum field theories and feyncop is validated using a combinatorial identity on the Hopf algebra of graphs.
NASA Astrophysics Data System (ADS)
Khudaverdian, H. M.
2014-03-01
We consider differential operators acting on densities of arbitrary weights on manifold M identifying pencils of such operators with operators on algebra of densities of all weights. This algebra can be identified with the special subalgebra of functions on extended manifold . On one hand there is a canonical lift of projective structures on M to affine structures on extended manifold . On the other hand the restriction of algebra of all functions on extended manifold to this special subalgebra of functions implies the canonical scalar product. This leads in particular to classification of second order operators with use of Kaluza-Klein-like mechanisms.
Handheld Computer Algebra Systems in the Pre-Algebra Classroom
ERIC Educational Resources Information Center
Gantz, Linda Ann Galofaro
2010-01-01
This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Philip, Bobby; Chartier, Dr Timothy
2012-01-01
methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coecient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem.
Statecharts Via Process Algebra
NASA Technical Reports Server (NTRS)
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumpsmore » and an anisotropy in one part.« less
Palenik, Mark C.; Dunlap, Brett I.
2015-07-28
Despite the fundamental importance of electron density in density functional theory, perturbations are still usually dealt with using Hartree-Fock-like orbital equations known as coupled-perturbed Kohn-Sham (CPKS). As an alternative, we develop a perturbation theory that solves for the perturbed density directly, removing the need for CPKS. This replaces CPKS with a true Hohenberg-Kohn density perturbation theory. In CPKS, the perturbed density is found in the basis of products of occupied and virtual orbitals, which becomes ever more over-complete as the size of the orbital basis set increases. In our method, the perturbation to the density is expanded in terms of a series of density basis functions and found directly. It is possible to solve for the density in such a way that it makes the total energy stationary even if the density basis is incomplete.
Numerical stability in problems of linear algebra.
NASA Technical Reports Server (NTRS)
Babuska, I.
1972-01-01
Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.
The Algebra of Complex Numbers.
ERIC Educational Resources Information Center
LePage, Wilbur R.
This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…
Algebraic Squares: Complete and Incomplete.
ERIC Educational Resources Information Center
Gardella, Francis J.
2000-01-01
Illustrates ways of using algebra tiles to give students a visual model of competing squares that appear in algebra as well as in higher mathematics. Such visual representations give substance to the symbolic manipulation and give students who do not learn symbolically a way of understanding the underlying concepts of completing the square. (KHR)
ERIC Educational Resources Information Center
Buerman, Margaret
2007-01-01
Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…
Online Algebraic Tools for Teaching
ERIC Educational Resources Information Center
Kurz, Terri L.
2011-01-01
Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)
Condensing Algebra for Technical Mathematics.
ERIC Educational Resources Information Center
Greenfield, Donald R.
Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-week…
Algebraic Thinking in Adult Education
ERIC Educational Resources Information Center
Manly, Myrna; Ginsburg, Lynda
2010-01-01
In adult education, algebraic thinking can be a sense-making tool that introduces coherence among mathematical concepts for those who previously have had trouble learning math. Further, a modeling approach to algebra connects mathematics and the real world, demonstrating the usefulness of math to those who have seen it as just an academic…
Linear Algebra and Image Processing
ERIC Educational Resources Information Center
Allali, Mohamed
2010-01-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
ERIC Educational Resources Information Center
Instructional Objectives Exchange, Los Angeles, CA.
A complete set of behavioral objectives for first-year algebra taught in any of grades 8 through 12 is presented. Three to six sample test items and answers are provided for each objective. Objectives were determined by surveying the most used secondary school algebra textbooks. Fourteen major categories are included: (1) whole numbers--operations…
Exploring Algebraic Patterns through Literature.
ERIC Educational Resources Information Center
Austin, Richard A.; Thompson, Denisse R.
1997-01-01
Presents methods for using literature to develop algebraic thinking in an environment that connects algebra to various situations. Activities are based on the book "Anno's Magic Seeds" with additional resources listed. Students express a constant function, exponential function, and a recursive function in their own words as well as writing about…
Learning Algebra from Worked Examples
ERIC Educational Resources Information Center
Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.
2014-01-01
For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is…
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.
Quantum hypercomputation based on the dynamical algebra \\mathfrak{su}(1\\, 1)
NASA Astrophysics Data System (ADS)
Sicard, A.; Ospina, J.; Vélez, M.
2006-10-01
An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is presented. The method that was used was to replace the Weyl Heisenberg algebra by other dynamical algebra of low dimension that admits infinite-dimensional irreducible representations with naturally defined generalized coherent states. We have selected the Lie algebra \\mathfrak{su}(1,1) , because this algebra possesses the necessary characteristics to realize the hypercomputation and also because such algebra has been identified as the dynamical algebra associated with many relatively simple quantum systems. In addition to an algebraic adaptation of KHQA over the algebra \\mathfrak{su}(1,1) , we presented an adaptation of KHQA over some concrete physical referents: the infinite square well, the infinite cylindrical well, the perturbed infinite cylindrical well, the Pöschl Teller potentials, the Holstein Primakoff system and the Laguerre oscillator. We conclude that it is possible to have many physical systems within condensed matter and quantum optics in which it is possible to consider an implementation of KHQA.
Element Agglomeration Algebraic Multilevel Monte-Carlo Library
2015-02-19
ElagMC is a parallel C++ library for Multilevel Monte Carlo simulations with algebraically constructed coarse spaces. ElagMC enables Multilevel variance reduction techniques in the context of general unstructured meshes by using the specialized element-based agglomeration techniques implemented in ELAG (the Element-Agglomeration Algebraic Multigrid and Upscaling Library developed by U. Villa and P. Vassilevski and currently under review for public release). The ElabMC library can support different type of deterministic problems, including mixed finite element discretizations of subsurface flow problems.
Element Agglomeration Algebraic Multilevel Monte-Carlo Library
2015-02-19
ElagMC is a parallel C++ library for Multilevel Monte Carlo simulations with algebraically constructed coarse spaces. ElagMC enables Multilevel variance reduction techniques in the context of general unstructured meshes by using the specialized element-based agglomeration techniques implemented in ELAG (the Element-Agglomeration Algebraic Multigrid and Upscaling Library developed by U. Villa and P. Vassilevski and currently under review for public release). The ElabMC library can support different type of deterministic problems, including mixed finite element discretizationsmore » of subsurface flow problems.« less
On Some Algebraic and Combinatorial Properties of Dunkl Elements
NASA Astrophysics Data System (ADS)
Kirillov, Anatol N.
2013-06-01
We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements and Schubert calculus, in Advances in Geometry (eds. J.-S. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, Boston, 1995), pp. 147-182, A. Postnikov, On a quantum version of Pieri's formula, in Advances in Geometry (eds. J.-S. Brylinski, R. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, 1995), pp. 371-383 and A. N. Kirillov and T. Maenor, A Note on Quantum K-Theory of Flag Varieties, preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [E. Mukhin, V. Tarasov and A. Varchenko, Bethe Subalgebras of the Group Algebra of the Symmetric Group, preprint arXiv:1004.4248]. Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra in a connection with the values of the β-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan-Robbins polytope.
On Some Algebraic and Combinatorial Properties of Dunkl Elements
NASA Astrophysics Data System (ADS)
Kirillov, Anatol N.
2012-11-01
We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements and Schubert calculus, in Advances in Geometry (eds. J.-S. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, Boston, 1995), pp. 147-182, A. Postnikov, On a quantum version of Pieri's formula, in Advances in Geometry (eds. J.-S. Brylinski, R. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, 1995), pp. 371-383 and A. N. Kirillov and T. Maenor, A Note on Quantum K-Theory of Flag Varieties, preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [E. Mukhin, V. Tarasov and A. Varchenko, Bethe Subalgebras of the Group Algebra of the Symmetric Group, preprint arXiv:1004.4248]. Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra in a connection with the values of the β-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan-Robbins polytope.
Miranda, Alex S.; Zanchin, Vilson T.
2006-03-15
We study in detail the quasinormal modes of linear gravitational perturbations of plane-symmetric anti-de Sitter black holes. The wave equations are obtained by means of the Newman-Penrose formalism and the Chandrasekhar transformation theory. We show that oscillatory modes decay exponentially with time such that these black holes are stable against gravitational perturbations. Our numerical results show that in the large (small) black hole regime the frequencies of the ordinary quasinormal modes are proportional to the horizon radius r{sub +} (wave number k). The frequency of the purely damped mode is very close to the algebraically special frequency in the small horizon limit, and goes as ik{sup 2}/3r{sub +} in the opposite limit. This result is confirmed by an analytical method based on the power series expansion of the frequency in terms of the horizon radius. The same procedure applied to the Schwarzschild anti-de Sitter spacetime proves that the purely damped frequency goes as i(l-1)(l+2)/3r{sub +}, where l is the quantum number characterizing the angular distribution. Finally, we study the limit of high overtones and find that the frequencies become evenly spaced in this regime. The spacing of the frequency per unit horizon radius seems to be a universal quantity, in the sense that it is independent of the wave number, perturbation parity, and black hole size.
Automated Lattice Perturbation Theory
Monahan, Christopher
2014-11-01
I review recent developments in automated lattice perturbation theory. Starting with an overview of lattice perturbation theory, I focus on the three automation packages currently "on the market": HiPPy/HPsrc, Pastor and PhySyCAl. I highlight some recent applications of these methods, particularly in B physics. In the final section I briefly discuss the related, but distinct, approach of numerical stochastic perturbation theory.
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)
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.
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.
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.
Dynamical semigroup for unbounded repeated perturbation of an open system
NASA Astrophysics Data System (ADS)
Tamura, Hiroshi; Zagrebnov, Valentin A.
2016-02-01
We consider a dynamical semigroup for unbounded Kossakowski-Lindblad-Davies generator corresponding to evolution of an open system for a tuned repeated harmonic perturbation. For this evolution, we prove the existence of uniquely determined minimal trace-preserving strongly continuous dynamical semigroups on the space of states. The corresponding dual W∗-dynamical system is shown to be unital quasi-free and completely positive automorphisms of the canonical commutation relation-algebra.
Using Homemade Algebra Tiles To Develop Algebra and Prealgebra Concepts.
ERIC Educational Resources Information Center
Leitze, Annette Ricks; Kitt, Nancy A.
2000-01-01
Describes how to use homemade tiles, sketches, and the box method to reach a broader group of students for successful algebra learning. Provides a list of concepts appropriate for such an approach. (KHR)
Distance geometry and geometric algebra
NASA Astrophysics Data System (ADS)
Dress, Andreas W. M.; Havel, Timothy F.
1993-10-01
As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18)
Loop Virasoro Lie conformal algebra
Wu, Henan Chen, Qiufan; Yue, Xiaoqing
2014-01-15
The Lie conformal algebra of loop Virasoro algebra, denoted by CW, is introduced in this paper. Explicitly, CW is a Lie conformal algebra with C[∂]-basis (L{sub i} | i∈Z) and λ-brackets [L{sub i} {sub λ} L{sub j}] = (−∂−2λ)L{sub i+j}. Then conformal derivations of CW are determined. Finally, rank one conformal modules and Z-graded free intermediate series modules over CW are classified.
Transient dynamics of perturbations in astrophysical disks
NASA Astrophysics Data System (ADS)
Razdoburdin, D. N.; Zhuravlev, V. V.
2015-11-01
We review some aspects of a major unsolved problem in understanding astrophysical (in particular, accretion) disks: whether the disk interiors can be effectively viscous in spite of the absence of magnetorotational instability. A rotational homogeneous inviscid flow with a Keplerian angular velocity profile is spectrally stable, making the transient growth of perturbations a candidate mechanism for energy transfer from regular motion to perturbations. Transient perturbations differ qualitatively from perturbation modes and can grow substantially in shear flows due to the nonnormality of their dynamical evolution operator. Because the eigenvectors of this operator, also known as perturbation modes, are not pairwise orthogonal, they can mutually interfere, resulting in the transient growth of their linear combinations. Physically, a growing transient perturbation is a leading spiral whose branches are shrunk as a result of the differential rotation of the flow. We discuss in detail the transient growth of vortex shearing harmonics in the spatially local limit, as well as methods for identifying the optimal (fastest growth) perturbations. Special attention is given to obtaining such solutions variationally by integrating the respective direct and adjoint equations forward and backward in time. The presentation is intended for experts new to the subject.
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.
Forms and algebras in (half-)maximal supergravity theories
NASA Astrophysics Data System (ADS)
Howe, Paul; Palmkvist, Jakob
2015-05-01
The forms in D-dimensional (half-)maximal supergravity theories are discussed for 3 ≤ D ≤ 11. Superspace methods are used to derive consistent sets of Bianchi identities for all the forms for all degrees, and to show that they are soluble and fully compatible with supersymmetry. The Bianchi identities determine Lie superalgebras that can be extended to Borcherds superalgebras of a special type. It is shown that any Borcherds superalgebra of this type gives the same form spectrum, up to an arbitrary degree, as an associated Kac-Moody algebra. For maximal supergravity up to D-form potentials, this is the very extended Kac-Moody algebra E 11. It is also shown how gauging can be carried out in a simple fashion by deforming the Bianchi identities by means of a new algebraic element related to the embedding tensor. In this case the appropriate extension of the form algebra is a truncated version of the so-called tensor hierarchy algebra.
Integrable perturbations of conformal field theories and Yetter-Drinfeld modules
Bücher, David; Runkel, Ingo
2014-11-15
In this paper we relate a problem in representation theory — the study of Yetter-Drinfeld modules over certain braided Hopf algebras — to a problem in two-dimensional quantum field theory, namely, the identification of integrable perturbations of a conformal field theory. A prescription that parallels Lusztig's construction allows one to read off the quantum group governing the integrable symmetry. As an example, we illustrate how the quantum group for the loop algebra of sl(2) appears in the integrable structure of the perturbed uncompactified and compactified free boson.
Frame independent cosmological perturbations
Prokopec, Tomislav; Weenink, Jan E-mail: j.g.weenink@uu.nl
2013-09-01
We compute the third order gauge invariant action for scalar-graviton interactions in the Jordan frame. We demonstrate that the gauge invariant action for scalar and tensor perturbations on one physical hypersurface only differs from that on another physical hypersurface via terms proportional to the equation of motion and boundary terms, such that the evolution of non-Gaussianity may be called unique. Moreover, we demonstrate that the gauge invariant curvature perturbation and graviton on uniform field hypersurfaces in the Jordan frame are equal to their counterparts in the Einstein frame. These frame independent perturbations are therefore particularly useful in relating results in different frames at the perturbative level. On the other hand, the field perturbation and graviton on uniform curvature hypersurfaces in the Jordan and Einstein frame are non-linearly related, as are their corresponding actions and n-point functions.
G-equivariant φ-coordinated quasi modules for quantum vertex algebras
NASA Astrophysics Data System (ADS)
Li, Haisheng
2013-05-01
This is a paper in a series to study quantum vertex algebras and their relations with various quantum algebras. In this paper, we introduce a notion of T-type quantum vertex algebra and a notion of G-equivariant ϕ-coordinated quasi module for a T-type quantum vertex algebra with an automorphism group G. We refine and extend several previous results and we obtain a commutator formula for G-equivariant ϕ-coordinated quasi modules. As an illustrating example, we study a special case of the deformed Virasoro algebra {V}ir_{p,q} with q = -1, to which we associate a Clifford vertex superalgebra and its G-equivariant ϕ-coordinated quasi modules.
Sixth SIAM conference on applied linear algebra: Final program and abstracts. Final technical report
1997-12-31
Linear algebra plays a central role in mathematics and applications. The analysis and solution of problems from an amazingly wide variety of disciplines depend on the theory and computational techniques of linear algebra. In turn, the diversity of disciplines depending on linear algebra also serves to focus and shape its development. Some problems have special properties (numerical, structural) that can be exploited. Some are simply so large that conventional approaches are impractical. New computer architectures motivate new algorithms, and fresh ways to look at old ones. The pervasive nature of linear algebra in analyzing and solving problems means that people from a wide spectrum--universities, industrial and government laboratories, financial institutions, and many others--share an interest in current developments in linear algebra. This conference aims to bring them together for their mutual benefit. Abstracts of papers presented are included.
G-equivariant {phi}-coordinated quasi modules for quantum vertex algebras
Li, Haisheng
2013-05-15
This is a paper in a series to study quantum vertex algebras and their relations with various quantum algebras. In this paper, we introduce a notion of T-type quantum vertex algebra and a notion of G-equivariant {phi}-coordinated quasi module for a T-type quantum vertex algebra with an automorphism group G. We refine and extend several previous results and we obtain a commutator formula for G-equivariant {phi}-coordinated quasi modules. As an illustrating example, we study a special case of the deformed Virasoro algebra Vir{sub p,q} with q=-1, to which we associate a Clifford vertex superalgebra and its G-equivariant {phi}-coordinated quasi modules.
Sequential products on effect algebras
NASA Astrophysics Data System (ADS)
Gudder, Stan; Greechie, Richard
2002-02-01
A sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The properties of sequential products on Hilbert space effect algebras are discussed. For a general SEA, relationships between sequential independence, coexistence and compatibility are given. It is shown that the sharp elements of a SEA form an orthomodular poset. The sequential center of a SEA is discussed and a characterization of when the sequential center is isomorphic to a fuzzy set system is presented. It is shown that the existence, of a sequential product is a strong restriction that eliminates many effect algebras from being SEA's. For example, there are no finite nonboolean SEA's, A measure of sharpness called the sharpness index is studied. The existence of horizontal sums of SEA's is characterized and examples of horizontal sums and tensor products are presented.
Curvature calculations with spacetime algebra
Hestenes, D.
1986-06-01
A new method for calculating the curvature tensor is developed and applied to the Scharzschild case. The method employs Clifford algebra and has definite advantages over conventional methods using differential forms or tensor analysis.
GCD, LCM, and Boolean Algebra?
ERIC Educational Resources Information Center
Cohen, Martin P.; Juraschek, William A.
1976-01-01
This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)
Cartooning in Algebra and Calculus
ERIC Educational Resources Information Center
Moseley, L. Jeneva
2014-01-01
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
NASA Technical Reports Server (NTRS)
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
Semiclassical states on Lie algebras
Tsobanjan, Artur
2015-03-15
The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.
