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
Lie 3-ALGEBRA and Super-Affinization of Split-Octonions
NASA Astrophysics Data System (ADS)
Carrión, Hector L.; Giardino, Sergio
The purpose of this study is to extend the concept of a generalized Lie 3-algebra, known to the divisional algebra of the octonions 𝕆, to split-octonions 𝕊𝕆, which is non-divisional. This is achieved through the unification of the product of both of the algebras in a single operation. Accordingly, a notational device is introduced to unify the product of both algebras. We verify that 𝕊𝕆 is a Malcev algebra and we recalculate known relations for the structure constants in terms of the introduced structure tensor. Finally we construct the manifestly supersymmetric {N} = 1{ SO} affine superalgebra. An application of the split Lie 3-algebra for a Bagger and Lambert gauge theory is also discussed.
NASA Astrophysics Data System (ADS)
McRae, Robert
2016-08-01
For a finite-dimensional simple Lie algebra {{g}}, we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra {{widehat{{g}}}} at a fixed level {ℓin{N}} with a certain tensor category of finite-dimensional {{g}}-modules. More precisely, the category of level ℓ standard {{widehat{{g}}}}-modules is the module category for the simple vertex operator algebra {L_{widehat{{g}}}(ℓ, 0)}, and as is well known, this category is equivalent as an abelian category to {{D}({g},ℓ)}, the category of finite-dimensional modules for the Zhu's algebra {A{(L_{widehat{{g}}}(ℓ, 0))}}, which is a quotient of {U({g})}. Our main result is a direct construction using Knizhnik-Zamolodchikov equations of the associativity isomorphisms in {{D}({g},ℓ)} induced from the associativity isomorphisms constructed by Huang and Lepowsky in {{L_{widehat{{g}}}(ℓ, 0) - {mod}}}. This construction shows that {{D}({g},ℓ)} is closely related to the Drinfeld category of {U({g})}[[h
NASA Astrophysics Data System (ADS)
Roytenberg, Dmitry
2007-11-01
A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and certain "up to homotopy" structures on the complex; for strictly skew-symmetric Lie 2-algebras these are L∞-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer-Cartan equation in some differential graded Lie algebras and L∞-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples.
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).
Computing Matrix Representations of Filiform Lie Algebras
NASA Astrophysics Data System (ADS)
Ceballos, Manuel; Núñez, Juan; Tenorio, Ángel F.
In this paper, we compute minimal faithful unitriangular matrix representations of filiform Lie algebras. To do it, we use the nilpotent Lie algebra, g_n, formed of n ×n strictly upper-triangular matrices. More concretely, we search the lowest natural number n such that the Lie algebra g_n contains a given filiform Lie algebra, also computing a representative of this algebra. All the computations in this paper have been done using MAPLE 9.5.
Bicovariant quantum algebras and quantum Lie algebras
NASA Astrophysics Data System (ADS)
Schupp, Peter; Watts, Paul; Zumino, Bruno
1993-10-01
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun(mathfrak{G}_q ) to U q g, given by elements of the pure braid group. These operators—the “reflection matrix” Y≡L + SL - being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for Y in SO q (N).
Post-Lie Algebras and Isospectral Flows
NASA Astrophysics Data System (ADS)
Ebrahimi-Fard, Kurusch; Lundervold, Alexander; Mencattini, Igor; Munthe-Kaas, Hans Z.
2015-11-01
In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical R-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation.
Almost split real forms for hyperbolic Kac Moody Lie algebras
NASA Astrophysics Data System (ADS)
Ben Messaoud, Hechmi
2006-11-01
A Borel Tits theory was developed for almost split forms of symmetrizable Kac Moody Lie algebras. In this paper, we look to almost split real forms and their restricted root systems for symmetrizable hyperbolic Kac Moody Lie algebras. We establish a complete list of these forms, in terms of their Satake Tits index, for the strictly hyperbolic ones and for those which are obtained as (hyperbolic) canonical Lorentzian extensions of affine Lie algebras. These forms are of particular interest in theoretical physics because of their connection to supergravity theories.
Lie algebras of classical and stochastic electrodynamics
NASA Astrophysics Data System (ADS)
Neto, J. J. Soares; Vianna, J. D. M.
1994-03-01
The Lie algebras associated with infinitesimal symmetry transformations of third-order differential equations of interest to classical electrodynamics and stochastic electrodynamics have been obtained. The structure constants for a general case are presented and the Lie algebra for each particular application is easily achieved. By the method used here it is not necessary to know the explicit expressions of the infinitesimal generators in order to determine the structure constants of the Lie algebra.
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.``
Symbolic Lie algebras manipulations using COMMON LISP
NASA Astrophysics Data System (ADS)
Cecchini, R.; Tarlini, M.
1989-01-01
We present a description and an implementation of a program in COMMON LISP to perform symbolic computations in a given Lie algebra. Using the general definitions of vector space Lie algebra and enveloping algebra, the program is able to compute commutators, to evaluate similarity transformations and the general Baker-Campbell-Hausdorff formula. All the computations are exact, including numerical coefficients. For the interactive user an optional menu facility and online help are available. LISP knowledge is unnecessary.
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
Conformal field theory on affine Lie groups
Clubok, Kenneth Sherman
1996-04-01
Working directly on affine Lie groups, we construct several new formulations of the WZW model, the gauged WZW model, and the generic affine-Virasoro action. In one formulation each of these conformal field theories (CFTs) is expressed as a one-dimensional mechanical system whose variables are coordinates on the affine Lie group. When written in terms of the affine group element, this formulation exhibits a two-dimensional WZW term. In another formulation each CFT is written as a two-dimensional field theory, with a three- dimensional WZW term, whose fields are coordinates on the affine group. On the basis of these equivalent formulations, we develop a translation dictionary in which the new formulations on the affine Lie group are understood as mode formulations of the conventional formulations on the Lie group. Using this dictionary, we also express each CFT as a three-dimensional field theory on the Lie group with a four-dimensional WZW term. 36 refs.
SAYD Modules over Lie-Hopf Algebras
NASA Astrophysics Data System (ADS)
Rangipour, Bahram; Sütlü, Serkan
2012-11-01
In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.
Classification of filiform Lie algebras of order 3
NASA Astrophysics Data System (ADS)
Navarro, Rosa María
2016-12-01
Lie algebras of order 3 constitute a generalization of Lie algebras and superalgebras. Throughout this paper the classification problem of filiform Lie algebras of order 3 is considered and therefore this work is a continuation papers seen in the literature. We approach this classification by extending Vergne's result for filiform Lie algebras and by considering algebras of order 3 of high nilindex. We find the expression of the law to which any elementary filiform Lie algebra of order 3 is isomorphic.
NASA Astrophysics Data System (ADS)
Sati, Hisham; Schreiber, Urs
2017-03-01
We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie ( p + 1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie ( p+1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.
Riemannian manifolds as Lie-Rinehart algebras
NASA Astrophysics Data System (ADS)
Pessers, Victor; van der Veken, Joeri
2016-07-01
In this paper, we show how Lie-Rinehart algebras can be applied to unify and generalize the elementary theory of Riemannian geometry. We will first review some necessary theory on a.o. modules, bilinear forms and derivations. We will then translate some classical theory on Riemannian geometry to the setting of Rinehart spaces, a special kind of Lie-Rinehart algebras. Some generalized versions of classical results will be obtained, such as the existence of a unique Levi-Civita connection, inducing a Levi-Civita connection on a submanifold, and the construction of spaces with constant sectional curvature.
Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Makhlouf, Abdenacer; Silvestrov, Sergei
2010-04-01
The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom-Nambu-Lie algebras obtained using this construction.
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.
Lie algebras and linear differential equations.
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Capability and Schur multiplier of a pair of Lie algebras
NASA Astrophysics Data System (ADS)
Johari, Farangis; Parvizi, Mohsen; Niroomand, Peyman
2017-04-01
The aim of this work is to find some criteria for detecting the capability of a pair of Lie algebras. We characterize the exact structure of all pairs of capable Lie algebras in the class of abelian and Heisenberg ones. Among the other results, we also give some exact sequences on the Schur multiplier and exterior product of Lie algebras.
Vector fields and nilpotent Lie algebras
NASA Technical Reports Server (NTRS)
Grayson, Matthew; Grossman, Robert
1987-01-01
An infinite-dimensional family of flows E is described with the property that the associated dynamical system: x(t) = E(x(t)), where x(0) is a member of the set R to the Nth power, is explicitly integrable in closed form. These flows E are of the form E = E1 + E2, where E1 and E2 are the generators of a nilpotent Lie algebra, which is either free, or satisfies some relations at a point. These flows can then be used to approximate the flows of more general types of dynamical systems.
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.
Metric Lie 3-algebras in Bagger-Lambert theory
NASA Astrophysics Data System (ADS)
de Medeiros, Paul; Figueroa-O'Farrill, José; Méndez-Escobar, Elena
2008-08-01
We recast physical properties of the Bagger-Lambert theory, such as shift-symmetry and decoupling of ghosts, the absence of scale and parity invariance, in Lie 3-algebraic terms, thus motivating the study of metric Lie 3-algebras and their Lie algebras of derivations. We prove a structure theorem for metric Lie 3-algebras in arbitrary signature showing that they can be constructed out of the simple and one-dimensional Lie 3-algebras iterating two constructions: orthogonal direct sum and a new construction called a double extension, by analogy with the similar construction for Lie algebras. We classify metric Lie 3-algebras of signature (2, p) and study their Lie algebras of derivations, including those which preserve the conformal class of the inner product. We revisit the 3-algebraic criteria spelt out at the start of the paper and select those algebras with signature (2, p) which satisfy them, as well as indicate the construction of more general metric Lie 3-algebras satisfying the ghost-decoupling criterion.
Spin and wedge representations of infinite-dimensional Lie algebras and groups
Kac, Victor G.; Peterson, Dale H.
1981-01-01
We suggest a purely algebraic construction of the spin representation of an infinite-dimensional orthogonal Lie algebra (sections 1 and 2) and a corresponding group (section 4). From this we deduce a construction of all level-one highest-weight representations of orthogonal affine Lie algebras in terms of creation and annihilation operators on an infinite-dimensional Grassmann algebra (section 3). We also give a similar construction of the level-one representations of the general linear affine Lie algebra in an infinite-dimensional “wedge space.” Along these lines we construct the corresponding representations of the universal central extension of the group SLn(k[t,t-1]) in spaces of sections of line bundles over infinite-dimensional homogeneous spaces (section 5). PMID:16593029
Lie algebraic methods for particle tracking calculations
Douglas, D.R.; Dragt, A.J.
1983-08-01
A study of the nonlinear stability of an accelerator or storage ring lattice typically includes particle tracking simulations. Such simulations trace rays through linear and nonlinear lattice elements by numerically evaluating linear matrix or impulsive nonlinear transformations. Using the mathematical tools of Lie groups and algebras, one may construct a formalism which makes explicit use of Hamilton's equations and which allows the description of groups of linear and nonlinear lattice elements by a single transformation. Such a transformation will be exactly canonical and will describe finite length linear and nonlinear elements through third (octupole) order. It is presently possible to include effects such as fringing fields and potentially possible to extend the formalism to include nonlinearities of higher order, multipole errors, and magnet misalignments. We outline this Lie algebraic formalism and its use in particle tracking calculations. A computer code, MARYLIE, has been constructed on the basis of this formalism. We describe the use of this program for tracking and provide examples of its application. 6 references, 3 figures.
Lie symmetry algebra of one-dimensional nonconservative dynamical systems
NASA Astrophysics Data System (ADS)
Liu, Cui-Mei; Wu, Run-Heng; Fu, Jing-Li
2007-09-01
Lie symmetry algebra of linear nonconservative dynamical systems is studied in this paper. By using 1-1 mapping, the Lie point and Lie contact symmetry algebras are obtained from two independent solutions of the one-dimensional linear equations of motion.
The structure of split regular BiHom-Lie algebras
NASA Astrophysics Data System (ADS)
Calderón, Antonio J.; Sánchez, José M.
2016-12-01
We introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. We show that an arbitrary split regular BiHom-Lie algebra L is of the form L = U +∑jIj with U a linear subspace of a fixed maximal abelian subalgebra H and any Ij a well described (split) ideal of L, satisfying [Ij ,Ik ] = 0 if j ≠ k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its simple ideals.
Lie-algebraic solutions of the type IIB matrix model
NASA Astrophysics Data System (ADS)
Chatzistavrakidis, Athanasios
2011-11-01
A systematic search for Lie-algebra solutions of the type IIB matrix model is performed. Our survey is based on the classification of all Lie algebras for dimensions up to five and of all nilpotent Lie algebras of dimension six. It is shown that Lie-type solutions of the equations of motion of the type IIB matrix model exist and they correspond to certain nilpotent and solvable Lie algebras. Their representation in terms of Hermitian matrices is discussed in detail. These algebras give rise to certain noncommutative spaces for which the corresponding star products are provided. Finally the issue of constructing quantized compact nilmanifolds and solvmanifolds based on the above algebras is addressed.
Capable n-Lie algebras and the classification of nilpotent n-Lie algebras with s(A) = 3
NASA Astrophysics Data System (ADS)
Darabi, Hamid; Saeedi, Farshid; Eshrati, Mehdi
2016-12-01
Darabi et al. (2016) associate to each d-dimensional nilpotent n-Lie algebra A, a number s(A) where s(A) =(d-1/n) + n - 1 - dim M(A) and M(A) denotes the multiplier of A. They prove that s(A) is non-negative and classify all nilpotent n-Lie algebras A for which s(A) = 0 , 1 , 2. In this paper, we will classify all capable n-Lie algebras with 1-dimensional derived subalgebra and apply it to obtain a classification of those nilpotent n-Lie algebras satisfying s(A) = 3.
Lie algebra type noncommutative phase spaces are Hopf algebroids
NASA Astrophysics Data System (ADS)
Meljanac, Stjepan; Škoda, Zoran; Stojić, Martina
2016-11-01
For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.
Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups
NASA Astrophysics Data System (ADS)
Batat, W.; Onda, K.
2017-04-01
We study algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are solvsolitons. In particular, we obtain new solitons on G2, G5, and G6, and we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not be algebraic Ricci solitons.
Structure of classical affine and classical affine fractional W-algebras
Suh, Uhi Rinn
2015-01-15
We introduce a classical BRST complex (See Definition 3.2.) and show that one can construct a classical affine W-algebra via the complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible λ-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we describe explicit forms of free generators and compute λ-brackets between them. Provided some assumptions on a classical affine fractional W-algebra, we find an infinite sequence of integrable systems related to the algebra, using the generalized Drinfel’d and Sokolov reduction.
Relativity symmetries and Lie algebra contractions
Cho, Dai-Ning; Kong, Otto C.W.
2014-12-15
We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n) symmetry as an isometry on an m+n dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n) preserving a symmetry of the same type at dimension m+n−1, e.g. a G(m,n−1), together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4), which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3) may be relevant to real physics.
Quantum integrable systems related to lie algebras
NASA Astrophysics Data System (ADS)
Olshanetsky, M. A.; Perelomov, A. M.
1983-03-01
Some quantum integrable finite-dimensional systems related to Lie algebras are considered. This review continues the previous review of the same authors [83] devoted to the classical aspects of these systems. The dynamics of some of these systems is closely related to free motion in symmetric spaces. Using this connection with the theory of symmetric spaces some results such as the forms of spectra, wave functions, S-matrices, quantum integrals of motion are derived. In specific cases the considered systems describe the one-dimensional n-body systems interacting pairwise via potentials g2v( q) of the following 5 types: vI( q) = q-2, vII( q) = sinh-2q, vIII( q) = sin-2q, v IV(q) = P(q) , vV( q) = q-2 + ω2q2. Here P(q) is the Weierstrass function, so that the first three cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbour potential exp( qjqj+ 1 ) is moreover considered. This review presents from a general and universal point of view results obtained mainly over the past fifteen years. Besides, it contains some new results both of physical and mathematical interest.
The Dixmier Map for Nilpotent Super Lie Algebras
NASA Astrophysics Data System (ADS)
Herscovich, Estanislao
2012-07-01
In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by {Prim({U}({g}))} the set of (graded) primitive ideals of the enveloping algebra {{U}({g})} of a nilpotent Lie superalgebra {{g}} and {{A}d0} the adjoint group of {{g}0}, we prove that the usual Dixmier map for nilpotent Lie algebras can be naturally extended to the context of nilpotent super Lie algebras, i.e. there exists a bijective map I : {g}0^{*}/{A}d0 rightarrow Prim({U}({g})) defined by sending the equivalence class [ λ] of a functional λ to a primitive ideal I( λ) of {{U}({g})}, and which coincides with the Dixmier map in the case of nilpotent Lie algebras. Moreover, the construction of the previous map is explicit, and more or less parallel to the one for Lie algebras, a major difference with a previous approach ( cf. [18]). One key fact in the construction is the existence of polarizations for super Lie algebras, generalizing the concept defined for Lie algebras. As a corollary of the previous description, we obtain the isomorphism {{U}({g})/I(λ) ˜eq Cliffq(k) ⊗ Ap(k)}, where {(p,q) = (dim({g}0/{g}0^{λ})/2,dim({g}1/{g}1^{λ}))}, we get a direct construction of the maximal ideals of the underlying algebra of {{U}({g})} and also some properties of the stabilizers of the primitive ideals of {{U}({g})}.
Direct determination of the underlying Lie algebra in nonlinear optics
NASA Astrophysics Data System (ADS)
Arnold, J. M.
1991-01-01
It is shown that the equations of resonant nonlinear optics can be studied entirely within the framework of an underlying Lie algebra, in which the 2x2 su(2) Hamiltonian and density matrices of the quantum mechanical description of the atomic system transform directly to the 2x2 sl(2,R) matrices of the Ablowitz-Kaup-Newell-Segur (AKNS) scheme, and the AKNS eigenvalue is introduced naturally as a free parameter. The Lie algebra sl(2,R) is also the symmetry algebra of transformations between equivalence classes of AKNS systems under SL(2,R) gauge transformations. The Lie algebra formalism condenses much algebraic manipulation, and provides a natural basis for the perturbation theory of "nearly integrable" nonlinear wave systems.
Poisson and symplectic structures on Lie algebras. I
NASA Astrophysics Data System (ADS)
Alekseevsky, D. V.; Perelomov, A. M.
1997-06-01
The purpose of this paper is to describe a new class of Poisson and symplectic structures on Lie algebras. This gives a new class of solutions of the classical Yang-Baxter equation. The class of elementary Lie algebras is defined and the Poisson and symplectic structures for them are described. The algorithm is given for description of all closed 2-forms and of symplectic structures on any Lie algebra G, which is decomposed into semidirect sum of elementary subalgebras. Using these results we obtain the description of closed 2-forms and symplectic forms (if they exist) on the Borel subalgebra B(G) of semisimple Lie algebra G. As a byproduct, we get description of the second cohomology group H2( B( G)).
Lie Conformal Algebra Cohomology and the Variational Complex
NASA Astrophysics Data System (ADS)
de Sole, Alberto; Kac, Victor G.
2009-12-01
We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a {mathfrak{g}}-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators.
On the intersection of irreducible components of the space of finite-dimensional Lie algebras
Gorbatsevich, Vladimir V
2012-07-31
The irreducible components of the space of n-dimensional Lie algebras are investigated. The properties of Lie algebras belonging to the intersection of all the irreducible components of this kind are studied (these Lie algebras are said to be basic or founding Lie algebras). It is proved that all Lie algebras of this kind are nilpotent and each of these Lie algebras has an Abelian ideal of codimension one. Specific examples of founding Lie algebras of arbitrary dimension are described and, to describe the Lie algebras in general, we state a conjecture. The concept of spectrum of a Lie algebra is considered and some of the most elementary properties of the spectrum are studied. Bibliography: 6 titles.
Differential graded Lie algebras and singularities of level sets of momentum mappings
NASA Astrophysics Data System (ADS)
Goldman, William M.; Millson, John J.
1990-08-01
The germ of an analytic variety X at a point x∈ X is said to be quadratic if it is bi-analytically isomorphic to the germ of a cone defined by a system of homogeneous quadratic equations at the origin. Arms, Marsden and Moncrief show in [2] that under certain conditions the analytic germ of a level set of a momentum mapping is quadratic. We discuss related ideas in a more algebraic context by associating to an affine Hamiltonian action a differential graded Lie algebra, which in the presence of an invariant positive complex structure, is formal in the sence of [5].
Representations and module-extensions of 3-hom-Lie algebras
NASA Astrophysics Data System (ADS)
Liu, Yan; Chen, Liangyun; Ma, Yao
2015-12-01
In this paper, we study the representations and module-extensions of 3-hom-Lie algebras. We show that a linear map between 3-hom-Lie algebras is a morphism if and only if its graph is a hom subalgebra and show that the set of derivations of a 3-hom-Lie algebra is a Lie algebra. Moreover, we introduce the definition of Tθ-extensions and Tθ∗ -extensions of 3-hom-Lie algebras in terms of modules, providing the necessary and sufficient conditions for a 2 k-dimensional metric 3-hom-Lie algebra to be isometric to a Tθ∗ -extension.
BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras
NASA Astrophysics Data System (ADS)
Graziani, Giacomo; Makhlouf, Abdenacer; Menini, Claudia; Panaite, Florin
2015-10-01
A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α,β\\colon A→ A such that α (a)(bc)=(ab)β (c), for all a, b, cin A. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).
Hom Gel'fand-Dorfman bialgebras and Hom-Lie conformal algebras
Yuan, Lamei
2014-04-15
The aim of this paper is to introduce the notions of Hom Gel'fand-Dorfman bialgebra and Hom-Lie conformal algebra. In this paper, we give four constructions of Hom Gel'fand-Dorfman bialgebras. Also, we provide a general construction of Hom-Lie conformal algebras from Hom-Lie algebras. Finally, we prove that a Hom Gel'fand-Dorfman bialgebra is equivalent to a Hom-Lie conformal algebra of degree 2.
Yau, Stephen S.-T.
1983-01-01
A natural mapping from the set of complex analytic isolated hypersurface singularities to the set of finite dimensional Lie algebras is first defined. It is proven that the image under this natural mapping is contained in the set of solvable Lie algebras. This approach gives rise to a continuous inequivalent family of finite dimensional representations of a solvable Lie algebra. PMID:16593401
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.
Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras
NASA Astrophysics Data System (ADS)
Sun, Bing; Chen, Liangyun
2015-12-01
In this paper, we introduce the concepts of Rota-Baxter operators and differential operators with weights on a multiplicative n-ary Hom-algebra. We then focus on Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras and show that they can be derived from Rota-Baxter Hom-Lie algebras, Hom-preLie algebras and Rota-Baxter commutative Hom-associative algebras. We also explore the connections between these Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras.
Lie algebra lattices and strings on T-folds
NASA Astrophysics Data System (ADS)
Satoh, Yuji; Sugawara, Yuji
2017-02-01
We study the world-sheet conformal field theories for T-folds systematically based on the Lie algebra lattices representing the momenta of strings. The fixed point condition required for the T-duality twist restricts the possible Lie algebras. When the T-duality acts as a simple chiral reflection, one is left with the four cases, A 1 , D 2 r , E 7 , E 8, among the simple simply-laced algebras. From the corresponding Englert-Neveu lattices, we construct the modular invariant partition functions for the T-fold CFTs in bosonic string theory. Similar construction is possible also by using Euclidean even self-dual lattices. We then apply our formulation to the T-folds in the E 8 × E 8 heterotic string theory. Incorporating non-trivial phases for the T-duality twist, we obtain, as simple examples, a class of modular invariant partition functions parametrized by three integers. Our construction includes the cases which are not reduced to the free fermion construction.
Rational solutions of CYBE for simple compact real Lie algebras
NASA Astrophysics Data System (ADS)
Pop, Iulia; Stolin, Alexander
2007-04-01
In [A.A. Stolin, On rational solutions of Yang-Baxter equation for sl(n), Math. Scand. 69 (1991) 57-80; A.A. Stolin, On rational solutions of Yang-Baxter equation. Maximal orders in loop algebra, Comm. Math. Phys. 141 (1991) 533-548; A. Stolin, A geometrical approach to rational solutions of the classical Yang-Baxter equation. Part I, in: Walter de Gruyter & Co. (Ed.), Symposia Gaussiana, Conf. Alg., Berlin, New York, 1995, pp. 347-357] a theory of rational solutions of the classical Yang-Baxter equation for a simple complex Lie algebra g was presented. We discuss this theory for simple compact real Lie algebras g. We prove that up to gauge equivalence all rational solutions have the form X(u,v)={Ω}/{u-v}+t1∧t2+⋯+t∧t2n, where Ω denotes the quadratic Casimir element of g and {ti} are linearly independent elements in a maximal torus t of g. The quantization of these solutions is also emphasized.
A quantum affine algebra for the deformed Hubbard chain
NASA Astrophysics Data System (ADS)
Beisert, Niklas; Galleas, Wellington; Matsumoto, Takuya
2012-09-01
The integrable structure of the one-dimensional Hubbard model is based on Shastry's R-matrix and the Yangian of a centrally extended \\mathfrak {sl}(2|2) superalgebra. Alcaraz and Bariev have shown that the model admits an integrable deformation whose R-matrix has recently been found. This R-matrix is of trigonometric type and here we derive its underlying exceptional quantum affine algebra. We also show how the algebra reduces to the above-mentioned Yangian and to the conventional quantum affine \\mathfrak {sl}(2|2) algebra in two special limits.
Paal, Eugen; Virkepu, Jueri
2009-05-15
Operadic Lax representations for the harmonic oscillator are used to construct the dynamical deformations of three-dimensional (3D) real Lie algebras in the Bianchi classification. It is shown that the energy conservation of the harmonic oscillator is related to the Jacobi identities of the dynamically deformed algebras. Based on this observation, it is proved that the dynamical deformations of 3D real Lie algebras in the Bianchi classification over the harmonic oscillator are Lie algebras.
NASA Astrophysics Data System (ADS)
Miao, Qian; Hu, Xiao-Rui; Chen, Yong
2014-02-01
We present a Maple computer algebra package, ONEOptimal, which can calculate one-dimensional optimal system of finite dimensional Lie algebra for nonlinear equations automatically based on Olver's theory. The core of this theory is viewing the Killing form of the Lie algebra as an invariant for the adjoint representation. Some examples are given to demonstrate the validity and efficiency of the program.
Classical integrable finite-dimensional systems related to Lie algebras
NASA Astrophysics Data System (ADS)
Olshanetsky, M. A.; Perelomov, A. M.