NASA Astrophysics Data System (ADS)
Lannes, A.; Teunissen, P. J. G.
2011-05-01
The first objective of this paper is to show that some basic concepts used in global navigation satellite systems (GNSS) are similar to those introduced in Fourier synthesis for handling some phase calibration problems. In experimental astronomy, the latter are at the heart of what is called `phase closure imaging.' In both cases, the analysis of the related structures appeals to the algebraic graph theory and the algebraic number theory. For example, the estimable functions of carrier-phase ambiguities, which were introduced in GNSS to correct some rank defects of the undifferenced equations, prove to be `closure-phase ambiguities:' the so-called `closure-delay' (CD) ambiguities. The notion of closure delay thus generalizes that of double difference (DD). The other estimable functional variables involved in the phase and code undifferenced equations are the receiver and satellite pseudo-clock biases. A related application, which corresponds to the second objective of this paper, concerns the definition of the clock information to be broadcasted to the network users for their precise point positioning (PPP). It is shown that this positioning can be achieved by simply having access to the satellite pseudo-clock biases. For simplicity, the study is restricted to relatively small networks. Concerning the phase for example, these biases then include five components: a frequency-dependent satellite-clock error, a tropospheric satellite delay, an ionospheric satellite delay, an initial satellite phase, and an integer satellite ambiguity. The form of the PPP equations to be solved by the network user is then similar to that of the traditional PPP equations. As soon as the CD ambiguities are fixed and validated, an operation which can be performed in real time via appropriate decorrelation techniques, estimates of these float biases can be immediately obtained. No other ambiguity is to be fixed. The satellite pseudo-clock biases can thus be obtained in real time. This is
Aspects of QCD current algebra on a null plane
NASA Astrophysics Data System (ADS)
Beane, S. R.; Hobbs, T. J.
2016-09-01
Consequences of QCD current algebra formulated on a light-like hyperplane are derived for the forward scattering of vector and axial-vector currents on an arbitrary hadronic target. It is shown that current algebra gives rise to a special class of sum rules that are direct consequences of the independent chiral symmetry that exists at every point on the two-dimensional transverse plane orthogonal to the lightlike direction. These sum rules are obtained by exploiting the closed, infinite-dimensional algebra satisfied by the transverse moments of null-plane axial-vector and vector charge distributions. In the special case of a nucleon target, this procedure leads to the Adler-Weisberger, Gerasimov-Drell-Hearn, Cabibbo-Radicati and Fubini-Furlan-Rossetti sum rules. Matching to the dispersion-theoretic language which is usually invoked in deriving these sum rules, the moment sum rules are shown to be equivalent to algebraic constraints on forward S-matrix elements in the Regge limit.
Noise Reduction in High-Throughput Gene Perturbation Screens
Technology Transfer Automated Retrieval System (TEKTRAN)
Motivation: Accurate interpretation of perturbation screens is essential for a successful functional investigation. However, the screened phenotypes are often distorted by noise, and their analysis requires specialized statistical analysis tools. The number and scope of statistical methods available...
Hopf algebras and topological recursion
NASA Astrophysics Data System (ADS)
Esteves, João N.
2015-11-01
We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).
2005-04-11
The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.
PREFACE: Algebra, Geometry, and Mathematical Physics 2010
NASA Astrophysics Data System (ADS)
Stolin, A.; Abramov, V.; Fuchs, J.; Paal, E.; Shestopalov, Y.; Silvestrov, S.
2012-02-01
This proceedings volume presents results obtained by the participants of the 6th Baltic-Nordic workshop 'Algebra, Geometry, and Mathematical Physics (AGMP-6)' held at the Sven Lovén Centre for Marine Sciences in Tjärnö, Sweden on October 25-30, 2010. The Baltic-Nordic Network AGMP 'Algebra, Geometry, and Mathematical Physics' http://www.agmp.eu was created in 2005 on the initiative of two Estonian universities and two Swedish universities: Tallinn University of Technology represented by Eugen Paal (coordinator of the network), Tartu University represented by Viktor Abramov, Lund University represented by Sergei Silvestrov, and Chalmers University of Technology and the University of Gothenburg represented by Alexander Stolin. The goal was to promote international and interdisciplinary cooperation between scientists and research groups in the countries of the Baltic-Nordic region in mathematics and mathematical physics, with special emphasis on the important role played by algebra and geometry in modern physics, engineering and technologies. The main activities of the AGMP network consist of a series of regular annual international workshops, conferences and research schools. The AGMP network also constitutes an important educational forum for scientific exchange and dissimilation of research results for PhD students and Postdocs. The network has expanded since its creation, and nowadays its activities extend beyond countries in the Baltic-Nordic region to universities in other European countries and participants from elsewhere in the world. As one of the important research-dissimilation outcomes of its activities, the network has a tradition of producing high-quality research proceedings volumes after network events, publishing them with various international publishers. The PDF also contains the following: List of AGMP workshops and other AGMP activities Main topics discussed at AGMP-6 Review of AGMP-6 proceedings Acknowledgments List of Conference Participants
NASA Astrophysics Data System (ADS)
Kuzmin, Dmitri; Möller, Matthias; Gurris, Marcel
Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCT-like algebraic flux correction schemes for the Euler equations of gas dynamics. In particular, we discuss the construction of artificial viscosity operators, the choice of variables to be limited, and the transformation of antidiffusive fluxes. An a posteriori control mechanism is implemented to make the limiter failsafe. The numerical treatment of initial and boundary conditions is discussed in some detail. The initialization is performed using an FCT-constrained L 2 projection. The characteristic boundary conditions are imposed in a weak sense, and an approximate Riemann solver is used to evaluate the fluxes on the boundary. We also present an unconditionally stable semi-implicit time-stepping scheme and an iterative solver for the fully discrete problem. The results of a numerical study indicate that the nonlinearity and non-differentiability of the flux limiter do not inhibit steady state convergence even in the case of strongly varying Mach numbers. Moreover, the convergence rates improve as the pseudo-time step is increased.
NASA Astrophysics Data System (ADS)
Rong, Shu-Jun; Liu, Qiu-Yu
2012-04-01
The puma model on the basis of the Lorentz and CPT violation may bring an economical interpretation to the conventional neutrinos oscillation and part of the anomalous oscillations. We study the effect of the perturbation to the puma model. In the case of the first-order perturbation which keeps the (23) interchange symmetry, the mixing matrix element Ue3 is always zero. The nonzero mixing matrix element Ue3 is obtained in the second-order perturbation that breaks the (23) interchange symmetry.
Nonnumeric Computer Applications to Algebra, Trigonometry and Calculus.
ERIC Educational Resources Information Center
Stoutemyer, David R.
1983-01-01
Described are computer program packages requiring little or no knowledge of computer programing for college algebra, calculus, and abstract algebra. Widely available computer algebra systems are listed. (MNS)
Fifth SIAM conference on applied linear algebra. Final report
Lewis, J.G.; Gilbert, J.R.; Parlett, B.N.
1994-11-16
The SIAM Conferences on Applied Linear Algebra are the centerpiece of activities for the SIAG on Linear Algebra. They are held every three years and bring together a diverse group of applied linear algebraists, representing industry, government and academics in both matrix theory and matrix computations. This sequence of conferences has two related goals: (1) to be useful and interesting to linear algebraists of every area of specialization, and, (2) to develop and expose connections among problems in different areas. Many aspects of the 1994 conference were carefully chosen to enhance interchange between the various groups and yet still provide a solid focus on specialities. The organizing committee adopted a new meeting structure to resolve the conflict between these two goals at earlier meetings in the series. We have prepared this report for others who may wish to consider our structure as an alternative to more traditional arrangements.
NASA Technical Reports Server (NTRS)
Byrnes, C. I.
1980-01-01
It is noted that recent work by Kamen (1979) on the stability of half-plane digital filters shows that the problem of the existence of a feedback law also arises for other Banach algebras in applications. This situation calls for a realization theory and stabilizability criteria for systems defined over Banach for Frechet algebra A. Such a theory is developed here, with special emphasis placed on the construction of finitely generated realizations, the existence of coprime factorizations for T(s) defined over A, and the solvability of the quadratic optimal control problem and the associated algebraic Riccati equation over A.
Virasoro algebra in the KN algebra; Bosonic string with fermionic ghosts on Riemann surfaces
Koibuchi, H. )
1991-10-10
In this paper the bosonic string model with fermionic ghosts is considered in the framework of the KN algebra. The authors' attentions are paid to representations of KN algebra and a Clifford algebra of the ghosts. The authors show that a Virasoro-like algebra is obtained from KN algebra when KN algebra has certain antilinear anti-involution, and that it is isomorphic to the usual Virasoro algebra. The authors show that there is an expected relation between a central charge of this Virasoro-like algebra and an anomaly of the combined system.
NASA Astrophysics Data System (ADS)
Pogrebkov, A. K.
2016-06-01
We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.
Diagrammatic perturbation theory applied to the ground state of the water molecule
NASA Technical Reports Server (NTRS)
Silver, D. M.; Wilson, S.
1977-01-01
The diagrammatic many-body perturbation theory is applied to the ground state of the water molecule within the algebraic approximation. Using four different basis sets, the total energy, the equilibrium OH bond length, and the equilibrium HOH bond angle are examined. The latter is found to be a particularly sensitive test of the convergence of perturbation expansions. Certain third-order results, which incorporate all two-, three-, and four-body effects, show evidence of good convergence properties.
Re-construction of polynomial potentials with a perturbation-interpolation constraint
NASA Astrophysics Data System (ADS)
Znojil, Miloslav
For all the confining and partially solvable (so called "quasi-solvable", QS) Coulomb plus polynomial oscillators H x= H [QS] and their small perturbations, quantum inversion amounts to the solution of certain nonlinear (Magyari) algebraic equations. Here, strong perturbations of the form H = left( {1 - λ } right)H^{left[ {QS_1 } right]} + λ H^{left[ {QS_2 } right]} are considered and shown tractable along the same lines.
Knots, BPS States, and Algebraic Curves
NASA Astrophysics Data System (ADS)
Garoufalidis, Stavros; Kucharski, Piotr; Sułkowski, Piotr
2016-08-01
We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.
Knots, BPS States, and Algebraic Curves
NASA Astrophysics Data System (ADS)
Garoufalidis, Stavros; Kucharski, Piotr; Sułkowski, Piotr
2016-07-01
We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.
Ternary generalization of Heisenberg's algebra
NASA Astrophysics Data System (ADS)
Kerner, Richard
2015-06-01
A concise study of ternary and cubic algebras with Z3 grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, S3, and its abelian subgroup Z3 may play an important role in quantum physics. We show then how most of important algebras with Z2 grading can be generalized with ternary composition laws combined with a Z3 grading. We investigate in particular a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, , one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra. An analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator, and some properties of its eigenfunctions are briefly discussed.
Beyond Dirac - a Unified Algebra
NASA Astrophysics Data System (ADS)
Lundberg, Wayne R.
2001-10-01
A introductory insight will be shared regarding a 'separation of variables' approach to understanding the relationship between QCD and the origins of cosmological and particle mass. The discussion will then build upon work presented at DFP 2000, focussing on the formal basis for using 3x3x3 matrix algebra as it underlies and extends Dirac notation. A set of restrictions are established which break the multiple symmetries of the 3x3x3 matrix algebra, yielding Standard Model QCD objects and interactions. It will be shown that the 3x3x3 matrix representation unifies the algebra of strong and weak (and by extension, electromagnetic) interactions. A direct correspondence to string theoretic objects is established by considering the string to be partitioned in thirds. Rubik's cube is used as a graphical means of handling algebraic manipulation of 3x3x3 algebra. Further, its potential utility for advancing pedagogical methods through active engagement is discussed. A simulated classroom exercize will be conducted.
Petrov type of linearly perturbed type-D spacetimes
NASA Astrophysics Data System (ADS)
Araneda, Bernardo; Dotti, Gustavo
2015-10-01
We show that a spacetime satisfying the linearized vacuum Einstein equations around a type-D background is generically of type I, and that the splittings of the principal null directions (PNDs) and of the degenerate eigenvalue of the Weyl tensor are non-analytic functions of the perturbation parameter of the metric. This provides a gauge-invariant characterization of the effect of the perturbation on the underlying geometry, without appealing to differential curvature invariants. This is of particular interest for the Schwarzschild solution, for which there are no signatures of the even perturbations on the algebraic curvature invariants. We also show that, unlike the general case, the unstable even modes of the Schwarzschild naked singularity deform the Weyl tensor into a type-II one.
Matter perturbations in scaling cosmology
NASA Astrophysics Data System (ADS)
Fuño, A. Romero; Hipólito-Ricaldi, W. S.; Zimdahl, W.
2016-04-01
A suitable nonlinear interaction between dark matter with an energy density ρM and dark energy with an energy density ρX is known to give rise to a non-canonical scaling ρM ∝ ρXa-ξ, where ξ is a parameter which generally deviates from ξ = 3. Here, we present a covariant generalization of this class of models and investigate the corresponding perturbation dynamics. The resulting matter power spectrum for the special case of a time-varying Lambda model is compared with data from the Sloan Digital Sky Survey (SDSS) DR9 catalogue (Ahn et al.). We find a best-fitting value of ξ = 3.25 which corresponds to a decay of dark matter into the cosmological term. Our results are compatible with the Lambda Cold Dark Matter model at the 2σ confidence level.
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.
Generalized Galilean algebras and Newtonian gravity
NASA Astrophysics Data System (ADS)
González, N.; Rubio, G.; Salgado, P.; Salgado, S.
2016-04-01
The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.
Computer Algebra Systems in Undergraduate Instruction.
ERIC Educational Resources Information Center
Small, Don; And Others
1986-01-01
Computer algebra systems (such as MACSYMA and muMath) can carry out many of the operations of calculus, linear algebra, and differential equations. Use of them with sketching graphs of rational functions and with other topics is discussed. (MNS)
Motivating Activities that Lead to Algebra
ERIC Educational Resources Information Center
Menon, Ramakrishnan
2004-01-01
Four activities consisting of puzzles are introduced, which help students to recognize the strength of algebraic generalizations. They also assist them to comprehend algebraic concepts, and enable them to develop their individual puzzles and games.
Scalable Parallel Algebraic Multigrid Solvers
Bank, R; Lu, S; Tong, C; Vassilevski, P
2005-03-23
The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.
Computational triadic algebras of signs
Zadrozny, W.
1996-12-31
We present a finite model of Peirce`s ten classes of signs. We briefly describe Peirce`s taxonomy of signs; we prove that any finite collection of signs can be extended to a finite algebra of signs in which all interpretants are themselves being interpreted; and we argue that Peirce`s ten classes of signs can be defined using constraints on algebras of signs. The paper opens the possibility of defining multimodal cognitive agents using Peirce`s classes of signs, and is a first step towards building a computational logic of signs based on Peirce`s taxonomies.
Non-perturbative String Theory from Water Waves
Iyer, Ramakrishnan; Johnson, Clifford V.; Pennington, Jeffrey S.; /SLAC
2012-06-14
We use a combination of a 't Hooft limit and numerical methods to find non-perturbative solutions of exactly solvable string theories, showing that perturbative solutions in different asymptotic regimes are connected by smooth interpolating functions. Our earlier perturbative work showed that a large class of minimal string theories arise as special limits of a Painleve IV hierarchy of string equations that can be derived by a similarity reduction of the dispersive water wave hierarchy of differential equations. The hierarchy of string equations contains new perturbative solutions, some of which were conjectured to be the type IIA and IIB string theories coupled to (4, 4k ? 2) superconformal minimal models of type (A, D). Our present paper shows that these new theories have smooth non-perturbative extensions. We also find evidence for putative new string theories that were not apparent in the perturbative analysis.
ERIC Educational Resources Information Center
Star, Jon R.; Rittle-Johnson, Bethany
2009-01-01
Competence in algebra is increasingly recognized as a critical milestone in students' middle and high school years. The transition from arithmetic to algebra is a notoriously difficult one, and improvements in algebra instruction are greatly needed (National Research Council, 2001). Algebra historically has represented students' first sustained…
NASA Technical Reports Server (NTRS)
Criminale, W. O.; Lasseigne, D. G.; Jackson, T. L.
1995-01-01
An initial value approach is used to examine the dynamics of perturbations introduced into a vortex under strain. Both the basic vortex considered and the perturbations are taken as fully three-dimensional. An explicit solution for the time evolution of the vorticity perturbations is given for arbitrary initial vorticity. Analytical solutions for the resulting velocity components are found when the initial vorticity is assumed to be localized. For more general initial vorticity distributions, the velocity components are determined numerically. It is found that the variation in the radial direction of the initial vorticity disturbance is the most important factor influencing the qualitative behavior of the solutions. Transient growth in the magnitude of the velocity components is found to be directly attributable to the compactness of the initial vorticity.
Spatial-Operator Algebra For Robotic Manipulators
NASA Technical Reports Server (NTRS)
Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.
1991-01-01
Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.
The weak Hopf algebras related to generalized Kac-Moody algebra
Wu Zhixiang
2006-06-15
We define a kind of quantized enveloping algebra of a generalized Kac-Moody algebra G by adding a generator J satisfying J{sup m}=J{sup m-1} for some integer m. We denote this algebra by wU{sub q}{sup {tau}}(G). This algebra is a weak Hopf algebra if and only if m=2. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usually quantum envelope algebra U{sub q}(G) of a generalized Kac-Moody algebra G.
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…
UCSMP Algebra. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2007
2007-01-01
"University of Chicago School Mathematics Project (UCSMP) Algebra," designed to increase students' skills in algebra, is appropriate for students in grades 7-10, depending on the students' incoming knowledge. This one-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and…
Graphing Calculator Use in Algebra Teaching
ERIC Educational Resources Information Center
Dewey, Brenda L.; Singletary, Ted J.; Kinzel, Margaret T.
2009-01-01
This study examines graphing calculator technology availability, characteristics of teachers who use it, teacher attitudes, and how use reflects changes to algebra curriculum and instructional practices. Algebra I and Algebra II teachers in 75 high school and junior high/middle schools in a diverse region of a northwestern state were surveyed.…
New family of Maxwell like algebras
NASA Astrophysics Data System (ADS)
Concha, P. K.; Durka, R.; Merino, N.; Rodríguez, E. K.
2016-08-01
We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.
Build an Early Foundation for Algebra Success
ERIC Educational Resources Information Center
Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela
2016-01-01
Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…
A Balancing Act: Making Sense of Algebra
ERIC Educational Resources Information Center
Gavin, M. Katherine; Sheffield, Linda Jensen
2015-01-01
For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…
Difficulties in Initial Algebra Learning in Indonesia
ERIC Educational Resources Information Center
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-01-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…
Teaching Strategies to Improve Algebra Learning
ERIC Educational Resources Information Center
Zbiek, Rose Mary; Larson, Matthew R.
2015-01-01
Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…
Lessons for Algebraic Thinking. Grades K-2.
ERIC Educational Resources Information Center
von Rotz, Leyani; Burns, Marilyn
Algebra is one of the top priorities of mathematics instruction for the elementary and middle grades. This book is designed to help K-2 teachers meet the challenge of making algebra an integral part of their mathematics instruction and realize both what to teach and how to teach central algebraic concepts. Classroom-tested lessons help teachers…
Unifying the Algebra for All Movement
ERIC Educational Resources Information Center
Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.
2015-01-01
There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…
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…
Inequalities, Assessment and Computer Algebra
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in…
ERIC Educational Resources Information Center
Bosse, Michael J.; Ries, Heather; Chandler, Kayla
2012-01-01
Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…
Implementing Change in College Algebra
ERIC Educational Resources Information Center
Haver, William E.
2007-01-01
In this paper, departments are urged to consider implementing the type of changes proposed in Beyond Crossroads in College Algebra. The author of this paper is chair of the Curriculum Renewal Across the First Two Years (CRAFTY) Committee of the Mathematical Association of America. The committee has members from two-year colleges, four-year…
Algebraic Activities Aid Discovery Lessons
ERIC Educational Resources Information Center
Wallace-Gomez, Patricia
2013-01-01
After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…
Fuzzy-algebra uncertainty assessment
Cooper, J.A.; Cooper, D.K.
1994-12-01
A significant number of analytical problems (for example, abnormal-environment safety analysis) depend on data that are partly or mostly subjective. Since fuzzy algebra depends on subjective operands, we have been investigating its applicability to these forms of assessment, particularly for portraying uncertainty in the results of PRA (probabilistic risk analysis) and in risk-analysis-aided decision-making. Since analysis results can be a major contributor to a safety-measure decision process, risk management depends on relating uncertainty to only known (not assumed) information. The uncertainties due to abnormal environments are even more challenging than those in normal-environment safety assessments; and therefore require an even more judicious approach. Fuzzy algebra matches these requirements well. One of the most useful aspects of this work is that we have shown the potential for significant differences (especially in perceived margin relative to a decision threshold) between fuzzy assessment and probabilistic assessment based on subtle factors inherent in the choice of probability distribution models. We have also shown the relation of fuzzy algebra assessment to ``bounds`` analysis, as well as a description of how analyses can migrate from bounds analysis to fuzzy-algebra analysis, and to probabilistic analysis as information about the process to be analyzed is obtained. Instructive examples are used to illustrate the points.
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.
Algebra for All. Research Brief
ERIC Educational Resources Information Center
Bleyaert, Barbara
2009-01-01
The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless national…
ERIC Educational Resources Information Center
Oishi, Lindsay
2011-01-01
"Solve for x." While many people first encountered this enigmatic instruction in high school, the last 20 years have seen a strong push to get students to take algebra in eighth grade or even before. Today, concerns about the economy highlight a familiar worry: American eighth-graders trailed their peers in five Asian countries on the 2007 TIMSS…
Exploring Algebraic Misconceptions with Technology
ERIC Educational Resources Information Center
Sakow, Matthew; Karaman, Ruveyda
2015-01-01
Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…
An Introduction to Algebraic Multigrid
Falgout, R D
2006-04-25
Algebraic multigrid (AMG) solves linear systems based on multigrid principles, but in a way that only depends on the coefficients in the underlying matrix. The author begins with a basic introduction to AMG methods, and then describes some more recent advances and theoretical developments
Elementary Algebra Connections to Precalculus
ERIC Educational Resources Information Center
Lopez-Boada, Roberto; Daire, Sandra Arguelles
2013-01-01
This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…
Adventures in Flipping College Algebra
ERIC Educational Resources Information Center
Van Sickle, Jenna
2015-01-01
This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…
Kinds of Knowledge in Algebra.
ERIC Educational Resources Information Center
Lewis, Clayton
Solving equations in elementary algebra requires knowledge of the permitted operations, and knowledge of what operation to use at a given point in the solution process. While just these kinds of knowledge would be adequate for an ideal solver, human solvers appear to need and use other kinds of knowledge. First, many errors seem to indicate that…
Algebra, Home Mortgages, and Recessions
ERIC Educational Resources Information Center
Mariner, Jean A. Miller; Miller, Richard A.
2009-01-01
The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…
Algebra from Chips and Chopsticks
ERIC Educational Resources Information Center
Yun, Jeong Oak; Flores, Alfinio
2012-01-01
Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…
Algebraic methods in system theory
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Willems, J. C.; Willsky, A. S.
1975-01-01
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
Principals + Algebra (- Fear) = Instructional Leadership
ERIC Educational Resources Information Center
Carver, Cynthia L.
2010-01-01
Recent state legislation in Michigan mandates that all graduating seniors successfully pass algebra I and II. Numerous initiatives have been enacted to help mathematics teachers meet this challenge, yet school principals have had little preparation for the necessary curricular and instructional changes. To address this unmet need, university-based…
Experts Question California's Algebra Edict
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
Business leaders from important sectors of the American economy have been urging schools to set higher standards in math and science--and California officials, in mandating that 8th graders be tested in introductory algebra, have responded with one of the highest such standards in the land. Still, many California educators and school…
Primordial fluctuations from deformed quantum algebras
Day, Andrew C.; Brown, Iain A.; Seahra, Sanjeev S. E-mail: ibrown@astro.uio.no
2014-03-01
We study the implications of deformed quantum algebras for the generation of primordial perturbations from slow-roll inflation. Specifically, we assume that the quantum commutator of the inflaton's amplitude and momentum in Fourier space gets modified at energies above some threshold M{sub *}. We show that when the commutator is modified to be a function of the momentum only, the problem of solving for the post-inflationary spectrum of fluctuations is formally equivalent to solving a one-dimensional Schr and quot;odinger equation with a time dependent potential. Depending on the class of modification, we find results either close to or significantly different from nearly scale invariant spectra. For the former case, the power spectrum is characterized by step-like behaviour at some pivot scale, where the magnitude of the jump is O(H{sup 2}/M{sub *}{sup 2}). (H is the inflationary Hubble parameter.) We use our calculated power spectra to generate predictions for the cosmic microwave background and baryon acoustic oscillations, hence demonstrating that certain types of deformations are incompatible with current observations.
Primordial fluctuations from deformed quantum algebras
NASA Astrophysics Data System (ADS)
Day, Andrew C.; Brown, Iain A.; Seahra, Sanjeev S.
2014-03-01
We study the implications of deformed quantum algebras for the generation of primordial perturbations from slow-roll inflation. Specifically, we assume that the quantum commutator of the inflaton's amplitude and momentum in Fourier space gets modified at energies above some threshold Mstar. We show that when the commutator is modified to be a function of the momentum only, the problem of solving for the post-inflationary spectrum of fluctuations is formally equivalent to solving a one-dimensional Schr"odinger equation with a time dependent potential. Depending on the class of modification, we find results either close to or significantly different from nearly scale invariant spectra. For the former case, the power spectrum is characterized by step-like behaviour at some pivot scale, where the magnitude of the jump is Script O(H2/Mstar2). (H is the inflationary Hubble parameter.) We use our calculated power spectra to generate predictions for the cosmic microwave background and baryon acoustic oscillations, hence demonstrating that certain types of deformations are incompatible with current observations.
Cosmological perturbations in antigravity
NASA Astrophysics Data System (ADS)
Oltean, Marius; Brandenberger, Robert
2014-10-01
We compute the evolution of cosmological perturbations in a recently proposed Weyl-symmetric theory of two scalar fields with oppositely signed conformal couplings to Einstein gravity. It is motivated from the minimal conformal extension of the standard model, such that one of these scalar fields is the Higgs while the other is a new particle, the dilaton, introduced to make the Higgs mass conformally symmetric. At the background level, the theory admits novel geodesically complete cyclic cosmological solutions characterized by a brief period of repulsive gravity, or "antigravity," during each successive transition from a big crunch to a big bang. For simplicity, we consider scalar perturbations in the absence of anisotropies, with potential set to zero and without any radiation. We show that despite the necessarily wrong-signed kinetic term of the dilaton in the full action, these perturbations are neither ghostlike nor tachyonic in the limit of strongly repulsive gravity. On this basis, we argue—pending a future analysis of vector and tensor perturbations—that, with respect to perturbative stability, the cosmological solutions of this theory are viable.
The Exocenter of a Generalized Effect Algebra
NASA Astrophysics Data System (ADS)
Foulis, David J.; Pulmannová, Sylvia
2011-12-01
Elements of the exocenter of a generalized effect algebra (GEA) correspond to decompositions of the GEA as a direct sum and thus the exocenter is a generalization to GEAs of the center of an effect algebra. The exocenter of a GEA is shown to be a boolean algebra, and the notion of a hull mapping for an effect algebra is generalized to a hull system for a GEA. We study Dedekind orthocompleteness of GEAs and extend to GEAs the notion of a centrally orthocomplete effect algebra.
Array algebra estimation in signal processing
NASA Astrophysics Data System (ADS)
Rauhala, U. A.
A general theory of linear estimators called array algebra estimation is interpreted in some terms of multidimensional digital signal processing, mathematical statistics, and numerical analysis. The theory has emerged during the past decade from the new field of a unified vector, matrix and tensor algebra called array algebra. The broad concepts of array algebra and its estimation theory cover several modern computerized sciences and technologies converting their established notations and terminology into one common language. Some concepts of digital signal processing are adopted into this language after a review of the principles of array algebra estimation and its predecessors in mathematical surveying sciences.
Beyond Unitary Parasupersymmetry from the Viewpoint of h3 and h4 Heisenberg Algebras
NASA Astrophysics Data System (ADS)
Chenaghlou, A.; Fakhri, H.
Using the partition of the number p-1 into p-1 real parts which are not equal with each other necessarily, we develop the unitary parasupersymmetry algebra of arbitrary order p so that the well-known Rubakov-Spiridonov-Khare parasupersymmetry becomes a special case of the developed one. It is shown that the developed algebra is realized by simple harmonic oscillator and Landau problem on a flat surface with the symmetries of h3 and h4 Heisenberg-Lie algebras. For this new parasupersymmetry, the well-known unitary condition is violated, however, unitarity of the corresponding algebra is structurally conserved. Moreover, the components of the bosonic Hamiltonian operator are derived as functions from the mean value of the partition numbers with their label weight function.
A High Order Perturbation Analysis of the Sitnikov Problem
NASA Astrophysics Data System (ADS)
Hagel, Johannes; Lhotka, Christoph
2005-09-01
The Sitnikov problem is one of the most simple cases of the elliptic restricted three body system. A massless body oscillates along a line (z) perpendicular to a plane (x,y) in which two equally massive bodies, called primary masses, perform Keplerian orbits around their common barycentre with a given eccentricity e. The crossing point of the line of motion of the third mass with the plane is equal to the centre of gravity of the entire system. In spite of its simple geometrical structure, the system is nonlinear and explicitly time dependent. It is globally non integrable and therefore represents an interesting application for advanced perturbative methods. In the present work a high order perturbation approach to the problem was performed, by using symbolic algorithms written in Mathematica. Floquet theory was used to derive solutions of the linearized equation up to 17th order in e. In this way precise analytical expressions for the stability of the system were obtained. Then, applying the Courant and Snyder transformation to the nonlinear equation, algebraic solutions of seventh order in z and e were derived using the method of Poincaré Lindstedt. The enormous amount of necessary computations were performed by extensive use of symbolic programming. We developed automated and highly modularized algorithms in order to master the problem of ordering an increasing number of algebraic terms originating from high order perturbation theory.
Effective Lagrangians and Current Algebra in Three Dimensions
NASA Astrophysics Data System (ADS)
Ferretti, Gabriele
In this thesis we study three dimensional field theories that arise as effective Lagrangians of quantum chromodynamics in Minkowski space with signature (2,1) (QCD3). In the first chapter, we explain the method of effective Langrangians and the relevance of current algebra techniques to field theory. We also provide the physical motivations for the study of QCD3 as a toy model for confinement and as a theory of quantum antiferromagnets (QAF). In chapter two, we derive the relevant effective Lagrangian by studying the low energy behavior of QCD3, paying particular attention to how the global symmetries are realized at the quantum level. In chapter three, we show how baryons arise as topological solitons of the effective Lagrangian and also show that their statistics depends on the number of colors as predicted by the quark model. We calculate mass splitting and magnetic moments of the soliton and find logarithmic corrections to the naive quark model predictions. In chapter four, we drive the current algebra of the theory. We find that the current algebra is a co -homologically non-trivial generalization of Kac-Moody algebras to three dimensions. This fact may provide a new, non -perturbative way to quantize the theory. In chapter five, we discuss the renormalizability of the model in the large-N expansion. We prove the validity of the non-renormalization theorem and compute the critical exponents in a specific limiting case, the CP^ {N-1} model with a Chern-Simons term. Finally, chapter six contains some brief concluding remarks.
Filiform Lie algebras of order 3
Navarro, R. M.
2014-04-15
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.
Atomic effect algebras with compression bases
Caragheorgheopol, Dan; Tkadlec, Josef
2011-01-15
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.
Atomic effect algebras with compression bases
NASA Astrophysics Data System (ADS)
Caragheorgheopol, Dan; Tkadlec, Josef
2011-01-01
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.
Topology and perturbation theory
NASA Astrophysics Data System (ADS)
Manjavidze, J.
2000-08-01
This paper contains description of the fields nonlinear modes successive quantization scheme. It is shown that the path integrals for absorption part of amplitudes are defined on the Dirac (δ-like) functional measure. This permits arbitrary transformation of the functional integral variables. New form of the perturbation theory achieved by mapping the quantum dynamics in the space WG of the (action, angle)-type collective variables. It is shown that the transformed perturbation theory contributions are accumulated exactly on the boundary ∂WG. Abilities of the developed formalism are illustrated by the Coulomb problem. This model is solved in the WC=(angle, angular momentum, Runge-Lentz vector) space and the reason of its exact integrability is emptiness of ∂WC.
Intermolecular perturbation theory
NASA Astrophysics Data System (ADS)
Hayes, I. C.; Hurst, G. J. B.; Stone, A. J.
The new intermolecular perturbation theory described in the preceding papers is applied to some van der Waals molecules. HeBe is used as a test case, and the perturbation method converges well at interatomic distances down to about 4 a0, giving results in excellent agreement with supermolecule calculations. ArHF and ArHCl have been studied using large basis sets, and the results agree well with experimental data. The ArHX configuration is favoured over the ArXH configuration mainly because of larger polarization and charge-transfer contributions. In NeH2 the equilibrium geometry is determined by a delicate balance between opposing effects; with a double-zeta-polarization basis the correct configuration is predicted.
Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type
NASA Astrophysics Data System (ADS)
Khongsap, Ta; Wang, Weiqiang
2009-01-01
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
2003-06-03
The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran codemore » is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.« less
Differential perturbations and D{sub {infinity}}-differential modules
Lapin, S V
2001-12-31
In the present paper the notions of a D{sub {infinity}}-differential and a D{sub {infinity}}-differential module are introduced, which are, respectively, homotopically invariant analogues of the differential and the chain complex. Basic homotopic properties of D{sub {infinity}}-differentials and D{sub {infinity}}-differential modules are established. The connection between the Gugenheim-Lambe-Stasheff theory of differential perturbations in homological algebra and the construction of a D{sub {infinity}}-differential module is considered.
Single axioms for Boolean algebra.
McCune, W.
2000-06-30
Explicit single axioms are presented for Boolean algebra in terms of (1) the Sheffer stroke; (2) disjunction and negation; (3) disjunction, conjunction, and negation; and (4) disjunction, conjunction, negation, 0, and 1. It was previously known that single axioms exist for these systems, but the procedures to generate them are exponential, producing huge equations. Automated deduction techniques were applied to find axioms of lengths 105, 131, 111, and 127, respectively, each with six variables.
The algebras of large N matrix mechanics
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
Alternative algebraic approaches in quantum chemistry
Mezey, Paul G.
2015-01-22
Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.
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.
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.
Curvature perturbation and waterfall dynamics in hybrid inflation
Abolhasani, Ali Akbar; Firouzjahi, Hassan; Sasaki, Misao E-mail: firouz@mail.ipm.ir
2011-10-01
We investigate the parameter spaces of hybrid inflation model with special attention paid to the dynamics of waterfall field and curvature perturbations induced from its quantum fluctuations. Depending on the inflaton field value at the time of phase transition and the sharpness of the phase transition inflation can have multiple extended stages. We find that for models with mild phase transition the induced curvature perturbation from the waterfall field is too large to satisfy the COBE normalization. We investigate the model parameter space where the curvature perturbations from the waterfall quantum fluctuations vary between the results of standard hybrid inflation and the results obtained here.
Walendziak, Andrzej
2015-01-01
The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained. PMID:26125050
Lax operator algebras and integrable systems
NASA Astrophysics Data System (ADS)
Sheinman, O. K.
2016-02-01
A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. Bibliography: 51 titles.
Algebra: A Challenge at the Crossroads of Policy and Practice
ERIC Educational Resources Information Center
Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.
2011-01-01
The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…
ERIC Educational Resources Information Center
Clarke, Jacqueline
2001-01-01
Presents unique ways to create special rituals that recognize individual students' achievements and milestones. Ideas include throwing a send-off party for a student who is moving; holding monthly birthday luncheons; choosing an ambassador to accompany new students around school; and making a lost tooth container that students can use to safely…
Superstring perturbation theory and Ramond-Ramond backgrounds
Berenstein, D.; Leigh, R.G.