1981-05-01
During the last few years many dynamical systems have been identified, that are completely integrable or even such to allow an explicit solution of the equations of motion. Some of these systems have the form of classical one-dimensional many-body problems with pair interactions; others are more general. All of them are related to Lie algebras, and in all known cases the property of integrability results from the presence of higher (hidden) symmetries. This review presents from a general and universal viewpoint the results obtained in this field during the last few years. Besides it contains some new results both of physical and mathematical interest. The main focus is on the one-dimensional models of n particles interacting pairwise via potentials V( q) = g2ν( q) of the following 5 types: ν I(q)=q -2, ν II(q)=a -2sinh2(aq), ν III(q)=a 2/ sin2(aq), ν IV=a 2P(aq), , ν V(q)=q -2+ω 2q 2. Here P( q) is the Weierstrass function, so that the first 3 cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbor potential, gj2exp[- a( qj- qj+1 )], is moreover considered. Various generalizations of these models, naturally suggested by their association with Lie algebras, are also treated.
Finite-Dimensional Lie Algebras for Fast Diffeomorphic Image Registration.
Zhang, Miaomiao; Fletcher, P Thomas
2015-01-01
This paper presents a fast geodesic shooting algorithm for diffeomorphic image registration. We first introduce a novel finite-dimensional Lie algebra structure on the space of bandlimited velocity fields. We then show that this space can effectively represent initial velocities for diffeomorphic image registration at much lower dimensions than typically used, with little to no loss in registration accuracy. We then leverage the fact that the geodesic evolution equations, as well as the adjoint Jacobi field equations needed for gradient descent methods, can be computed entirely in this finite-dimensional Lie algebra. The result is a geodesic shooting method for large deformation metric mapping (LDDMM) that is dramatically faster and less memory intensive than state-of-the-art methods. We demonstrate the effectiveness of our model to register 3D brain images and compare its registration accuracy, run-time, and memory consumption with leading LDDMM methods. We also show how our algorithm breaks through the prohibitive time and memory requirements of diffeomorphic atlas building.
Unified algebraic method to non-Hermitian systems with Lie algebraic linear structure
NASA Astrophysics Data System (ADS)
Zhang, Hong-Biao; Jiang, Guang-Yuan; Wang, Gang-Cheng
2015-07-01
We suggest a generic algebraic method to solve non-Hermitian Hamiltonian systems with Lie algebraic linear structure. Such method can not only unify the non-Hermitian Hamiltonian and the Hermitian Hamiltonian with the same structure but also keep self-consistent between similarity transformation and unitary transformation. To clearly reveal the correctness and physical meaning of such algebraic method, we illustrate our method with two different types of non-Hermitian Hamiltonians: the non-Hermitian Hamiltonian with Heisenberg algebraic linear structure and the non-Hermitian Hamiltonian with su(1, 1) algebraic linear structure. We obtain energy eigenvalues and the corresponding eigenstates of non-Hermitian forced harmonic oscillator with Heisenberg algebra structure via a proper similarity transformation. We also calculate the eigen-problems of general non-Hermitian Hamiltonian with su(1, 1) structure in terms of the similarity transformation. As an application, we focus on studying the non-Hermitian single-mode squeezed and coherent harmonic oscillator system and find that such similarity transformation associated with this model is in fact gauge-like transformation for simple harmonic oscillator.
NASA Astrophysics Data System (ADS)
Krylyuk, Ya S.
1985-02-01
The maximal dimension is computed for irreducible representations of the Hamiltonian Lie p-algebra and the special Lie p-algebra of an even number of variables over an algebraically closed field of characteristic p>3.Bibliography: 11 titles.
A Cohomological Proof that Real Representations of Semisimple Lie Algebras Have Q-Forms
NASA Astrophysics Data System (ADS)
Morris, Dave Witte
2015-04-01
A Lie algebra g_Q over Q is said to be R-universal if every homomorphism from g_Q to gl(n,R) is conjugate to a homomorphism into gl(n,Q) (for every n). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an R-universal Q-form. We also provide a classification of the R-universal Lie algebras that are semisimple.
One-parameter formal deformations of Hom-Lie-Yamaguti algebras
NASA Astrophysics Data System (ADS)
Ma, Yao; Chen, Liangyun; Lin, Jie
2015-01-01
This paper studies one-parameter formal deformations of Hom-Lie-Yamaguti algebras. The first, second, and third cohomology groups on Hom-Lie-Yamaguti algebras extending ones on Lie-Yamaguti algebras are provided. It is proved that first and second cohomology groups are suitable to the deformation theory involving infinitesimals, equivalent deformations, and rigidity. However, the third cohomology group is not suitable for the obstructions.
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.
Yu, Zhang; Zhang, Yufeng
2009-01-15
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.
Dedekind's η-function and the cohomology of infinite dimensional Lie algebras
Garland, Howard
1975-01-01
We compute the cohomology of certain infinite dimensional Lie algebras which are subalgebras of Lie algebras introduced by Moody and Kac. We note a relation between our results and the cohomology of loop spaces of compact groups. Finally, we derive, by Euler-Poincaré, identities of Macdonald for powers of the Dedekind η-function. PMID:16592258
Generating functions and multiplicity formulas: The case of rank two simple Lie algebras
NASA Astrophysics Data System (ADS)
Fernández Núñez, José; García Fuertes, Wifredo; Perelomov, Askold M.
2015-09-01
A procedure is described that makes use of the generating function of characters to obtain a new generating function H giving the multiplicities of each weight in all the representations of a simple Lie algebra. The way to extract from H explicit multiplicity formulas for particular weights is explained and the results corresponding to rank two simple Lie algebras are shown.
On the Lie Symmetry Algebras of the Stationary Schrödinger and Pauli Equations
NASA Astrophysics Data System (ADS)
Boldyreva, M. N.; Magazev, A. A.
2017-02-01
A general method for constructing first-order symmetry operators for the stationary Schrödinger and Pauli equations is proposed. It is proven that the Lie algebra of these symmetry operators is a one-dimensional extension of some subalgebra of an e(3) algebra. We also assemble a classification of stationary electromagnetic fields for which the Schrödinger (or Pauli) equation admits a Lie algebra of first-order symmetry operators.
A few Lie algebras and their applications for generating integrable hierarchies of evolution types
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Feng, Binlu
2011-08-01
A Lie algebra consisting of 3 × 3 matrices is introduced, whose induced Lie algebra by using an inverted linear transformation is obtained as well. As for application examples, we obtain a unified integrable model of the integrable couplings of the AKNS hierarchy, the D-AKNS hierarchy and the TD hierarchy as well as their induced integrable hierarchies. These integrable couplings are different from those results obtained before. However, the Hamiltonian structures of the integrable couplings cannot be obtained by using the quadratic-form identity or the variational identity. For solving the problem, we construct a higher-dimensional subalgebra R and its reduced algebra Q of the Lie algebra A2 by decomposing the induced Lie algebra and then again making some linear combinations. The subalgebras of the Lie algebras R and Q do not satisfy the relation ( G=G1⊕G2,[G1,G2]⊂G2), but we can deduce integrable couplings, which indicates that the above condition is not necessary to generate integrable couplings. As for application example, an expanding integrable model of the AKNS hierarchy is obtained whose Hamiltonian structure is generated by the trace identity. Finally, we give another Lie algebras which can be decomposed into two simple Lie subalgebras for which a nonlinear integrable coupling of the classical Boussinesq-Burgers (CBB) hierarchy is obtained.
A new approach to tolerance analysis method based onthe screw and the Lie Algebra of Lie Group
NASA Astrophysics Data System (ADS)
Zhai, X. C.; Du, Q. G.; Wang, W. X.; Wen, Q.; Liu, B. S.; Sun, Z. Q.
2016-11-01
Tolerance analysis refers to the process of establishing mapping relations between tolerance features and the target feature along the dimension chain. Traditional models for tolerance analysis are all based on rigid body kinematics, and they adopt the Homogeneous Transformation Matrix to describe feature variation and accumulation. However, those models can hardly reveal the nature of feature variations. This paper proposes a new tolerance analysis method based on the screw and the Lie Algebra of Lie Group, which describes feature variation as the screw motion, and completely maps the twist, an element of the Lie Algebra, to the Lie Group that represents the feature configuration space. Thus, the analysis can be conducted in a more succinct and direct way. In the end, the method is applied in an example and proven to be robust and effective.
Zhao, Shouwei
2011-06-01
A Lie algebraic condition for global exponential stability of linear discrete switched impulsive systems is presented in this paper. By considering a Lie algebra generated by all subsystem matrices and impulsive matrices, when not all of these matrices are Schur stable, we derive new criteria for global exponential stability of linear discrete switched impulsive systems. Moreover, simple sufficient conditions in terms of Lie algebra are established for the synchronization of nonlinear discrete systems using a hybrid switching and impulsive control. As an application, discrete chaotic system's synchronization is investigated by the proposed method.
NASA Astrophysics Data System (ADS)
Zhao, Shouwei
2011-06-01
A Lie algebraic condition for global exponential stability of linear discrete switched impulsive systems is presented in this paper. By considering a Lie algebra generated by all subsystem matrices and impulsive matrices, when not all of these matrices are Schur stable, we derive new criteria for global exponential stability of linear discrete switched impulsive systems. Moreover, simple sufficient conditions in terms of Lie algebra are established for the synchronization of nonlinear discrete systems using a hybrid switching and impulsive control. As an application, discrete chaotic system's synchronization is investigated by the proposed method.
NASA Astrophysics Data System (ADS)
Dobrev, V. K.
2014-05-01
In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier '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 introduced recently 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 E7(7) which is parabolically related to the CLA E7(-25). 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). Further we consider the algebras sl(2n, Bbb R) and for n = 2k the algebras su* (4k) which are parabolically related to the CLA su(n,n). Further we consider the algebras sp(r,r) which are parabolically related to the CLA sp(2r, Bbb R). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14),
Lie Algebraic Discussions for Time-Inhomogeneous Linear Birth-Death Processes with Immigration
NASA Astrophysics Data System (ADS)
Ohkubo, Jun
2014-10-01
Analytical solutions for time-inhomogeneous linear birth-death processes with immigration are derived. While time-inhomogeneous linear birth-death processes without immigration have been studied by using a generating function approach, the processes with immigration are here analyzed by Lie algebraic discussions. As a result, a restriction for time-inhomogeneity of the birth-death process is understood from the viewpoint of the finiteness of the dimensionality of the Lie algebra.
On an approach for computing the generating functions of the characters of simple Lie algebras
NASA Astrophysics Data System (ADS)
Fernández Núñez, José; García Fuertes, Wifredo; Perelomov, Askold M.
2014-04-01
We describe a general approach to obtain the generating functions of the characters of simple Lie algebras which is based on the theory of the quantum trigonometric Calogero-Sutherland model. We show how the method works in practice by means of a few examples involving some low rank classical algebras.
Abedi-Fardad, J.; Rezaei-Aghdam, A.; Haghighatdoost, Gh.
2014-05-15
We construct integrable and superintegrable Hamiltonian systems using the realizations of four dimensional real Lie algebras as a symmetry of the system with the phase space R{sup 4} and R{sup 6}. Furthermore, we construct some integrable and superintegrable Hamiltonian systems for which the symmetry Lie group is also the phase space of the system.
Higher powers of analytical operators and associated ∗-Lie algebras
NASA Astrophysics Data System (ADS)
Ettaieb, Aymen; Khalifa, Narjess Turki; Ouerdiane, Habib; Rguigui, Hafedh
2016-06-01
We introduce a new product of two test functions denoted by f□g (where f and g in the Schwartz space 𝒮(ℝ)). Based on the space of entire functions with θ-exponential growth of minimal type, we define a new family of infinite dimensional analytical operators using the holomorphic derivative and its adjoint. Using this new product f□g, such operators give us a new representation of the centerless Virasoro-Zamolodchikov-ω∞∗-Lie algebras (in particular the Witt algebra) by using analytical renormalization conditions and by taking the test function f as any Hermite function. Replacing the classical pointwise product f ṡ g of two test functions f and g by f□g, we prove the existence of new ∗-Lie algebras as counterpart of the classical powers of white noise ∗-Lie algebra, the renormalized higher powers of white noise (RHPWN) ∗-Lie algebra and the second quantized centerless Virasoro-Zamolodchikov-ω∞∗-Lie algebra.
SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System
NASA Astrophysics Data System (ADS)
Choi, J. R.; Nahm, I. H.
2007-01-01
Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola-Kanai oscillator. The probability density of these coherent states for the Caldirola-Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed.
NASA Astrophysics Data System (ADS)
Fernández Núñez, J.; García Fuertes, W.; Perelomov, A. M.
2005-07-01
The quantum trigonometric Calogero-Sutherland models related to Lie algebras admit a parametrization in which the dynamical variables are the characters of the fundamental representations of the algebra. We develop here this approach for the case of the exceptional Lie algebra E6.
NASA Astrophysics Data System (ADS)
Guzzo, H.; Hernández, I.; Sánchez-Valenzuela, O. A.
2014-09-01
Finite dimensional semisimple real Lie superalgebras are described via finite dimensional semisimple complex Lie superalgebras. As an application of these results, finite dimensional real Lie superalgebras mathfrak {m}=mathfrak {m}_0 oplus mathfrak {m}_1 for which mathfrak {m}_0 is a simple Lie algebra are classified up to isomorphism.
Hidden symmetries and Lie algebra structures from geometric and supergravity Killing spinors
NASA Astrophysics Data System (ADS)
Açık, Özgür; Ertem, Ümit
2016-08-01
We consider geometric and supergravity Killing spinors and the spinor bilinears constructed out of them. The spinor bilinears of geometric Killing spinors correspond to the antisymmetric generalizations of Killing vector fields which are called Killing-Yano forms. They constitute a Lie superalgebra structure in constant curvature spacetimes. We show that the Dirac currents of geometric Killing spinors satisfy a Lie algebra structure up to a condition on 2-form spinor bilinears. We propose that the spinor bilinears of supergravity Killing spinors give way to different generalizations of Killing vector fields to higher degree forms. It is also shown that those supergravity Killing forms constitute a Lie algebra structure in six- and ten-dimensional cases. For five- and eleven-dimensional cases, the Lie algebra structure depends on an extra condition on supergravity Killing forms.
Analysis of higher order optical aberrations in the SLC final focus using Lie Algebra techniques
Walker, N.J.; Irwin, J.; Woodley, M.
1993-04-01
The SLC final focus system is designed to have an overall demagnification of 30:1, with a {beta} at the interaction point ({beta}*) of 5 mm, and an energy band pass of {approximately}0.4%. Strong sextupole pairs are used to cancel the large chromaticity which accrues primarily from the final triplet. Third-order aberrations limit the performance of the system, the dominating terms being U{sub 1266} and U{sub 3466} terms (in the notation of K. Brown). Using Lie Algebra techniques, it is possible to analytically calculate the soave of these terms in addition to understanding their origin. Analytical calculations (using Lie Algebra packages developed in the Mathematica language) are presented of the bandwidth and minimum spot size as a function of divergence at the interaction point (IP). Comparisons of the analytical results from the Lie Algebra maps and results from particle tracking (TURTLE) are also presented.
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Tam, Honwah
2016-03-01
In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016
Nonlinear Two-Mode Squeezed Vacuum States as Realization of SU(1,1) Lie Algebra
NASA Astrophysics Data System (ADS)
Abd Al-Kader, Gamal Mohamed; Fahmy Obada, Abdel-Shafy
2008-05-01
Nonlinear extensions of the two-mode squeezed vacuum states (NTMSVS's) are constructed and special cases of these states are discussed. We have constructed the NTMSVS's realization of SU(1,1) Lie algebra. Two cases of the definition are considered for unitary and non-unitary deformation operator functions. Some nonclassical properties of these states are discussed.
Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms
NASA Astrophysics Data System (ADS)
Fu, Chih-Hao; Krasnov, Kirill
2017-01-01
Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work.
On the generating function of weight multiplicities for the representations of the Lie algebra C2
NASA Astrophysics Data System (ADS)
Fernández-Núñez, José; García-Fuertes, Wifredo; Perelomov, Askold M.
2015-04-01
We use the generating function of the characters of C2 to obtain a generating function for the multiplicities of the weights entering in the irreducible representations of that simple Lie algebra. From this generating function, we derive some recurrence relations among the multiplicities and a simple graphical recipe to compute them.
Description of a class of superstring compactifications related to semi-simple Lie algebras
NASA Astrophysics Data System (ADS)
Markushevich, D. G.; Olshanetsky, M. A.; Perelomov, A. M.
1987-06-01
A class of vacuum configurations in the superstring theory obtained by compactification of physical dimensions from ten to four is constructed. The compactification scheme involves taking quotients of tori of semisimple Lie algebras by finite symmetry group actions. The complete list of such configurations arising from actions by a Coxeter transformation is given. Some topological invariants having physical interpretations are calculated.
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.
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].
Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups
Guedes, Carlos; Oriti, Daniele; Raasakka, Matti
2013-08-15
The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a generalized notion of (non-commutative) Fourier transform, different from standard harmonic analysis, has been recently developed, and found several applications, especially in the quantum gravity literature. We show that this algebra representation can be defined on the sole basis of a quantization map of the classical Poisson algebra, and identify the conditions for its existence. In particular, the corresponding non-commutative star-product carried by this representation is obtained directly from the quantization map via deformation quantization. We then clarify under which conditions a unitary intertwiner between such algebra representation and the usual group representation can be constructed giving rise to the non-commutative plane waves and consequently, the non-commutative Fourier transform. The compact groups U(1) and SU(2) are considered for different choices of quantization maps, such as the symmetric and the Duflo map, and we exhibit the corresponding star-products, algebra representations, and non-commutative plane waves.
Galitski, Victor
2011-07-15
We propose a Lie-algebraic duality approach to analyze nonequilibrium evolution of closed dynamical systems and thermodynamics of interacting quantum lattice models (formulated in terms of Hubbard-Stratonovich dynamical systems). The first part of the paper utilizes a geometric Hilbert-space-invariant formulation of unitary time evolution, where a quantum Hamiltonian is viewed as a trajectory in an abstract Lie algebra, while the sought-after evolution operator is a trajectory in a dynamic group, generated by the algebra via exponentiation. The evolution operator is uniquely determined by the time-dependent dual generators that satisfy a system of differential equations, dubbed here dual Schroedinger-Bloch equations, which represent a viable alternative to the conventional Schroedinger formulation. These dual Schroedinger-Bloch equations are derived and analyzed on a number of specific examples. It is shown that deterministic dynamics of a closed classical dynamical system occurs as action of a symmetry group on a classical manifold and is driven by the same dual generators as in the corresponding quantum problem. This represents quantum-to-classical correspondence. In the second part of the paper, we further extend the Lie-algebraic approach to a wide class of interacting many-particle lattice models. A generalized Hubbard-Stratonovich transform is proposed and it is used to show that the thermodynamic partition function of a generic many-body quantum lattice model can be expressed in terms of traces of single-particle evolution operators governed by the dynamic Hubbard-Stratonovich fields. The corresponding Hubbard-Stratonovich dynamical systems are generally nonunitary, which yields a number of notable complications, including breakdown of the global exponential representation. Finally, we derive Hubbard-Stratonovich dynamical systems for the Bose-Hubbard model and a quantum spin model and use the Lie-algebraic approach to obtain new nonperturbative dual
Lie algebraic structures of (1+1)-dimensional Lax integrable systems
Chen, D.; Zhang, D.
1996-11-01
An approach of constructing isospectral flows {ital K}{sub {ital l}}, nonisospectral flows {sigma}{sub {ital k}} and their implicit representations of a general Lax integrable system is proposed. By introducing product function matrices, it is shown that the two sets of flows and of related symmetries both constitute infinite-dimensional Lie algebras with respect to the commutator [{center_dot},{center_dot}] given in this paper. Algebraic properties for some well-known integrable systems such as the AKNS system, the generalized Harry Dym system, and the {ital n}-wave interaction system are obtained as particular examples. {copyright} {ital 1996 American Institute of Physics.}
The Hom-Yang-Baxter equation and Hom-Lie algebras
Yau, Donald
2011-05-15
Motivated by recent work on Hom-Lie algebras, a twisted version of the Yang-Baxter equation, called the Hom-Yang-Baxter equation (HYBE), was introduced by Yau [J. Phys. A 42, 165202 (2009)]. In this paper, several more classes of solutions of the HYBE are constructed. Some of the solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the Jones-Conway polynomial, and Yetter-Drinfel'd modules. Under some invertibility conditions, we construct a new infinite sequence of solutions of the HYBE from a given one.
a Perspective on the Magic Square and the "special Unitary" Realization of Real Simple Lie Algebras
NASA Astrophysics Data System (ADS)
Santander, Mariano
2013-07-01
This paper contains the last part of the minicourse "Spaces: A Perspective View" delivered at the IFWGP2012. The series of three lectures was intended to bring the listeners from the more naive and elementary idea of space as "our physical Space" (which after all was the dominant one up to the 1820s) through the generalization of the idea of space which took place in the last third of the 19th century. That was a consequence of first the discovery and acceptance of non-Euclidean geometry and second, of the views afforded by the works of Riemann and Klein and continued since then by many others, outstandingly Lie and Cartan. Here we deal with the part of the minicourse which centers on the classification questions associated to the simple real Lie groups. We review the original introduction of the Magic Square "á la Freudenthal", putting the emphasis in the role played in this construction by the four normed division algebras ℝ, ℂ, ℍ, 𝕆. We then explore the possibility of understanding some simple real Lie algebras as "special unitary" over some algebras 𝕂 or tensor products 𝕂1 ⊗ 𝕂2, and we argue that the proper setting for this construction is not to confine only to normed division algebras, but to allow the split versions ℂ‧, ℍ‧, 𝕆‧ of complex, quaternions and octonions as well. This way we get a "Grand Magic Square" and we fill in all details required to cover all real forms of simple real Lie algebras within this scheme. The paper ends with the complete lists of all realizations of simple real Lie algebras as "special unitary" (or only unitary when n = 2) over some tensor product of two *-algebras 𝕂1, 𝕂2, which in all cases are obtained from ℝ, ℂ, ℂ‧, ℍ, ℍ‧, 𝕆, 𝕆‧ as sets, endowing them with a *-conjugation which usually but not always is the natural complex, quaternionic or octonionic conjugation.
NASA Astrophysics Data System (ADS)
Kurnyavko, O. L.; Shirokov, I. V.
2016-07-01
We offer a method for constructing invariants of the coadjoint representation of Lie groups that reduces this problem to known problems of linear algebra. This method is based on passing to symplectic coordinates on the coadjoint representation orbits, which play the role of local coordinates on those orbits. The corresponding transition functions are their parametric equations. Eliminating the symplectic coordinates from the transition functions, we can obtain the complete set of invariants. The proposed method allows solving the problem of constructing invariants of the coadjoint representation for Lie groups with an arbitrary dimension and structure.
Bifurcation diagram and the discriminant of a spectral curve of integrable systems on Lie algebras
Konyaev, Andrei Yu
2010-11-11
A bifurcation diagram is a stratified (in general, nonclosed) set and is one of the efficient tools of studying the topology of the Liouville foliation. In the framework of the present paper, the coincidence of the closure of a bifurcation diagram {Sigma}-bar of the moment map defined by functions obtained by the method of argument shift with the closure of the discriminant D-bar{sub z} of a spectral curve is proved for the Lie algebras sl(n+1), sp(2n), so(2n+1), and g{sub 2}. Moreover, it is proved that these sets are distinct for the Lie algebra so(2n). Bibliography: 22 titles.
Lie algebras for systems with mixed spectra. I. The scattering Pöschl-Teller potential
NASA Astrophysics Data System (ADS)
Frank, Alejandro; Wolf, Kurt Bernardo
1985-05-01
Starting from an N-body quantum space, we consider the Lie-algebraic framework where the Pöschl-Teller Hamiltonian, - 1/2 ∂2χ +c sech2 χ+s csch2 χ, is the single sp(2,R) Casimir operator. The spectrum of this system is mixed: it contains a finite number of negative-energy bound states and a positive-energy continuum of free states; it is identified with the Clebsch-Gordan series of the D+×D- representation coupling. The wave functions are the sp(2,R) Clebsch-Gordan coefficients of that coupling in the parabolic basis. Using only Lie-algebraic techniques, we find the asymptotic behavior of these wave functions; for the special pure-trough potential (s=0) we derive thus the transmission and reflection amplitudes of the scattering matrix.
The Wheeler-DeWitt Equation in Filćhenkov Model: The Lie Algebraic Approach
NASA Astrophysics Data System (ADS)
Panahi, H.; Zarrinkamar, S.; Baradaran, M.
2016-11-01
The Wheeler-DeWitt equation in Filćhenkov model with terms related to strings, dust, relativistic matter, bosons and fermions, and ultra stiff matter is solved in a quasi-exact analytical manner via the Lie algebraic approach. In the calculations, using the representation theory of sl(2), the general (N+1)-dimensional matrix equation is constructed whose determinant yields the solutions of the problem.
Calculus structure on the Lie conformal algebra complex and the variational complex
NASA Astrophysics Data System (ADS)
De Sole, Alberto; Hekmati, Pedram; Kac, Victor G.
2011-05-01
We construct a calculus structure on the Lie conformal algebra cochain complex. By restricting to degree one chains, we recover the structure of a {mathfrak g}-complex introduced in [A. De Sole and V. G. Kac, Commun. Math. Phys. 292, 667 (2009), 10.1007/s00220-009-0886-1]. A special case of this construction is the variational calculus, for which we provide explicit formulas.
Scattering and bound state Green's functions on a plane via so(2,1) Lie algebra
Borges, P. F.; Boschi-Filho, H.; Vaidya, A. N.
2006-11-15
We calculate the Green's functions for the particle-vortex system, for two anyons on a plane with and without a harmonic regulator and in a uniform magnetic field. These Green's functions which describe scattering or bound states (depending on the specific potential in each case) are obtained exactly using an algebraic method related to the SO(2,1) Lie group. From these Green's functions we obtain the corresponding wave functions and for the bound states we also find the energy spectra.
Lie{endash}Poisson deformation of the Poincar{acute e} algebra
Stern, A. |
1996-04-01
We find a one-parameter family of quadratic Poisson structures on {bold R}{sup 4}{times}SL(2,{ital C}) which satisfies the properties: (a) that it reduces to the standard Poincar{acute e} algebra for a particular limiting value of the parameter (which we associate with the {open_quote}{open_quote}canonical limit{close_quote}{close_quote}), as well as, (b) that it is preserved under the Lie{endash}Poisson action of the Lorentz group (and the Lie{endash}Poisson transformations reduce to canonical ones in the canonical limit). As with the Poincar{acute e} algebra, our deformed Poincar{acute e} algebra has two Casimir functions which correspond to {open_quote}{open_quote}mass{close_quote}{close_quote} and {open_quote}{open_quote}spin.{close_quote}{close_quote} The constant mass and spin surfaces in {bold R}{sup 4}{times}SL(2,{ital C}) define symplectic leaves which we parametrize with space{endash}time coordinates, momenta, and spin. We thereby obtain realizations of the deformed Poincar{acute e} algebra for both spinning and spinless particles. The formalism can be applied for finding a one-parameter family of canonically inequivalent descriptions of the photon. {copyright} {ital 1996 American Institute of Physics.}
NASA Astrophysics Data System (ADS)
Campoamor-Stursberg, R.