1999-11-01
We consider perturbative type II superstring theory in the covariant NSR formalism in the presence of NSNS and RR backgrounds. A concrete example that we have in mind is the geometry of D3-branes which in the near-horizon region is AdS{sub 5}{times}S{sub 5}, although our methods may be applied to other backgrounds as well. We show how conformal invariance of the string path integral is maintained order by order in the number of holes. This procedure makes uses of the Fischler-Susskind mechanism to build up the background geometry. A simple formal expression is given for a {sigma}-model Lagrangian. This suggests a perturbative expansion in 1/g{sup 2}N and 1/N. As applications, we consider at leading order the mixing of RR and NSNS states, and the realization of the spacetime supersymmetry algebra. {copyright} {ital 1999} {ital The American Physical Society}
Universal perturbative explicitly correlated basis set incompleteness correction
NASA Astrophysics Data System (ADS)
Torheyden, Martin; Valeev, Edward F.
2009-11-01
Basis set incompleteness error for an arbitrary approximate electronic wave function is robustly reduced using a second-order perturbative correction into a basis of explicitly correlated, internally contracted geminal functions. The Hylleraas functional for the second-order energy correction is evaluated algebraically involving at most a four-electron reduced density matrix and four-electron integrals. By using the R12 technology in combination with screening approximations such a correction only requires a two-electron reduced density matrix and two-electron integrals. Preliminary investigations of potential energy surfaces of hydrogen fluoride and nitrogen molecules at the multireference configuration interaction singles and doubles indicate that with the perturbative correction only an aug-cc-pVDZ basis is necessary to compute correlation energies of an aug-cc-pVQZ quality, or better. The proposed correction, dubbed [2]R12, can in principle be combined with any single reference and multireference method in use today.
Discrete reductive perturbation technique
Levi, Decio; Petrera, Matteo
2006-04-15
We expand a partial difference equation (P{delta}E) on multiple lattices and obtain the P{delta}E which governs its far field behavior. The perturbative-reductive approach is here performed on well-known nonlinear P{delta}Es, both integrable and nonintegrable. We study the cases of the lattice modified Korteweg-de Vries (mKdV) equation, the Hietarinta equation, the lattice Volterra-Kac-Van Moerbeke equation and a nonintegrable lattice KdV equation. Such reductions allow us to obtain many new P{delta}Es of the nonlinear Schroedinger type.
Baryon chiral perturbation theory
NASA Astrophysics Data System (ADS)
Scherer, S.
2012-03-01
We provide a short introduction to the one-nucleon sector of chiral perturbation theory and address the issue of power counting and renormalization. We discuss the infrared regularization and the extended on-mass-shell scheme. Both allow for the inclusion of further degrees of freedom beyond pions and nucleons and the application to higher-loop calculations. As applications we consider the chiral expansion of the nucleon mass to order Script O(q6) and the inclusion of vector and axial-vector mesons in the calculation of nucleon form factors. Finally, we address the complex-mass scheme for describing unstable particles in effective field theory.
ERIC Educational Resources Information Center
Cope, Eric W.; Suppes, Patrick
2002-01-01
Examined student performance in distance computer-based calculus and linear algebra courses offered by Stanford University to pre-college students as part of their Education Program for Gifted youth (EPGY). Puts special emphasis on modeling student performance over time and on capturing long-term trend effects using stochastic and nonlinear…
Calculus structure on the Lie conformal algebra complex and the variational complex
De Sole, Alberto; Hekmati, Pedram; Kac, Victor G.
2011-05-15
We construct a calculus structure on the Lie conformal algebra cochain complex. By restricting to degree one chains, we recover the structure of a g-complex introduced in [A. De Sole and V. G. Kac, Commun. Math. Phys. 292, 667 (2009)]. A special case of this construction is the variational calculus, for which we provide explicit formulas.
Resummation of divergent perturbation series: Application to the vibrational states of H2CO molecule
NASA Astrophysics Data System (ADS)
Duchko, A. N.; Bykov, A. D.
2015-10-01
Large-order Rayleigh-Schrödinger perturbation theory (RSPT) is applied to the calculation of anharmonic vibrational energy levels of H2CO molecule. We use the model of harmonic oscillators perturbed by anharmonic terms of potential energy. Since the perturbation series typically diverge due to strong couplings, we apply the algebraic approximation technique because of its effectiveness shown earlier by Goodson and Sergeev [J. Chem. Phys. 110, 8205 (1999); ibid. 124, 094111 (2006)] and in our previous articles [A. D. Bykov et al. Opt. Spectrosc. 114, 396 (2013); ibid. 116, 598 (2014)]. To facilitate the resummation of terms contributing to perturbed states, when resonance mixing between states is especially strong and perturbation series diverge very quick, we used repartition of the Hamiltonian by shifting the normal mode frequencies. Energy levels obtained by algebraic approximants were compared with the results of variational calculation. It was found that for low energy states (up to ˜5000 cm-1), algebraic approximants gave accurate values of energy levels, which were in excellent agreement with the variational method. For highly excited states, strong and multiple resonances complicate series resummation, but a suitable change of normal mode frequencies allows one to reduce the resonance mixing and to get accurate energy levels. The theoretical background of the problem of RSPT series divergence is discussed along with its numerical analysis. For these purposes, the vibrational energy is considered as a function of a complex perturbation parameter. Layout and classification of its singularities allow us to model the asymptotic behavior of the perturbation series and prove the robustness of the algorithm.
Coverings of topological semi-abelian algebras
NASA Astrophysics Data System (ADS)
Mucuk, Osman; Demir, Serap
2016-08-01
In this work, we study on a category of topological semi-abelian algebras which are topological models of given an algebraic theory T whose category of models is semi-abelian; and investigate some results on the coverings of topological models of such theories yielding semi-abelian categories. We also consider the internal groupoid structure in the semi-abelian category of T-algebras, and give a criteria for the lifting of internal groupoid structure to the covering groupoids.
Stability of algebraically unstable dispersive flows
NASA Astrophysics Data System (ADS)
King, Kristina; Zaretzky, Paula; Weinstein, Steven; Cromer, Michael; Barlow, Nathaniel
2015-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to a class of partial differential equations describing wave propagation in dispersive media. There are several morphological differences between algebraically growing disturbances and the exponentially growing wave packets inherent to classical linear stability analysis, and these are elucidated in this study.
Explicit travelling waves and invariant algebraic curves
NASA Astrophysics Data System (ADS)
Gasull, Armengol; Giacomini, Hector
2015-06-01
We introduce a precise definition of algebraic travelling wave solution of n-th order partial differential equations and prove that the only algebraic travelling waves solutions for the celebrated Fisher-Kolmogorov equation are the ones found in 1979 by Ablowitz and Zeppetella. This question is equivalent to study when an associated one-parameter family of planar ordinary differential systems has invariant algebraic curves.
Finite-dimensional simple graded algebras
Bahturin, Yu A; Zaicev, M V; Sehgal, S K
2008-08-31
Let R be a finite-dimensional algebra over an algebraically closed field F graded by an arbitrary group G. In the paper it is proved that if the characteristic of F is zero or does not divide the order of any finite subgroup of G, then R is graded simple if and only if it is isomorphic to a matrix algebra over a finite-dimensional graded skew field. Bibliography: 24 titles.
NASA Astrophysics Data System (ADS)
Manerowska, Anna; Nieznański, Edward; Mulawka, Jan
2013-10-01
Our aim is to present the algebra of concepts in two formal languages. First, after introducing a primary relation between concepts, which is subsumption, we shall specify in a language that uses quantifiers, the Boolean algebra of general concepts. Next, we shall note down the same algebra in simplified non-quantifying language, in order to use it as basis for two specific implementations, i.e. to create the Boolean algebras of deontic concepts and axiological concepts.
Representations of Super Yang-Mills Algebras
NASA Astrophysics Data System (ADS)
Herscovich, Estanislao
2013-06-01
We study in this article the representation theory of a family of super algebras, called the super Yang-Mills algebras, by exploiting the Kirillov orbit method à la Dixmier for nilpotent super Lie algebras. These super algebras are an extension of the so-called Yang-Mills algebras, introduced by A. Connes and M. Dubois-Violette in (Lett Math Phys 61(2):149-158, 2002), and in fact they appear as a "background independent" formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras {{Cliff}q(k) ⊗ Ap(k)}, for p ≥ 3, or p = 2 and q ≥ 2, appear as a quotient of all super Yang-Mills algebras, for n ≥ 3 and s ≥ 1. This provides thus a family of representations of the super Yang-Mills algebras.
Difficulties in initial algebra learning in Indonesia
NASA Astrophysics Data System (ADS)
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-12-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.
Multicloning and Multibroadcasting in Operator Algebras
NASA Astrophysics Data System (ADS)
Kaniowski, Krzysztof; Lubnauer, Katarzyna; Łuczak, Andrzej
2015-12-01
We investigate multicloning and multibroadcasting in the general operator algebra framework in arbitrary dimension, generalizing thus results obtained in this framework for simple cloning and broadcasting.
On Realization of Generalized Effect Algebras
NASA Astrophysics Data System (ADS)
Paseka, Jan
2012-12-01
A well-known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice L(H) of all closed subspaces of a separable complex Hilbert space. We show that a generalized effect algebra is representable in the operator generalized effect algebra G(H) of effects of a complex Hilbert space H iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Riečanová and Zajac. Further, any operator generalized effect algebra G(H) possesses an order determining set of generalized states.
Banach Algebras Associated to Lax Pairs
NASA Astrophysics Data System (ADS)
Glazebrook, James F.
2015-04-01
Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall- Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose-Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand-Naimark-Segal construction. Instrumental is the Plücker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan-Lie-Banach algebras structures can be compared.
Type-Decomposition of an Effect Algebra
NASA Astrophysics Data System (ADS)
Foulis, David J.; Pulmannová, Sylvia
2010-10-01
Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.
NASA Astrophysics Data System (ADS)
Chajda, Ivan
2014-10-01
Commutative BCI-algebras can be considered as semilattices whose sections are equipped with certain involutions. A similar view can be applied to commutative BCK-algebras. However, for general BCK-algebras a certain construction was settled by the author and J. Kühr (Miskolc Math. Notes 8:11-21, 2007) showing that they can be considered as structures essentially weaker than semilattices but still with certain involutions in sections. The aim of this paper is to involve a similar approach for BCI-algebras.
N=2 central charge bounds from 2 d chiral algebras
NASA Astrophysics Data System (ADS)
Lemos, Madalena; Liendo, Pedro
2016-04-01
We study protected correlation functions in N=2 SCFT whose description is captured by a two-dimensional chiral algebra. Our analysis implies a new analytic bound for the c-anomaly as a function of the flavor central charge k, valid for any theory with a flavor symmetry. Combining our result with older bounds in the literature puts strong constraints on the parameter space of N=2 theories. In particular, it singles out a special set of models whose value of c is uniquely fixed once k is given. This set includes the canonical rank one N=2 SCFTs given by Kodaira's classification.
Automorphism groups of composition algebras and quark models
Bjerregard, P.A.; Gonzalez, C.M.
1996-12-01
In this the authors study the automorphisms and derivations of real composition algebras with a view to its physical interpretations. They obtain canonical forms with a special stress in the four and eight dimensional cases. Also, using this description, they work with two mathematical models which describe some particles with certain observables in a surprising way. A first model, split g{sub 2}, describes two observables for three quarks, their antiquarks, and eight mesons combining the quarks involved. A second one, so(4,4) {circle_plus} so(2,2), describes all the observables for all quarks (u, d, s, c, b and t).
Generalized Lotka—Volterra systems connected with simple Lie algebras
NASA Astrophysics Data System (ADS)
Charalambides, Stelios A.; Damianou, Pantelis A.; Evripidou, Charalambos A.
2015-06-01
We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type An for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type An, we produce new integrable Hamiltonian systems.
ERIC Educational Resources Information Center
Ozgun-Koca, S. Ash
2010-01-01
Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This…
Results of Using Algebra Tiles as Meaningful Representations of Algebra Concepts.
ERIC Educational Resources Information Center
Sharp, Janet M.
Mathematical meanings can be developed when individuals construct translations between algebra symbol systems and physical systems that represent one another. Previous research studies indicated (1) few high school students connect whole number manipulations to algebraic manipulations and (2) students who encounter algebraic ideas through…
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.
Canonical floquet perturbation theory
NASA Astrophysics Data System (ADS)
Pohlen, David J.
1992-12-01
Classical Floquet theory is examined in order to generate a canonical transformation to modal variables for periodic system. This transformation is considered canonical if the periodic matrix of eigenvectors is symplectic at the initial time. Approaches for symplectic normalization of the eigenvectors had to be examined for each of the different Poincare eigenvalue cases. Particular attention was required in the degenerate case, which depended on the solution of a generalized eigenvector. Transformation techniques to ensure real modal variables and real periodic eigenvectors were also needed. Periodic trajectories in the restricted three-body case were then evaluated using the canonical Floquet solution. The system used for analyses is the Sun-Jupiter system. This system was especially useful since it contained two of the more difficult Poincare eigenvalue cases, the degenerate case and the imaginary eigenvalue case. The perturbation solution to the canonical modal variables was examined using both an expansion of the Hamiltonian and using a representation that was considered exact. Both methods compared quite well for small perturbations to the initial condition. As expected, the expansion solution failed first due to truncation after the third order term of the expansion.
Maximizing algebraic connectivity in interconnected networks
NASA Astrophysics Data System (ADS)
Shakeri, Heman; Albin, Nathan; Darabi Sahneh, Faryad; Poggi-Corradini, Pietro; Scoglio, Caterina
2016-03-01
Algebraic connectivity, the second eigenvalue of the Laplacian matrix, is a measure of node and link connectivity on networks. When studying interconnected networks it is useful to consider a multiplex model, where the component networks operate together with interlayer links among them. In order to have a well-connected multilayer structure, it is necessary to optimally design these interlayer links considering realistic constraints. In this work, we solve the problem of finding an optimal weight distribution for one-to-one interlayer links under budget constraint. We show that for the special multiplex configurations with identical layers, the uniform weight distribution is always optimal. On the other hand, when the two layers are arbitrary, increasing the budget reveals the existence of two different regimes. Up to a certain threshold budget, the second eigenvalue of the supra-Laplacian is simple, the optimal weight distribution is uniform, and the Fiedler vector is constant on each layer. Increasing the budget past the threshold, the optimal weight distribution can be nonuniform. The interesting consequence of this result is that there is no need to solve the optimization problem when the available budget is less than the threshold, which can be easily found analytically.
The Relative Lie Algebra Cohomology of the Weil Representation
NASA Astrophysics Data System (ADS)
Ralston, Jacob
We study the relative Lie algebra cohomology of so(p,q) with values in the Weil representation piof the dual pair Sp(2k, R) x O(p,q ). Using the Fock model defined in Chapter 2, we filter this complex and construct the associated spectral sequence. We then prove that the resulting spectral sequence converges to the relative Lie algebra cohomology and has E0 term, the associated graded complex, isomorphic to a Koszul complex, see Section 3.4. It is immediate that the construction of the spectral sequence of Chapter 3 can be applied to any reductive subalgebra g ⊂ sp(2k(p + q), R). By the Weil representation of O( p,|q), we mean the twist of the Weil representation of the two-fold cover O(pq)[special character omitted] by a suitable character. We do this to make the center of O(pq)[special character omitted] act trivially. Otherwise, all relative Lie algebra cohomology groups would vanish, see Proposition 4.10.2. In case the symplectic group is large relative to the orthogonal group (k ≥ pq), the E 0 term is isomorphic to a Koszul complex defined by a regular sequence, see 3.4. Thus, the cohomology vanishes except in top degree. This result is obtained without calculating the space of cochains and hence without using any representation theory. On the other hand, in case k < p, we know the Koszul complex is not that of a regular sequence from the existence of the class ϕkq of Kudla and Millson, see te{KM2}, a nonzero element of the relative Lie algebra cohomology of degree kq. For the case of SO0(p, 1) we compute the cohomology groups in these remaining cases, namely k < p. We do this by first computing a basis for the relative Lie algebra cochains and then splitting the complex into a sum of two complexes, each of whose E0 term is then isomorphic to a Koszul complex defined by a regular sequence. This thesis is adapted from the paper, [BMR], this author wrote with his advisor John Millson and Nicolas Bergeron of the University of Paris.
The Algebra of Lexical Semantics
NASA Astrophysics Data System (ADS)
Kornai, András
The current generative theory of the lexicon relies primarily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the generative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of describing discrepancies between syntactic type (lexical category) and semantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR.
Strengthening Effect Algebras in a Logical Perspective: Heyting-Wajsberg Algebras
NASA Astrophysics Data System (ADS)
Konig, Martinvaldo
2014-10-01
Heyting effect algebras are lattice-ordered pseudoboolean effect algebras endowed with a pseudocomplementation that maps on the center (i.e. Boolean elements). They are the algebraic counterpart of an extension of both Łukasiewicz many-valued logic and intuitionistic logic. We show that Heyting effect algebras are termwise equivalent to Heyting-Wajsberg algebras where the two different logical implications are defined as primitive operators. We prove this logic to be decidable, to be strongly complete and to have the deduction-detachment theorem.
Automorphisms and Derivations of the Insertion-Elimination Algebra and Related Graded Lie Algebras
NASA Astrophysics Data System (ADS)
Ondrus, Matthew; Wiesner, Emilie
2016-07-01
This paper addresses several structural aspects of the insertion-elimination algebra {mathfrak{g}}, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of {mathfrak{g}}, the automorphism group of {mathfrak{g}}, the derivation Lie algebra of {mathfrak{g}}, and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.
Discrete Newtonian cosmology: perturbations
NASA Astrophysics Data System (ADS)
Ellis, George F. R.; Gibbons, Gary W.
2015-03-01
In a previous paper (Gibbons and Ellis 2014 Discrete Newtonian cosmology Class. Quantum Grav. 31 025003), we showed how a finite system of discrete particles interacting with each other via Newtonian gravitational attraction would lead to precisely the same dynamical equations for homothetic motion as in the case of the pressure-free Friedmann-Lemaître-Robertson-Walker cosmological models of general relativity theory, provided the distribution of particles obeys the central configuration equation. In this paper we show that one can obtain perturbed such Newtonian solutions that give the same linearized structure growth equations as in the general relativity case. We also obtain the Dmitriev-Zel’dovich equations for subsystems in this discrete gravitational model, and show how it leads to the conclusion that voids have an apparent negative mass.
Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations
Wackerbauer, R. ); Huebler, A. . Center for Complex Systems Research); Mayer-Kress, G. California Univ., Santa Cruz, CA . Dept. of Mathematics)
1991-07-25
The relation between the parameters of a differential equation and corresponding discrete maps are becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. A new iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODE's with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is polynomial and if the ODE has fixed point in the origin. Approximations of different orders respectively of the rest term are investigated for several nonlinear systems. 31 refs., 16 figs.
Algebraic versus Exponential Decoherence in Dissipative Many-Particle Systems.
Cai, Zi; Barthel, Thomas
2013-10-11
The interplay between dissipation and internal interactions in quantum many-body systems gives rise to a wealth of novel phenomena. Here we investigate spin-1/2 chains with uniform local couplings to a Markovian environment using the time-dependent density matrix renormalization group. For the open XXZ model, we discover that the decoherence time diverges in the thermodynamic limit. The coherence decay is then algebraic instead of exponential. This is due to a vanishing gap in the spectrum of the corresponding Liouville superoperator and can be explained on the basis of a perturbative treatment. In contrast, decoherence in the open transverse-field Ising model is found to be always exponential. In this case, the internal interactions can both facilitate and impede the environment-induced decoherence. PMID:24160582
AdS perturbations, isometries, selection rules and the Higgs oscillator
NASA Astrophysics Data System (ADS)
Evnin, Oleg; Nivesvivat, Rongvoram
2016-01-01
Dynamics of small-amplitude perturbations in the global anti-de Sitter (AdS) spacetime is restricted by selection rules that forbid effective energy transfer between certain sets of normal modes. The selection rules arise algebraically because some integrals of products of AdS mode functions vanish. Here, we reveal the relation of these selection rules to AdS isometries. The formulation we discover through this systematic approach is both simpler and stronger than what has been reported previously. In addition to the selection rule considerations, we develop a number of useful representations for the global AdS mode functions, with connections to algebraic structures of the Higgs oscillator, a superintegrable system describing a particle on a sphere in an inverse cosine-squared potential, where the AdS isometries play the role of a spectrum-generating algebra.
Realizations of conformal current-type Lie algebras
Pei Yufeng; Bai Chengming
2010-05-15
In this paper we obtain the realizations of some infinite-dimensional Lie algebras, named 'conformal current-type Lie algebras', in terms of a two-dimensional Novikov algebra and its deformations. Furthermore, Ovsienko and Roger's loop cotangent Virasoro algebra, which can be regarded as a nice generalization of the Virasoro algebra with two space variables, is naturally realized as an affinization of the tensor product of a deformation of the two-dimensional Novikov algebra and the Laurent polynomial algebra. These realizations shed new light on various aspects of the structure and representation theory of the corresponding infinite-dimensional Lie algebras.
Is Algebra Really Difficult for All Students?
ERIC Educational Resources Information Center
Egodawatte, Gunawardena
2009-01-01
Research studies have shown that students encounter difficulties in transitioning from arithmetic to algebra. Errors made by high school students were analyzed for patterns and their causes. The origins of errors were: intuitive assumptions, failure to understand the syntax of algebra, analogies with other familiar symbol systems such as the…
Some Applications of Algebraic System Solving
ERIC Educational Resources Information Center
Roanes-Lozano, Eugenio
2011-01-01
Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact solve"…
A Technology-Intensive Approach to Algebra.
ERIC Educational Resources Information Center
Heid, M. Kathleen; Zbiek, Rose Mary
1995-01-01
Computer-Intensive Algebra (CIA) focuses on the use of technology to help develop a rich understanding of fundamental algebraic concepts in real-world settings using computing tools for easy access to numerical, graphical, and symbolic representations of mathematical ideas. (MKR)
An Inquiry-Based Linear Algebra Class
ERIC Educational Resources Information Center
Wang, Haohao; Posey, Lisa
2011-01-01
Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…
Algebra in the Early Years? Yes!
ERIC Educational Resources Information Center
Taylor-Cox, Jennifer
2003-01-01
Suggests ways early years educators can begin teaching young children to think algebraically and prepare them for success in algebra. Focuses on ways to promote mathematical patterns, mathematical situations and structures, models of quantitative relationship, and change. Describes how first-graders used real object representations to better…
Algebraic Thinking: A Problem Solving Approach
ERIC Educational Resources Information Center
Windsor, Will
2010-01-01
Algebraic thinking is a crucial and fundamental element of mathematical thinking and reasoning. It initially involves recognising patterns and general mathematical relationships among numbers, objects and geometric shapes. This paper will highlight how the ability to think algebraically might support a deeper and more useful knowledge, not only of…
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.
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.
Cartan calculus on quantum Lie algebras
Schupp, P.; Watts, P.; Zumino, B.
1993-12-09
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions au into one big algebra, the ``Cartan Calculus.``
Low Performers Found Unready to Take Algebra
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
As state and school leaders across the country push to have more students take algebra in 8th grade, a new study argues that middle schoolers struggling the most in math are being enrolled in that course despite being woefully unprepared. "The Misplaced Math Student: Lost in Eighth Grade Algebra," scheduled for release by the Brookings Institution…
An algebraic approach to the scattering equations
NASA Astrophysics Data System (ADS)
Huang, Rijun; Rao, Junjie; Feng, Bo; He, Yang-Hui
2015-12-01
We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.
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…
Success in Algebra among Community College Students
ERIC Educational Resources Information Center
Reyes, Czarina
2010-01-01
College algebra is a required course for most majors, but is viewed by many as a gatekeeper course for degree completion by students. With almost half a million students taking college algebra each year, faculty are experimenting with new course lengths of time that might result in higher success, completion, and retention rates for college…
Using the Internet To Investigate Algebra.
ERIC Educational Resources Information Center
Sherwood, Walter
The lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approach algebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…
Algebraic Geodesics on Three-Dimensional Quadrics
NASA Astrophysics Data System (ADS)
Kai, Yue
2015-12-01
By Hamilton-Jacobi method, we study the problem of algebraic geodesics on the third-order surface. By the implicit function theorem, we proved the existences of the real geodesics which are the intersections of two algebraic surfaces, and we also give some numerical examples.
Algebraic Formulas for Areas between Curves.
ERIC Educational Resources Information Center
Gabai, Hyman
1982-01-01
Korean secondary school students preparing for college learn about a simple algebraic formula for area bounded by a parabola and line. The approach does not seem well-known among American students. It is noted that, while the formula derivations rely on integration, algebra students could use the formulas without proofs. (MP)
Classical and quantum Kummer shape algebras
NASA Astrophysics Data System (ADS)
Odzijewicz, A.; Wawreniuk, E.
2016-07-01
We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras, here called Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, are also presented.
Young Mathematicians at Work: Constructing Algebra
ERIC Educational Resources Information Center
Fosnot, Catherine Twomey; Jacob, Bill
2010-01-01
This book provides a landscape of learning that helps teachers recognize, support, and celebrate their students' capacity to structure their worlds algebraically. It identifies the models, contexts, and landmarks that facilitate algebraic thinking in young students and provides insightful and practical methods for teachers, math supervisors, and…
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…
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.
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…
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…
Fusion rule algebras from graph theory
NASA Astrophysics Data System (ADS)
Caselle, M.; Ponzano, G.
1989-06-01
We describe a new class of fusion algebras related to graph theory which bear intriguing connections with group algebras. The structure constants and the matrix S, which diagonalizes the fusion rules, are explicitly computed in terms of SU(2) coupling coefficients.
NINTH YEAR MATHEMATICS. COURSE I, ALGEBRA.
ERIC Educational Resources Information Center
New York State Education Dept., Albany.
THIS GUIDE OUTLINES THE MINIMUM MATERIAL FOR WHICH STUDENTS OF NINTH YEAR MATHEMATICS - COURSE 1 - ALGEBRA WERE HELD RESPONSIBLE ON THE REGENTS EXAMINATIONS BEGINNING IN JUNE, 1966. THE REPORT ALSO PRESENTS THE SCOPE AND CONTENT OF THE ALGEBRA COURSE AND POSSIBLE SUGGESTIONS FOR TEACHING THE MATERIAL TO STUDENTS. (RP)
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…
The Structural Algebra Option: A Discussion Paper.
ERIC Educational Resources Information Center
Kirshner, David
The goal of this paper is to renew interest in the structural option to algebra instruction. Concern for the usual secondary school algebra curriculum related to simplifying expressions, solving equations, and rationalizing numerators and denominators is viewed from three pedagogical approaches: (1) structural approach, (2) empirical approach, and…
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…
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)
Loop realizations of quantum affine algebras
Cautis, Sabin; Licata, Anthony
2012-12-15
We give a simplified description of quantum affine algebras in their loop presentation. This description is related to Drinfeld's new realization via halves of vertex operators. We also define an idempotent version of the quantum affine algebra which is suitable for categorification.
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. PMID:21769158
Perturbed effects at radiation physics
NASA Astrophysics Data System (ADS)
Külahcı, Fatih; Şen, Zekâi
2013-09-01
Perturbation methodology is applied in order to assess the linear attenuation coefficient, mass attenuation coefficient and cross-section behavior with random components in the basic variables such as the radiation amounts frequently used in the radiation physics and chemistry. Additionally, layer attenuation coefficient (LAC) and perturbed LAC (PLAC) are proposed for different contact materials. Perturbation methodology provides opportunity to obtain results with random deviations from the average behavior of each variable that enters the whole mathematical expression. The basic photon intensity variation expression as the inverse exponential power law (as Beer-Lambert's law) is adopted for perturbation method exposition. Perturbed results are presented not only in terms of the mean but additionally the standard deviation and the correlation coefficients. Such perturbation expressions provide one to assess small random variability in basic variables.
Algebraic Apect of Helicities in Hadron Physics
NASA Astrophysics Data System (ADS)
An, Murat; Ji, Chueng
2015-04-01
We examined the relation of polarization vectors and spinors of (1 , 0) ⊕(0 , 1) representation of Lorentz group in Clifford algebra Cl1 , 3 , their relation with standard algebra, and properties of these spinors. Cl1 , 3 consists of different grades:e.g. the first and the second grades represent (1 / 2 , 1 / 2) and (1 , 0) ⊕(0 , 1) representation of spin groups respectively with 4 and 6 components. However, these Clifford numbers are not the helicity eigenstates and thus we transform them into combinations of helicity eigenstates by expressing them as spherical harmonics. We relate the spin-one polarization vectors and (1 , 0) ⊕(0 , 1) spinors under one simple transformation with the spin operators. We also link our work with Winnberg's work of a superfield of a spinors of Clifford algebra by giving a physical meaning to Grassmann variables and discuss how Grassman algebra is linked with Clifford algebra.
Massopust, P.R.
1997-08-01
All solutions of an in its angular coordinates continuously perturbed Laplace-Beltrami equation in the open unit ball IB{sup n+2} {contained_in} IR{sup n+2}, n {ge} 1, are characterized. Moreover, it is shown that such pertubations yield distributional boundary values which are different from, but algebraically and topologically equivalent to, the hyperfunctions of Lions & Magenes. This is different from the case of radially perturbed Laplace-Beltrami operators (cf. [7]) where one has stability of distributional boundary values under such perturbations.
Algebraic K-theory, K-regularity, and -duality of -stable C ∗-algebras
NASA Astrophysics Data System (ADS)
Mahanta, Snigdhayan
2015-12-01
We develop an algebraic formalism for topological -duality. More precisely, we show that topological -duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C ∗-algebra . Then we show that there is a functorial topological K-theory symmetric spectrum construction on the category of separable C ∗-algebras, such that is an algebra over this operad; moreover, is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C ∗-algebras. We also show that -stable C ∗-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of a x+ b-semigroup C ∗-algebras coming from number theory and that of -stabilized noncommutative tori.
Higher gauge theories from Lie n-algebras and off-shell covariantization
NASA Astrophysics Data System (ADS)
Carow-Watamura, Ursula; Heller, Marc Andre; Ikeda, Noriaki; Kaneko, Yukio; Watamura, Satoshi
2016-07-01
We analyze higher gauge theories in various dimensions using a supergeometric method based on a differential graded symplectic manifold, called a QP-manifold, which is closely related to the BRST-BV formalism in gauge theories. Extensions of the Lie 2-algebra gauge structure are formulated within the Lie n-algebra induced by the QP-structure. We find that in 5 and 6 dimensions there are special extensions of the gauge algebra. In these cases, a restriction of the gauge symmetry by imposing constraints on the auxiliary gauge fields leads to a covariantized theory. As an example we show that we can obtain an off-shell covariantized higher gauge theory in 5 dimensions, which is similar to the one proposed in [1].
ALGEBRA: ALgorithm for the heterogeneous dosimetry based on GEANT4 for BRAchytherapy.
Afsharpour, H; Landry, G; D'Amours, M; Enger, S; Reniers, B; Poon, E; Carrier, J-F; Verhaegen, F; Beaulieu, L
2012-06-01
Task group 43 (TG43)-based dosimetry algorithms are efficient for brachytherapy dose calculation in water. However, human tissues have chemical compositions and densities different than water. Moreover, the mutual shielding effect of seeds on each other (interseed attenuation) is neglected in the TG43-based dosimetry platforms. The scientific community has expressed the need for an accurate dosimetry platform in brachytherapy. The purpose of this paper is to present ALGEBRA, a Monte Carlo platform for dosimetry in brachytherapy which is sufficiently fast and accurate for clinical and research purposes. ALGEBRA is based on the GEANT4 Monte Carlo code and is capable of handling the DICOM RT standard to recreate a virtual model of the treated site. Here, the performance of ALGEBRA is presented for the special case of LDR brachytherapy in permanent prostate and breast seed implants. However, the algorithm is also capable of handling other treatments such as HDR brachytherapy. PMID:22572100
ALGEBRA: ALgorithm for the heterogeneous dosimetry based on GEANT4 for BRAchytherapy
NASA Astrophysics Data System (ADS)
Afsharpour, H.; Landry, G.; D'Amours, M.; Enger, S.; Reniers, B.; Poon, E.; Carrier, J.-F.; Verhaegen, F.; Beaulieu, L.
2012-06-01
Task group 43 (TG43)-based dosimetry algorithms are efficient for brachytherapy dose calculation in water. However, human tissues have chemical compositions and densities different than water. Moreover, the mutual shielding effect of seeds on each other (interseed attenuation) is neglected in the TG43-based dosimetry platforms. The scientific community has expressed the need for an accurate dosimetry platform in brachytherapy. The purpose of this paper is to present ALGEBRA, a Monte Carlo platform for dosimetry in brachytherapy which is sufficiently fast and accurate for clinical and research purposes. ALGEBRA is based on the GEANT4 Monte Carlo code and is capable of handling the DICOM RT standard to recreate a virtual model of the treated site. Here, the performance of ALGEBRA is presented for the special case of LDR brachytherapy in permanent prostate and breast seed implants. However, the algorithm is also capable of handling other treatments such as HDR brachytherapy.
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
Resumming the string perturbation series
NASA Astrophysics Data System (ADS)
Grassi, Alba; Mariño, Marcos; Zakany, Szabolcs
2015-05-01
We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local , which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.
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.
On dark energy isocurvature perturbation
Liu, Jie; Zhang, Xinmin; Li, Mingzhe E-mail: limz@nju.edu.cn
2011-06-01
Determining the equation of state of dark energy with astronomical observations is crucially important to understand the nature of dark energy. In performing a likelihood analysis of the data, especially of the cosmic microwave background and large scale structure data the dark energy perturbations have to be taken into account both for theoretical consistency and for numerical accuracy. Usually, one assumes in the global fitting analysis that the dark energy perturbations are adiabatic. In this paper, we study the dark energy isocurvature perturbation analytically and discuss its implications for the cosmic microwave background radiation and large scale structure. Furthermore, with the current astronomical observational data and by employing Markov Chain Monte Carlo method, we perform a global analysis of cosmological parameters assuming general initial conditions for the dark energy perturbations. The results show that the dark energy isocurvature perturbations are very weakly constrained and that purely adiabatic initial conditions are consistent with the data.
Computing singularities of perturbation series
Kvaal, Simen; Jarlebring, Elias; Michiels, Wim
2011-03-15
Many properties of current ab initio approaches to the quantum many-body problem, both perturbational and otherwise, are related to the singularity structure of the Rayleigh-Schroedinger perturbation series. A numerical procedure is presented that in principle computes the complete set of singularities, including the dominant singularity which limits the radius of convergence. The method approximates the singularities as eigenvalues of a certain generalized eigenvalue equation which is solved using iterative techniques. It relies on computation of the action of the Hamiltonian matrix on a vector and does not rely on the terms in the perturbation series. The method can be useful for studying perturbation series of typical systems of moderate size, for fundamental development of resummation schemes, and for understanding the structure of singularities for typical systems. Some illustrative model problems are studied, including a helium-like model with {delta}-function interactions for which Moeller-Plesset perturbation theory is considered and the radius of convergence found.
Phase Boundaries in Algebraic Conformal QFT
NASA Astrophysics Data System (ADS)
Bischoff, Marcel; Kawahigashi, Yasuyuki; Longo, Roberto; Rehren, Karl-Henning
2016-02-01
We study the structure of local algebras in relativistic conformal quantum field theory with phase boundaries. Phase boundaries are instances of a more general notion of boundaries that give rise to a variety of algebraic structures. These can be formulated in a common framework originating in Algebraic QFT, with the principle of Einstein Causality playing a prominent role. We classify the phase boundary conditions by the centre of a certain universal construction, which produces a reducible representation in which all possible boundary conditions are realized. For a large class of models, the classification reproduces results obtained in a different approach by Fuchs et al. before.
Toward robust scalable algebraic multigrid solvers.