2016-06-01
A functional realization of the Lie algebra s l (" separators=" 3 , R) as a Vessiot-Guldberg-Lie algebra of second order differential equation (SODE) Lie systems is proposed. It is shown that a minimal Vessiot-Guldberg-Lie algebra L V G is obtained from proper subalgebras of s l (" separators=" 3 , R) for each of the SODE Lie systems of this type by particularization of one functional and two scalar parameters of the s l (" separators=" 3 , R) -realization. The relation between the various Vessiot-Guldberg-Lie algebras by means of a limiting process in the scalar parameters further allows to define a notion of contraction of SODE Lie systems.
Lie super-bialgebra structures on super W-algebra SW(3/2, 3/2)
NASA Astrophysics Data System (ADS)
Yang, Hengyun; Sun, Jiancai
2017-04-01
Super W-algebra SW(3/2,3/2) is the supersymmetric extension of Galilean conformal algebra in 2d. In this paper, we obtain that super W-algebra SW(3/2,3/2) admits only triangular coboundary Lie super-bialgebra structures and the proof is mainly based on the computation of the first cohomology group of SW(3/2,3/2) with coefficients in the tensor product of its adjoint module.
NASA Astrophysics Data System (ADS)
Fernández Núñez, J.; García Fuertes, W.; Perelomov, A. M.
2003-11-01
We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra D4 in terms of a set of Weyl-invariant variables, namely, the characters of the fundamental representations of the Lie algebra. This parametrization allows us to solve for the energy eigenfunctions of the theory and to study properties of the system of orthogonal polynomials associated with them such as recurrence relations and generating functions.
Joint geometric and photometric direct image registration based on Lie algebra parameterization
NASA Astrophysics Data System (ADS)
Li, Chenxi; Shi, Zelin; Liu, Yunpeng
2016-10-01
In this paper, we consider direct image registration problem which estimate the geometric and photometric transformations between two images. The efficient second-order minimization method (ESM) is based on a second-order Taylor series of image differences without computing the Hessian under brightness constancy assumption. This can be done due to the fact that the considered geometric transformations is Lie group and can be parameterized by its Lie algebra. In order to deal with lighting changes, we extend ESM to the compositional dual efficient second-order minimization method (CDESM). In our approach, the photometric transformations is parameterized by its Lie algebra with compositional operation, which is similar to that of geometric transformations. Our algorithm can give a second-order approximation of image differences with respect to geometric and photometric parameters. The geometric and photometric parameters are simultaneously obtained by non-linear least-square optimization. Our algorithm preserves the advantages of the original ESM method which has high convergence rate and large capture radius. Experimental results show that our algorithm is more robust to lighting changes and has higher registration accuracy compared to previous algorithms.
Laplace operators of infinite-dimensional Lie algebras and theta functions.
Kac, V G
1984-01-01
Until recently, the generalized Casimir operator constructed by Kac [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70] has been the only known element of the center of a completion of the enveloping algebra of a Kac-Moody algebra. It has been conjectured [Deodhar, V. V., Gabber, O. & Kac, V. G. (1982) Adv. Math. 45, 92-116], however, that the image of the Harish-Chandra homomorphism contains all theta functions defined on the interior of the complexified Tits cone and hence separates the orbits of the Weyl group. Developing the ideas of Feigin and Fuchs [Feigin, B. L. & Fuchs, D. B. (1983) Dokl. Akad. Nauk SSSR 269, 1057-1060], I prove this conjecture. Another application of this method is the Chevalley type restriction theorem for simple finite-dimensional Lie superalgebras.
Laplace operators of infinite-dimensional Lie algebras and theta functions
Kac, Victor G.
1984-01-01
Until recently, the generalized Casimir operator constructed by Kac [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70] has been the only known element of the center of a completion of the enveloping algebra of a Kac-Moody algebra. It has been conjectured [Deodhar, V. V., Gabber, O. & Kac, V. G. (1982) Adv. Math. 45, 92-116], however, that the image of the Harish-Chandra homomorphism contains all theta functions defined on the interior of the complexified Tits cone and hence separates the orbits of the Weyl group. Developing the ideas of Feigin and Fuchs [Feigin, B. L. & Fuchs, D. B. (1983) Dokl. Akad. Nauk SSSR 269, 1057-1060], I prove this conjecture. Another application of this method is the Chevalley type restriction theorem for simple finite-dimensional Lie superalgebras. PMID:16593411
Laplace Operators of Infinite-Dimensional Lie Algebras and Theta Functions
NASA Astrophysics Data System (ADS)
Kac, Victor G.
1984-01-01
Until recently, the generalized Casimir operator constructed by Kac [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70] has been the only known element of the center of a completion of the enveloping algebra of a Kac-Moody algebra. It has been conjectured [Deodhar, V. V., Gabber, O. & Kac, V. G. (1982) Adv. Math. 45, 92-116], however, that the image of the Harish-Chandra homomorphism contains all theta functions defined on the interior of the complexified Tits cone and hence separates the orbits of the Weyl group. Developing the ideas of Feigin and Fuchs [Feigin, B. L. & Fuchs, D. B. (1983) Dokl. Akad. Nauk SSSR 269, 1057-1060], I prove this conjecture. Another application of this method is the Chevalley type restriction theorem for simple finite-dimensional Lie superalgebras.
Einstein billiards and overextensions of finite-dimensional simple Lie algebras
NASA Astrophysics Data System (ADS)
Damour, Thibault; de Buyl, Sophie; Henneaux, Marc; Schomblond, Christiane
2002-08-01
In recent papers, it has been shown that (i) the dynamics of theories involving gravity can be described, in the vicinity of a spacelike singularity, as a billiard motion in a region of hyperbolic space bounded by hyperplanes; and (ii) that the relevant billiard has remarkable symmetry properties in the case of pure gravity in d+1 spacetime dimensions, or supergravity theories in 10 or 11 spacetime dimensions, for which it turns out to be the fundamental Weyl chamber of the Kac-Moody algebras AEd, E10, BE10 or DE10 (depending on the model). We analyse in this paper the billiards associated to other theories containing gravity, whose toroidal reduction to three dimensions involves coset models G/H (with G maximally non compact). We show that in each case, the billiard is the fundamental Weyl chamber of the (indefinite) Kac-Moody ``overextension'' (or ``canonical lorentzian extension'') of the finite-dimensional Lie algebra that appears in the toroidal compactification to 3 spacetime dimensions. A remarkable feature of the billiard properties, however, is that they do not depend on the spacetime dimension in which the theory is analyzed and hence are rather robust, while the symmetry algebra that emerges in the toroidal dimensional reduction is dimension-dependent.
Explicit generators in rectangular affine W-algebras of type A
NASA Astrophysics Data System (ADS)
Arakawa, Tomoyuki; Molev, Alexander
2016-10-01
We produce in an explicit form free generators of the affine W-algebra of type A associated with a nilpotent matrix whose Jordan blocks are of the same size. This includes the principal nilpotent case and we thus recover the quantum Miura transformation of Fateev and Lukyanov.
Explicit generators in rectangular affine W-algebras of type A
NASA Astrophysics Data System (ADS)
Arakawa, Tomoyuki; Molev, Alexander
2017-01-01
We produce in an explicit form free generators of the affine W-algebra of type A associated with a nilpotent matrix whose Jordan blocks are of the same size. This includes the principal nilpotent case and we thus recover the quantum Miura transformation of Fateev and Lukyanov.
Exceptional quantum subgroups for the rank two Lie algebras B{sub 2} and G{sub 2}
Coquereaux, R.; Rais, R.; Tahri, E. H.
2010-09-15
Exceptional modular invariants for the Lie algebras B{sub 2} (at levels 2, 3, 7, and 12) and G{sub 2} (at levels 3 and 4) can be obtained from conformal embeddings. We determine the associated algebras of quantum symmetries and discover or recover, as a by-product, the graphs describing exceptional quantum subgroups of type B{sub 2} or G{sub 2} that encode their module structure over the associated fusion category. Global dimensions are given.
On the frames of spaces of finite-dimensional Lie algebras of dimension at most 6
Gorbatsevich, V V
2014-05-31
In this paper, the frames of spaces of complex n-dimensional Lie algebras (that is, the intersections of all irreducible components of these spaces) are studied. A complete description of the frames and their projectivizations for n ≤ 6 is given. It is also proved that for n ≤ 6 the projectivizations of these spaces are simply connected. Bibliography: 7 titles.
Graphical tensor product reduction scheme for the Lie algebras so(5) = sp(2) , su(3) , and g(2)
NASA Astrophysics Data System (ADS)
Vlasii, N. D.; von Rütte, F.; Wiese, U.-J.
2016-08-01
We develop in detail a graphical tensor product reduction scheme, first described by Antoine and Speiser, for the simple rank 2 Lie algebras so(5) = sp(2) , su(3) , and g(2) . This leads to an efficient practical method to reduce tensor products of irreducible representations into sums of such representations. For this purpose, the 2-dimensional weight diagram of a given representation is placed in a "landscape" of irreducible representations. We provide both the landscapes and the weight diagrams for a large number of representations for the three simple rank 2 Lie algebras. We also apply the algebraic "girdle" method, which is much less efficient for calculations by hand for moderately large representations. Computer code for reducing tensor products, based on the graphical method, has been developed as well and is available from the authors upon request.
On boundary fusion and functional relations in the Baxterized affine Hecke algebra
Babichenko, A.; Regelskis, V.
2014-04-15
We construct boundary type operators satisfying fused reflection equation for arbitrary representations of the Baxterized affine Hecke algebra. These operators are analogues of the fused reflection matrices in solvable half-line spin chain models. We show that these operators lead to a family of commuting transfer matrices of Sklyanin type. We derive fusion type functional relations for these operators for two families of representations.
NASA Astrophysics Data System (ADS)
Fu, Yuchen; Shelley-Abrahamson, Seth
2016-06-01
We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using R-matrices for U_q(sl_N). Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.
SU(1,1) Lie algebraic approach for the evolution of the quantum inflationary universe
NASA Astrophysics Data System (ADS)
Choi, Jeong Ryeol
2013-03-01
Quantum behavior of scalar fields and vacuum energy density in the inflationary universe are investigated using SU(1,1) Lie algebraic approach. Wave functions describing the evolution of scalar fields that have been thought to have driven cosmic inflation are identified in several possible quantum states at the early stage of the universe, such as the Fock state, the Glauber coherent state, and the SU(1,1) coherent states. In particular, we focus in this research on two important classes of the SU(1,1) coherent states, which are the so-called even and odd coherent states and the Perelomov coherent state. It is shown in the spatially flat universe driven by a single scalar field that the probability densities in all these states have converged to the origin (ϕ = 0, where ϕ is the scalar field) as time goes by. This outcome implies that the vacuum energy density characterized by the scalar field dissipates with time. The probability density in the matter-dominated era converged more rapidly than that in the radiation-dominated era. Hence, we can confirm that the progress of dissipation for the vacuum energy density became faster as the matter era began after the end of the early dominance of radiation. This consequence is, indeed, in agreement with the results of our previous researches in cosmology (for example, see [Chin. Phys. C 35 (2011) 233] and references there in).
NASA Astrophysics Data System (ADS)
Choi, Jeong Ryeol; Choi, Seong Soo
We realized SU(1,1) Lie algebra in terms of the appropriate SU(1,1) generators for the time-dependent quadratic Hamiltonian system perturbed by a singularity. Exact quantum states of the system are investigated using SU(1,1) Lie algebra. Various expectation values in two kinds of the generalized SU(1,1) coherent states, that is, BG coherent states and Perelomov coherent states are derived. We applied our study to the CKOPS (Caldirola-Kanai oscillator perturbed by a singularity). Due to the damping constant γ, the probability density of the SU(1,1) coherent states for the CKOPS converged to the center with time. The time evolution of the probability density in SU(1,1) coherent states for the CKOPS are very similar to the classical trajectory.
Kozlov, I K
2014-04-30
In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra so(4), which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration, the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank 0 are determined, the bifurcations of Liouville tori are described, and the loop molecules are computed for all singular points of the bifurcation diagrams. It follows from the obtained results that some topological properties of the classical Kovalevskaya case can be obtained from the corresponding properties of the considered integrable case on the Lie algebra so(4) by taking a natural limit. Bibliography: 21 titles.
NASA Astrophysics Data System (ADS)
Akbarzadeh, Rasoul; Haghighatdoost, Ghorbanali
2015-05-01
In 2001, A.V. Borisov, I. S.Mamaev, and V.V. Sokolov discovered a new integrable case on the Lie algebra so(4). This system coincides with the Poincaré equations on the Lie algebra so(4), which describe the motion of a body with cavities filled with an incompressible vortex fluid. Moreover, the Poincaré equations describe the motion of a four-dimensional gyroscope. In this paper topological properties of this system are studied. In particular, for the system under consideration the bifurcation diagrams of the momentum mapping are constructed and all Fomenko invariants are calculated. Thereby, a classification of isoenergy surfaces for this system up to the rough Liouville equivalence is obtained.
A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Wu, Li-Xin; Rui, Wen-Juan
2015-05-01
With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding (2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation (BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing (2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the (2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the (2+1)-dimensional AKNS equation (also called the Davey-Stewartson hierarchy), a kind of (2+1)-dimensional Schrödinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new (2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the (2+1)-dimensional integrable coupling, which is further reduced to the standard (2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known (1+1)-dimensional AKNS hierarchy, the (1+1)-dimensional nonlinear Schrödinger equation are all special cases of the (2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the (2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated. Supported by the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), the National Natural Science Foundation of China under Grant No. 11371361, the Fundamental Research Funds for the Central Universities (2013XK03), and the Natural Science Foundation
Contraction-based classification of supersymmetric extensions of kinematical lie algebras
Campoamor-Stursberg, R.; Rausch de Traubenberg, M.
2010-02-15
We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra leads to a hierarchy similar in structure to the classical Bacry-Levy-Leblond classification.
Super-Lie n-algebra extensions, higher WZW models and super-p-branes with tensor multiplet fields
NASA Astrophysics Data System (ADS)
Fiorenza, Domenico; Sati, Hisham; Schreiber, Urs
2015-12-01
We formalize higher-dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L∞-extension theory and how the resulting "extended (super-)space-times" formalize spacetimes containing σ-model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super-p-brane spectrum of superstring/M-theory is realized this way, including the pure σ-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional (11D) spacetime with an M2-brane condensate turns out to be the "M-theory super-Lie algebra". We also observe that in this formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11D sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string theory. Finally we give the non-perturbative description of all this by higher WZW-type σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology.
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Tam, Honwah; Wu, Lixin
2015-10-01
We deduce a set of integrable equations under the framework of zero curvature equations and obtain two sets of integrable soliton equations, which can be reduced to some new integrable equations including the generalised nonlinear Schrödinger (NLS) equation. Under the case where the isospectral functions are one-order polynomials in the parameter λ, we generate a set of rational integrable equations, which are reduced to the loop soliton equation. Under the case where the derivative λt of the spectral parameter λ is a quadratic algebraic curve in λ, we derive a set of variable-coefficient integrable equations. In addition, we discretise a pair of isospectral problems introduced through the Lie algebra given by us for which a set of new semi-discrete nonlinear equations are available; furthermore, the semi-discrete MKdV equation and the Hirota lattice equation are followed to produce, respectively. Finally, we apply the Lie algebra to introduce a set of operator Lax pairs with an operator, and then through the Tu scheme and the binomial-residue representation method proposed by us, we generate a 2+1-dimensional integrable hierarchy of evolution equations, which reduces to a generalised 2+1-dimensional Davey-Stewartson (DS) equation.
NASA Astrophysics Data System (ADS)
Matone, Marco
2016-11-01
Recently it has been introduced an algorithm for the Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are 13 types of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, X, Y of the Cartan-Weyl basis, we find W, linear combination of X, Y, such that exp (X) exp (Y)=exp (W). The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of the root system. Furthermore, if X, Y, and Z are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, W, a linear combination of X, Y and Z, such that exp (X) exp (Y) exp (Z)=exp (W). It turns out that the relevant commutator algebras are type 1c-i, type 4 and type 5. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper.
Bethe subalgebras in affine Birman-Murakami-Wenzl algebras and flat connections for q-KZ equations
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Kirillov, A. N.; Tarasov, V. O.
2016-05-01
Commutative sets of Jucys-Murphy elements for affine braid groups of {A}(1),{B}(1),{C}(1),{D}(1) types were defined. Construction of R-matrix representations of the affine braid group of type {C}(1) and its distinguished commutative subgroup generated by the {C}(1)-type Jucys-Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik-Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the {C}(1)-type Jucys-Murphy elements. We specify our general construction to the case of the Birman-Murakami-Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl-Cherednik elements {Y}\\prime {{s}} in the double affine Hecke algebra of type A. Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.
Adjoint affine fusion and tadpoles
NASA Astrophysics Data System (ADS)
Urichuk, Andrew; Walton, Mark A.
2016-06-01
We study affine fusion with the adjoint representation. For simple Lie algebras, elementary and universal formulas determine the decomposition of a tensor product of an integrable highest-weight representation with the adjoint representation. Using the (refined) affine depth rule, we prove that equally striking results apply to adjoint affine fusion. For diagonal fusion, a coefficient equals the number of nonzero Dynkin labels of the relevant affine highest weight, minus 1. A nice lattice-polytope interpretation follows and allows the straightforward calculation of the genus-1 1-point adjoint Verlinde dimension, the adjoint affine fusion tadpole. Explicit formulas, (piecewise) polynomial in the level, are written for the adjoint tadpoles of all classical Lie algebras. We show that off-diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non-dominant representations.
Generalized Pascal's triangles and singular elements of modules of Lie algebras
NASA Astrophysics Data System (ADS)
Lyakhovsky, V. D.; Postnova, O. V.
2015-10-01
We consider the problem of determining the multiplicity function m_ξ ^{{ ⊗ ^p}ω } in the tensor power decomposition of a module of a semisimple algebra g into irreducible submodules. For this, we propose to pass to the corresponding decomposition of a singular element Ψ((L g ω )⊗p) of the module tensor power into singular elements of irreducible submodules and formulate the problem of determining the function M_ξ ^{{ ⊗ ^p}ω }. This function satisfies a system of recurrence relations that corresponds to the procedure for multiplying modules. To solve this problem, we introduce a special combinatorial object, a generalized (g,ω) pyramid, i.e., a set of numbers ( p, { mi})g,ω satisfying the same system of recurrence relations. We prove that M_ξ ^{{ ⊗ ^p}ω } can be represented as a linear combination of the corresponding ( p, { mi})g,ω. We illustrate the obtained solution with several examples of modules of the algebras sl(3) and so(5).
NASA Astrophysics Data System (ADS)
Trell, Erik; Edeagu, Samuel; Animalu, Alexander
2017-01-01
From a brief recapitulation of the foundational works of Marius Sophus Lie and Herrmann Günther Grassmann, and including missing African links, a rhapsodic survey is made of the straight line of extension and existence that runs as the very fibre of generation and creation throughout Nature's all utterances, which must therefore ultimately be the web of Reality itself of which the Arts and Sciences are interpreters on equal explorer terms. Assuming their direct approach, the straight line and its archaic and algebraic and artistic bearings and convolutions have been followed towards their inner reaches, which earlier resulted in a retrieval of the baryon and meson elementary particles and now equally straightforward the electron geodesics and the organic build of the periodic system of the elements.
Abdalla, M. Sebawe; Elkasapy, A.I.
2010-08-15
In this paper we consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be isotropic and the magnetic field is applied along the z-axis. The canonical transformation is invoked to remove the interaction term and the system is reduced to a model containing the second harmonic generation. Two classes of the real and complex quadratic invariants (constants of motion) are obtained. We have employed the Lie algebraic technique to find the most general solution for the wave function for both real and complex invariants. Some discussions related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself are also given.
NASA Astrophysics Data System (ADS)
Khan, Mayukh; Teo, Jeffrey; Hughes, Taylor
2015-03-01
Non-abelian anyons exhibit exotic braiding statistics which can be utilized to realize a universal topological quantum computer. In this work we focus on Fibonacci anyons which occur in Z3 Read Rezayi fractional quantum hall states. Traditionally they have been constructed using su(2)3 / u (1) coset theories. We introduce conformal field theories(CFTs) of exceptional and non-simply laced Lie Algebras at level 1, for example G2 ,F4 which host Fibonacci anyons. We realize these CFT's concretely on the 1d gapless edge of an anisotropic 2d system built out of coupled, interacting Luttinger wires. Interactions are introduced within a bundle of wires to fractionalize the original chiral bosons into different sectors. Next, we couple these sectors to get the desired topological phase in the bulk. The 2d bulk of the stack is gapped by backscattering terms between counterpropagating modes on different bundles. The emergence of this topological phase can be interpreted using techniques of anyon condensation . We also explicitly construct the Kac Moody algebra on the edge CFT using original bosonic degrees of freedom.We acknowledge support from NSF CAREER DMR-1351895(TH) and Simons Foundation (JT).
NASA Astrophysics Data System (ADS)
Grishin, A. V.; Pchelintsev, S. V.
2016-12-01
We study the centre of a relatively free associative algebra F(n) with the identity [x_1,\\dots,x_n]=0 of Lie nilpotency of degree n= 5,6 over a field of characteristic 0. It is proved that the core Z^*(F(5)) of the algebra F(5) (the sum of all ideals of F(5) contained in its centre) is generated as a T-ideal by the weak Hall polynomial [[x,y]2,y]. It is also proved that every proper central polynomial of F(5) is contained in the sum of Z^*(F(5)) and the T-space generated by [[x,y]2, z] and the commutator [x_1,\\dots, x_4] of degree 4. This implies that the centre of F(5) is contained in the T-ideal generated by the commutator of degree 4. Similar results are obtained for F(6); in particular, it is proved that the core Z*(F(6)) is generated as a T-ideal by the commutator of degree 5. Bibliography: 15 titles.
Ibarra-Sierra, V.G.; Sandoval-Santana, J.C.; Cardoso, J.L.; Kunold, A.
2015-11-15
We discuss the one-dimensional, time-dependent general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraic approach. Such method consists in finding a set of generators that form a closed Lie algebra in terms of which it is possible to express a quantum Hamiltonian and therefore the evolution operator. The evolution operator is then the starting point to obtain the propagator as well as the explicit form of the Heisenberg picture position and momentum operators. First, the set of generators forming a closed Lie algebra is identified for the general quadratic Hamiltonian. This algebra is later extended to study the Hamiltonian of a charged particle in electromagnetic fields exploiting the similarities between the terms of these two Hamiltonians. These results are applied to the solution of five different examples: the linear potential which is used to introduce the Lie algebraic method, a radio frequency ion trap, a Kanai–Caldirola-like forced harmonic oscillator, a charged particle in a time dependent magnetic field, and a charged particle in constant magnetic field and oscillating electric field. In particular we present exact analytical expressions that are fitting for the study of a rotating quadrupole field ion trap and magneto-transport in two-dimensional semiconductor heterostructures illuminated by microwave radiation. In these examples we show that this powerful method is suitable to treat quadratic Hamiltonians with time dependent coefficients quite efficiently yielding closed analytical expressions for the propagator and the Heisenberg picture position and momentum operators. -- Highlights: •We deal with the general quadratic Hamiltonian and a particle in electromagnetic fields. •The evolution operator is worked out through the Lie algebraic approach. •We also obtain the propagator and Heisenberg picture position and momentum operators. •Analytical expressions for a
NASA Astrophysics Data System (ADS)
Fernández Núñez, J.; García Fuertes, W.; Perelomov, A. M.
2008-02-01
In a previous paper, we studied the characters and Clebsch-Gordan series for the exceptional Lie algebra E7 by relating them to the quantum trigonometric Calogero-Sutherland Hamiltonian with the coupling constant κ = 1. We now extend that approach to the case of an arbitrary coupling constant.
NASA Astrophysics Data System (ADS)
Wu, Ming-Zhong; Bai, Cheng-Ming
2015-06-01
A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie algebras as an analogue of a Lie bialgebra. They can also be regarded as a “compatible version” of Lie bialgebras, that is, a pair of Lie bialgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the “compatible version” of the corresponding properties of Lie bialgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in Lie algebras. Furthermore, a notion of compatible pre-Lie algebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie algebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov. Supported by National Natural Science Foundation of China under Grant Nos. 11271202, 11221091, 11425104 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20120031110022
Singh, S; Modi, S; Bagga, D; Kaur, P; Shankar, L R; Khushu, S
2013-03-01
The present study aimed to investigate whether brain morphological differences exist between adult hypothyroid subjects and age-matched controls using voxel-based morphometry (VBM) with diffeomorphic anatomic registration via an exponentiated lie algebra algorithm (DARTEL) approach. High-resolution structural magnetic resonance images were taken in ten healthy controls and ten hypothyroid subjects. The analysis was conducted using statistical parametric mapping. The VBM study revealed a reduction in grey matter volume in the left postcentral gyrus and cerebellum of hypothyroid subjects compared to controls. A significant reduction in white matter volume was also found in the cerebellum, right inferior and middle frontal gyrus, right precentral gyrus, right inferior occipital gyrus and right temporal gyrus of hypothyroid patients compared to healthy controls. Moreover, no meaningful cluster for greater grey or white matter volume was obtained in hypothyroid subjects compared to controls. Our study is the first VBM study of hypothyroidism in an adult population and suggests that, compared to controls, this disorder is associated with differences in brain morphology in areas corresponding to known functional deficits in attention, language, motor speed, visuospatial processing and memory in hypothyroidism.
NASA Astrophysics Data System (ADS)
Mikhalev, A. V.; Pinchuk, I. A.
2005-06-01
The structure of Steinberg conformal algebras is studied; these are analogues of Steinberg groups (algebras, superalgebras).A Steinberg conformal algebra is defined as an abstract algebra by a system of generators and relations between the generators. It is proved that a Steinberg conformal algebra is the universal central extension of the corresponding conformal Lie algebra; the kernel of this extension is calculated.
NASA Astrophysics Data System (ADS)
Letellier, Christophe; Amaral, Gleison F. V.; Aguirre, Luis A.
2007-06-01
The characterization of chaotic attractors has been a widely addressed problem and there are now many different techniques to define their nature in a rather accurate way, at least in the case of a three-dimensional system. Nevertheless, the link between the structure of the ordinary differential equations and the topology of their solutions is still missing and only a few necessary conditions on the algebraic structure are known today. By using a feedback circuit analysis, it is shown that it is possible to identify the relevant terms of the equations, that is, the terms that really contribute to the structure of the phase portrait. Such analysis also provides some guidelines for constructing piecewise affine models. Moreover, equivalence classes can be defined on the basis of the active feedback circuits involved.
NASA Astrophysics Data System (ADS)
Sergeev, A. N.