Waisman, Haim; Schroder, Jacob; Olson, Luke; Hiriyur, Badri; Gaidamour, Jeremie; Siefert, Christopher; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen
2010-10-01
This talk highlights some multigrid challenges that arise from several application areas including structural dynamics, fluid flow, and electromagnetics. A general framework is presented to help introduce and understand algebraic multigrid methods based on energy minimization concepts. Connections between algebraic multigrid prolongators and finite element basis functions are made to explored. It is shown how the general algebraic multigrid framework allows one to adapt multigrid ideas to a number of different situations. Examples are given corresponding to linear elasticity and specifically in the solution of linear systems associated with extended finite elements for fracture problems.
Algebraic method for finding equivalence groups
NASA Astrophysics Data System (ADS)
Bihlo, Alexander; Dos Santos Cardoso-Bihlo, Elsa; Popovych, Roman O.
2015-06-01
The algebraic method for computing the complete point symmetry group of a system of differential equations is extended to finding the complete equivalence group of a class of such systems. The extended method uses the knowledge of the corresponding equivalence algebra. Two versions of the method are presented, where the first involves the automorphism group of this algebra and the second is based on a list of its megaideals. We illustrate the megaideal-based version of the method with the computation of the complete equivalence group of a class of nonlinear wave equations with applications in nonlinear elasticity.
The nth root of sequential effect algebras
NASA Astrophysics Data System (ADS)
Shen, Jun; Wu, Junde
2010-06-01
In 2005, Gudder [Int. J. Theor. Phys. 44, 2219 (2005)] presented 25 problems of sequential effect algebras, the 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique? In this paper, we show that for each given positive integer n >1, there is a sequential effect algebra such that the nth root of its some element c is not unique, and the nth root of c is not the kth root of c (k
On computational complexity of Clifford algebra
NASA Astrophysics Data System (ADS)
Budinich, Marco
2009-05-01
After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced by Budinich and Budinich ["A spinorial formulation of the maximum clique problem of a graph," J. Math. Phys. 47, 043502 (2006)].
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.
Contractions of affine Kac-Moody algebras
NASA Astrophysics Data System (ADS)
Daboul, J.; Daboul, C.; de Montigny, M.
2008-08-01
I review our recent work on contractions of affine Kac-Moody algebras (KMA) and present new results. We study generalized contractions of KMA with respect to their twisted and untwisted KM subalgebras. As a concrete example, we discuss contraction of D(1)4 and D(3)4, based on Z3-grading. We also describe examples of 'level-dependent' contractions, which are based on Z-gradings of KMA. Our work generalizes the Inönü-Wigner contraction of P. Majumdar in several directions. We also give an algorithm for constructing Kac-Moody-like algebras hat g for any Lie algebra g.
Statistical anisotropy of the curvature perturbation from vector field perturbations
Dimopoulos, Konstantinos; Karciauskas, Mindaugas; Lyth, David H.; Rodriguez, Yeinzon E-mail: m.karciauskas@lancaster.ac.uk E-mail: yeinzon.rodriguez@uan.edu.co
2009-05-15
The {delta}N formula for the primordial curvature perturbation {zeta} is extended to include vector as well as scalar fields. Formulas for the tree-level contributions to the spectrum and bispectrum of {zeta} are given, exhibiting statistical anisotropy. The one-loop contribution to the spectrum of {zeta} is also worked out. We then consider the generation of vector field perturbations from the vacuum, including the longitudinal component that will be present if there is no gauge invariance. Finally, the {delta}N formula is applied to the vector curvaton and vector inflation models with the tensor perturbation also evaluated in the latter case.
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.
Alternative algebras admitting derivations with invertible values and invertible derivations
NASA Astrophysics Data System (ADS)
Kaygorodov, I. B.; Popov, Yu S.
2014-10-01
We prove an analogue of the Bergen-Herstein-Lanski theorem for alternative algebras: describe all alternative algebras that admit derivations with invertible values. We also prove an analogue of Moens' theorem for alternative algebras (a finite-dimensional alternative algebra over a field of characteristic zero is nilpotent if and only if it admits an invertible Leibniz derivation).
Spinor-vector supersymmetry algebra in three dimensions
NASA Astrophysics Data System (ADS)
Shima, Kazunari; Tsuda, Motomu
2006-06-01
We focus on a spin-3/2 supersymmetry (SUSY) algebra of Baaklini in D = 3 and explicitly show a nonlinear realization of the SUSY algebra. The unitary representation of the spin-3/2 SUSY algebra is discussed and compared with the ordinary (spin-1/2) SUSY algebra.
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.
Lie bialgebra structures on the Schroedinger-Virasoro Lie algebra
Han Jianzhi; Su Yucai; Li Junbo
2009-08-15
In this paper we shall investigate Lie bialgebra structures on the Schroedinger-Virasoro algebra L. We found out that not all Lie bialgebra structures on the Schroedinger-Virasoro algebra are triangular coboundary, which is different from the related known results of some other Lie algebras related to the Virasoro algebra.
Study of dose perturbation parameters for eye shielding in megavoltage photon beam therapy
Das, I.J.; Kase, K.R.; Fitzgerald, T.J.; Ligon, D.A. )
1990-08-01
Shielding blocks are frequently used to minimize dose and shield sensitive organs in radiation therapy. The blocks, which are made of high atomic number materials, produce significant dose perturbations in megavoltage photon beams. The effects of these perturbations are studied with special interest in the eye shielding in the treatment of head and neck malignancies. Optimum parameters for the treatment are suggested.
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.
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)
Top Element Problem and Macneille Completions of Generalized Effect Algebras
NASA Astrophysics Data System (ADS)
RieČanová, Z.; Kalina, M.
2014-10-01
Effect algebras (EAs), introduced by D. J. Foulis and M. K. Bennett, as common generalizations of Boolean algebras, orthomodular lattices and MV-algebras, are nondistributive algebraic structures including unsharp elements. Their unbounded versions, called generalized effect algebras, are posets which may have or may have not an EA-MacNeille completion, or cannot be embedded into any complete effect algebra. We give a necessary and sufficient condition for a generalized effect algebra to have an EA-MacNeille completion. Some examples are provided.
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…
Clifford Algebras in Symplectic Geometry and Quantum Mechanics
NASA Astrophysics Data System (ADS)
Binz, Ernst; de Gosson, Maurice A.; Hiley, Basil J.
2013-04-01
The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C 0,2. This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, {F}a of the Euclidean three-space. This enables us to construct a Poisson Clifford algebra, ℍ F , of a finite dimensional phase space which will carry the dynamics. The quantum dynamics appears as a realisation of ℍ F in terms of a Clifford algebra consisting of Hermitian operators.
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
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.
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.
Excision in algebraic K-theory and Karoubi's conjecture.
Suslin, A A; Wodzicki, M
1990-12-15
We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130
Excision in algebraic K-theory and Karoubi's conjecture.
Suslin, A A; Wodzicki, M
1990-01-01
We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130
On algebraic endomorphisms of the Einstein gyrogroup
NASA Astrophysics Data System (ADS)
Molnár, Lajos; Virosztek, Dániel
2015-08-01
We describe the structure of all continuous algebraic endomorphisms of the open unit ball B of ℝ3 equipped with the Einstein velocity addition. We show that any nonzero such transformation originates from an orthogonal linear transformation on ℝ3.
Clifford algebras and physical and engineering sciences
NASA Astrophysics Data System (ADS)
Furui, Sadataka
2013-10-01
Clifford algebra in physical and engineering science are studied. Roles of triality symmetry of Cartan's spinor in axial anomaly of particle physics and quaternion and octonion in the memristic circuits are discussed.
Positive basis for surface skein algebras
Thurston, Dylan Paul
2014-01-01
We show that the twisted SL2 skein algebra of a surface has a natural basis (the bracelets basis) that is positive, in the sense that the structure constants for multiplication are positive integers. PMID:24982193
Lisa's Lemonade Stand: Exploring Algebraic Ideas.
ERIC Educational Resources Information Center
Billings, Esther M. H.; Lakatos, Tracy
2003-01-01
Presents an activity, "Lisa's Lemonade Stand," that actively engages students in algebraic thinking as they analyze change by investigating relationships between variables and gain experience describing and representing these relationships graphically. (YDS)
Lima Beans, Paper Cups, and Algebra.
ERIC Educational Resources Information Center
Loewen, A. C.
1991-01-01
An activity in which students use manipulative materials to help solve simple algebraic equations using the operations of adding inverses, removing opposites, and sharing equally is presented. Directions, examples, the rationale, and cautions are included. (KR)
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.
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)
Semigroups And Computer Algebra In Discrete Structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2010-10-01
Some concepts in semigroup theory are interpreted in discrete structures such as finite lattices, binary relations, and finite semilattices. An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. By analogy with the linear spaces in the linear algebra semilattice homomorphisms, isomorphisms, projections on Boolean vector spaces are defined and some properties of them are investigated in detail. Maps, corresponding to them in the linear algebra, are connected with matrices and their pseudouinverse. Important properties of these maps, which are essential for solving linear systems, remain the same in the Boolean vector spaces. Stochastic experiments using the maps defined and computer algebra methods have been made for solving linear equations Ax = b. The Hamming distance between b and the projection p(b) = Ax of b is equal or close to the least possible one, if the system has no solutions.
Cosmological perturbations during radion stabilization
NASA Astrophysics Data System (ADS)
Ashcroft, P. R.; van de Bruck, C.; Davis, A.-C.
2005-01-01
We consider the evolution of cosmological perturbations during radion stabilization, which we assume to happen after a period of inflation in the early universe. Concentrating on the Randall-Sundrum brane world scenario, we find that, if matter is present both on the positive and negative tension branes, the coupling of the radion to matter fields could have significant impact on the evolution of the curvature perturbation and on the production of entropy perturbations. We investigate both the case of a long-lived and a short-lived radion and outline similarities and differences to the curvaton scenario.
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.
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.
Twisted Logarithmic Modules of Vertex Algebras
NASA Astrophysics Data System (ADS)
Bakalov, Bojko
2016-07-01
Motivated by logarithmic conformal field theory and Gromov-Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted fields involve the logarithm of the formal variable. We develop the theory of such twisted modules and, in particular, derive a Borcherds identity and commutator formula for them. We investigate in detail the examples of affine and Heisenberg vertex algebras.
Edge covers and independence: Algebraic approach
NASA Astrophysics Data System (ADS)
Kalinina, E. A.; Khitrov, G. M.; Pogozhev, S. V.
2016-06-01
In this paper, linear algebra methods are applied to solve some problems of graph theory. For ordinary connected graphs, edge coverings and independent sets are considered. Some results concerning minimum edge covers and maximum matchings are proved with the help of linear algebraic approach. The problem of finding a maximum matching of a graph is fundamental both practically and theoretically, and has numerous applications, e.g., in computational chemistry and mathematical chemistry.
Filtering Algebraic Multigrid and Adaptive Strategies
Nagel, A; Falgout, R D; Wittum, G
2006-01-31
Solving linear systems arising from systems of partial differential equations, multigrid and multilevel methods have proven optimal complexity and efficiency properties. Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner. This paper proposes a variant of Wagner's method with substantially improved robustness properties. The method is used in an adaptive, self-correcting framework and tested numerically.
Sharply Dominating MV-Effect Algebras
NASA Astrophysics Data System (ADS)
Kalina, Martin; Olejček, Vladimír; Paseka, Jan; Riečanová, Zdenka
2011-04-01
Some open questions on Archimedean atomic MV-effect algebras are answered. Namely we prove that there are Archimedean atomic MV-effect algebras which are not sharply dominating. Equivalently, they don't have a basic decomposition of elements. Moreover, if their set of sharp elements (their center) is a complete lattice then they need not be complete lattices. The existence of infinite orthogonal sums of their elements is discussed.
Stability of Lie groupoid C∗-algebras
NASA Astrophysics Data System (ADS)
Debord, Claire; Skandalis, Georges
2016-07-01
In this paper we generalize a theorem of M. Hilsum and G. Skandalis stating that the C∗-algebra of any foliation of nonzero dimension is stable. Precisely, we show that the C∗-algebra of a Lie groupoid is stable whenever the groupoid has no orbit of dimension zero. We also prove an analogous theorem for singular foliations for which the holonomy groupoid as defined by I. Androulidakis and G. Skandalis is not Lie in general.
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
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.
One-Equation Algebraic Model Of Turbulence
NASA Technical Reports Server (NTRS)
Baldwin, B. S.; Barth, T. J.
1993-01-01
One-equation model of turbulence based on standard equations of k-epsilon model of turbulence, where k is turbulent energy and e is rate of dissipation of k. Derivation of one-equation model motivated partly by inaccuracies of flows computed by some Navier-Stokes-equations-solving algorithms incorporating algebraic models of turbulence. Satisfies need to avoid having to determine algebraic length scales.
The arithmetic theory of algebraic groups
NASA Astrophysics Data System (ADS)
Platonov, V. P.
1982-06-01
CONTENTS Introduction § 1. Arithmetic groups § 2. Adèle groups § 3. Tamagawa numbers § 4. Approximations in algebraic groups § 5. Class numbers and class groups of algebraic groups § 6. The genus problem in arithmetic groups § 7. Classification of maximal arithmetic subgroups § 8. The congruence problem § 9. Groups of rational points over global fields § 10. Galois cohomology and the Hasse principle § 11. Cohomology of arithmetic groups References
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”.
Clifford algebras and Hestenes spinors
NASA Astrophysics Data System (ADS)
Lounesto, Pertti
1993-09-01
This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within the real Clifford algebra Cl 1,3 ≃ M2 (H). Hestenes invented first in 1966 his ideal spinorsφ in Cl_{1,3 _2}^1 (1 - γ _{03} ) and later 1967/75 he recognized the importance of his operator spinors ψ ∈ Cl{/1,3 + } ≃ M2 (C). This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decomposition C ⊗ Cl 1,3 and not in C ⊗ M4 (R)=M4 (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor. Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ψ†γ0ψ=0, ψ†γ0γ0123ψ=0. The null case is thoroughly scrutinized for the first time with a new concept called boomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation. Projection operators P±, Σ± are resettled among their new relatives in End(Cl 1,3 ). Finally, a new mapping, called tilt, is introduced to enable a transition from Cl 1,3 to the (graded) opposite algebra Cl 3,1 without resorting to complex numbers, that is, not using a replacement γμ → iγμ.
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)}.
Causal compensated perturbations in cosmology
NASA Technical Reports Server (NTRS)
Veeraraghavan, Shoba; Stebbins, Albert
1990-01-01
A theoretical framework is developed to calculate linear perturbations in the gravitational and matter fields which arise causally in response to the presence of stiff matter sources in a FRW cosmology. It is shown that, in order to satisfy energy and momentum conservation, the gravitational fields of the source must be compensated by perturbations in the matter and gravitational fields, and the role of such compensation in containing the initial inhomogeneities in their subsequent evolution is discussed. A complete formal solution is derived in terms of Green functions for the perturbations produced by an arbitrary source in a flat universe containing cold dark matter. Approximate Green function solutions are derived for the late-time density perturbations and late-time gravitational waves in a universe containing a radiation fluid. A cosmological energy-momentum pseudotensor is defined to clarify the nature of energy and momentum conservation in the expanding universe.
Isocurvature perturbations in extra radiation
Kawasaki, Masahiro; Miyamoto, Koichi; Nakayama, Kazunori; Sekiguchi, Toyokazu E-mail: miyamone@icrr.u-tokyo.ac.jp E-mail: oyokazu.sekiguchi@nagoya-u.jp
2012-02-01
Recent cosmological observations, including measurements of the CMB anisotropy and the primordial helium abundance, indicate the existence of an extra radiation component in the Universe beyond the standard three neutrino species. In this paper we explore the possibility that the extra radiation has isocurvatrue fluctuations. A general formalism to evaluate isocurvature perturbations in the extra radiation is provided in the mixed inflaton-curvaton system, where the extra radiation is produced by the decay of both scalar fields. We also derive constraints on the abundance of the extra radiation and the amount of its isocurvature perturbation. Current observational data favors the existence of an extra radiation component, but does not indicate its having isocurvature perturbation. These constraints are applied to some particle physics motivated models. If future observations detect isocurvature perturbations in the extra radiation, it will give us a hint to the origin of the extra radiation.
Robust stability under additive perturbations
NASA Technical Reports Server (NTRS)
Bhaya, A.; Desoer, C. A.
1985-01-01
A MIMO linear time-invariant feedback system 1S(P,C) is considered which is assumed to be U-stable. The plant P is subjected to an additive perturbation Delta P which is proper but not necessarily stable. It is proved that the perturbed system is U-stable if and only if Delta P(I + Q x Delta P) exp -1 is U-stable.
Duchko, A. N.; Bykov, A. D.
2015-10-21
Large-order Rayleigh–Schrödinger perturbation theory (RSPT) is applied to the calculation of anharmonic vibrational energy levels of H{sub 2}CO molecule. We use the model of harmonic oscillators perturbed by anharmonic terms of potential energy. Since the perturbation series typically diverge due to strong couplings, we apply the algebraic approximation technique because of its effectiveness shown earlier by Goodson and Sergeev [J. Chem. Phys. 110, 8205 (1999); ibid. 124, 094111 (2006)] and in our previous articles [A. D. Bykov et al. Opt. Spectrosc. 114, 396 (2013); ibid. 116, 598 (2014)]. To facilitate the resummation of terms contributing to perturbed states, when resonance mixing between states is especially strong and perturbation series diverge very quick, we used repartition of the Hamiltonian by shifting the normal mode frequencies. Energy levels obtained by algebraic approximants were compared with the results of variational calculation. It was found that for low energy states (up to ∼5000 cm{sup −1}), algebraic approximants gave accurate values of energy levels, which were in excellent agreement with the variational method. For highly excited states, strong and multiple resonances complicate series resummation, but a suitable change of normal mode frequencies allows one to reduce the resonance mixing and to get accurate energy levels. The theoretical background of the problem of RSPT series divergence is discussed along with its numerical analysis. For these purposes, the vibrational energy is considered as a function of a complex perturbation parameter. Layout and classification of its singularities allow us to model the asymptotic behavior of the perturbation series and prove the robustness of the algorithm.