1985-02-01
Let T be the tensor algebra of the identity representation of the Lie superalgebras in the series \\mathfrak{Gl} and Q. The method of Weyl is used to construct a correspondence between the irreducible representations (respectively, the irreducible projective representations) of the symmetric group and the irreducible \\mathfrak{Gl}-(respectively, Q-) submodules of T. The properties of the representations are studied on the basis of this correspondence. A formula is given for the characters on the irreducible Q-submodules of T.Bibliography: 8 titles.
Linder, Mats; Ranganathan, Anirudh; Brinck, Tore
2013-02-12
We present a structure-based parametrization of the Linear Interaction Energy (LIE) method and show that it allows for the prediction of absolute protein-ligand binding energies. We call the new model "Adapted" LIE (ALIE) because the α and β coefficients are defined by system-dependent descriptors and do therefore not require any empirical γ term. The best formulation attains a mean average deviation of 1.8 kcal/mol for a diverse test set and depends on only one fitted parameter. It is robust with respect to additional fitting and cross-validation. We compare this new approach with standard LIE by Åqvist and co-workers and the LIE + γSASA model (initially suggested by Jorgensen and co-workers) against in-house and external data sets and discuss their applicabilities.
D p-branes, NS5-branes and U-duality from nonabelian (2,0) theory with Lie 3-algebra
NASA Astrophysics Data System (ADS)
Honma, Yoshinori; Ogawa, Morirou; Shiba, Shotaro
2011-04-01
We derive the super Yang-Mills action of D p-branes on a torus T p-4 from the nonabelian (2, 0) theory with Lie 3-algebra [1]. Our realization is based on Lie 3-algebra with pairs of Lorentzian metric generators. The resultant theory then has negative norm modes, but it results in a unitary theory by setting VEV's of these modes. This procedure corresponds to the torus compactification, therefore by taking a transformation which is equivalent to T-duality, the D p-brane action is obtained. We also study type IIA/IIB NS5brane and Kaluza-Klein monopole systems by taking other VEV assignments. Such various compactifications can be realized in the nonabelian (2, 0) theory, since both longitudinal and transverse directions can be compactified, which is different from the BLG theory. We finally discuss U-duality among these branes, and show that most of the moduli parameters in U-duality group are recovered. Especially in D5-brane case, the whole U-duality relation is properly reproduced.
Affine group formulation of the Standard Model coupled to gravity
Chou, Ching-Yi; Ita, Eyo; Soo, Chopin
2014-04-15
In this work we apply the affine group formalism for four dimensional gravity of Lorentzian signature, which is based on Klauder’s affine algebraic program, to the formulation of the Hamiltonian constraint of the interaction of matter and all forces, including gravity with non-vanishing cosmological constant Λ, as an affine Lie algebra. We use the hermitian action of fermions coupled to gravitation and Yang–Mills theory to find the density weight one fermionic super-Hamiltonian constraint. This term, combined with the Yang–Mills and Higgs energy densities, are composed with York’s integrated time functional. The result, when combined with the imaginary part of the Chern–Simons functional Q, forms the affine commutation relation with the volume element V(x). Affine algebraic quantization of gravitation and matter on equal footing implies a fundamental uncertainty relation which is predicated upon a non-vanishing cosmological constant. -- Highlights: •Wheeler–DeWitt equation (WDW) quantized as affine algebra, realizing Klauder’s program. •WDW formulated for interaction of matter and all forces, including gravity, as affine algebra. •WDW features Hermitian generators in spite of fermionic content: Standard Model addressed. •Constructed a family of physical states for the full, coupled theory via affine coherent states. •Fundamental uncertainty relation, predicated on non-vanishing cosmological constant.
An analogue of Wagner's theorem for decompositions of matrix algebras
NASA Astrophysics Data System (ADS)
Ivanov, D. N.
2004-12-01
Wagner's celebrated theorem states that a finite affine plane whose collineation group is transitive on lines is a translation plane. The notion of an orthogonal decomposition (OD) of a classically semisimple associative algebra introduced by the author allows one to draw an analogy between finite affine planes of order n and ODs of the matrix algebra M_n(\\mathbb C) into a sum of subalgebras conjugate to the diagonal subalgebra. These ODs are called WP-decompositions and are equivalent to the well-known ODs of simple Lie algebras of type A_{n-1} into a sum of Cartan subalgebras. In this paper we give a detailed and improved proof of the analogue of Wagner's theorem for WP-decompositions of the matrix algebra of odd non-square order an outline of which was earlier published in a short note in "Russian Math. Surveys" in 1994. In addition, in the framework of the theory of ODs of associative algebras, based on the method of idempotent bases, we obtain an elementary proof of the well-known Kostrikin-Tiep theorem on irreducible ODs of Lie algebras of type A_{n-1} in the case where n is a prime-power.
NASA Astrophysics Data System (ADS)
Jafarizadeh, M. A.; Majarshin, A. Jalili; Fouladi, N.
2016-11-01
In order to investigate negative parity states, it is necessary to consider negative parity-bosons additionally to the usual s- and d-bosons. The dipole and octupole degrees of freedom are essential to describe the observed low-lying collective states with negative parity. An extended interacting boson model (IBM) that describes pairing interactions among s, p, d and f-boson based on affine SU(1, 1) Lie algebra in the quantum phase transition (QPT) field, such as spd-IBM, sdf-IBM and spdf-IBM, is composed based on algebraic structure. In this paper, a solvable extended transitional Hamiltonian based on affine SU(1, 1) Lie algebra is proposed to describe low-lying positive and negative parity states between the spherical and deformed gamma-unstable shape. Three model of new algebraic solution for even-even nuclei are introduced. Numerical extraction to low-lying energy levels and transition rates within the control parameters of this evaluated Hamiltonian are presented for various N values. We reproduced the positive and negative parity states and our calculations suggest that the results of spdf-IBM are better than spd-IBM and sdf-IBM in this literature. By reproducing the experimental results, the method based on signature of the phase transition such as level crossing in the lowest excited states is used to provide a better description of Ru isotopes in this transitional region.
Central extensions of Lax operator algebras
NASA Astrophysics Data System (ADS)
Schlichenmaier, M.; Sheinman, O. K.
2008-08-01
Lax operator algebras were introduced by Krichever and Sheinman as a further development of Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this paper local cocycles and associated almost-graded central extensions of Lax operator algebras are classified. It is shown that in the case when the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives.
Control systems on Lie groups.
NASA Technical Reports Server (NTRS)
Jurdjevic, V.; Sussmann, H. J.
1972-01-01
The controllability properties of systems which are described by an evolution equation in a Lie group are studied. The revelant Lie algebras induced by a right invariant system are singled out, and the basic properties of attainable sets are derived. The homogeneous case and the general case are studied, and results are interpreted in terms of controllability. Five examples are given.
Parastatistics Algebras and Combinatorics
NASA Astrophysics Data System (ADS)
Popov, T.
2005-03-01
We consider the algebras spanned by the creation parafermionic and parabosonic operators which give rise to generalized parastatistics Fock spaces. The basis of such a generalized Fock space can be labelled by Young tableaux which are combinatorial objects. By means of quantum deformations a nice combinatorial structure of the algebra of the plactic monoid that lies behind the parastatistics is revealed.
NASA Astrophysics Data System (ADS)
Abd El-Wahab, N. H.; Abdel Rady, A. S.; Osman, Abdel-Nasser A.; Salah, Ahmed
2015-10-01
In this paper, a model is introduced to investigate the interaction between a three-level atom and one-mode of the radiation field. The atomic motion and the classical homogenous gravitational field are taken into consideration. For this purpose, we first introduce a set of new atomic operators obeying an su(3) algebraic structure to derive an effective Hamiltonian for the system under consideration. By solving the Schrödinger equation in the interaction picture, the exact solution is given when the atom and the field are initially prepared in excited state and coherent state, respectively. The influences of the gravity parameter on the collapses-revivals phenomena, the atomic momentum diffusion, the Mandel Q-parameter, the normal squeezing phenomena and the coherent properties for the considered system are examined. It is found that the gravity parameter has important effects on the properties of these phenomena.
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
Discrete Minimal Surface Algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Hoppe, Jens
2010-05-01
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
NASA Astrophysics Data System (ADS)
Dobrev, V. K.
2013-01-01
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 Script G and Script G' that have the same complexification and possess maximal parabolic subalgebras with the same complexification.
Yangian of the Queer Lie Superalgebra
NASA Astrophysics Data System (ADS)
Nazarov, Maxim
Consider the complex matrix Lie superalgebra with the standard generators , where . Define an involutory automorphism η of by . The twisted polynomial current Lie superalgebra
Temme, F P
2004-03-01
The physics of dual group scalar invariants (SIs) as (Lie algebraic) group measures (L-GMs) and its significance to non-Abelian NMR spin systems motivates this overview of uniform general-2n [AX](2n) spin evolution, which represents an extensive addendum to Corio's earlier (essentially restricted) view of Abelian spin system SU(2)-based SI-cardinalities. The [Formula: see text] values in [J. Magn. Reson., 134 (1998) 131] arise from strictly linear recoupled time-reversal invariance (TRI) models. In contrast, here we discuss the physical significance of an alternative polyhedral combinatorics approach to democratic recoupling (DR), a property inherent in both the TRI and statistical sampling. Recognition of spin ensemble SIs as being L-GMs over isomorphic algebras is invaluable in many DR-based NMR problems. Various [AX]n model spin systems, including the [AX]3 bis odd-odd parity spin system, are examined as direct applications of these L-GM- and combinatorial-based SI ideas. Hence in place of /SI/=15 (implied by Corio's [Formula: see text] approach), the bis 3-fold spin system cardinality is seen now as constrained to a single invariant on an isomorphic product algebra under L-GMs, in accord with the subspectral analysis of Jones et al. [Canad. J. Chem., 43 (1965) 683]. The group projective ideas cited here for DR (as cf. to graph theoretic views) apply to highly degenerate non-Abelian problems. Over dual tensorial bases, they define models of spin dynamical evolution whose (SR) quasiparticle superboson carrier (sub)spaces are characterised by SIs acting as explicit auxiliary labels [Physica, A198 (1993) 245; J. Math. Chem., 31 (2002) 281]. A deeper [Formula: see text] network-based view of spin-alone space developed in Balasubramanian's work [J. Chem. Phys., 78 (1983) 6358] is especially important, (e.g.) in the study of spin waves [J. Math. Chem., 31 (2002) 363]. Beyond the specific NMR SIs derived here, there are DR applications where a sporadic, still higher, 2
Generalization of n-ary Nambu algebras and beyond
Ataguema, H.; Makhlouf, A.; Silvestrov, S.
2009-08-15
The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.
Algebraic Nonlinear Collective Motion
NASA Astrophysics Data System (ADS)
Troupe, J.; Rosensteel, G.
1998-11-01
Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real numberΛ. TheΛ=0 solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear group action on Euclidean space transforms a certain family of deformed droplets among themselves. For positiveΛ, the droplets have a neck that becomes more pronounced asΛincreases; for negativeΛ, the droplets contain a spherical bubble of radius |Λ|1/3. The nonlinear vector field algebra is extended to the nonlinear general collective motion algebra gcm(3) which includes the inertia tensor. The quantum algebraic models of nonlinear nuclear collective motion are given by irreducible unitary representations of the nonlinear gcm(3) Lie algebra. These representations model fissioning isotopes (Λ>0) and bubble and two-fluid nuclei (Λ<0).
Cohomology of Heisenberg Lie superalgebras
NASA Astrophysics Data System (ADS)
Bai, Wei; Liu, Wende
2017-02-01
Suppose the ground field to be algebraically closed and of characteristic different from 2 and 3. All Heisenberg Lie superalgebras consist of two super-versions of the Heisenberg Lie algebras, 𝔥2m,n and 𝔟𝔞n with m a non-negative integer and n a positive integer. The space of a "classical" Heisenberg Lie superalgebra 𝔥2m,n is the direct sum of a superspace with a non-degenerate anti-supersymmetric even bilinear form and a one-dimensional space of values of this form constituting the even center. The other super-analog of the Heisenberg Lie algebra, 𝔟𝔞n, is constructed by means of a non-degenerate anti-supersymmetric odd bilinear form with values in the one-dimensional odd center. In this paper, we study the cohomology of 𝔥2m,n and 𝔟𝔞n with coefficients in the trivial module by using the Hochschild-Serre spectral sequences relative to a suitable ideal. In the characteristic zero case, for any Heisenberg Lie superalgebra, we determine completely the Betti numbers and associative superalgebra structures for their cohomology. In the characteristic p > 3 case, we determine the associative superalgebra structure for the divided power cohomology of 𝔟𝔞n and we also make an attempt to determine the divided power cohomology of 𝔥2m,n by computing it in a low-dimensional case.
Merzel, Avraham; Ritov, Ilana; Kareev, Yaakov; Avrahami, Judith
2015-01-01
Do we feel bound by our own misrepresentations? Does one act of cheating compel the cheater to make subsequent choices that maintain the false image even at a cost? To answer these questions we employed a two-task paradigm such that in the first task the participants could benefit from false reporting of private observations whereas in the second they could benefit from making a prediction in line with their actual, rather than their previously reported observations. Thus, for those participants who inflated their report during the first task, sticking with that report for the second task was likely to lead to a loss, whereas deviating from it would imply that they had lied. Data from three experiments (total N = 116) indicate that, having lied, participants were ready to suffer future loss rather than admit, even if implicitly, that they had lied.
The affine structure of gravitational theories: Symplectic groups and geometry
NASA Astrophysics Data System (ADS)
Capozziello, Salvatore; Cirilo-Lombardo, D. J.; de Laurentis, Mariafelicia
2014-09-01
We give a geometrical description of gravitational theories from the viewpoint of symmetries and affine structure. We show how gravity, considered as a gauge theory, can be consistently achieved by the nonlinear realization of the conformal-affine group in an indirect manner: due to the partial isomorphism between CA(3, 1) and the centrally extended Sp( 8), we perform a nonlinear realization of the centrally extended (CE)Sp( 8) in its semi-simple version. In particular, starting from the bundle structure of gravity, we derive the conformal-affine Lie algebra and then, by the nonlinear realization, we define the coset field transformations, the Cartan forms and the inverse Higgs constraints. Finally, we discuss the geometrical Lagrangians where all the information on matter fields and their interactions can be contained.
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…
M2 to D2 and vice versa by 3-Lie and Lie bialgebra
NASA Astrophysics Data System (ADS)
Aali-Javanangrouh, M.; Rezaei-Aghdam, A.
2016-11-01
Using the concept of a 3-Lie bialgebra, which has recently been defined in arXiv:1604.04475, we construct a Bagger-Lambert-Gustavson (BLG) model for the M2-brane on a Manin triple of a special 3-Lie bialgebra. Then by using the correspondence and the relation between those 3-Lie bialgebra with Lie bialgebra, we reduce this model to an N=(4,4) WZW model (D2-brane), such that its algebraic structure is a Lie bialgebra with one 2-cocycle. In this manner by using the correspondence of the 3-Lie bialgebra and Lie bialgebra (for this special 3-Lie algebra) one can construct the M2-brane from a D2-brane and vice versa.
Integrable systems on semidirect product Lie groups
NASA Astrophysics Data System (ADS)
Capriotti, S.; Montani, H.
2014-05-01
We study integrable systems on the semidirect product of a Lie group and its Lie algebra as the representation space of the adjoint action. Regarding the tangent bundle of a Lie group as phase space endowed with this semidirect product Lie group structure, we construct a class of symplectic submanifolds equipped with a Dirac bracket on which integrable systems (in the Adler-Kostant-Symes sense) are naturally built through collective dynamics. In doing so, we address other issues such as factorization, Poisson-Lie structures and dressing actions. We show that the procedure becomes recursive for some particular Hamilton functions, giving rise to a tower of nested integrable systems.
Symmetry algebras of linear differential equations
NASA Astrophysics Data System (ADS)
Shapovalov, A. V.; Shirokov, I. V.
1992-07-01
The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.
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.
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.
Novikov algebras with associative bilinear forms
NASA Astrophysics Data System (ADS)
Zhu, Fuhai; Chen, Zhiqi
2007-11-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. The goal of this paper is to study Novikov algebras with non-degenerate associative symmetric bilinear forms, which we call quadratic Novikov algebras. Based on the classification of solvable quadratic Lie algebras of dimension not greater than 4 and Novikov algebras in dimension 3, we show that quadratic Novikov algebras up to dimension 4 are commutative. Furthermore, we obtain the classification of transitive quadratic Novikov algebras in dimension 4. But we find that not every quadratic Novikov algebra is commutative and give a non-commutative quadratic Novikov algebra in dimension 6.
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.
Classification of central extensions of Lax operator algebras
NASA Astrophysics Data System (ADS)
Schlichenmaier, Martin
2008-11-01
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
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.
NASA Astrophysics Data System (ADS)
Durka, R.
2017-04-01
The S-expansion framework is analyzed in the context of a freedom in closing the multiplication tables for the abelian semigroups. Including the possibility of the zero element in the resonant decomposition, and associating the Lorentz generator with the semigroup identity element, leads to a wide class of the expanded Lie algebras introducing interesting modifications to the gauge gravity theories. Among the results, we find all the Maxwell algebras of type {{B}m} , {{C}m} , and the recently introduced {{D}m} . The additional new examples complete the resulting generalization of the bosonic enlargements for an arbitrary number of the Lorentz-like and translational-like generators. Some further prospects concerning enlarging the algebras are discussed, along with providing all the necessary constituents for constructing the gravity actions based on the obtained results.
Symplectic Clifford Algebraic Field Theory.
NASA Astrophysics Data System (ADS)
Dixon, Geoffrey Moore
We develop a mathematical framework on which is built a theory of fermion, scalar, and gauge vector fields. This field theory is shown to be equivalent to the original Weinberg-Salam model of weak and electromagnetic interactions, but since the new framework is more rigid than that on which the original Weinberg-Salam model was built, a concomitant reduction in the number of assumptions lying outside of the framework has resulted. In particular, parity violation is actually hiding within our framework, and with little difficulty we are able to manifest it. The mathematical framework upon which we build our field theory is arrived at along two separate paths. The first is by the marriage of a Clifford algebra and a Lie superalgebra, the result being called a super Clifford algebra. The second is by providing a new characterization for a Clifford algebra employing its generators and a symmetric array of metric coefficients. Subsequently we generalize this characterization to the case of an antisymmetric array of metric coefficients, and we call the algebra which results a symplectic Clifford algebra. It is upon one of these that we build our field theory, and it is shown that this symplectic Clifford algebra is a particular subalgebra of a super Clifford algebra. The final ingredient is the operation of bracketing which involves treating the elements of our algebra as endomorphisms of a particular inner product space, and employing this space and its inner product to provide us with maps from our algebra to the reals. It is this operation which enables us to manifest the parity violation hiding in our algebra.
Contractions of affine spherical varieties
Arzhantsev, I V
1999-08-31
The language of filtrations and contractions is used to describe the class of G-varieties obtainable as the total spaces of the construction of contraction applied to affine spherical varieties, which is well-known in invariant theory. These varieties are local models for arbitrary affine G-varieties of complexity 1 with a one-dimensional categorical quotient. As examples, reductive algebraic semigroups and three-dimensional SL{sub 2}-varieties are considered.
Numerical linear algebra algorithms and software
NASA Astrophysics Data System (ADS)
Dongarra, Jack J.; Eijkhout, Victor
2000-11-01
The increasing availability of advanced-architecture computers has a significant effect on all spheres of scientific computation, including algorithm research and software development in numerical linear algebra. Linear algebra - in particular, the solution of linear systems of equations - lies at the heart of most calculations in scientific computing. This paper discusses some of the recent developments in linear algebra designed to exploit these advanced-architecture computers. We discuss two broad classes of algorithms: those for dense, and those for sparse matrices.
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.
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…
Representations of some quantum tori Lie subalgebras
Jiang, Jingjing; Wang, Song
2013-03-15
In this paper, we define the q-analog Virasoro-like Lie subalgebras in x{sub {infinity}}=a{sub {infinity}}(b{sub {infinity}}, c{sub {infinity}}, d{sub {infinity}}). The embedding formulas into x{sub {infinity}} are introduced. Irreducible highest weight representations of A(tilde sign){sub q}, B(tilde sign){sub q}, and C(tilde sign){sub q}-series of the q-analog Virasoro-like Lie algebras in terms of vertex operators are constructed. We also construct the polynomial representations of the A(tilde sign){sub q}, B(tilde sign){sub q}, C(tilde sign){sub q}, and D(tilde sign){sub q}-series of the q-analog Virasoro-like Lie algebras.
Symmetry algebra of a generalized anisotropic harmonic oscillator
NASA Technical Reports Server (NTRS)
Castanos, O.; Lopez-Pena, R.
1993-01-01
It is shown that the symmetry Lie algebra of a quantum system with accidental degeneracy can be obtained by means of the Noether's theorem. The procedure is illustrated by considering a generalized anisotropic two dimensional harmonic oscillator, which can have an infinite set of states with the same energy characterized by an u(1,1) Lie algebra.
Cartan-Weyl 3-algebras and the BLG theory. I: classification of Cartan-Weyl 3-algebras
NASA Astrophysics Data System (ADS)
Chu, Chong-Sun
2010-10-01
As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H I and a number of step generators E α that are characterized by a root space of non-degenerate one-forms α. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.
Differential geometry on Hopf algebras and quantum groups
Watts, Paul
1994-12-15
The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined.
Generalized conformal realizations of Kac-Moody algebras
NASA Astrophysics Data System (ADS)
Palmkvist, Jakob
2009-01-01
We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n =1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3×3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n =2. Moreover, we obtain their infinite-dimensional extensions for n ≥3. In the case of 2×2 matrices, the resulting Lie algebras are of the form so(p +n,q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q).
Invariant differential operators for non-compact Lie groups: Summary of su(4,4) multiplets
NASA Astrophysics Data System (ADS)
Dobrev, V. K.
2017-03-01
The present paper is part of the project of systematic construction of invariant differential operators of noncompact semisimple Lie algebras. Here we give a summary of all multiplets containing physically relevant representations including the minimal ones for the algebra su(4, 4). Due to the recently established parabolic relations the results are valid also for the algebras sl(8, R) and su*(8)
Boyko, Vyacheslav M.; Popovych, Roman O.; Shapoval, Nataliya M.
2013-01-01
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach. PMID:23564972
Boyko, Vyacheslav M; Popovych, Roman O; Shapoval, Nataliya M
2013-01-01
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
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.…
Dispersion Operators Algebra and Linear Canonical Transformations
NASA Astrophysics Data System (ADS)
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-04-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
Dispersion Operators Algebra and Linear Canonical Transformations
NASA Astrophysics Data System (ADS)
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-02-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
Dynamical algebras for Poeschl-Teller Hamiltonian hierarchies
Kuru, S.; Negro, J.
2009-12-15
The dynamical algebras of the trigonometric and hyperbolic symmetric Poeschl-Teller Hamiltonian hierarchies are obtained. A kind of discrete-differential realizations of these algebras are found which are isomorphic to so(3, 2) Lie algebras. In order to get them, first the relation between ladder and factor operators is investigated. In particular, the action of the ladder operators on normalized eigenfunctions is found explicitly. Then, the whole dynamical algebras are generated in a straightforward way.
ERIC Educational Resources Information Center
DePaulo, Bella; And Others
1980-01-01
Discusses several studies of whether and how well humans can detect lies. Examines the accuracy of such persons as well as the process of how they actually detect lies, how they think they detect lies, and whether the actual and perceived processes of lie detection correspond to one another. (JMF)
The Hamiltonian of the quantum trigonometric Calogero-Sutherland model in the exceptional algebra E8
NASA Astrophysics Data System (ADS)
Fernández Núñez, J.; García Fuertes, W.; Perelomov, A. M.
2009-01-01
We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model for the Lie algebra E8 and coupling constant κ by using the fundamental irreducible characters of the algebra as dynamical independent variables.
Bilinear forms on fermionic Novikov algebras
NASA Astrophysics Data System (ADS)
Chen, Zhiqi; Zhu, Fuhai
2007-05-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian super-operator in a super-variable. In this paper, we show that there is a remarkable geometry on fermionic Novikov algebras with non-degenerate invariant symmetric bilinear forms, which we call pseudo-Riemannian fermionic Novikov algebras. They are related to pseudo-Riemannian Lie algebras. Furthermore, we obtain a procedure to classify pseudo-Riemannian fermionic Novikov algebras. As an application, we give the classification in dimension <=4. Motivated by the one in dimension 4, we construct some examples in high dimensions.
NASA Astrophysics Data System (ADS)
Minguzzi, E.
2017-03-01
We investigate spacetimes whose light cones could be anisotropic. We prove the equivalence of the structures: (a) Lorentz-Finsler manifold for which the mean Cartan torsion vanishes, (b) Lorentz-Finsler manifold for which the indicatrix (observer space) at each point is a convex hyperbolic affine sphere centered on the zero section, and (c) pair given by a spacetime volume and a sharp convex cone distribution. The equivalence suggests to describe (affine sphere) spacetimes with this structure, so that no algebraic-metrical concept enters the definition. As a result, this work shows how the metric features of spacetime emerge from elementary concepts such as measure and order. Non-relativistic spacetimes are obtained replacing proper spheres with improper spheres, so the distinction does not call for group theoretical elements. In physical terms, in affine sphere spacetimes the light cone distribution and the spacetime measure determine the motion of massive and massless particles (hence the dispersion relation). Furthermore, it is shown that, more generally, for Lorentz-Finsler theories non-differentiable at the cone, the lightlike geodesics and the transport of the particle momentum over them are well defined, though the curve parametrization could be undefined. Causality theory is also well behaved. Several results for affine sphere spacetimes are presented. Some results in Finsler geometry, for instance in the characterization of Randers spaces, are also included.
ERIC Educational Resources Information Center
Heyman, Gail D.; Luu, Diem H.; Lee, Kang
2009-01-01
The present set of studies identifies the phenomenon of "parenting by lying", in which parents lie to their children as a means of influencing their emotional states and behaviour. In Study 1, undergraduates (n = 127) reported that their parents had lied to them while maintaining a concurrent emphasis on the importance of honesty. In Study 2 (n =…
NASA Technical Reports Server (NTRS)
Mulligan, Jeffrey B.
2017-01-01
A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.
Affine coherent states and Toeplitz operators
NASA Astrophysics Data System (ADS)
Hutníková, Mária; Hutník, Ondrej
2012-06-01
We study a parameterized family of Toeplitz operators in the context of affine coherent states based on the Calderón reproducing formula (= resolution of unity on L_2( {R})) and the specific admissible wavelets (= affine coherent states in L_2( {R})) related to Laguerre functions. Symbols of such Calderón-Toeplitz operators as individual coordinates of the affine group (= upper half-plane with the hyperbolic geometry) are considered. In this case, a certain class of pseudo-differential operators, their properties and their operator algebras are investigated. As a result of this study, the Fredholm symbol algebras of the Calderón-Toeplitz operator algebras for these particular cases of symbols are described. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
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.
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”.
Dynamical systems and quantum bicrossproduct algebras
NASA Astrophysics Data System (ADS)
Arratia, Oscar; del Olmo, Mariano A.
2002-06-01
We present a unified study of some aspects of quantum bicrossproduct algebras of inhomogeneous Lie algebras, such as Poincaré, Galilei and Euclidean in N dimensions. The action associated with the bicrossproduct structure allows us to obtain a nonlinear action over a new group linked to the translations. This new nonlinear action associates a dynamical system with each generator which is the object of our study.
Relation of deformed nonlinear algebras with linear ones
NASA Astrophysics Data System (ADS)
Nowicki, A.; Tkachuk, V. M.
2014-01-01
The relation between nonlinear algebras and linear ones is established. For a one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to a linear one with three operators. We also establish the relation between the Lie algebra of total angular momentum and corresponding nonlinear one. This relation gives a possibility to simplify and to solve the eigenvalue problem for the Hamiltonian in a nonlinear case using the reduction of this problem to the case of linear algebra. It is demonstrated in an example of a harmonic oscillator.
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.
Heyman, Gail D.; Luu, Diem H.; Lee, Kang
2010-01-01
The present set of studies identifies the phenomenon of `parenting by lying', in which parents lie to their children as a means of influencing their emotional states and behaviour. In Study 1, undergraduates (n = 127) reported that their parents had lied to them while maintaining a concurrent emphasis on the importance of honesty. In Study 2 (n = 127), parents reported lying to their children and considered doing so to be acceptable under some circumstances, even though they also reported teaching their children that lying is unacceptable. As compared to European American parents, Asian American parents tended to hold a more favourable view of lying to children for the purpose of promoting behavioural compliance. PMID:20930948
ERIC Educational Resources Information Center
Capani, Antonio; De Dominicis, Gabriel
This paper proposes a model for a general interface between people and Computer Algebra Systems (CAS). The main features in the CAS interface are data navigation and the possibility of accessing powerful remote machines. This model is based on the idea of session management, in which the main engine of the tool enables interactions with the…
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.
Quantum Q systems: from cluster algebras to quantum current algebras
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Kedem, Rinat
2017-02-01
This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the A_r quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593-2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra U_{√{q}}({n}[u,u^{-1}])subset U_{√{q}}(widehat{{sl}}_2), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97-152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.
The automorphisms of Novikov algebras in low dimensions
NASA Astrophysics Data System (ADS)
Bai, Chengming; Meng, Daoji
2003-07-01
Novikov algebras were introduced in connection with Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. They also correspond to a class of vertex algebras. An automorphism of a Novikov algebra is a linear isomorphism varphi satisfying varphi(xy) = varphi(x)varphi(y) which keeps the algebraic structure. The set of automorphisms of a Novikov algebra is a Lie group whose Lie algebra is just the Novikov algebra's derivation algebra. The theory of automorphisms plays an important role in the study of Novikov algebras. In this paper, we study the automorphisms of Novikov algebras. We get some results on their properties and classification in low dimensions. These results are fundamental in a certain sense, and they will serve as a guide for further development. Moreover, we apply these results to classify Gel'fand-Dorfman bialgebras and Novikov-Poisson algebras. These results also can be used to study certain phase spaces and geometric classical r-matrices.
ERIC Educational Resources Information Center
Mondry, Henrietta; Taylor, John R.
1992-01-01
It is argued that any attempt to explicate the notion of lying can proceed only on a prior understanding of the notion of truth. Two Russian words for "truth" and two for "lie" are examined, and various dimensions of meanings of the pairs are discussed. (17 references) (LB)
ERIC Educational Resources Information Center
Vermillion, Marti
1985-01-01
Lying is a symptom of a much broader problem. Primary motivations are need for acceptance, fear of punishment, and desire for attention. Children learn about honesty through observation, both directly and indirectly. Admitting mistakes, especially to children, is invaluable and can help break the lying syndrome. (MT)
Towards a cladistics of double Yangians and elliptic algebras*
NASA Astrophysics Data System (ADS)
Arnaudon, D.; Avan, J.; Frappat, L.; Ragoucy, E.; Rossi, M.
2000-09-01
A self-contained description of algebraic structures, obtained by combinations of various limit procedures applied to vertex and face sl(2) elliptic quantum affine algebras, is given. New double Yangian structures of dynamical type are defined. Connections between these structures are established. A number of them take the form of twist-like actions. These are conjectured to be evaluations of universal twists.
The Taylor spectrum and transversality for a Heisenberg algebra of operators
Dosi, Anar A
2010-05-11
A problem on noncommutative holomorphic functional calculus is considered for a Banach module over a finite-dimensional nilpotent Lie algebra. As the main result, the transversality property of algebras of noncommutative holomorphic functions with respect to the Taylor spectrum is established for a family of bounded linear operators generating a Heisenberg algebra. Bibliography: 25 titles.
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.
Breathing difficulty - lying down
Waking at night short of breath; Paroxysmal nocturnal dyspnea; PND; Difficulty breathing while lying down; Orthopnea ... obstructive pulmonary disease (COPD) Cor pulmonale Heart failure ... conditions that lead to it) Panic disorder Sleep apnea Snoring
On modality and complexity of affine embeddings
Arzhantsev, I V
2001-08-31
Let G be a reductive algebraic group and let H be a reductive subgroup of G. The modality of a G-variety X is the largest number of the parameters in a continuous family of G-orbits in X. A precise formula for the maximum value of the modality over all affine embeddings of the homogeneous space G/H is obtained.
Generalized Bifuzzy Lie Subalgebras
Alshehri, Noura
2013-01-01
We introduce the concept of (γ, δ)-bifuzzy Lie subalgebra, where γ, δ are any two of {∈, q, ∈∨q, ∈∧q} with γ ≠ ∈∧q, by using belongs to relation (∈) and quasi-coincidence with relation (q) between bifuzzy points and bifuzzy sets and discuss some of its properties. Then we introduce bifuzzy soft Lie subalgebras and investigate some of their properties. PMID:24489499
2010-01-01
This article reports two worldwide studies of stereotypes about liars. These studies are carried out in 75 different countries and 43 different languages. In Study 1, participants respond to the open-ended question “How can you tell when people are lying?” In Study 2, participants complete a questionnaire about lying. These two studies reveal a dominant pan-cultural stereotype: that liars avert gaze. The authors identify other common beliefs and offer a social control interpretation. PMID:20976033
Lie algebras of conservation laws of variational ordinary differential equations
NASA Astrophysics Data System (ADS)
Fiorani, Emanuele; Spiro, Andrea
2015-02-01
We establish a new version of the first Noether Theorem, according to which the (equivalence classes of) first integrals of given Euler-Lagrange equations in one independent variable are in exact one-to-one correspondence with the (equivalence classes of) vector fields satisfying two simple geometric conditions, namely they simultaneously preserve the holonomy distribution of the jets space and the action from which the Euler-Lagrange equations are derived.
Lie algebra contractions on two-dimensional hyperboloid
Pogosyan, G. S. Yakhno, A.
2010-03-15
The Inoenue-Wigner contraction from the SO(2, 1) group to the Euclidean E(2) and E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the four corresponding two-dimensional homogeneous spaces: two-dimensional hyperboloids and two-dimensional Euclidean and pseudo-Euclidean spaces. We show how the nine systems of coordinates on the two-dimensional hyperboloids contracted to the four systems of coordinates on E{sub 2} and eight on E{sub 1,1}. The text was submitted by the authors in English.
NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
Digital Maps, Matrices and Computer Algebra
ERIC Educational Resources Information Center
Knight, D. G.
2005-01-01
The way in which computer algebra systems, such as Maple, have made the study of complex problems accessible to undergraduate mathematicians with modest computational skills is illustrated by some large matrix calculations, which arise from representing the Earth's surface by digital elevation models. Such problems are often considered to lie in…
Lie antialgebras: cohomology and representations
Ovsienko, V.
2008-11-18
We describe the main algebraic and geometric properties of the class of algebras introduced in [1]. We discuss their origins in symplectic geometry and associative algebra, and the notions of cohomology and representations. We formulate classification theorems and give a number of examples.
Supersymmetry in physics: an algebraic overview
Ramond, P.
1983-01-01
In 1970, while attempting to generalize the Veneziano model (string model) to include fermions, I introduced a new algebraic structure which turned out to be a graded Lie algebra; it was used as a spectrum-generating algebra. This approach was soon after generalized to include interactions, yielding a complete model of fermions and boson (RNS model). In an unrelated work in the Soviet Union, it was shown how to generalize the Poincare group to include fermionic charges. However it was not until 1974 that an interacting field theory invariant under the Graded Poincare group in 3 + 1 dimensions was built (WZ model). Supersymmetric field theories turned out to have less divergent ultraviolet behavior than non-supersymmetric field theories. Gravity was generalized to include supersymmetry, to a theory called supergravity. By now many interacting local field theories exhibiting supersymmetry have been built and studied from 1 + 1 to 10 + 1 dimensions. Supersymmetric local field theories in less than 9 + 1 dimensions, can be understood as limits of multilocal (string) supersymmetric theories, in 9 + 1 dimensions. On the other hand, graded Lie algebras have been used in non-relativistic physics as approximate symmetries of Hamiltonians. The most striking such use so far helps comparing even and odd nuclei energy levels. It is believed that graded Lie algebras can be used whenever paired and unpaired fermions excitations can coexist. In this overview of a tremendously large field, I will only survey finite graded Lie algebras and their representations. For non-relativistic applications, all of GLA are potentially useful, while for relativistic applications, only these which include the Poincare group are to be considered.
Algebraic solutions for two-level pairing model in IBM-2 and IVBM
NASA Astrophysics Data System (ADS)
Jalili-Majarshin, A.; Jafarizadeh, M. A.; Fouladi, N.
2016-09-01
In this paper the affine SU(1,1) approach is applied to numerically solve two pairing problems. A dynamical symmetry limit of the two-fluid interacting boson model-2 (IBM-2) and of the interacting vector boson model (IVBM) defined through the chains U_{π}(6) ⊗ U_{ν}(6) supset SO_{π}(5)⊗ SO_{ν}(5) supset SO_{π}(3) ⊗ SO_{ν}(3) supset SO(3) and U(6) supset U_{π}(3) ⊗ U_{ν}(3) supset SO_{π}(3) ⊗ SO_{ν}(3) supset SO(3) are introduced, respectively. The quantum phase transition between spherical and γ-soft shapes in medium-mass nuclei is analyzed using U(5) leftrightarrow SO(6) transitional nuclei in IBM-2 and one case U_{π}(3) ⊗ U_{ν}(3) leftrightarrow SO(6) transitional nuclei in IVBM found by using an infinite dimensional algebraic method based on affine SU(1,1) Lie algebra. The calculated energy spectra, energy ratio and energy staggering of Mo isotopes are compared with experimental results. The interplay between phase transitions and configuration mixing of intruder excitations between spherical vibrations and the γ-soft shapes in Mo isotopes is succinctly addressed and displays fingerprints of the transitional dynamical symmetry E(5).
Derive Workshop Matrix Algebra and Linear Algebra.
ERIC Educational Resources Information Center
Townsley Kulich, Lisa; Victor, Barbara
This document presents the course content for a workshop that integrates the use of the computer algebra system Derive with topics in matrix and linear algebra. The first section is a guide to using Derive that provides information on how to write algebraic expressions, make graphs, save files, edit, define functions, differentiate expressions,…
Seron, X
2014-10-01
The issue of lying occurs in neuropsychology especially when examinations are conducted in a forensic context. When a subject intentionally either presents non-existent deficits or exaggerates their severity to obtain financial or material compensation, this behaviour is termed malingering. Malingering is discussed in the general framework of lying in psychology, and the different procedures used by neuropsychologists to evidence a lack of collaboration at examination are briefly presented and discussed. When a lack of collaboration is observed, specific emphasis is placed on the difficulty in unambiguously establishing that this results from the patient's voluntary decision.
The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
NASA Astrophysics Data System (ADS)
Hallowell, Karl; Waldron, Andrew
2007-09-01
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R). These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra! was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.
Applications of Lie Group Integrators and Exponential Schemes
2007-11-02
Classical numerical ODE-solvers progress solution along straight lines. • Lie group integrators map a straight line in some other space (Lie algebra) to...term. Includes: NLS, Nonlinear heat equations , KdV , Allen–Cahn, Kuramoto–Sivashinsky, and many more. Unbounded L requires a form of implicit integrator...variational equation Mdzt + Kdzx = DzzS(z)dz It easily follows that this pair of solutions satisfies ∂tω(U, V ) + ∂xκ(U, V ) = 0 the symplectic conservation law
1996-02-01
The Department of Energy has prepared an Environmental Assessment (DOE/EA-1143) evaluating the construction, equipping and operation of the proposed Lied Transplant Center at the University of Nebraska Medical Center in Omaha, Nebraska. Based on the analysis in the EA, the DOE has determined that the proposed action does not constitute a major federal action significantly affecting the quality of the human environment within the meaning of the National Environmental Policy Act of 1969 (NEPA). Therefore, the preparation of an Environmental Statement in not required.
Police lie detection accuracy: the effect of lie scenario.
O'Sullivan, Maureen; Frank, Mark G; Hurley, Carolyn M; Tiwana, Jaspreet
2009-12-01
Although most people are not better than chance in detecting deception, some groups of police professionals have demonstrated significant lie detection accuracy. One reason for this difference may be that the types of lies police are asked to judge in scientific experiments often do not represent the types of lies they see in their profession. Across 23 studies, involving 31 different police groups in eight countries, police officers tested with lie detection scenarios using high stakes lies (i.e., the lie was personally involving and/or resulted in substantial rewards or punishments for the liar) were significantly more accurate than law enforcement officials tested with low stakes lies. Face validity and construct validity of various lie scenarios are differentiated.
[Diagnostic imaging of lying].
Lass, Piotr; Sławek, Jarosław; Sitek, Emilia; Szurowska, Edyta; Zimmermann, Agnieszka
2013-01-01
Functional diagnostic imaging has been applied in neuropsychology for more than two decades. Nowadays, the functional magnetic resonance (fMRI) seems to be the most important technique. Brain imaging in lying has been performed and discussed since 2001. There are postulates to use fMRI for forensic purposes, as well as commercially, e.g. testing the loyalty of employees, especially because of the limitations of traditional polygraph in some cases. In USA fMRI is performed in truthfulness/lying assessment by at least two commercial companies. Those applications are a matter of heated debate of practitioners, lawyers and specialists of ethics. The opponents of fMRI use for forensic purposes indicate the lack of common agreement on it and the lack of wide recognition and insufficient standardisation. Therefore it cannot serve as a forensic proof, yet. However, considering the development of MRI and a high failure rate of traditional polygraphy, forensic applications of MRI seem to be highly probable in future.
Correlation functions from a unified variational principle: Trial Lie groups
Balian, R.; Vénéroni, M.
2015-11-15
Time-dependent expectation values and correlation functions for many-body quantum systems are evaluated by means of a unified variational principle. It optimizes a generating functional depending on sources associated with the observables of interest. It is built by imposing through Lagrange multipliers constraints that account for the initial state (at equilibrium or off equilibrium) and for the backward Heisenberg evolution of the observables. The trial objects are respectively akin to a density operator and to an operator involving the observables of interest and the sources. We work out here the case where trial spaces constitute Lie groups. This choice reduces the original degrees of freedom to those of the underlying Lie algebra, consisting of simple observables; the resulting objects are labeled by the indices of a basis of this algebra. Explicit results are obtained by expanding in powers of the sources. Zeroth and first orders provide thermodynamic quantities and expectation values in the form of mean-field approximations, with dynamical equations having a classical Lie–Poisson structure. At second order, the variational expression for two-time correlation functions separates–as does its exact counterpart–the approximate dynamics of the observables from the approximate correlations in the initial state. Two building blocks are involved: (i) a commutation matrix which stems from the structure constants of the Lie algebra; and (ii) the second-derivative matrix of a free-energy function. The diagonalization of both matrices, required for practical calculations, is worked out, in a way analogous to the standard RPA. The ensuing structure of the variational formulae is the same as for a system of non-interacting bosons (or of harmonic oscillators) plus, at non-zero temperature, classical Gaussian variables. This property is explained by mapping the original Lie algebra onto a simpler Lie algebra. The results, valid for any trial Lie group, fulfill
Blue Lies and Police Placebos: The Moralities of Police Lying.
ERIC Educational Resources Information Center
Klockars, Carl B.
1984-01-01
The concession that the lie is preferred over force as a means of social control forms the basis for the morality of policy lying, i.e., in any situation in which police have a legitimate right to use force they acquire a moral right to achieve the same ends by lying. (RM)
Telling Lies: The Irrepressible Truth?
Williams, Emma J.; Bott, Lewis A.; Patrick, John; Lewis, Michael B.
2013-01-01
Telling a lie takes longer than telling the truth but precisely why remains uncertain. We investigated two processes suggested to increase response times, namely the decision to lie and the construction of a lie response. In Experiments 1 and 2, participants were directed or chose whether to lie or tell the truth. A colored square was presented and participants had to name either the true color of the square or lie about it by claiming it was a different color. In both experiments we found that there was a greater difference between lying and telling the truth when participants were directed to lie compared to when they chose to lie. In Experiments 3 and 4, we compared response times when participants had only one possible lie option to a choice of two or three possible options. There was a greater lying latency effect when questions involved more than one possible lie response. Experiment 5 examined response choice mechanisms through the manipulation of lie plausibility. Overall, results demonstrate several distinct mechanisms that contribute to additional processing requirements when individuals tell a lie. PMID:23573277
On the Structure of Graded Lie Superalgebras
NASA Astrophysics Data System (ADS)
Calderón Martín, Antonio J.; Sánchez Delgado, José M.
2012-08-01
We study the structure of graded Lie superalgebras with arbitrary dimension and over an arbitrary field 𝕂. We show that any of such algebras 𝔏 with a symmetric G-support is of the form 𝔏 = U + ∑jIj with U a subspace of 𝔏1 and any Ij a well-described graded ideal of 𝔏, satisfying [Ij, Ik] = 0 if j≠k. Under certain conditions, it is shown that 𝔏 = (⨁k∈K Ik) ⊕ (⨁q ∈Q Iq), where any Ik is a gr-simple graded ideal of 𝔏 and any Iq is a completely determined low-dimensional non-gr-simple graded ideal of 𝔏, satisfying [Iq, Iq'] = 0 for any q'∈Q with q ≠q'.
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…
ERIC Educational Resources Information Center
Miller, L. Diane; England, David A.
1989-01-01
Describes a study in a large metropolitan high school to ascertain what influence the use of regular writing in algebra classes would have on students' attitudes towards algebra and their skills in algebra. Reports the simpler and more direct the writing topics the better. (MVL)
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.
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…
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…
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.
Warneken, Felix; Orlins, Emily
2015-09-01
In this reply to Ceci, Burd, and Helm, we discuss future directions for developmental research to (1) study the motivations underlying white lies and (2) how to classify lies that reflect other-regard and self-interest simultaneously.
The algebra of supertraces for 2+1 super de Sitter gravity
NASA Technical Reports Server (NTRS)
Urrutia, L. F.; Waelbroeck, H.; Zertuche, F.
1993-01-01
The algebra of the observables for 2+1 super de Sitter gravity, for one genus of the spatial surface is calculated. The algebra turns out to be an infinite Lie algebra subject to non-linear constraints. The constraints are solved explicitly in terms of five independent complex supertraces. These variables are the true degrees of freedom of the system and their quantized algebra generates a new structure which is referred to as a 'central extension' of the quantum algebra SU(2)q.
How People Really Detect Lies.
ERIC Educational Resources Information Center
Park, Hee Sun; Levine, Timothy R.; McCornack, Steven A.; Morrison, Kelly; Ferrara, Merissa
2002-01-01
Considers that participants in previous deception detection experiments may not have had access to the types of information people most often use to detect real-life lies. Suggests that people most often rely on information from third parties and physical evidence when detecting lies, and that the detection of a lie is a process that takes days,…
Wakimoto realizations of current algebras: an explicit construction
de Boer, Jan; Feher, Laszlo
1996-11-12
A generalized Wakimoto realization of $\\widehat\\cal G_K$ can be associated with each parabolic subalgebra $\\cal P=(\\cal G_0 +\\cal G_+)$ of a simple Lie algebra $\\cal G$ according to an earlier proposal by Feigin and Frenkel. In this paper the proposal is made explicit by developing the construction of Wakimoto realizations from a simple but unconventional viewpoint. An explicit formula is derived for the Wakimoto current first at the Poisson bracket level by Hamiltonian symmetry reduction of the WZNW model. The quantization is then performed by normal ordering the classical formula and determining the required quantum correction for it to generate $\\widehat\\cal G_K$ by means of commutators. The affine-Sugawara stress-energy tensor is verified to have the expected quadratic form in the constituents, which are symplectic bosons belonging to $\\cal G_+$ and a current belonging to $\\cal G_0$. The quantization requires a choice of special polynomial coordinates on the big cell of the flag manifold $P\\backslash G$. The effect of this choice is investigated in detail by constructing quantum coordinate transformations. Finally, the explicit form of the screening charges for each generalized Wakimoto realization is determined, and some applications are briefly discussed.
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.
Current algebra formulation of M-theory based on E11 Kac-Moody algebra
NASA Astrophysics Data System (ADS)
Sugawara, Hirotaka
2017-02-01
Quantum M-theory is formulated using the current algebra technique. The current algebra is based on a Kac-Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the E11 Kac-Moody algebra that was shown recently1‑5 to contain all the ingredients of M-theory. Both the internal symmetry and the external Lorentz symmetry can be realized inside E11, so that, by constructing the current algebra of E11, I obtain both internal gauge theory and gravity theory. The energy-momentum tensor is constructed as the bilinear form of the currents, yielding a system of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way. The so-called “field-current identity” is built in and, for example, the gravitino field is itself a conserved supercurrent. One unanticipated outcome is that the quantum gravity equation is not identical to the one obtained from the Einstein-Hilbert action.
The algebra of the quantum nondegenerate three-dimensional Kepler-Coulomb potential
Tanoudis, Y.; Daskaloyannis, C.
2011-07-15
The classical generalized Kepler-Coulomb potential, introduced by Verrier and Evans, corresponds to a quantum superintegrable system, with quadratic and quartic integrals of motion. In this paper we show that the algebra of the integrals is a quadratic ternary algebra, i.e a quadratic extension of a Lie triple system.
NASA Astrophysics Data System (ADS)
Balagović, Martina
2015-03-01
We show that, under Drinfeld's degeneration (Proceedings of the International Congress of Mathematicians. American Mathematical Society, Providence, pp 798-820, 1987) of quantum loop algebras to Yangians, the trigonometric dynamical difference equations [Etingof and Varchenko (Adv Math 167:74-127, 2002)] for the quantum affine algebra degenerate to the trigonometric Casimir differential equations [Toledano Laredo (J Algebra 329:286-327, 2011)] for Yangians.
Debey, Evelyne; De Houwer, Jan; Verschuere, Bruno
2014-09-01
Cognitive models of deception focus on the conflict-inducing nature of the truth activation during lying. Here we tested the counterintuitive hypothesis that the truth can also serve a functional role in the act of lying. More specifically, we examined whether the construction of a lie can involve a two-step process, where the first step entails activating the truth, based upon which a lie response can be formulated in a second step. To investigate this hypothesis, we tried to capture the covert truth activation in a reaction-time based deception paradigm. Together with each question, we presented either the truth or lie response as distractors. If lying depends on the covert activation of the truth, deceptive responses would thus be facilitated by truth distractors relative to lie distractors. Our results indeed revealed such a "covert congruency" effect, both in errors and reaction times (Experiment 1). Moreover, stimulating participants to use the distractor information by increasing the proportion of truth distractor trials enlarged the "covert congruency" effects, and as such confirmed that the effects operate at a covert response level (Experiment 2). Our findings lend support to the idea that lying relies on a first step of truth telling, and call for a shift in theoretical thinking that highlights both the functional and interfering properties of the truth activation in the lying process.
Prediction of Algebraic Instabilities
NASA Astrophysics Data System (ADS)
Zaretzky, Paula; King, Kristina; Hill, Nicole; Keithley, Kimberlee; Barlow, Nathaniel; Weinstein, Steven; Cromer, Michael
2016-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A new methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to the governing differential equation (or dispersion relation) describing wave propagation in dispersive media. Several types of algebraic instabilities are explored in the context of both linear and nonlinear waves.
Theatres of the lie: 'crazy' deception and lying as drama.
Dongen, Els van
2002-08-01
In this article, the author argues that lying is drama, theatre, which brings about transition, reflection, reversal and involvement of the participants in the drama. By means of ethnographic data of a psychiatric ward, the author shows that lying of mental patients is not pathological, but a ritual of affliction. By using Turner's theory about rituals and performance and Goffman's theory about presentation of the self it will be showed that lying serves the redefinition of reciprocity and solidarity. With the help of Bakhtin's work on Rabelais, the author discusses the nature of the drama of the lie. It is concluded that a perspective on lying as theatre may be of use outside psychiatric wards and will occur in imbalanced power relationships.
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)
On Differential form Method to Find Lie Symmetries of two Types of Toda Lattices
NASA Astrophysics Data System (ADS)
Ding, Qi; Tian, Shou-Fu
2014-12-01
In this paper, we investigate Lie symmetries of the (1 + 1)-dimensional celebrated Toda lattice and the (2 + 1)-dimensional modified semidiscrete Toda lattice by using the extended Harrison and Estabrook's geometric approach. Two closed ideals written in terms of a set of differential forms are constructed for Toda lattices. Moreover, commutation relations of a Kac-Moody-Virasoro type Lie algebra are obtained by direct computation.
Nonstandard Methods in Lie Theory
ERIC Educational Resources Information Center
Goldbring, Isaac Martin
2009-01-01
In this thesis, we apply model theory to Lie theory and geometric group theory. These applications of model theory come via nonstandard analysis. In Lie theory, we use nonstandard methods to prove two results. First, we give a positive solution to the local form of Hilbert's Fifth Problem, which asks whether every locally euclidean local…
On Quantizable Odd Lie Bialgebras
NASA Astrophysics Data System (ADS)
Khoroshkin, Anton; Merkulov, Sergei; Willwacher, Thomas
2016-09-01
Motivated by the obstruction to the deformation quantization of Poisson structures in infinite dimensions, we introduce the notion of a quantizable odd Lie bialgebra. The main result of the paper is a construction of the highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so-called quantizable Poisson structures.
Lying despite telling the truth.
Wiegmann, Alex; Samland, Jana; Waldmann, Michael R
2016-05-01
According to the standard definition of lying an utterance counts as a lie if the agent believes the statement to be false. Thus, according to this view it is possible that a lie states something that happens to be true. This subjective view on lying has recently been challenged by Turri and Turri (2015) who presented empirical evidence suggesting that people only consider statements as lies that are objectively false (objective view). We argue that the presented evidence is in fact consistent with the standard subjective view if conversational pragmatics is taken into account. Three experiments are presented that directly test and support the subjective view. An additional experiment backs up our pragmatic hypothesis by using the uncontroversial case of making a promise.