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…
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…
ERIC Educational Resources Information Center
Gupta, Neil Kenneth
2004-01-01
Students often look for real-life situations where they can apply the concepts they learn. A project of measuring the speed of moving cars demonstrates that they learn communication skills, teamwork skills, and develop patience specially when work in a group with a common purpose.
Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms
NASA Astrophysics Data System (ADS)
Benayadi, Saïd; Makhlouf, Abdenacer
2014-02-01
The aim of this paper is to introduce and study quadratic Hom-Lie algebras, which are Hom-Lie algebras equipped with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the Double Extension Theory to this class of nonassociative algebras. Elements of Representation Theory for Hom-Lie algebras, including adjoint and coadjoint representations, are supplied with application to quadratic Hom-Lie algebras. Centerless involutive quadratic Hom-Lie algebras are characterized. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. Also, we establish a correspondence between the class of involutive quadratic Hom-Lie algebras and quadratic simple Lie algebras with symmetric involution.
Perturbation of eigenvalues of preconditioned Navier-Stokes operators
Elman, H.C.
1996-12-31
We study the sensitivity of algebraic eigenvalue problems associated with matrices arising from linearization and discretization of the steady-state Navier-Stokes equations. In particular, for several choices of preconditioners applied to the system of discrete equations, we derive upper bounds on perturbations of eigenvalues as functions of the viscosity and discretization mesh size. The bounds suggest that the sensitivity of the eigenvalues is at worst linear in the inverse of the viscosity and quadratic in the inverse of the mesh size, and that scaling can be used to decrease the sensitivity in some cases. Experimental results supplement these results and confirm the relatively mild dependence on viscosity. They also indicate a dependence on the mesh size of magnitude smaller than the analysis suggests.
Three-algebra for supermembrane and two-algebra for superstring
NASA Astrophysics Data System (ADS)
Lee, Kanghoon; Park, Jeong-Hyuck
2009-04-01
While string or Yang-Mills theories are based on Lie algebra or two-algebra structure, recent studies indicate that Script M-theory may require a one higher, three-algebra structure. Here we construct a covariant action for a supermembrane in eleven dimensions, which is invariant under global supersymmetry, local fermionic symmetry and worldvolume diffeomorphism. Our action is classically on-shell equivalent to the celebrated Bergshoeff-Sezgin-Townsend action. However, the novelty is that we spell the action genuinely in terms of Nambu three-brackets: All the derivatives appear through Nambu brackets and hence it manifests the three-algebra structure. Further the double dimensional reduction of our action gives straightforwardly to a type IIA string action featuring two-algebra. Applying the same method, we also construct a covariant action for type IIB superstring, leading directly to the IKKT matrix model.
NASA Astrophysics Data System (ADS)
Kimura, Yusuke
2015-07-01
It has been understood that correlation functions of multi-trace operators in SYM can be neatly computed using the group algebra of symmetric groups or walled Brauer algebras. On the other hand, such algebras have been known to construct 2D topological field theories (TFTs). After reviewing the construction of 2D TFTs based on symmetric groups, we construct 2D TFTs based on walled Brauer algebras. In the construction, the introduction of a dual basis manifests a similarity between the two theories. We next construct a class of 2D field theories whose physical operators have the same symmetry as multi-trace operators constructed from some matrices. Such field theories correspond to non-commutative Frobenius algebras. A matrix structure arises as a consequence of the noncommutativity. Correlation functions of the Gaussian complex multi-matrix models can be translated into correlation functions of the two-dimensional field theories.
NASA Astrophysics Data System (ADS)
Hamacher, K.
2010-09-01
Elastic network models in their different flavors have become useful models for the dynamics and functions of biomolecular systems such as proteins and their complexes. Perturbation to the interactions occur due to randomized and fixated changes (in molecular evolution) or designed modifications of the protein structures (in bioengineering). These perturbations are modifications in the topology and the strength of the interactions modeled by the elastic network models. We discuss how a naive approach to compute properties for a large number of perturbed structures and interactions by repeated diagonalization can be replaced with an identity found in linear algebra. We argue about the computational complexity and discuss the advantages of the protocol. We apply the proposed algorithm to the acetylcholinesterase, a well-known enzyme in neurobiology, and show how one can gain insight into the "breathing dynamics" of a structural funnel necessary for the function of the protein. The computational speed-up was a 60-fold increase in this example.
Perturbation spreading in many-particle systems: a random walk approach.
Zaburdaev, V; Denisov, S; Hänggi, P
2011-05-01
The propagation of an initially localized perturbation via an interacting many-particle Hamiltonian dynamics is investigated. We argue that the propagation of the perturbation can be captured by the use of a continuous-time random walk where a single particle is traveling through an active, fluctuating medium. Employing two archetype ergodic many-particle systems, namely, (i) a hard-point gas composed of two unequal masses and (ii) a Fermi-Pasta-Ulam chain, we demonstrate that the corresponding perturbation profiles coincide with the diffusion profiles of the single-particle Lévy walk approach. The parameters of the random walk can be related through elementary algebraic expressions to the physical parameters of the corresponding test many-body systems. PMID:21635077
Perturbation Spreading in Many-Particle Systems: A Random Walk Approach
NASA Astrophysics Data System (ADS)
Zaburdaev, V.; Denisov, S.; Hänggi, P.
2011-05-01
The propagation of an initially localized perturbation via an interacting many-particle Hamiltonian dynamics is investigated. We argue that the propagation of the perturbation can be captured by the use of a continuous-time random walk where a single particle is traveling through an active, fluctuating medium. Employing two archetype ergodic many-particle systems, namely, (i) a hard-point gas composed of two unequal masses and (ii) a Fermi-Pasta-Ulam chain, we demonstrate that the corresponding perturbation profiles coincide with the diffusion profiles of the single-particle Lévy walk approach. The parameters of the random walk can be related through elementary algebraic expressions to the physical parameters of the corresponding test many-body systems.
Algebraic theory of recombination spaces.
Stadler, P F; Wagner, G P
1997-01-01
A new mathematical representation is proposed for the configuration space structure induced by recombination, which we call "P-structure." It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourier decomposition of fitness landscapes into a superposition of "elementary landscapes." This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for P-structures. For binary string recombination, the elementary landscapes are exactly the p-spin functions (Walsh functions), that is, the same as the elementary landscapes of the string point mutation spaces (i.e., the hypercube). This supports the notion of a strong homomorphism between string mutation and recombination spaces. However, the effective nearest neighbor correlations on these elementary landscapes differ between mutation and recombination and among different recombination operators. On average, the nearest neighbor correlation is higher for one-point recombination than for uniform recombination. For one-point recombination, the correlations are higher for elementary landscapes with fewer interacting sites as well as for sites that have closer linkage, confirming the qualitative predictions of the Schema Theorem. We conclude that the algebraic approach to fitness landscape analysis can be extended to recombination spaces and provides an effective way to analyze the relative hardness of a landscape for a given recombination operator. PMID:10021760
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.
Perturbative double field theory on general backgrounds
NASA Astrophysics Data System (ADS)
Hohm, Olaf; Marques, Diego
2016-01-01
We develop the perturbation theory of double field theory around arbitrary solutions of its field equations. The exact gauge transformations are written in a manifestly background covariant way and contain at most quadratic terms in the field fluctuations. We expand the generalized curvature scalar to cubic order in fluctuations and thereby determine the cubic action in a manifestly background covariant form. As a first application we specialize this theory to group manifold backgrounds, such as S U (2 )≃S3 with H -flux. In the full string theory this corresponds to a Wess-Zumino-Witten background CFT. Starting from closed string field theory, the cubic action around such backgrounds has been computed before by Blumenhagen, Hassler, and Lüst. We establish precise agreement with the cubic action derived from double field theory. This result confirms that double field theory is applicable to arbitrary curved background solutions, disproving assertions in the literature to the contrary.
Quantum cosmological perturbations of multiple fluids
NASA Astrophysics Data System (ADS)
Peter, Patrick; Pinto-Neto, N.; Vitenti, Sandro D. P.
2016-01-01
The formalism to treat quantization and evolution of cosmological perturbations of multiple fluids is described. We first construct the Lagrangian for both the gravitational and matter parts, providing the necessary relevant variables and momenta leading to the quadratic Hamiltonian describing linear perturbations. The final Hamiltonian is obtained without assuming any equations of motions for the background variables. This general formalism is applied to the special case of two fluids, having in mind the usual radiation and matter mix which made most of our current Universe history. Quantization is achieved using an adiabatic expansion of the basis functions. This allows for an unambiguous definition of a vacuum state up to the given adiabatic order. Using this basis, we show that particle creation is well defined for a suitable choice of vacuum and canonical variables, so that the time evolution of the corresponding quantum fields is unitary. This provides constraints for setting initial conditions for an arbitrary number of fluids and background time evolution. We also show that the common choice of variables for quantization can lead to an ill-defined vacuum definition. Our formalism is not restricted to the case where the coupling between fields is small, but is only required to vary adiabatically with respect to the ultraviolet modes, thus paving the way to consistent descriptions of general models not restricted to single-field (or fluid).
Cosmological perturbations in a mimetic matter model
NASA Astrophysics Data System (ADS)
Matsumoto, Jiro; Odintsov, Sergei D.; Sushkov, Sergey V.
2015-03-01
We investigate the cosmological evolution of a mimetic matter model with arbitrary scalar potential. The cosmological reconstruction—which is the method for constructing a model for an arbitrary evolution of the scale factor—is explicitly performed for different choices of potential. The cases where the mimetic matter model shows the evolution as cold dark matter (CDM), the w CDM model, dark matter and dark energy with a dynamical O m (z ) [where O m (z )≡[(H (z )/H0)2-1 ]/[(1 +z )3-1 ] ], and phantom dark energy with a phantom-nonphantom crossing are presented in detail. The cosmological perturbations for such evolutions are studied in the mimetic matter model. For instance, the evolution behavior of the matter density contrast (which is different than the usual one, i.e., δ ¨+2 H δ ˙-κ2ρ δ /2 =0 ) is investigated. The possibility of a peculiar evolution of δ in the model under consideration is shown. Special attention is paid to the behavior of the matter density contrast near the future singularity, where the decay of perturbations may occur much earlier than the singularity.
Closure of the algebra of constraints for a nonprojectable Horava model
Bellorin, Jorge; Restuccia, Alvaro
2011-02-15
We perform the Hamiltonian analysis for a nonprojectable Horava model whose potential is composed of R and R{sup 2} terms. We show that Dirac's algorithm for the preservation of the constraints can be done in a closed way, hence the algebra of constraints for this model is consistent. The model has an extra, odd, scalar mode whose decoupling limit can be seen in a linear-order perturbative analysis on weakly varying backgrounds. Although our results for this model point in favor of the consistency of the Horava theory, the validity of the full nonprojectable theory still remains unanswered.
Role of division algebra in seven-dimensional gauge theory
NASA Astrophysics Data System (ADS)
Kalauni, Pushpa; Barata, J. C. A.
2015-03-01
The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang-Mills theory and field equation in seven-dimensional space.
Boundary Lax pairs from non-ultra-local Poisson algebras
Avan, Jean; Doikou, Anastasia
2009-11-15
We consider non-ultra-local linear Poisson algebras on a continuous line. Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or 'boundary' extensions. They are parametrized by a boundary scalar matrix and depend, in addition, on the choice of an antiautomorphism. The new algebras are the classical-linear counterparts of the known quadratic quantum boundary algebras. For any choice of parameters, the non-ultra-local contribution of the original Poisson algebra disappears. We also systematically construct the associated classical Lax pair. The classical boundary principal chiral model is examined as a physical example.
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.
NASA Astrophysics Data System (ADS)
Carlen, E. A.; Carvalho, M. C.; Orlandi, E.
1999-06-01
This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v 0 of a front that is sufficiently small in L 2 norm, and sufficiently localized that ∫ x 2 v 0( x)2 dx<∞, yields a solution that relaxes to another front, selected by a conservation law, in the L 1 norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L 2 norm and the rate of decrease of the excess free energy. Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.
Acoustooptic linear algebra processors - Architectures, algorithms, and applications
NASA Technical Reports Server (NTRS)
Casasent, D.
1984-01-01
Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.
Permutation centralizer algebras and multimatrix invariants
NASA Astrophysics Data System (ADS)
Mattioli, Paolo; Ramgoolam, Sanjaye
2016-03-01
We introduce a class of permutation centralizer algebras which underly the combinatorics of multimatrix gauge-invariant observables. One family of such noncommutative algebras is parametrized by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of two-matrix models. The structure of the algebra, notably its dimension, its center and its maximally commuting subalgebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The center of the algebra allows efficient computation of a sector of multimatrix correlators. These generate the counting of a certain class of bicoloured ribbon graphs with arbitrary genus.
Spinor representations of affine Lie algebras
Frenkel, I. B.
1980-01-01
Let [unk] be an infinite-dimensional Kac-Moody Lie algebra of one of the types Dl+1(2), Bl(1), or Dl(1). These algebras are characterized by the property that an elimination of any endpoint of their Dynkin diagrams gives diagrams of types Bl or Dl of classical orthogonal Lie algebras. We construct two representations of a Lie algebra [unk], which we call spinor representations, following the analogy with the classical case. We obtain that every spinor representation is either irreducible or has two irreducible components. This provides us with an explicit construction of fundamental representations of [unk], two for the type Dl+1(2), three for Bl(1), and four for Dl(1). We note the profound connection of our construction with quantum field theory—in particular, with fermion fields. Comparing the character formulas of our representations with another construction of the fundamental representations of Kac-Moody Lie algebras of types Al(1), Dl(1), El(1), we obtain classical Jacobi identities and addition formulas for elliptic θ-functions. PMID:16592912
The kinematic algebras from the scattering equations
NASA Astrophysics Data System (ADS)
Monteiro, Ricardo; O'Connell, Donal
2014-03-01
We study kinematic algebras associated to the recently proposed scattering equations, which arise in the description of the scattering of massless particles. In particular, we describe the role that these algebras play in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex which is associated to each solution of those equations. We also identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.
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.
Fang, Ferric C.
2014-01-01
As the body of scientific knowledge in a discipline increases, there is pressure for specialization. Fields spawn subfields that then become entities in themselves that promote further specialization. The process by which scientists join specialized groups has remarkable similarities to the guild system of the middle ages. The advantages of specialization of science include efficiency, the establishment of normative standards, and the potential for greater rigor in experimental research. However, specialization also carries risks of monopoly, monotony, and isolation. The current tendency to judge scientific work by the impact factor of the journal in which it is published may have roots in overspecialization, as scientists are less able to critically evaluate work outside their field than before. Scientists in particular define themselves through group identity and adopt practices that conform to the expectations and dynamics of such groups. As part of our continuing analysis of issues confronting contemporary science, we analyze the emergence and consequences of specialization in science, with a particular emphasis on microbiology, a field highly vulnerable to balkanization along microbial phylogenetic boundaries, and suggest that specialization carries significant costs. We propose measures to mitigate the detrimental effects of scientific specialism. PMID:24421049
ERIC Educational Resources Information Center
Braswell, Ray, Ed.
This document contains the following papers on special needs instruction and technology: (1) "Hawaii Special Education Teacher Induction" (Kalena Oliva and Quinn Avery); (2) "The Impact of Group v Individual Use of Hypermedia-Based Instruction" (Lewis R. Johnson, Louis P. Semrau, and Gail E. Fitzgerald); (3) "Assistive Technology Meets…
Gravitational waves from perturbed stars
NASA Astrophysics Data System (ADS)
Ferrari, V.
2011-12-01
Non radial oscillations of neutron stars are associated with the emission of gravitational waves. The characteristic frequencies of these oscillations can be computed using the theory of stellar perturbations, and they are shown to carry detailed information on the internal structure of the emitting source. Moreover, they appear to be encoded in various radiative processes, as for instance, in the tail of the giant flares of Soft Gamma Repeaters. Thus, their determination is central to the theory of stellar perturbation. A viable approach to the problem consists in formulating this theory as a problem of resonant scattering of gravitational waves incident on the potential barrier generated by the spacetime curvature. This approach discloses some unexpected correspondences between the theory of stellar perturbations and the theory of quantum mechanics, and allows us to predict new relativistic effects.
Gravitational waves from perturbed stars
NASA Astrophysics Data System (ADS)
Ferrari, V.
2011-03-01
Non radial oscillations of neutron stars are associated with the emission of gravitational waves. The characteristic frequencies of these oscillations can be computed using the theory of stellar perturbations, and they are shown to carry detailed information on the internal structure of the emitting source. Moreover, they appear to be encoded in various radiative processes, as for instance in the tail of the giant flares of Soft Gamma Repeaters. Thus, their determination is central to the theory of stellar perturbation. A viable approach to the problem consists in formulating this theory as a problem of resonant scattering of gravitational waves incident on the potential barrier generated by the spacetime curvature. This approach discloses some unexpected correspondences between the theory of stellar perturbations and the theory of quantum mechanics, and allows us to predict new relativistic effects.
Poulsen, P; Kuklo, R M
2001-03-01
We have previously reported the degree of attenuation and perturbation by a Cu jet passing through Comp B explosive. Similar tests have now been performed with high explosive (HE) targets having CJ pressures higher than and lower than the CJ pressure of Comp B. The explosives were LX-14 and TNT, respectively. We found that the measured exit velocity of the jet where it transitions from perturbed to solid did not vary significantly as a function of HE type for each HE thickness. The radial momentum imparted to the perturbed jet segment did vary as a function of HE type, however, and we report the radial spreading of the jet and the penetration of a downstream target as a function of HE type and thickness.
Perturbed motion at small eccentricities
NASA Astrophysics Data System (ADS)
Emel'yanov, N. V.
2015-09-01
In the study of the motion of planets and moons, it is often necessary to have a simple approximate analytical motion model, which takes into account major perturbations and preserves almost the same accuracy at long time intervals. A precessing ellipse model is used for this purpose. In this paper, it is shown that for small eccentricities this model of the perturbed orbit does not correspond to body motion characteristics. There is perturbed circular motion with a constant zero mean anomaly. The corresponding solution satisfies the Lagrange equations with respect to Keplerian orbital elements. There are two families of solutions with libration and circulation changes in the mean anomaly close to this particular solution. The paper shows how the eccentricity and mean anomaly change in these solutions. Simple analytical models of the motion of the four closest moons of Jupiter consistent with available ephemerides are proposed, which in turn are obtained by the numerical integration of motion equations and are refined by observations.