Group discussion improves lie detection
Klein, Nadav; Epley, Nicholas
2015-01-01
Groups of individuals can sometimes make more accurate judgments than the average individual could make alone. We tested whether this group advantage extends to lie detection, an exceptionally challenging judgment with accuracy rates rarely exceeding chance. In four experiments, we find that groups are consistently more accurate than individuals in distinguishing truths from lies, an effect that comes primarily from an increased ability to correctly identify when a person is lying. These experiments demonstrate that the group advantage in lie detection comes through the process of group discussion, and is not a product of aggregating individual opinions (a “wisdom-of-crowds” effect) or of altering response biases (such as reducing the “truth bias”). Interventions to improve lie detection typically focus on improving individual judgment, a costly and generally ineffective endeavor. Our findings suggest a cheap and simple synergistic approach of enabling group discussion before rendering a judgment. PMID:26015581
On the Primitive Ideal spaces of the C(*) -algebras of graphs
NASA Astrophysics Data System (ADS)
Bates, Teresa
2005-11-01
We characterise the topological spaces which arise as the primitive ideal spaces of the Cuntz-Krieger algebras of graphs satisfying condition (K): directed graphs in which every vertex lying on a loop lies on at least two loops. We deduce that the spaces which arise as Prim;C(*(E)) are precisely the spaces which arise as the primitive ideal spaces of AF-algebras. Finally, we construct a graph wt{E} from E such that C(*(wt{E})) is an AF-algebra and Prim;C(*(E)) and Prim;C(*(wt{E})) are homeomorphic.
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…
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
Learning Activity Package, Algebra.
ERIC Educational Resources Information Center
Evans, Diane
A set of ten teacher-prepared Learning Activity Packages (LAPs) in beginning algebra and nine in intermediate algebra, these units cover sets, properties of operations, number systems, open expressions, solution sets of equations and inequalities in one and two variables, exponents, factoring and polynomials, relations and functions, radicals,…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
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…
Chen Famin; Wu Yongshi
2010-11-15
We present a superspace formulation of the D=3, N=4, 5 superconformal Chern-Simons Matter theories, with matter supermultiplets valued in a symplectic 3-algebra. We first construct an N=1 superconformal action and then generalize a method used by Gaitto and Witten to enhance the supersymmetry from N=1 to N=5. By decomposing the N=5 supermultiplets and the symplectic 3-algebra properly and proposing a new superpotential term, we construct the N=4 superconformal Chern-Simons matter theories in terms of two sets of generators of a (quaternion) symplectic 3-algebra. The N=4 theories can also be derived by requiring that the supersymmetry transformations are closed on-shell. The relationship between the 3-algebras, Lie superalgebras, Lie algebras, and embedding tensors (proposed in [E. A. Bergshoeff, O. Hohm, D. Roest, H. Samtleben, and E. Sezgin, J. High Energy Phys. 09 (2008) 101.]) is also clarified. The general N=4, 5 superconformal Chern-Simons matter theories in terms of ordinary Lie algebras can be re-derived in our 3-algebra approach. All known N=4, 5 superconformal Chern-Simons matter theories can be recovered in the present superspace formulation for super-Lie algebra realization of symplectic 3-algebras.
Detecting true lies: police officers' ability to detect suspects' lies.
Mann, Samantha; Vrij, Aldert; Bull, Ray
2004-02-01
Ninety-nine police officers, not identified in previous research as belonging to groups that are superior in lie detection, attempted to detect truths and lies told by suspects during their videotaped police interviews. Accuracy rates were higher than those typically found in deception research and reached levels similar to those obtained by specialized lie detectors in previous research. Accuracy was positively correlated with perceived experience in interviewing suspects and with mentioning cues to detecting deceit that relate to a suspect's story. Accuracy was negatively correlated with popular stereotypical cues such as gaze aversion and fidgeting. As in previous research, accuracy and confidence were not significantly correlated, but the level of confidence was dependent on whether officers judged actual truths or actual lies and on the method by which confidence was measured.
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.
Baxter Operator and Archimedean Hecke Algebra
NASA Astrophysics Data System (ADS)
Gerasimov, A.; Lebedev, D.; Oblezin, S.
2008-12-01
In this paper we introduce Baxter integral {mathcal{Q}} -operators for finite-dimensional Lie algebras {mathfrak{gl}_{ell+1}} and {mathfrak{so}_{2ell+1}} . Whittaker functions corresponding to these algebras are eigenfunctions of the {mathcal{Q}}-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G = GL( ℓ + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter {mathcal{Q}}-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter {mathcal{Q}}-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra {mathcal {H}(G(mathbb{R}), K)} , K being a maximal compact subgroup of G. Finally we stress an analogy between {mathcal{Q}}-operators and certain elements of the non-Archimedean Hecke algebra {mathcal {H}(G(mathbb{Q}_p),G(mathbb{Z}_p))}.
Historical Techniques of Lie Detection
Vicianova, Martina
2015-01-01
Since time immemorial, lying has been a part of everyday life. For this reason, it has become a subject of interest in several disciplines, including psychology. The purpose of this article is to provide a general overview of the literature and thinking to date about the evolution of lie detection techniques. The first part explores ancient methods recorded circa 1000 B.C. (e.g., God’s judgment in Europe). The second part describes technical methods based on sciences such as phrenology, polygraph and graphology. This is followed by an outline of more modern-day approaches such as FACS (Facial Action Coding System), functional MRI, and Brain Fingerprinting. Finally, after the familiarization with the historical development of techniques for lie detection, we discuss the scope for new initiatives not only in the area of designing new methods, but also for the research into lie detection itself, such as its motives and regulatory issues related to deception. PMID:27247675
Similarity analysis of differential equations by Lie group.
NASA Technical Reports Server (NTRS)
Na, T. Y.; Hansen, A. G.
1971-01-01
Methods for transforming partial differential equations into forms more suitable for analysis and solution are investigated. The idea of Lie's infinitesimal contact transformation group is introduced to develop a systematic method which involves mostly algebraic manipulations. A thorough presentation of the application of this general method to the problem of similarity analysis in a broader sense - namely, the similarity between partial and ordinary differential equations, boundary value and initial value problems, and nonlinear and linear equations - is given with new and very general methods evolved for deriving the possible groups of transformations.
Representations of centrally extended Lie superalgebra psl(2|2)
Matsumoto, Takuya; Molev, Alexander
2014-09-15
The symmetries provided by representations of the centrally extended Lie superalgebra psl(2|2) are known to play an important role in the spin chain models originated in the planar anti-de Sitter/conformal field theory correspondence and one-dimensional Hubbard model. We give a complete description of finite-dimensional irreducible representations of this superalgebra thus extending the work of Beisert which deals with a generic family of representations. Our description includes a new class of modules with degenerate eigenvalues of the central elements. Moreover, we construct explicit bases in all irreducible representations by applying the techniques of Mickelsson–Zhelobenko algebras.
The κ-(A)dS quantum algebra in (3 + 1) dimensions
NASA Astrophysics Data System (ADS)
Ballesteros, Ángel; Herranz, Francisco J.; Musso, Fabio; Naranjo, Pedro
2017-03-01
The quantum duality principle is used to obtain explicitly the Poisson analogue of the κ-(A)dS quantum algebra in (3 + 1) dimensions as the corresponding Poisson-Lie structure on the dual solvable Lie group. The construction is fully performed in a kinematical basis and deformed Casimir functions are also explicitly obtained. The cosmological constant Λ is included as a Poisson-Lie group contraction parameter, and the limit Λ → 0 leads to the well-known κ-Poincaré algebra in the bicrossproduct basis. A twisted version with Drinfel'd double structure of this κ-(A)dS deformation is sketched.
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
-Moody algebras, Virasoro algebras etc. The exploitation of these mathematical structures inevitably leads to a deeper understanding of the physical systems. This issue provides some further progress in the investigations of the algebraic structures, such as Lie groups and Lie algebras, quantum groups, algebroids, etc, which have always played an important role in the development of the field. Quantum groups, for instance, have given an algebraic shape to the kinematics of the quantum inverse scattering method and these ideas are developed further in this issue. Some contributions focus on integrable systems with boundaries, which are interesting in their own right from a formal point of view as they exhibit some peculiarities which cannot be found within systems with periodic boundary conditions. The reflection equations and underlying quantum group covariant algebras allow for meaningful generalisations of results found in integrable scattering theories. Meanwhile the off shell structures have also been developed further and the first examples for form factor calculations, ultimately leading to correlation functions, are presented in this issue. Non-Hermitian Hamiltonian systems have already featured for some time in the context of integrable models, as for instance in the form of affine Toda field theories with a complex coupling constant or the Yang-Lee model. However, a systematic study of such types of models has only been initiated recently. It is now well understood that the reality of the spectrum of these models can be attributed either to the unbroken PT-symmetry of the entire system or to its pseudo(quasi)-Hermiticity. In reverse, one may take these concepts as starting points for the construction of new types of models, such as integrable ones which are the central topic of this special issue. We gratefully acknowledge the financial support provided by Clay Mathematics Institute, the Group of Mathematical Physics of the University of Lisbon, the Gulbenkian Foundation
Algebraic invariants for homotopy types
NASA Astrophysics Data System (ADS)
Blanc, David
1999-11-01
We define a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the [Pi]-algebra [pi][low asterisk]X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract [Pi]-algebra can be realized as the homotopy [Pi]-algebra of a space.
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,…
Polytope expansion of Lie characters and applications
Walton, Mark A.
2013-12-15
The weight systems of finite-dimensional representations of complex, simple Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way. Since lattice polytopes are relatively simple, this decomposition is useful, in addition to being more economical than the decomposition into single weights. An expansion of characters into polytope sums follows from the polytope decomposition of weight systems. We study this polytope expansion here. A new, general formula is given for the polytope sums involved. The combinatorics of the polytope expansion are analyzed; we point out that they are reduced from those of the Weyl character formula (described by the Kostant partition function) in an optimal way. We also show that the weight multiplicities can be found easily from the polytope multiplicities, indicating explicitly the equivalence of the two descriptions. Finally, we demonstrate the utility of the polytope expansion by showing how polytope multiplicities can be used in the calculation of tensor product decompositions, and subalgebra branching rules.
Pseudo-Riemannian Novikov algebras
NASA Astrophysics Data System (ADS)
Chen, Zhiqi; Zhu, Fuhai
2008-08-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.
NASA Astrophysics Data System (ADS)
Markarian, Nikita
2017-03-01
We introduce Weyl n-algebras and show how their factorization complex may be used to define invariants of manifolds. In the appendix, we heuristically explain why these invariants must be perturbative Chern-Simons invariants.
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)
Jordan Algebraic Quantum Categories
NASA Astrophysics Data System (ADS)
Graydon, Matthew; Barnum, Howard; Ududec, Cozmin; Wilce, Alexander
2015-03-01
State cones in orthodox quantum theory over finite dimensional complex Hilbert spaces enjoy two particularly essential features: homogeneity and self-duality. Orthodox quantum theory is not, however, unique in that regard. Indeed, all finite dimensional formally real Jordan algebras -- arenas for generalized quantum theories with close algebraic kinship to the orthodox theory -- admit homogeneous self-dual positive cones. We construct categories wherein these theories are unified. The structure of composite systems is cast from universal tensor products of the universal C*-algebras enveloping ambient spaces for the constituent state cones. We develop, in particular, a notion of composition that preserves the local distinction of constituent systems in quaternionic quantum theory. More generally, we explicitly derive the structure of hybrid quantum composites with subsystems of arbitrary Jordan algebraic type.
Accounting Equals Applied Algebra.
ERIC Educational Resources Information Center
Roberts, Sondra
1997-01-01
Argues that students should be given mathematics credits for completing accounting classes. Demonstrates that, although the terminology is different, the mathematical concepts are the same as those used in an introductory algebra class. (JOW)
Minimum deformations of commutative algebra and linear group GL(n)
NASA Astrophysics Data System (ADS)
Zupnik, B. M.
1993-06-01
In the algebra of formal series M q ( x i ), the relations of generalized commutativity that preserve the tensor I q grading and depend on parameters q(i, k) are considered. A norm of the differential calculus on M q consistent with the I q grading is chosen. A new construction of a symmetrized tensor product of algebras of the type M q ( x i ) and a corresponding definition of the minimally deformed linear group QGL(n) and Lie algebra qgl(n) are proposed. A study is made of the connection of QGL(n) and qgl(n) with the special matrix algebra Mat( n, Q), which consists of matrices with noncommuting elements. The deformed determinant in the algebra Mat( n, Q) is defined. The exponential mapping in the algebra Mat( n, Q) is considered on the basis of the Campbell-Hausdorff formula.
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
Hidden symmetries of the Higgs oscillator and the conformal algebra
NASA Astrophysics Data System (ADS)
Evnin, Oleg; Nivesvivat, Rongvoram
2017-01-01
We give a solution to the long-standing problem of constructing the generators of hidden symmetries of the quantum Higgs oscillator, a particle on a d-sphere moving in a central potential varying as the inverse cosine-squared of the polar angle. This superintegrable system is known to possess a rich algebraic structure, including a hidden SU(d) symmetry that can be deduced from classical conserved quantities and degeneracies of the quantum spectrum. The quantum generators of this SU(d) have not been constructed thus far, except at d = 2, and naive quantization of classical conserved quantities leads to deformed Lie algebras with quadratic terms in the commutation relations. The nonlocal generators we obtain here satisfy the standard su(d) Lie algebra, and their construction relies on a recently discovered realization of the conformal algebra, which contains a complete set of raising and lowering operators for the Higgs oscillator. This operator structure has emerged from a relation between the Higgs oscillator Schrödinger equation and the Klein-Gordon equation in Anti-de Sitter spacetime. From such a point-of-view, constructing the hidden symmetry generators reduces to manipulations within the abstract conformal algebra so(d, 2).
Algebraic methods for the solution of some linear matrix equations
NASA Technical Reports Server (NTRS)
Djaferis, T. E.; Mitter, S. K.
1979-01-01
The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.
On an algebraic approach to the Kratzer oscillator
NASA Astrophysics Data System (ADS)
Mikulski, Damian; Molski, Marcin; Konarski, Jerzy
2009-08-01
The ladder operators for the Kratzer-Fues oscillator have been derived within the algebraic approach. The method is extended to include the rotating Kratzer-Fues oscillator. For these operators, SU(2) Lie algebra has been constructed. The results obtained differ significantly from those recently derived by Setare and Karimi (2007 Phys. Scr.75 90-3). We have shown that in their study the ladder operators and the solutions of the Schrödinger equations with the Kratzer potential have no physical meaning.
ERIC Educational Resources Information Center
Cai, Jinfa
2014-01-01
Drawing on evidence from the Longitudinal Investigation of the Effect of Curriculum on Algebra Learning (LieCal) Project, issues related to mathematics curriculum reform and student learning are discussed. The LieCal Project was designed to longitudinally investigate the impact of a reform mathematics curriculum called the Connected Mathematics…
Gauging the Carroll algebra and ultra-relativistic gravity
NASA Astrophysics Data System (ADS)
Hartong, Jelle
2015-08-01
It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincaré algebra. Recently it has been shown that gauging the centrally extended Galilei algebra, known as the Bargmann algebra, leads to a geometrical framework that when made dynamical gives rise to Hořava-Lifshitz gravity. Here we consider the case where we contract the Poincaré algebra by sending the speed of light to zero leading to the Carroll algebra. We show how this algebra can be gauged and we construct the most general affine connection leading to the geometry of so-called Carrollian space-times. Carrollian space-times appear for example as the geometry on null hypersurfaces in a Lorentzian space-time of one dimension higher. We also construct theories of ultra-relativistic (Carrollian) gravity in 2+1 dimensions with dynamical exponent z < 1 including cases that have anisotropic Weyl invariance for z = 0.
Algebraic mesh quality metrics
KNUPP,PATRICK
2000-04-24
Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally-invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. the singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. Condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Combined metrics for shape and volume, shape-volume-orientation are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to non-simplical elements. A series of numerical tests verify the theoretical properties of the metrics defined.
NASA Astrophysics Data System (ADS)
Cherniha, Roman; King, John R.; Kovalenko, Sergii
2016-07-01
Complete descriptions of the Lie symmetries of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity in one and two space dimensions are obtained. A surprisingly rich set of Lie symmetry algebras depending on the form of diffusivity and source (sink) in the equations is derived. It is established that there exists a subclass in 1-D space admitting an infinite-dimensional Lie algebra of invariance so that it is linearisable. A special power-law diffusivity with a fixed exponent, which leads to wider Lie invariance of the equations in question in 2-D space, is also derived. However, it is shown that the diffusion equation without a source term (which often arises in applications and is sometimes called the Perona-Malik equation) possesses no rich variety of Lie symmetries depending on the form of gradient-dependent diffusivity. The results of the Lie symmetry classification for the reduction to lower dimensionality, and a search for exact solutions of the nonlinear 2-D equation with power-law diffusivity, also are included.
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
ERIC Educational Resources Information Center
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
Adaptive Algebraic Multigrid Methods
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
Algebra for Gifted Third Graders.
ERIC Educational Resources Information Center
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
A Holistic Approach to Algebra.
ERIC Educational Resources Information Center
Barbeau, Edward J.
1991-01-01
Described are two examples involving recursive mathematical sequences designed to integrate a holistic approach to learning algebra. These examples promote pattern recognition with algebraic justification, full class participation, and mathematical values that can be transferred to other situations. (MDH)
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.
Deformation of supersymmetric and conformal quantum mechanics through affine transformations
NASA Technical Reports Server (NTRS)
Spiridonov, Vyacheslav
1993-01-01
Affine transformations (dilatations and translations) are used to define a deformation of one-dimensional N = 2 supersymmetric quantum mechanics. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are q-isospectral, i.e. the spectrum of one can be obtained from another (with possible exception of the lowest level) by q(sup 2)-factor scaling. This construction allows easily to rederive a special self-similar potential found by Shabat and to show that for the latter a q-deformed harmonic oscillator algebra of Biedenharn and Macfarlane serves as the spectrum generating algebra. A general class of potentials related to the quantum conformal algebra su(sub q)(1,1) is described. Further possibilities for q-deformation of known solvable potentials are outlined.
A dielectric affinity microbiosensor
NASA Astrophysics Data System (ADS)
Huang, Xian; Li, Siqi; Schultz, Jerome S.; Wang, Qian; Lin, Qiao
2010-01-01
We present an affinity biosensing approach that exploits changes in dielectric properties of a polymer due to its specific, reversible binding with an analyte. The approach is demonstrated using a microsensor comprising a pair of thin-film capacitive electrodes sandwiching a solution of poly(acrylamide-ran-3-acrylamidophenylboronic acid), a synthetic polymer with specific affinity to glucose. Binding with glucose induces changes in the permittivity of the polymer, which can be measured capacitively for specific glucose detection, as confirmed by experimental results at physiologically relevant concentrations. The dielectric affinity biosensing approach holds the potential for practical applications such as long-term continuous glucose monitoring.
Heegaard, Niels H H
2009-06-01
The journal Electrophoresis has greatly influenced my approaches to biomolecular affinity studies. The methods that I have chosen as my main tools to study interacting biomolecules--native gel and later capillary zone electrophoresis--have been the topic of numerous articles in Electrophoresis. Below, the role of the journal in the development and dissemination of these techniques and applications reviewed. Many exhaustive reviews on affinity electrophoresis and affinity CE have been published in the last few years and are not in any way replaced by the present deliberations that are focused on papers published by the journal.
Unconscious processes improve lie detection.
Reinhard, Marc-André; Greifeneder, Rainer; Scharmach, Martin
2013-11-01
The capacity to identify cheaters is essential for maintaining balanced social relationships, yet humans have been shown to be generally poor deception detectors. In fact, a plethora of empirical findings holds that individuals are only slightly better than chance when discerning lies from truths. Here, we report 5 experiments showing that judges' ability to detect deception greatly increases after periods of unconscious processing. Specifically, judges who were kept from consciously deliberating outperformed judges who were encouraged to do so or who made a decision immediately; moreover, unconscious thinkers' detection accuracy was significantly above chance level. The reported experiments further show that this improvement comes about because unconscious thinking processes allow for integrating the particularly rich information basis necessary for accurate lie detection. These findings suggest that the human mind is not unfit to distinguish between truth and deception but that this ability resides in previously overlooked processes.
An algebra of reversible computation.
Wang, Yong
2016-01-01
We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules: basic reversible processes algebra, algebra of reversible communicating processes, recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.
Can lies be detected unconsciously?
Moi, Wen Ying; Shanks, David R.
2015-01-01
People are typically poor at telling apart truthful and deceptive statements. Based on the Unconscious Thought Theory, it has been suggested that poor lie detection arises from the intrinsic limitations of conscious thinking and can be improved by facilitating the contribution of unconscious thought (UT). In support of this hypothesis, Reinhard et al. (2013) observed improved lie detection among participants engaging in UT. The present study aimed to replicate this UT advantage using a similar experimental procedure but with an important improvement in a key control condition. Specifically, participants judged the truthfulness of eight video recordings in three thinking modes: immediately after watching them or after a period of unconscious or conscious deliberation. Results from two experiments (combined N = 226) failed to reveal a significant difference in lie detection accuracy between the thinking modes, even after efforts were made to facilitate the occurrence of an UT advantage in Experiment 2. The results imply that the UT advantage in deception detection is not a robust phenomenon. PMID:26379575
Cohomology, cocyles and the current algebra for the nonlinear σ-model
NASA Astrophysics Data System (ADS)
Fujiwara, Takanori; Kitakado, Shinsaku; Nonoyama, Tatsuhiko
1985-05-01
Using the idea of cohomology defined for the Lie algebra of gauge transformations, we examine the extension of the current algebra for the system of the gauged nonlinear σ-model. An anomalous term in the current commutation relation is constructed and shown to be equivalent to that arising in the gauged nonlinear σ-model with the Wess-Zumino term. The relation with the anomalous Schwinger term given by Faddeev is also discussed.
Cremmer-Gervais r-Matrices and the Cherednik Algebras of Type GL 2
NASA Astrophysics Data System (ADS)
Johnson, Garrett
2010-11-01
We give an interpretation of the Cremmer-Gervais r-matrices for {mathfrak{sl}_n} in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL 2 on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer-Gervais quantization in terms of the corresponding double affine Hecke algebra.
NASA Astrophysics Data System (ADS)
Gültekin, Kemal
2016-03-01
In this study, we give a thorough analysis of a general affine gravity with torsion. After a brief exposition of the affine gravities considered by Eddington and Schrödinger, we construct and analyze different affine gravities based on the determinants of the Ricci tensor, the torsion tensor, the Riemann tensor, and their combinations. In each case we reduce equations of motion to their simplest forms and give a detailed analysis of their solutions. Our analyses lead to the construction of the affine connection in terms of the curvature and torsion tensors. Our solutions of the dynamical equations show that the curvature tensors at different points are correlated via non-local, exponential rescaling factors determined by the torsion tensor.
Lectin affinity electrophoresis.
Kobayashi, Yuka
2014-01-01
An interaction or a binding event typically changes the electrophoretic properties of a molecule. Affinity electrophoresis methods detect changes in the electrophoretic pattern of molecules (mainly macromolecules) that occur as a result of biospecific interactions or complex formation. Lectin affinity electrophoresis is a very effective method for the detection and analysis of trace amounts of glycobiological substances. It is particularly useful for isolating and separating the glycoisomers of target molecules. Here, we describe a sensitive technique for the detection of glycoproteins separated by agarose gel-lectin affinity electrophoresis that uses antibody-affinity blotting. The technique is tested using α-fetoprotein with lectin (Lens culinaris agglutinin and Phaseolus vulgaris agglutinin)-agarose gels.
Coherent States for Landau Levels: Algebraic and Thermodynamical Properties
NASA Astrophysics Data System (ADS)
Aremua, Isiaka; Hounkonnou, Mahouton Norbert; Baloïtcha, Ezinvi
2015-10-01
This work describes coherent states for a physical system governed by a Hamiltonian operator, in two-dimensional space, of spinless charged particles subject to a perpendicular magnetic field B, coupled with a harmonic potential. The underlying su(1 , 1) Lie algebra and Barut-Girardello coherent states are constructed and discussed. Then, the Berezin-Klauder-Toeplitz quantization, also known as coherent state (or anti-Wick) quantization, is discussed. The thermodynamics of such a quantum gas system is elaborated and analyzed.
Geometrical description of algebraic structures: Applications to Quantum Mechanics
Carinena, J. F.; Ibort, A.; Marmo, G.; Morandi, G.
2009-05-06
Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to quantum mechanics. We will concentrate our attention into quantum theories and we will show how to use in a systematic way the transition from algebraic to geometrical structures to explore their geometry, mainly its Jordan-Lie structure.
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.…
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…
Algebraic Thinking through Origami.
ERIC Educational Resources Information Center
Higginson, William; Colgan, Lynda
2001-01-01
Describes the use of paper folding to create a rich environment for discussing algebraic concepts. Explores the effect that changing the dimensions of two-dimensional objects has on the volume of related three-dimensional objects. (Contains 13 references.) (YDS)
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…
Affine Equivalence of Quartic Monomial Rotation Symmetric Boolean Functions in Prime Power Dimension
2015-01-27
multivariate polynomial, called the algebraic normal form (ANF) f(x1, . . . , xn) = a0 + ∑ 1≤i≤n aixi + ∑ 1≤i<j≤n aijxixj + · · ·+ a12...nx1x2...xn, where the coefficients a0, aij , . . . , a12...n ∈ F2. The maximum number of variables in a monomial is called the ( algebraic ) degree, and it is...called affine functions. An affine function with constant term equal to zero is called a linear function. Define Preprint submitted to Elsevier
Superalgebra realization of the 3-algebras in N=6, 8 Chern-Simons-matter theories
NASA Astrophysics Data System (ADS)
Chen, Fa-Min
2012-01-01
We use superalgebras to realize the 3-algebras used to construct N=6, 8 Chern-Simons-matter (CSM) theories. We demonstrate that the superalgebra realization of the 3-algebras provides a unified framework for classifying the gauge groups of the Nge 5 theories based on 3-algebras. Using this realization, we rederive the ordinary Lie algebra construction of the general N=6 CSM theory from its 3-algebra counterpart and reproduce all known examples as well. In particular, we explicitly construct the Nambu 3-bracket in terms of a double graded commutator of PSU(2|2). The N=8 theory of Bagger, Lambert and Gustavsson (BLG) with SO(4) gauge group is constructed by using several different ways. A quantization scheme for the 3-brackets is proposed by promoting the double graded commutators as quantum mechanical double graded commutators.
Nonlinear W∞ algebras from nonlinear integrable deformations of self dual gravity
NASA Astrophysics Data System (ADS)
Castro, Carlos
1995-02-01
A proposal for constructing a universal nonlinear Ŵ∞ algebra is made as the symmetry algebra of a rotational Killing-symmetry reduction of the nonlinear perturbations of Moyal-integrable deformations of D = 4 self dual gravity (IDSDG). This is attained upon the construction of a nonlinear bracket based on nonlinear gauge theories associated with infinite dimensional Lie algebras. A quantization and supersymmetrization program can also be carried out. The relevance to the Kadomtsev-Petviashvili hierarchy, 2D dilaton gravity, quantum gravity and black hole physics is discussed in the concluding remarks.
On the Virasoro Structure of Symmetry Algebras of Nonlinear Partial Differential Equations
NASA Astrophysics Data System (ADS)
Güngör, Faruk
2006-01-01
We discuss Lie algebras of the Lie symmetry groups of two generically non-integrable equations in one temporal and two space dimensions arising in different contexts. The first is a generalization of the KP equation and contains 9 arbitrary functions of one and two arguments. The second one is a system of PDEs that depend on some physical parameters. We require that these PDEs are invariant under a Kac-Moody-Virasoro algebra. This leads to several limitations on the coefficients (either functions or parameters) under which equations are prime candidates for being integrable.
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.
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.
NASA Astrophysics Data System (ADS)
Hartwig, J. T.; Stokman, J. V.
2013-02-01
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.
Hom-Lie superalgebra structures on infinite-dimensional simple Lie superalgebras of vector fields
NASA Astrophysics Data System (ADS)
Yuan, Jixia; Sun, Liping; Liu, Wende
2014-10-01
This paper considers eight series of infinite-dimensional simple Lie superalgebras of vector fields over a field of characteristic zero. It is proved that Hom-Lie superalgebra structures on these Lie superalgebras must be scalars. As a consequence, multiplicative Hom-Lie superalgebra structures on these Lie superalgebras must be zero or the identity automorphisms.
Algebraic approach to electronic spectroscopy and dynamics.
Toutounji, Mohamad
2008-04-28
Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a(+). While exp(a(+)) translates coherent states, exp(a(+)a(+)) operation on coherent states has always been a challenge, as a(+) has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated
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.
Littelmann path model for geometric crystals, Whittaker functions on Lie groups and Brownian motion
NASA Astrophysics Data System (ADS)
Chhaibi, Reda
2013-02-01
Generally speaking, this thesis focuses on the interplay between the representations of Lie groups and probability theory. It subdivides into essentially three parts. In a first rather algebraic part, we construct a path model for geometric crystals in the sense of Berenstein and Kazhdan, for complex semi-simple Lie groups. We will mainly describe the algebraic structure, its natural morphisms and parameterizations. The theory of total positivity will play a particularly important role. Then, we anticipate on the probabilistic part by exhibiting a canonical measure on geometric crystals. It uses as ingredients the superpotential for the flag manifold and a measure invariant under the crystal actions. The image measure under the weight map plays the role of Duistermaat-Heckman measure. Its Laplace transform defines Whittaker functions, providing an interesting formula for all Lie groups. Then it appears clearly that Whittaker functions are to geometric crystals, what characters are to combinatorial crystals. The Littlewood-Richardson rule is also exposed. Finally we present the probabilistic approach that allows to find the canonical measure. It is based on the fundamental idea that the Wiener measure will induce the adequate measure on the algebraic structures through the path model. In the last chapter, we show how our geometric model degenerates to the continuous classical Littelmann path model and thus recover known results. For example, the canonical measure on a geometric crystal of highest weight degenerates into a uniform measure on a polytope, and recovers the parameterizations of continuous crystals.
[Lie, whacking lie and pseudologia phantastica--pathological lying in factitious disorder].
Haapasalo, Jaana
2014-01-01
Pseudologia phantastica refers to chronic pathological lying without a clear motive. It is a symptom in the factitious disorder in adults, Munchausen syndrome and an illness made up for or inflicted on a child. Child abuse is often involved. Patients making up or causing symptoms for themselves of their child may have been exposed to similar behavior as a child. Some of them have received care and attention only through an illness. Pseudologia phantastica may then in adulthood be directed to making up or causing illnesses for oneself or another person.
The Primitive Spectrum of a Basic Classical Lie Superalgebra
NASA Astrophysics Data System (ADS)
Coulembier, Kevin
2016-12-01
We prove Conjecture 5.7 in Coulembier and Musson (Math. J., arXiv:1409.2532), describing all inclusions between primitive ideals for the general linear superalgebra in terms of the {Ext1}-quiver of simple highest weight modules. For arbitrary basic classical Lie superalgebras, we formulate two types of Kazhdan-Lusztig quasi-orders on the dual of the Cartan subalgebra, where one corresponds to the above conjecture. Both orders can be seen as generalisations of the left Kazhdan-Lusztig order on Hecke algebras and are related to categorical braid group actions. We prove that the primitive spectrum is always described by one of the orders, obtaining for the first time a description of the inclusions. We also prove that the two orders are identical if category O admits `enough' abstract Kazhdan-Lusztig theories. In particular, they are identical for the general linear superalgebra, concluding the proof of the conjecture.
Invariants and labels for Lie-Poisson Systems
Thiffeault, J.L.; Morrison, P.J.
1998-04-01
Reduction is a process that uses symmetry to lower the order of a Hamiltonian system. The new variables in the reduced picture are often not canonical: there are no clear variables representing positions and momenta, and the Poisson bracket obtained is not of the canonical type. Specifically, we give two examples that give rise to brackets of the noncanonical Lie-Poisson form: the rigid body and the two-dimensional ideal fluid. From these simple cases, we then use the semidirect product extension of algebras to describe more complex physical systems. The Casimir invariants in these systems are examined, and some are shown to be linked to the recovery of information about the configuration of the system. We discuss a case in which the extension is not a semidirect product, namely compressible reduced MHD, and find for this case that the Casimir invariants lend partial information about the configuration of the system.
Detecting lies in children and adults.
Edelstein, Robin S; Luten, Tanya L; Ekman, Paul; Goodman, Gail S
2006-02-01
In this study, observers' abilities to detect lies in children and adults were examined. Adult participants observed videotaped interviews of both children and adults either lying or telling the truth about having been touched by a male research assistant. As hypothesized, observers detected children's lies more accurately than adults' lies; however, adults' truthful statements were detected more accurately than were children's. Further analyses revealed that observers were biased toward judging adults' but not children's statements as truthful. Finally, consistent with the notion that there are stable individual differences in the ability to detect lies, observers who were highly accurate in detecting children's lies were similarly accurate in detecting adults' lies. Implications of these findings for understanding lie-detection accuracy are discussed, as are potential applications to the forensic context.
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.
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
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®,…
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.
Clifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras
NASA Astrophysics Data System (ADS)
Catto, Sultan; Gürcan, Yasemin; Khalfan, Amish; Kurt, Levent; Kato La, V.
2016-10-01
We discuss a construction scheme for Clifford numbers of arbitrary dimension. The scheme is based upon performing direct products of the Pauli spin and identity matrices. Conjugate fermionic algebras can then be formed by considering linear combinations of the Clifford numbers and the Hermitian conjugates of such combinations. Fermionic algebras are important in investigating systems that follow Fermi-Dirac statistics. We will further comment on the applications of Clifford algebras to Fueter analyticity, twistors, color algebras, M-theory and Leech lattice as well as unification of ancient and modern geometries through them.
Some applications of Lie groups in astrodynamics
NASA Technical Reports Server (NTRS)
Jackson, A. A.
1983-01-01
Differential equations that arise in astrodynamics are examined from the standpoint of Lie group theory. A summary of the Lie method is given for first degree differential equations. The Kepler problem in Hamiltonian form is treated by this method. Extension of the Lie method to optimal trajectories is outlined.
Emergence of Lying in Very Young Children
ERIC Educational Resources Information Center
Evans, Angela D.; Lee, Kang
2013-01-01
Lying is a pervasive human behavior. Evidence to date suggests that from the age of 42 months onward, children become increasingly capable of telling lies in various social situations. However, there is limited experimental evidence regarding whether very young children will tell lies spontaneously. The present study investigated the emergence of…
Empirical Evidence for a Typology of Lies.
ERIC Educational Resources Information Center
Hample, Dale
A study was conducted to test the empirical merit of R. M. Chisholm's and T. D. Feehan's proposed typology of deception: (1) commission versus omission (lies of commission are those where the liar contributes causally to the receiver's believing the lie, perhaps by telling the falsehood; lies of omission would occur if the liar could have…
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.
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.
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…
Applications of algebraic grid generation
NASA Technical Reports Server (NTRS)
Eiseman, Peter R.; Smith, Robert E.
1990-01-01
Techniques and applications of algebraic grid generation are described. The techniques are univariate interpolations and transfinite assemblies of univariate interpolations. Because algebraic grid generation is computationally efficient, the use of interactive graphics in conjunction with the techniques is advocated. A flexible approach, which works extremely well in an interactive environment, called the control point form of algebraic grid generation is described. The applications discussed are three-dimensional grids constructed about airplane and submarine configurations.
Low-dimensional filiform Lie superalgebras
NASA Astrophysics Data System (ADS)
Navarro, R. M.
2016-10-01
The present work is regarding filiform Lie superalgebras which is an important type of nilpotent Lie superalgebras. In general, classifying nilpotent Lie superalgebras is at present an open and unsolved problem. Throughout the present work we contribute to the resolution of this wide problem by classifying filiform Lie superalgebras of low dimensions, in particular less or equal to 7. Furthermore we would establish a method that could be applied to obtain similar results for higher dimensions. Thus, this method would mainly consist in using infinitesimal deformations of the model filiform Lie superalgebra.
[Psychopathological study of lie motif in schizophrenia].
Otsuka, Koichiro; Kato, Satoshi
2006-01-01
The theme of a statement is called "lie motif" by the authors when schizophrenic patients say "I have lied to anybody". We tried to analyse of the psychopathological characteristics and anthropological meanings of the lie motifs in schizophrenia, which has not been thematically examined until now, based on 4 cases, and contrasting with the lie motif (Lügenmotiv) in depression taken up by A. Kraus (1989). We classified the lie motifs in schizophrenia into the following two types: a) the past directive lie motif: the patients speak about their real lie regarding it as a 'petty fault' in their distant past with self-guilty feeling, b) the present directive lie motif: the patients say repeatedly 'I have lied' (about their present speech and behavior), retreating from their previous commitments. The observed false confessions of innocent fault by the patients seem to belong to the present directed lie motif. In comparison with the lie motif in depression, it is characteristic for the lie motif in schizophrenia that the patients feel themselves to already have been caught out by others before they confess the lie. The lie motif in schizophrenia seems to come into being through the attribution process of taking the others' blame on ones' own shoulders, which has been pointed out to be common in the guilt experience in schizophrenia. The others' blame on this occasion is due to "the others' gaze" in the experience of the initial self-centralization (i.e. non delusional self-referential experience) in the early stage of schizophrenia (S. Kato 1999). The others' gaze is supposed to bring about the feeling of amorphous self-revelation which could also be regarded as the guilt feeling without content, to the patients. When the guilt feeling is bound with a past concrete fault, the patients tell the past directive lie motif. On the other hand, when the patients cannot find a past fixed content, and feel their present actions as uncertain and experience them as lies, the
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Learning to lie: effects of practice on the cognitive cost of lying.
Van Bockstaele, B; Verschuere, B; Moens, T; Suchotzki, Kristina; Debey, Evelyne; Spruyt, Adriaan
2012-01-01
Cognitive theories on deception posit that lying requires more cognitive resources than telling the truth. In line with this idea, it has been demonstrated that deceptive responses are typically associated with increased response times and higher error rates compared to truthful responses. Although the cognitive cost of lying has been assumed to be resistant to practice, it has recently been shown that people who are trained to lie can reduce this cost. In the present study (n = 42), we further explored the effects of practice on one's ability to lie by manipulating the proportions of lie and truth-trials in a Sheffield lie test across three phases: Baseline (50% lie, 50% truth), Training (frequent-lie group: 75% lie, 25% truth; control group: 50% lie, 50% truth; and frequent-truth group: 25% lie, 75% truth), and Test (50% lie, 50% truth). The results showed that lying became easier while participants were trained to lie more often and that lying became more difficult while participants were trained to tell the truth more often. Furthermore, these effects did carry over to the test phase, but only for the specific items that were used for the training manipulation. Hence, our study confirms that relatively little practice is enough to alter the cognitive cost of lying, although this effect does not persist over time for non-practiced items.
Hybrid Topological Lie-Hamiltonian Learning in Evolving Energy Landscapes
NASA Astrophysics Data System (ADS)
Ivancevic, Vladimir G.; Reid, Darryn J.
2015-11-01
In this Chapter, a novel bidirectional algorithm for hybrid (discrete + continuous-time) Lie-Hamiltonian evolution in adaptive energy landscape-manifold is designed and its topological representation is proposed. The algorithm is developed within a geometrically and topologically extended framework of Hopfield's neural nets and Haken's synergetics (it is currently designed in Mathematica, although with small changes it could be implemented in Symbolic C++ or any other computer algebra system). The adaptive energy manifold is determined by the Hamiltonian multivariate cost function H, based on the user-defined vehicle-fleet configuration matrix W, which represents the pseudo-Riemannian metric tensor of the energy manifold. Search for the global minimum of H is performed using random signal differential Hebbian adaptation. This stochastic gradient evolution is driven (or, pulled-down) by `gravitational forces' defined by the 2nd Lie derivatives of H. Topological changes of the fleet matrix W are observed during the evolution and its topological invariant is established. The evolution stops when the W-topology breaks down into several connectivity-components, followed by topology-breaking instability sequence (i.e., a series of phase transitions).
Quantization and harmonic analysis on nilpotent Lie groups
Wildberger, N.J.
1983-01-01
Weyl Quantization is a procedure for associating a function on which the canonical commutation relations are realized. If G is a simply-connected, connected nilpotent Lie group with Lie algebra g and dual g/sup */, it is shown how to inductively construct symplectic isomorphisms between every co-adjoint orbit O and the bundle in Hilbert Space for some m. Weyl Quantization can then be used to associate to each orbit O a unitary representation rho/sub 0/ of G, recovering the classification of the unitary dual by Kirillov. It is used to define a geometric Fourier transform, F : L/sup 1/(G) ..-->.. functions on g/sup */, and it is shown that the usual operator-valued Fourier transform can be recovered from F, characters are inverse Fourier transforms of invariant measures on orbits, and matrix coefficients are inverse Fourier transforms of non-invariant measures supported on orbits. Realizations of the representations rho/sub 0/ in subspaces of L/sup 2/(O) are obtained.. Finally, the kernel function is computed for the upper triangular unipotent group and one other example.
Koszul information geometry and Souriau Lie group thermodynamics
Barbaresco, Frédéric
2015-01-13
The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from 'Characteristic Functions', was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of 'Information Geometry' theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean 'Moment map' by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. These elements has been developed by author in [10][11].
On spaces of commuting elements in Lie groups
NASA Astrophysics Data System (ADS)
Cohen, Frederick R.; Stafa, Mentor
2016-11-01
The main purpose of this paper is to introduce a method to stabilize certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group $G$, namely $Hom(\\mathbb Z^n,G)$. We show that this stabilized space of homomorphisms decomposes after suspending once with summands which can be reassembled, in a sense to be made precise below, into the individual spaces $Hom(\\mathbb Z^n,G)$ after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces $Hom(\\mathbb Z^n,G)$, with the order of the Weyl group inverted.
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
Algebraic construction of a Nambu bracket for the two-dimensional vorticity equation.
Sommer, M; Brazda, K; Hantel, M
2011-08-29
So far fluid mechanical Nambu brackets have mainly been given on an intuitive basis. Alternatively an algorithmic construction of such a bracket for the two-dimensional vorticity equation is presented here. Starting from the Lie-Poisson form and its algebraic properties it is shown how the Nambu representation can be explicitly constructed as the continuum limit from the structure preserving Zeitlin discretization.
A study on neural learning on manifold foliations: the case of the Lie group SU(3).
Fiori, Simone
2008-04-01
Learning on differential manifolds may involve the optimization of a function of many parameters. In this letter, we deal with Riemannian-gradient-based optimization on a Lie group, namely, the group of unitary unimodular matrices SU(3). In this special case, subalgebras of the associated Lie algebra su(3) may be individuated by computing pair-wise Gell-Mann matrices commutators. Subalgebras generate subgroups of a Lie group, as well as manifold foliation. We show that the Riemannian gradient may be projected over tangent structures to foliation, giving rise to foliation gradients. Exponentiations of foliation gradients may be computed in closed forms, which closely resemble Rodriguez forms for the special orthogonal group SO(3). We thus compare optimization by Riemannian gradient and foliation gradients.
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
Viterbi/algebraic hybrid decoder
NASA Technical Reports Server (NTRS)
Boyd, R. W.; Ingels, F. M.; Mo, C.
1980-01-01
Decoder computer program is hybrid between optimal Viterbi and optimal algebraic decoders. Tests have shown that hybrid decoder outperforms any strictly Viterbi or strictly algebraic decoder and effectively handles compound channels. Algorithm developed uses syndrome-detecting logic to direct two decoders to assume decoding load alternately, depending on real-time channel characteristics.
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.)
ERIC Educational Resources Information Center
1997
Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…
Elementary maps on nest algebras
NASA Astrophysics Data System (ADS)
Li, Pengtong
2006-08-01
Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.
Linear algebra and image processing
NASA Astrophysics Data System (ADS)
Allali, Mohamed
2010-09-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty.
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.)
Learning Algebra from Worked Examples
ERIC Educational Resources Information Center
Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.
2014-01-01
For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…
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…
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…
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.
Algebraic Functions for Recognition
1994-01-01
ftoinits won,’t ’I’ lie t riIiniea r im et h od requiriies at letast nini e corrtespon d- uised IFor re- lprol ect iou, b tut nevyerthlt’less. thle...as first applied w~ithI th li’Minimal ear equtat ioiis art’ sigtiificaiit or not . ’hit linhear comiiiia- niumiber of points ( ninie ) for sol vinug for
The roles of liar intention, lie content, and theory of mind in children's evaluation of lies.
Cheung, Him; Siu, Tik-Sze Carrey; Chen, Lan
2015-04-01
This study found that 7-, 9-, and 11-year-old children and young adults identified prosocial lies as lies less frequently and evaluated them less negatively than selfish lies (liar intention effect); lies about opinions were identified as lies less frequently and evaluated less negatively than those about reality (lie content effect). The lie content effect was more pronounced in the prosocial lies than in the selfish lies for both identification and evaluation. Overall, the older participants considered liar intention more than the younger participants in lie evaluation. For the child participants, second-order belief understanding correlated marginally with sensitivity to liar intention in the opinion lies, but not with content sensitivity. Finally, lie identification correlated with evaluation in the prosocial-opinion lies for all of the children. The independent effects of intention and content could potentially explain children's development in "white lie" understanding demonstrated in the literature. Although the content effect appears to stem from a more general concern for whether communication is about objective reality, the intention effect may involve theory of mind.
Effect of lie labelling on children's evaluation of selfish, polite, and altruistic lies.
Cheung, Him; Chan, Yawen; Tsui, Wan Chi Gigi
2016-09-01
This study investigates how 5- and 6-year-olds' evaluations of selfish, polite, and altruistic lies change as a result of whether these false statements are explicitly labelled as lies. We are also interested in how interpretive theory of mind may correlate with such evaluations with and without a lie label. Our results showed that labelling lowered children's evaluations for the polite and altruistic lies, but not for the selfish lies. Interpretive theory of mind correlated positively with the evaluation difference between the polite and altruistic lies and that between the selfish and altruistic lies in the label, but not in the non-label condition. Correlation between the selfish and altruistic lies and that between the polite and altruistic lies were stronger with than without labelling, after controlling for age, and verbal and non-verbal intelligence. We conclude that lie labelling biases children towards more negative evaluations for non-selfish lies and makes them see lies of different motives as more similar. If a lie label is applied, whether lies of different motives are still evaluated differently depends on interpretive theory of mind, which reflects the child's ability to represent and allow different interpretations of an ambiguous reality.
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)
Quantum algebra of N superspace
Hatcher, Nicolas; Restuccia, A.; Stephany, J.
2007-08-15
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.
Chen, J.; Safro, I.
2011-01-01
Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
NASA Astrophysics Data System (ADS)
Fernández Núñez, J.; García Fuertes, W.; Perelomov, A. M.
2005-10-01
We reexpress the quantum Calogero-Sutherland model for the Lie algebra E7 and the particular value of the coupling constant κ =1 by using the fundamental irreducible characters of the algebra as dynamical variables. For that, we need to develop a systematic procedure to obtain all the Clebsch-Gordan series required to perform the change of variables. We describe how the resulting quantum Hamiltonian operator can be used to compute more characters and Clebsch-Gordan series for this exceptional algebra.
Investigating Teacher Noticing of Student Algebraic Thinking
ERIC Educational Resources Information Center
Walkoe, Janet Dawn Kim
2013-01-01
Learning algebra is critical for students in the U.S. today. Algebra concepts provide the foundation for much advanced mathematical content. In addition, algebra serves as a gatekeeper to opportunities such as admission to college. Yet many students in the U.S. struggle in algebra classes. Researchers claim that one reason for these difficulties…
Example of a quantum field theory based on a nonlinear Lie algebra
Schoutens, K. . Inst. for Theoretical Physics); Sevrin, A. ); van Nieuwenhuizen, P. . Theory Div.)
1991-11-01
In this contribution to Tini Veltman's Festschrift we shall give a paedagogical account of our work on a new class of gauge theories called W gravities. They contain higher spin gauge fields, but the usual no-go theorems for interacting field theories with spins exceeding two do not apply since these theories are in two dimensions. It is, of course, well known that ghost-free interacting massless spin 2 fields ( the metric') are gauge fields, and correspond to the geometrical notion of general coordinate transformations in general relativity, but it is yet unknown what extension of these ideas is introduced by the presence of massless higher spin gauge fields. A parallel with supergravity may be drawn: there the presence of massless spin 3/2 fields (gravitinos) corresponds to local fermi-bose symmetries of which these gravitinos are the gauge fields. Their geometrical meaning becomes only clear if one introduces superspace (with bosonic and fermionic coordinates): they correspond to local transformations of the fermionic coordinates. For W gravity one might speculate on a kind of W-superspace with extra bosonic coordinates.
Example of a quantum field theory based on a nonlinear Lie algebra
Schoutens, K.; Sevrin, A.; van Nieuwenhuizen, P.
1991-11-01
In this contribution to Tini Veltman`s Festschrift we shall give a paedagogical account of our work on a new class of gauge theories called W gravities. They contain higher spin gauge fields, but the usual no-go theorems for interacting field theories with spins exceeding two do not apply since these theories are in two dimensions. It is, of course, well known that ghost-free interacting massless spin 2 fields (`the metric`) are gauge fields, and correspond to the geometrical notion of general coordinate transformations in general relativity, but it is yet unknown what extension of these ideas is introduced by the presence of massless higher spin gauge fields. A parallel with supergravity may be drawn: there the presence of massless spin 3/2 fields (gravitinos) corresponds to local fermi-bose symmetries of which these gravitinos are the gauge fields. Their geometrical meaning becomes only clear if one introduces superspace (with bosonic and fermionic coordinates): they correspond to local transformations of the fermionic coordinates. For W gravity one might speculate on a kind of W-superspace with extra bosonic coordinates.
The Lie Algebraic Structure of a Class of Finite Dimensional Nonlinear Filters.
1980-07-23
where w and v are independent unit variance vector Wiener processes, f and h are vector-valued functions, G is a matrix -valued function, and R > 0. The...I ... and n 1 (0), so that the canonical map v: A - 0 A/l. is injective. . 3 (vi) L is the semidirect sum [18] of A and the nilpotent ideal o Io...Kalman filter error covariance matrix ; (16) is obtained by selecting the corresponding components of the Riccatj. equation. Then 9, y , and y are seen
Computer-Aided Closure of the Lie Algebra Associated with a Nonlinear PDE,
1980-10-01
wave packets which are slowly varying on the space and time scales of the corresponding wavelengths and periods, yet sufficiently broad in Fourier space ...velocities. The solutions to Eqs. (1) ignoring spatial derivatives ( time -only) are given in standard plasma physic’ texts6. The time -only equations are...commutators. We add one element at a time , substituting all newly determined commutators into the Jacobi identities as we proceed. For the example which
Topological features of the Sokolov integrable case on the Lie algebra so(3,1)
Novikov, D V
2014-08-31
The integrable Sokolov case on so(3,1){sup ⋆} is investigated. This is a Hamiltonian system with two degrees of freedom, in which the Hamiltonian and the additional integral are homogeneous polynomials of degrees 2 and 4, respectively. It is an interesting feature of this system that connected components of common level surfaces of the Hamiltonian and the additional integral turn out to be noncompact. The critical points of the moment map and their indices are found, the bifurcation diagram is constructed, and the topology of noncompact level surfaces is determined, that is, the closures of solutions of the Sokolov system on so(3,1) are described. Bibliography: 24 titles.
Lie Group Techniques for Neural Learning
2005-01-03
Lie group techniques for Neural Learning Edinburgh June 2004 Elena Celledoni SINTEF Applied Mathematics, IMF-NTNU Lie group techniques for Neural...ORGANIZATION NAME(S) AND ADDRESS(ES) SINTEF Applied Mathematics, IMF-NTNU 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND
Developmental Changes in Ideas about Lying.
ERIC Educational Resources Information Center
Peterson, Candida C.; And Others
1983-01-01
Videotaped stories depicting deliberate lies and unintentionally untrue statements were presented to 200 subjects evenly divided into the following age groups: 5, 8, 9, 11 years, and adult. Definitions of lying were seen to change gradually over this age range. (Author/RH)
Some evidence for unconscious lie detection.
Ten Brinke, Leanne; Stimson, Dayna; Carney, Dana R
2014-05-01
To maximize survival and reproductive success, primates evolved the tendency to tell lies and the ability to accurately detect them. Despite the obvious advantage of detecting lies accurately, conscious judgments of veracity are only slightly more accurate than chance. However, findings in forensic psychology, neuroscience, and primatology suggest that lies can be accurately detected when less-conscious mental processes (as opposed to more-conscious mental processes) are used. We predicted that observing someone tell a lie would automatically activate cognitive concepts associated with deception, and observing someone tell the truth would activate concepts associated with truth. In two experiments, we demonstrated that indirect measures of deception detection are significantly more accurate than direct measures. These findings provide a new lens through which to reconsider old questions and approach new investigations of human lie detection.
Characteristics of the Eysenck Personality Questionnaire Lie Scale and of Extreme Lie Scorers.