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.
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.
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.
Thermal perturbation of the Sun
NASA Technical Reports Server (NTRS)
Twigg, L. W.; Endal, A. S.
1981-01-01
An investigation of thermal perturbations of the solar convective zone via changes in the mixing length parameter were carried out, with a view toward understanding the possible solar radius and luminosity changes cited in the literature. The results show that: (a) a single perturbation of alpha is probably not the cause of the solar radius change and (b) the parameter W = d lambda nR./d lambda nL. can not be characterized by a single value, as implied in recent work.
Determination of orbital drag perturbations caused by atmospheric effects
NASA Astrophysics Data System (ADS)
Sehnal, L.
Atmospheric perturbations of the elements of the artificial satellites orbits are determined using a special model of distribution and variations of the total density of the upper atmosphere between 200-500 km. The model includes diurnal and semi-annual density variations, variations with solar activity and geomagnetic index and latitudinal changes. The height profile is expressed by multiple exponential functions. The equations of motion are solved analytically.
PREFACE: Infinite Dimensional Algebras and their Applications to Quantum Integrable Systems
NASA Astrophysics Data System (ADS)
Fring, Andreas; Kulish, Petr P.; Manojlović, Nenad; Nagy, Zoltán; Nunes da Costa, Joana; Samtleben, Henning
2008-05-01
This special issue is centred around the workshop Infinite Dimensional Algebras and Quantum Integrable Systems II—IDAQUIS 2007, held at the University of Algarve, Faro, Portugal in July 2007. It was the second workshop in the IDAQUIS series following a previous meeting at the same location in 2003. The latest workshop gathered around forty experts in the field reviewing recent developments in the theory and applications of integrable systems in the form of invited lectures and in a number of contributions from the participants. All contributions contain significant new results or provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants are also included. The origins of the topic of this issue can be traced back a long way to the early investigations of completely integrable systems of classical mechanics in the fundamental papers by Euler, Lagrange, Jacobi, Liouville, Kowalevski and others. By the end of the nineteenth century all interesting examples seemed to have been exhausted. A revival in the study of integrable systems began with the development of the classical inverse scattering method, or the theory of solitons. Later developments led to the basic geometrical ideas of the theory, of which infinite dimensional algebras are a key ingredient. In a loose sense one may think that all integrable systems possess some hidden symmetry. In the quantum version of these systems the representation theory of these algebras may be exploited in the description of the structure of the Hilbert space of states. Modern examples of field theoretical systems such as conformal field theories, with the Liouville model being a prominent example, affine Toda field theories and the AdS/CFT correspondence are based on algebraic structures like quantum groups, modular doubles, global conformal invariance, Hecke algebras, Kac
Localization of Free Field Realizations of Affine Lie Algebras
NASA Astrophysics Data System (ADS)
Futorny, Vyacheslav; Grantcharov, Dimitar; Martins, Renato A.
2015-04-01
We use localization technique to construct new families of irreducible modules of affine Kac-Moody algebras. In particular, localization is applied to the first free field realization of the affine Lie algebra or, equivalently, to imaginary Verma modules.
On q-deformed infinite-dimensional n-algebra
NASA Astrophysics Data System (ADS)
Ding, Lu; Jia, Xiao-Yu; Wu, Ke; Yan, Zhao-Wen; Zhao, Wei-Zhong
2016-03-01
The q-deformation of the infinite-dimensional n-algebras is investigated. Based on the structure of the q-deformed Virasoro-Witt algebra, we derive a nontrivial q-deformed Virasoro-Witt n-algebra which is nothing but a sh-n-Lie algebra. Furthermore in terms of the pseud-differential operators, we construct the (co)sine n-algebra and the q-deformed S Diff (T2)n-algebra. We find that they are the sh-n-Lie algebras for the n even case. In terms of the magnetic translation operators, an explicit physical realization of the (co)sine n-algebra is given.
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.
Rees algebras, Monomial Subrings and Linear Optimization Problems
NASA Astrophysics Data System (ADS)
Dupont, Luis A.
2010-06-01
In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.
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)
Algebraic and analytic reconstruction methods for dynamic tomography.
Desbat, L; Rit, S; Clackdoyle, R; Mennessier, C; Promayon, E; Ntalampeki, S
2007-01-01
In this work, we discuss algebraic and analytic approaches for dynamic tomography. We present a framework of dynamic tomography for both algebraic and analytic approaches. We finally present numerical experiments. PMID:18002059
Upper bound for the length of commutative algebras
Markova, Ol'ga V
2009-12-31
By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field one means the least positive integer k such that the words of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, an upper bound for the length of a commutative algebra in terms of a function of two invariants of the algebra, the dimension and the maximal degree of the minimal polynomial for the elements of the algebra, is obtained. As a corollary, a formula for the length of the algebra of diagonal matrices over an arbitrary field is obtained. Bibliography: 8 titles.
Cosmological perturbation theory in 1+1 dimensions
NASA Astrophysics Data System (ADS)
McQuinn, Matthew; White, Martin
2016-01-01
Many recent studies have highlighted certain failures of the standard Eulerian-space cosmological perturbation theory (SPT). Its problems include (1) not capturing large-scale bulk flows [leading to an Script O( 1) error in the 1-loop SPT prediction for the baryon acoustic peak in the correlation function], (2) assuming that the Universe behaves as a pressureless, inviscid fluid, and (3) treating fluctuations on scales that are non-perturbative as if they were. Recent studies have highlighted the successes of perturbation theory in Lagrangian space or theories that solve equations for the effective dynamics of smoothed fields. Both approaches mitigate some or all of the aforementioned issues with SPT. We discuss these physical developments by specializing to the simplified 1D case of gravitationally interacting sheets, which allows us to substantially reduces the analytic overhead and still (as we show) maintain many of the same behaviors as in 3D. In 1D, linear-order Lagrangian perturbation theory ("the Zeldovich approximation") is exact up to shell crossing, and we prove that nth-order Eulerian perturbation theory converges to the Zeldovich approximation as narrow ∞. In no 1D cosmology that we consider (including a CDM-like case and power-law models) do these theories describe accurately the matter power spectrum on any mildly nonlinear scale. We find that theories based on effective equations are much more successful at describing the dynamics. Finally, we discuss many topics that have recently appeared in the perturbation theory literature such as beat coupling, the shift and smearing of the baryon acoustic oscillation feature, and the advantages of Fourier versus configuration space. Our simplified 1D case serves as an intuitive review of these perturbation theory results.
Piecewise lexsegment ideals in exterior algebras
NASA Astrophysics Data System (ADS)
Shakin, D. A.
2005-02-01
The problem of describing the Hilbert functions of homogeneous ideals of an exterior algebra over a field containing a fixed monomial ideal I is considered. For this purpose the notion of a piecewise lexsegment ideal in an exterior algebra is introduced generalizing the notion of a lexsegment ideal. It is proved that if I is a piecewise lexsegment ideal, then it is possible to describe the Hilbert functions of the homogeneous ideals containing I in a way similar to that suggested by Kruskal and Katona for the situation I=0. Moreover, a generalization of the extremal properties of lexsegment ideals is obtained (the inequality for the Betti numbers).
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.
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.
Algebraic properties of basic isohedral marked tilings
NASA Astrophysics Data System (ADS)
Greco, Gabriele H.
2006-05-01
In 1977 Grünbaum and Shephard described all possible 93 types of isohedral marked tilings of the plane; 46 of them are called basic, since their induced tile group is trivial. The aim of this paper is to give an algebraic description of all basic tilings. A purely algebraic characterization of the adjacency symmetries of tiles of the 46 basic tilings is presented. Moreover, 46 related abstract definitions of two-dimensional crystallographic groups supplement and extend those of the well-known book Generators and Relations for Discrete Groups by Coxeter and Moser.
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.
Algebraic models of flexible manufacturing systems
NASA Astrophysics Data System (ADS)
Leskin, Aleksei Alekseevich
Various aspects of the use of mathematical methods in the development of flexible manufacturing systems are examined. Attention is given to dynamical and structural models of flexible manufacturing systems developed by using methods of algebraic and differential geometry, topology, polynomial algebra, and extreme value problem theory. The principles of model integration are discussed, and approaches are proposed for solving problems related to the selection of flexible manufacturing equipment, real-time modeling of the manufacturing process, and optimization of local automation systems. The discussion is illustrated by examples.
Weak Lie symmetry and extended Lie algebra
Goenner, Hubert
2013-04-15
The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found ('extended Lie algebras') which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).
The early history of current Algebra
NASA Astrophysics Data System (ADS)
Pietschmann, Herbert
2011-07-01
The history of Current Algebra is reviewed up to the appearance of the Adler-Weisberger sum rule. Particular emphasis is given to the role of current algebra in the historical struggle in strong interaction physics of elementary particles between field theory and the S-matrix approach based on dispersion relations. The question as to whether some particles are truly fundamental or all hadrons are bound or resonant states of one another played an important role in this struggle and is thus also regarded.
Fock representations of exchange algebras with involution
Liguori, A.; Mintchev, M.; Rossi, M.
1997-06-01
An associative algebra scr(A){sub R} with exchange properties generalizing the canonical (anti)commutation relations is considered. We introduce a family of involutions in scr(A){sub R} and construct the relative Fock representations, examining the positivity of the metric. As an application of the general results, we rigorously prove unitarity of the scattering operator of integrable models in 1+1 space-time dimensions. In this context the possibility of adopting various involutions in the Zamolodchikov{endash}Faddeev algebra is also explored. {copyright} {ital 1997 American Institute of Physics.}
C∗-algebras of Penrose hyperbolic tilings
NASA Astrophysics Data System (ADS)
Oyono-Oyono, Hervé; Petite, Samuel
2011-02-01
Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the C∗-algebras and of the K-theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these C∗-algebras are traceless. Nevertheless, harmonic currents give rise to 3-cyclic cocycles and we discuss in this setting a higher-order version of the gap-labeling.
The Dirac equation and Hestenes' geometric algebra
NASA Astrophysics Data System (ADS)
Hamilton, J. Dwayne
1984-06-01
Hestenes' geometric algebra and Dirac spinors are reviewed and united into a common mathematical formalism, a unification that establishes the Dirac equation as being manifestly covariant under the Lorentz group, and one that needs no matrix representation of the Dirac algebra. New and simple methods of amplitude or ``trace'' calculations are then described. A number of problems are then considered within the context of the new approach, such as relativistic spin projections, new and covariant C and T-transformations and spinors for massless and Majorana fields.
Feynman identity: A special case. I
da Costa, G.A.
1997-02-01
There is an identity due to Feynman which relates graphs and closed curves on a lattice and it was used by Feynman in his combinatorial proof of Onsager{close_quote}s closed formula for the partition function of the two-dimensional Ising model. Long ago Sherman considered a special case of this identity and pointed out similarities with the Witt identity of Lie algebra theory. In this paper and following, we revisit this special case and solve some problems related with it. In particular, the weights are computed explicitly using paths and words and a direct connection with the Witt formula is found. {copyright} {ital 1997 American Institute of Physics.}
Mutations of the cluster algebra of type {A}_{1}^{(1)} and the periodic discrete Toda lattice
NASA Astrophysics Data System (ADS)
Nobe, Atsushi
2016-07-01
A direct connection between two sequences of points, one of which is generated by seed mutations of the cluster algebra of type {A}1(1) and the other by time evolutions of the periodic discrete Toda lattice, is explicitly given. In this construction, each of them is realized as an orbit of a QRT map, and specialization of the parameters in the maps and appropriate choices of the initial points relate them. The connection with the periodic discrete Toda lattice enables us a geometric interpretation of the seed mutations of the cluster algebra of type {A}1(1) as an addition of points on an elliptic curve.
NASA Astrophysics Data System (ADS)
Bartels, Jochen
2006-06-01
I summarize the present status of the AGK cutting rules in perturbative QCD. Particular attention is given to the application of the AGK analysis to diffraction and multiple scattering in DIS at HERA and to pp collisions at the LHC. I also discuss the bootstrap conditions which appear in pQCD.
VHS Movies: Perturbations for Morphogenesis.
ERIC Educational Resources Information Center
Holmes, Danny L.
This paper discusses the concept of a family system in terms of an interactive system of interrelated, interdependent parts and suggests that VHS movies can act as perturbations, i.e., change promoting agents, for certain dysfunctional family systems. Several distinct characteristics of a family system are defined with particular emphasis on…
Adaptation Strategies in Perturbed /s/
ERIC Educational Resources Information Center
Brunner, Jana; Hoole, Phil; Perrier, Pascal
2011-01-01
The purpose of this work is to investigate the role of three articulatory parameters (tongue position, jaw position and tongue grooving) in the production of /s/. Six normal speakers' speech was perturbed by a palatal prosthesis. The fricative was recorded acoustically and through electromagnetic articulography in four conditions: (1) unperturbed,…
Basics of QCD perturbation theory
Soper, D.E.
1997-06-01
This is an introduction to the use of QCD perturbation theory, emphasizing generic features of the theory that enable one to separate short-time and long-time effects. The author also covers some important classes of applications: electron-positron annihilation to hadrons, deeply inelastic scattering, and hard processes in hadron-hadron collisions. 31 refs., 38 figs.
Generalized perturbations in neutrino mixing
NASA Astrophysics Data System (ADS)
Liao, Jiajun; Marfatia, D.; Whisnant, K.
2015-10-01
We derive expressions for the neutrino mixing parameters that result from complex perturbations on (1) the Majorana neutrino mass matrix (in the basis of charged lepton mass eigenstates) and on (2) the charged lepton mass matrix, for arbitrary initial (unperturbed) mixing matrices. In the first case, we find that the phases of the elements of the perturbation matrix, and the initial values of the Dirac and Majorana phases, strongly impact the leading-order corrections to the neutrino mixing parameters and phases. For experimentally compatible scenarios wherein the initial neutrino mass matrix has μ -τ symmetry, we find that the Dirac phase can take any value under small perturbations. Similarly, in the second case, perturbations to the charged lepton mass matrix can generate large corrections to the mixing angles and phases of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. As an illustration of our generalized procedure, we apply it to a situation in which nonstandard scalar and nonstandard vector interactions simultaneously affect neutrino oscillations.
Seven topics in perturbative QCD
Buras, A.J.
1980-09-01
The following topics of perturbative QCD are discussed: (1) deep inelastic scattering; (2) higher order corrections to e/sup +/e/sup -/ annihilation, to photon structure functions and to quarkonia decays; (3) higher order corrections to fragmentation functions and to various semi-inclusive processes; (4) higher twist contributions; (5) exclusive processes; (6) transverse momentum effects; (7) jet and photon physics.
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…
Stages in the History of Algebra with Implications for Teaching
ERIC Educational Resources Information Center
Katz, Victor J.; Barton, Bill
2007-01-01
In this article, we take a rapid journey through the history of algebra, noting the important developments and reflecting on the importance of this history in the teaching of algebra in secondary school or university. Frequently, algebra is considered to have three stages in its historical development: the rhetorical stage, the syncopated stage,…
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…
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,…
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.
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…
Lessons for Algebraic Thinking. Grades 3-5.
ERIC Educational Resources Information Center
Wickett, Maryann; Kharas, Katharine; Burns, Marilyn
Algebra is one of the top priorities of mathematics instruction for the elementary and middle grades. This book is designed to help 3-5 teachers meet the challenge of making algebra an integral part of their mathematics instruction and realize both what to teach and how to teach central algebraic concepts. Classroom-tested lessons help teachers…
Lessons for Algebraic Thinking. Grades 6-8.
ERIC Educational Resources Information Center
Lawrence, Ann; Hennessy, Charlie
Algebra is one of the top priorities of mathematics instruction for the elementary and middle grades. This book is designed to help 6-8 teachers meet the challenge of making algebra an integral part of their mathematics instruction and realize both what to teach and how to teach central algebraic concepts. Classroom-tested lessons help teachers…
Ideas in Practice: Graphing Calculators in Beginning Algebra
ERIC Educational Resources Information Center
Martin, Aimee
2008-01-01
This paper reports on a project to improve Beginning Algebra students' understanding of basic algebraic concepts through fully integrated use of the TI-83 graphing calculator. The methodology incorporated an intervention case study including approximately 700 Beginning Algebra students at an open-door community college of 8,500 students in the…
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…
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…
Classical versus Computer Algebra Methods in Elementary Geometry
ERIC Educational Resources Information Center
Pech, Pavel
2005-01-01
Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…
ERIC Educational Resources Information Center
Zimmer, Phil
1986-01-01
Specialized publications such as "Opera News,""Gourmet," and "Forbes" can bring an institution's story to targeted audiences. The experiences of Chautauqua Institution are described. Some of the benefits of marketing articles to these publications are discussed. (MLW)
ERIC Educational Resources Information Center
Lavendel, Giuliana
1977-01-01
Discusses problems involved in maintaining special scientific or engineering libraries, including budget problems, remote storage locations, rental computer retrieval systems, protecting trade secrets, and establishing a magnetic tape library. (MLH)
Scale-adaptive tensor algebra for local many-body methods of electronic structure theory
Liakh, Dmitry I
2014-01-01
While the formalism of multiresolution analysis (MRA), based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent-particle level and, recently, second-order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many-particle) theory which is based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this paper, we present a formalism called scale-adaptive tensor algebra (SATA) which exploits an adaptive representation of tensors of many-body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability and reliability of local correlated many-body methods of electronic structure theory, especially those directly based on atomic orbitals (or any other localized basis functions).
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…
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…
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,…
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…
Algebra [Student's Individualized Career Source Package].
ERIC Educational Resources Information Center
Tingle, H. Burton
This is a volume of teacher-developed units to supplement the textbook in a first-year algebra course. The units consist mainly of statements of objectives and student worksheets with some examples and references to the textbook given as aids. Major topics covered are basic operations with signed rational numbers and with polynomials,…
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.