ERIC Educational Resources Information Center
Loo, Robert
1980-01-01
Results of statistical analyses suggest that high lie-scorers respond honestly, and that the Lie Scale for the Eysenck Personality Inventory may reflect a personality dimension of interest rather than an extraneous and undesirable factor to be eliminated. (Author)
The Prevalence of Lying in America: Three Studies of Self-Reported Lies
ERIC Educational Resources Information Center
Serota, Kim B.; Levine, Timothy R.; Boster, Franklin J.
2010-01-01
This study addresses the frequency and the distribution of reported lying in the adult population. A national survey asked 1,000 U.S. adults to report the number of lies told in a 24-hour period. Sixty percent of subjects report telling no lies at all, and almost half of all lies are told by only 5% of subjects; thus, prevalence varies widely and…
Asymptotic aspect of derivations in Banach algebras.
Roh, Jaiok; Chang, Ick-Soon
2017-01-01
We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.
The algebraic cluster model: Structure of 16O
NASA Astrophysics Data System (ADS)
Bijker, R.; Iachello, F.
2017-01-01
We discuss an algebraic treatment of four-body clusters which includes both continuous and discrete symmetries. In particular, tetrahedral configurations with Td symmetry are analyzed with respect to the energy spectrum, transition form factors and B (EL) values. It is concluded that the low-lying spectrum of 16O can be described by four α particles at the vertices of a regular tetrahedron, not as a rigid structure but rather a more floppy structure with relatively large rotation-vibration interactions and Coriolis forces.
The affine cohomology spaces and its applications
NASA Astrophysics Data System (ADS)
Fraj, Nizar Ben; Laraiedh, Ismail
2016-12-01
We compute the nth cohomology space of the affine Lie superalgebra 𝔞𝔣𝔣(1) on the (1,1)-dimensional real superspace with coefficient in a large class of 𝔞𝔣𝔣(1)-modules M. We apply our results to the module of weight densities and the module of linear differential operators acting on a superspace of weighted densities. This work is the generalization of a result by Basdouri et al. [The linear 𝔞𝔣𝔣(n|1)-invariant differential operators on weighted densities on the superspace ℝ1|n and 𝔞𝔣𝔣(n|1)-relative cohomology, Int. J. Geom. Meth. Mod. Phys. 10 (2013), Article ID: 1320004, 9 pp.
The oscillator model for the Lie superalgebra sh(2|2) and Charlier polynomials
Jafarov, E. I.; Van der Jeugt, J.
2013-10-15
We investigate an algebraic model for the quantum oscillator based upon the Lie superalgebra sh(2|2), known as the Heisenberg–Weyl superalgebra or “the algebra of supersymmetric quantum mechanics,” and its Fock representation. The model offers some freedom in the choice of a position and a momentum operator, leading to a free model parameter γ. Using the technique of Jacobi matrices, we determine the spectrum of the position operator, and show that its wavefunctions are related to Charlier polynomials C{sub n} with parameter γ{sup 2}. Some properties of these wavefunctions are discussed, as well as some other properties of the current oscillator model.
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.
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)
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.
Reflection symmetry detection using locally affine invariant edge correspondence.
Wang, Zhaozhong; Tang, Zesheng; Zhang, Xiao
2015-04-01
Reflection symmetry detection receives increasing attentions in recent years. The state-of-the-art algorithms mainly use the matching of intensity-based features (such as the SIFT) within a single image to find symmetry axes. This paper proposes a novel approach by establishing the correspondence of locally affine invariant edge-based features, which are superior to the intensity based in the aspects that it is insensitive to illumination variations, and applicable to textureless objects. The locally affine invariance is achieved by simple linear algebra for efficient and robust computations, making the algorithm suitable for detections under object distortions like perspective projection. Commonly used edge detectors and a voting process are, respectively, used before and after the edge description and matching steps to form a complete reflection detection pipeline. Experiments are performed using synthetic and real-world images with both multiple and single reflection symmetry axis. The test results are compared with existing algorithms to validate the proposed method.
A new family of algebras underlying the Rogers-Ramanujan identities and generalizations
Lepowsky, James; Wilson, Robert Lee
1981-01-01
The classical Rogers-Ramanujan identities have been interpreted by Lepowsky-Milne and the present authors in terms of the representation theory of the Euclidean Kac-Moody Lie algebra A1(1). Also, the present authors have introduced certain “vertex” differential operators providing a construction of A1(1) on its basic module, and Kac, Kazhdan, and we have generalized this construction to a general class of Euclidean Lie algebras. Starting from this viewpoint, we now introduce certain new algebras [unk]v which centralize the action of the principal Heisenberg subalgebra of an arbitrary Euclidean Lie algebra [unk] on a highest weight [unk]-module V. We state a general (tautological) Rogers-Ramanujan-type identity, which by our earlier theorem includes the classical identities, and we show that [unk]v can be used to reformulate the general identity. For [unk] = A1(1), we develop the representation theory of [unk]v in considerable detail, allowing us to prove our earlier conjecture that our general Rogers-Ramanujan-type identity includes certain identities of Gordon, Andrews, and Bressoud. In the process, we construct explicit bases of all of the standard and Verma modules of nonzero level for A1(1), with an explicit realization of A1(1) as operators in each case. The differential operator constructions mentioned above correspond to the trivial case [unk]v = (1) of the present theory. PMID:16593131
Coherent States for Hopf Algebras
NASA Astrophysics Data System (ADS)
Škoda, Zoran
2007-07-01
Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. If, in addition, the Hopf algebra has a left Haar integral, then a formula for noncommutative resolution of identity in terms of the family of coherent states holds. Examples come from quantum groups.
Multiplier operator algebras and applications
Blecher, David P.; Zarikian, Vrej
2004-01-01
The one-sided multipliers of an operator space X are a key to “latent operator algebraic structure” in X. We begin with a survey of these multipliers, together with several of the applications that they have had to operator algebras. We then describe several new results on one-sided multipliers, and new applications, mostly to one-sided M-ideals. PMID:14711990
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).
Kernel Affine Projection Algorithms
NASA Astrophysics Data System (ADS)
Liu, Weifeng; Príncipe, José C.
2008-12-01
The combination of the famed kernel trick and affine projection algorithms (APAs) yields powerful nonlinear extensions, named collectively here, KAPA. This paper is a follow-up study of the recently introduced kernel least-mean-square algorithm (KLMS). KAPA inherits the simplicity and online nature of KLMS while reducing its gradient noise, boosting performance. More interestingly, it provides a unifying model for several neural network techniques, including kernel least-mean-square algorithms, kernel adaline, sliding-window kernel recursive-least squares (KRLS), and regularization networks. Therefore, many insights can be gained into the basic relations among them and the tradeoff between computation complexity and performance. Several simulations illustrate its wide applicability.
On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions
NASA Astrophysics Data System (ADS)
Ballesteros, A.; Herranz, F. J.; Musso, F.
2014-09-01
Quantum deformations of (anti-)de Sitter (A)dS algebras in (2+1) dimensions are revisited, and several features of these quantum structures are reviewed. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the associated noncommutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishing cosmological constant) of all these structures leading to (2+1) quantum Poincare algebras and groups is simultaneously given by considering the cosmological constant as an explicit Lie algebra parameter in the (A)dS algebras. By making use of this classification, a three-parameter generalization of the K-deformation for the (2+1) (A)dS algebras and quantum spacetimes is given. Finally, the same problem is studied in (3+1) dimensions, where a two-parameter generalization of the κ-(A)dS deformation that preserves the space isotropy is found.
Affine group representation formalism for four-dimensional, Lorentzian, quantum gravity
NASA Astrophysics Data System (ADS)
Chou, Ching-Yi; Ita, Eyo E.; Soo, Chopin
2013-03-01
Within the context of the Ashtekar variables, the Hamiltonian constraint of four-dimensional pure general relativity with cosmological constant, Λ, is re-expressed as an affine algebra with the commutator of the imaginary part of the Chern-Simons functional, Q, and the positive-definite volume element. This demonstrates that the affine algebra quantization program of Klauder can indeed be applicable to the full Lorentzian signature theory of quantum gravity with non-vanishing cosmological constant, and it facilitates the construction of solutions to all of the constraints. Unitary, irreducible representations of the affine group exhibit a natural Hilbert space structure, and coherent states and other physical states can be generated from a fiducial state. It is also intriguing that formulation of the Hamiltonian constraint or the Wheeler-DeWitt equation as an affine algebra requires a non-vanishing cosmological constant, and a fundamental uncertainty relation of the form {Δ{V}/{< {V}> }Δ {Q}≥ 2π Λ L^2_{Planck} (wherein V is the total volume) may apply to all physical states of quantum gravity.
Statistics on Lie groups: A need to go beyond the pseudo-Riemannian framework
NASA Astrophysics Data System (ADS)
Miolane, Nina; Pennec, Xavier
2015-01-01
Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by others [5, 6, 7, 8, 9]. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is compatible with the group structure, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group G. The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.
Lie symmetry analysis of the Heisenberg equation
NASA Astrophysics Data System (ADS)
Zhao, Zhonglong; Han, Bo
2017-04-01
The Lie symmetry analysis is performed on the Heisenberg equation from the statistical physics. Its Lie point symmetries and optimal system of one-dimensional subalgebras are determined. The similarity reductions and invariant solutions are obtained. Using the multipliers, some conservation laws are obtained. We prove that this equation is nonlinearly self-adjoint. The conservation laws associated with symmetries of this equation are constructed by means of Ibragimov's method.
Lie theoretic aspects of the Riccati equation
NASA Technical Reports Server (NTRS)
Hermann, R.; Martin, C.
1977-01-01
Various features of the application of Lie theory to matrix Riccati equations, of basic importance in control and system theories, are discussed. Particular consideration is given to centralizer foliation, the Cartan decomposition, matrix Riccati equations as Lie systems on Grassmanians, local analysis near a zero point of a vector field, linearization in homogeneous space, the tangent bundle in terms of partitioned matrices, and stability properties of fixed points of Riccati vector fields.
Clustered Numerical Data Analysis Using Markov Lie Monoid Based Networks
NASA Astrophysics Data System (ADS)
Johnson, Joseph
2016-03-01
We have designed and build an optimal numerical standardization algorithm that links numerical values with their associated units, error level, and defining metadata thus supporting automated data exchange and new levels of artificial intelligence (AI). The software manages all dimensional and error analysis and computational tracing. Tables of entities verses properties of these generalized numbers (called ``metanumbers'') support a transformation of each table into a network among the entities and another network among their properties where the network connection matrix is based upon a proximity metric between the two items. We previously proved that every network is isomorphic to the Lie algebra that generates continuous Markov transformations. We have also shown that the eigenvectors of these Markov matrices provide an agnostic clustering of the underlying patterns. We will present this methodology and show how our new work on conversion of scientific numerical data through this process can reveal underlying information clusters ordered by the eigenvalues. We will also show how the linking of clusters from different tables can be used to form a ``supernet'' of all numerical information supporting new initiatives in AI.
Multipoint Lax operator algebras: almost-graded structure and central extensions
NASA Astrophysics Data System (ADS)
Schlichenmaier, M.
2014-05-01
Recently, Lax operator algebras appeared as a new class of higher genus current-type algebras. Introduced by Krichever and Sheinman, they were based on Krichever's theory of Lax operators on algebraic curves. These algebras are almost-graded Lie algebras of currents on Riemann surfaces with marked points (in-points, out-points and Tyurin points). In a previous joint article with Sheinman, the author classified the local cocycles and associated almost-graded central extensions in the case of one in-point and one out-point. It was shown that the almost-graded extension is essentially unique. In this article the general case of Lax operator algebras corresponding to several in- and out-points is considered. As a first step they are shown to be almost-graded. The grading is given by splitting the marked points which are non-Tyurin points into in- and out-points. Next, classification results both for local and bounded cocycles are proved. The uniqueness theorem for almost-graded central extensions follows. To obtain this generalization additional techniques are needed which are presented in this article. Bibliography: 30 titles.
Electron Affinity Calculations for Thioethers
NASA Technical Reports Server (NTRS)
Sulton, Deley L.; Boothe, Michael; Ball, David W.; Morales, Wilfredo
1997-01-01
Previous work indicated that polyphenyl thioethers possessed chemical properties, related to their electron affinities, which could allow them to function as vapor phase lubricants (VPL). Indeed, preliminary tribological tests revealed that the thioethers could function as vapor phase lubricants but not over a wide temperature and hertzian pressure range. Increasing the electron affinity of the thioethers may improve their VPL properties over this range. Adding a substituent group to the thioether will alter its electron affinity in many cases. Molecular orbital calculations were undertaken to determine the effect of five different substituent groups on the electron affinity of polyphenyl thioethers. It was found that the NO2, F, and I groups increased the thioethers electron affinity by the greatest amount. Future work will involve the addition of these groups to the thioethers followed by tribological testing to assess their VPL properties.
The algebra of two dimensional generalized Chebyshev-Koornwinder oscillator
NASA Astrophysics Data System (ADS)
Borzov, V. V.; Damaskinsky, E. V.
2014-10-01
In the previous works of Borzov and Damaskinsky ["Chebyshev-Koornwinder oscillator," Theor. Math. Phys. 175(3), 765-772 (2013)] and ["Ladder operators for Chebyshev-Koornwinder oscillator," in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillator-like system that is associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev-Koornwinder oscillator. In this paper, we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev-Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space H of functions that are defined on a region which is bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev-Koornwinder polynomials and these polynomials form an orthonormalized basis in the space H. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-07-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
Colored Quantum Algebra and Its Bethe State
NASA Astrophysics Data System (ADS)
Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun
2014-12-01
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.
Using Number Theory to Reinforce Elementary Algebra.
ERIC Educational Resources Information Center
Covillion, Jane D.
1995-01-01
Demonstrates that using the elementary number theory in algebra classes helps students to use acquired algebraic skills as well as helping them to more clearly understand concepts that are presented. Discusses factoring, divisibility rules, and number patterns. (AIM)
Algebraic orbifold conformal field theories
Xu, Feng
2000-01-01
The unitary rational orbifold conformal field theories in the algebraic quantum field theory and subfactor theory framework are formulated. Under general conditions, it is shown that the orbifold of a given unitary rational conformal field theory generates a unitary modular category. Many new unitary modular categories are obtained. It is also shown that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time. PMID:11106383
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.
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.
NASA Astrophysics Data System (ADS)
Zhang, Xuguang; Zhang, Yun; Zhang, Jie; Chen, Shengyong; Chen, Dan; Li, Xiaoli
2012-04-01
Toward the unsupervised clustering for color logo images corrupted by noise, we propose a novel framework in which the logo images are described by a model called singular values based region covariance matrices (SVRCM), and the mean shift algorithm is performed on Lie groups for clustering covariance matrices. To decrease the influence of noise, we choose the larger singular values, which can better represent the original image and discard the smaller singular values. Therefore, the chosen singular values are grouped and fused by a covariance matrix to form a SVRCM model that can represent the correlation and variance between different singular value features to enhance the discriminating ability of the model. In order to cluster covariance matrices, which do not lie on Euclidean space, the mean shift algorithm is performed on manifolds by iteratively transforming points between the Lie group and Lie algebra. Experimental results on 38 categories of logo images demonstrate the superior performance of the proposed method whose clustering rate can be achieved at 88.55%.
Applications of Algebraic Logic and Universal Algebra to Computer Science
1989-06-21
conference, with roughly equal representation from Mathematics and Computer Science . The conference consisted of eight invited lectures (60 minutes...each) and 26 contributed talks (20-40 minutes each). There was also a round-table discussion on the role of algebra and logic in computer science . Keywords
A Balancing Act: Making Sense of Algebra
ERIC Educational Resources Information Center
Gavin, M. Katherine; Sheffield, Linda Jensen
2015-01-01
For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…
Algebra? A Gate! A Barrier! A Mystery!
ERIC Educational Resources Information Center
Mathematics Educatio Dialogues, 2000
2000-01-01
This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…
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…
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…
Constraint-Referenced Analytics of Algebra Learning
ERIC Educational Resources Information Center
Sutherland, Scot M.; White, Tobin F.
2016-01-01
The development of the constraint-referenced analytics tool for monitoring algebra learning activities presented here came from the desire to firstly, take a more quantitative look at student responses in collaborative algebra activities, and secondly, to situate those activities in a more traditional introductory algebra setting focusing on…
Embedding Algebraic Thinking throughout the Mathematics Curriculum
ERIC Educational Resources Information Center
Vennebush, G. Patrick; Marquez, Elizabeth; Larsen, Joseph
2005-01-01
This article explores the algebra that can be uncovered in many middle-grades mathematics tasks that, on first inspection, do not appear to be algebraic. It shows connections to the other four Standards that occur in traditional algebra problems, and it offers strategies for modifying activities so that they can be used to foster algebraic…
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"…
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…
Teacher Actions to Facilitate Early Algebraic Reasoning
ERIC Educational Resources Information Center
Hunter, Jodie
2015-01-01
In recent years there has been an increased emphasis on integrating the teaching of arithmetic and algebra in primary school classrooms. This requires teachers to develop links between arithmetic and algebra and use pedagogical actions that facilitate algebraic reasoning. Drawing on findings from a classroom-based study, this paper provides an…
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…
Cyclic homology for Hom-associative algebras
NASA Astrophysics Data System (ADS)
Hassanzadeh, Mohammad; Shapiro, Ilya; Sütlü, Serkan
2015-12-01
In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.
A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation
Somma, Rolando D.
2016-06-01
In this paper, we present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for themore » quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.« less
A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation
NASA Astrophysics Data System (ADS)
Somma, Rolando D.
2016-06-01
We present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.
A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation
Somma, Rolando D.
2016-06-01
In this paper, we present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.
Lie, truth, lie: the role of task switching in a deception context.
Debey, Evelyne; Liefooghe, Baptist; De Houwer, Jan; Verschuere, Bruno
2015-05-01
A cornerstone of the task switching literature is the finding that task performance is typically slower and more error-prone when the task switches than when it repeats. So far, deception research has largely ignored that such cognitive switch costs should also emerge when switching between truth telling and lying, and may affect the cognitive cost of lying as reflected in higher prefrontal brain activity and slower and less accurate responding compared to truth telling. To get a grasp on the relative size of the switch costs associated with lying and truth telling, the current study had participants perform a reaction time-based deception task, in which they alternated between lying and telling the truth to yes/no questions that were related to activities performed in the lab (Experiment 1) or neutral autobiographical facts (Experiment 2). In both experiments, the error and reaction time switch costs were found to be equally large for switching from truth telling to lying and from lying to truth telling. This symmetry in switch costs can be explained from the hypothesis that lying requires a first step of truth telling, and demonstrates that task switching does not contribute to the cognitive cost of lying when the repetition/switch ratio is balanced. Theoretical and methodological implications are considered.
Why Do Lie-Catchers Fail? A Lens Model Meta-Analysis of Human Lie Judgments
ERIC Educational Resources Information Center
Hartwig, Maria; Bond, Charles F., Jr.
2011-01-01
Decades of research has shown that people are poor at detecting lies. Two explanations for this finding have been proposed. First, it has been suggested that lie detection is inaccurate because people rely on invalid cues when judging deception. Second, it has been suggested that lack of valid cues to deception limits accuracy. A series of 4…
Teaching the Truth about Lies to Psychology Students: The Speed Lying Task
ERIC Educational Resources Information Center
Pearson, Matthew R.; Richardson, Thomas A.
2013-01-01
To teach the importance of deception in everyday social life, an in-class activity called the "Speed Lying Task" was given in an introductory social psychology class. In class, two major research findings were replicated: Individuals detected deception at levels no better than expected by chance and lie detection confidence was unrelated…
NASA Astrophysics Data System (ADS)
Stoilova, N. I.; Van der Jeugt, J.
2016-04-01
A new, so called odd Gel’fand-Zetlin (GZ) basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra {gl}(n| n). The related GZ patterns are based upon the decomposition according to a particular chain of subalgebras of {gl}(n| n). This chain contains only genuine Lie superalgebras of type {gl}(k| l) with k and l nonzero (apart from the final element of the chain which is {gl}(1| 0)\\equiv {gl}(1)). Explicit expressions for a set of generators of the algebra on this GZ basis are determined. The results are extended to an explicit construction of a class of irreducible highest weight modules of the general linear Lie superalgebra {gl}(∞ | ∞ ).
NASA Astrophysics Data System (ADS)
Possieri, Corrado; Tornambè, Antonio
2015-05-01
The main goal of this paper is to compute a class of polynomial vector fields, whose associated dynamical system has a given affine variety as attractive and invariant set, a given point in such an affine variety as invariant and attractive and another given affine variety as invariant set, solving the application of this technique in the robotic area. This objective is reached by using some tools taken from algebraic geometry. Practical examples of how these vector fields can be computed are reported. Moreover, by using these techniques, two feedback control laws, respectively, for a unicycle-like mobile robot and for a car-like mobile robot, which make them move, within the workspace, approaching to a selected algebraic curve, are given.
Carry Groups: Abstract Algebra Projects
ERIC Educational Resources Information Center
Miller, Cheryl Chute; Madore, Blair F.
2004-01-01
Carry Groups are a wonderful collection of groups to introduce in an undergraduate Abstract Algebra course. These groups are straightforward to define but have interesting structures for students to discover. We describe these groups and give examples of in-class group projects that were developed and used by Miller.
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…
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…
Easing Students' Transition to Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2006-01-01
Traditionally, students learn arithmetic throughout their primary schooling, and this is seen as the ideal preparation for the learning of algebra in the junior secondary school. The four operations are taught and rehearsed in the early years and from this, it is assumed, "children will induce the fundamental structure of arithmetic" (Warren &…
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…
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.
Inequalities, Assessment and Computer Algebra
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary…
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…
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…
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.
Math Sense: Algebra and Geometry.
ERIC Educational Resources Information Center
Howett, Jerry
This book is designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four strands: (1) skill lessons…
Weaving Geometry and Algebra Together
ERIC Educational Resources Information Center
Cetner, Michelle
2015-01-01
When thinking about student reasoning and sense making, teachers must consider the nature of tasks given to students along with how to plan to use the tasks in the classroom. Students should be presented with tasks in a way that encourages them to draw connections between algebraic and geometric concepts. This article focuses on the idea that it…
Algebraic 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…
Teachers' Understanding of Algebraic Generalization
NASA Astrophysics Data System (ADS)
Hawthorne, Casey Wayne
Generalization has been identified as a cornerstone of algebraic thinking (e.g., Lee, 1996; Sfard, 1995) and is at the center of a rich conceptualization of K-8 algebra (Kaput, 2008; Smith, 2003). Moreover, mathematics teachers are being encouraged to use figural-pattern generalizing tasks as a basis of student-centered instruction, whereby teachers respond to and build upon the ideas that arise from students' explorations of these activities. Although more and more teachers are engaging their students in such generalizing tasks, little is known about teachers' understanding of generalization and their understanding of students' mathematical thinking in this domain. In this work, I addressed this gap, exploring the understanding of algebraic generalization of 4 exemplary 8th-grade teachers from multiple perspectives. A significant feature of this investigation is an examination of teachers' understanding of the generalization process, including the use of algebraic symbols. The research consisted of two phases. Phase I was an examination of the teachers' understandings of the underlying quantities and quantitative relationships represented by algebraic notation. In Phase II, I observed the instruction of 2 of these teachers. Using the lens of professional noticing of students' mathematical thinking, I explored the teachers' enacted knowledge of algebraic generalization, characterizing how it supported them to effectively respond to the needs and queries of their students. Results indicated that teachers predominantly see these figural patterns as enrichment activities, disconnected from course content. Furthermore, in my analysis, I identified conceptual difficulties teachers experienced when solving generalization tasks, in particular, connecting multiple symbolic representations with the quantities in the figures. Moreover, while the teachers strived to overcome the challenges of connecting different representations, they invoked both productive and unproductive
Explicit field realizations of W algebras
NASA Astrophysics Data System (ADS)
Wei, Shao-Wen; Liu, Yu-Xiao; Zhang, Li-Jie; Ren, Ji-Rong
2009-06-01
The fact that certain nonlinear W2,s algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize W2,s algebras from linear W1,2,s algebras. In this paper, we first construct the explicit field realizations of linear W1,2,s algebras with double scalar and double spinor, respectively. Then, after a change of basis, the realizations of W2,s algebras are presented. The results show that all these realizations are Romans-type realizations.
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.
On special classes of n-algebras
NASA Astrophysics Data System (ADS)
Vainerman, L.; Kerner, R.
1996-05-01
We define n-algebras as linear spaces on which the internal composition law involves n elements: m:V⊗n■V. It is known that such algebraic structures are interesting for their applications to problems of modern mathematical physics. Using the notion of a commutant of two subalgebras of an n-algebra, we distinguish certain classes of n-algebras with reasonable properties: semisimple, Abelian, nilpotent, solvable. We also consider a few examples of n-algebras of different types, and show their properties.
NASA Astrophysics Data System (ADS)
Hermann, Robert
1982-07-01
It is known that many of the Special Functions of mathematical physics appear as matrix elements of Lie group representations. This paper is concerned with a beginning attack on the converse problem, i.e., finding conditions that a given function be a matrix element. The methods used are based on a combination of ideas from system theory, functional analysis, Lie theory, differential algebra, and linear ordinary differential equation theory. A key idea is to attach a symbol as an element of a commutative algebra. In favorable cases, this symbol defines a Riemann surface, and a meromorphic differential form on that surface. The topological and analytical invariants attached to this form play a key role in system theory. The Lie algebras of the groups appear as linear differential operators on this Riemann surface. Finally, it is shown how the Picard-Vessiot-Infeld-Hull theory of factorization of linear differential operators leads to realization of many Special Functions as matrix representations of group representations.
NASA Astrophysics Data System (ADS)
Fernandez-Nunez, J.; Garcia-Fuertes, W.; Perelomov, A. M.
We re-express the quantum Calogero-Sutherland model for the Lie algebra $E_6$ and the particular value of the coupling constant $\\kappa=1$ by using the fundamental irreducible characters of the algebra as dynamical variables. For that, we need to develop a systematic procedure to obtain all the Clebsch-Gordan series required to perform the change of variables. We describe how the resulting quantum Hamiltonian operator can be used to compute more characters and Clebsch-Gordan series for this exceptional algebra.
Recursion and feedback in image algebra
NASA Astrophysics Data System (ADS)
Ritter, Gerhard X.; Davidson, Jennifer L.
1991-04-01
Recursion and feedback are two important processes in image processing. Image algebra, a unified algebraic structure developed for use in image processing and image analysis, provides a common mathematical environment for expressing image processing transforms. It is only recently that image algebra has been extended to include recursive operations [1]. Recently image algebra was shown to incorporate neural nets [2], including a new type of neural net, the morphological neural net [3]. This paper presents the relationship of the recursive image algebra to the field of fractions of the ring of matrices, and gives the two dimensional moving average filter as an example. Also, the popular multilayer perceptron with back propagation and a morphology neural network with learning rule are presented in image algebra notation. These examples show that image algebra can express these important feedback concepts in a succinct way.
Deformed Kac Moody and Virasoro algebras
NASA Astrophysics Data System (ADS)
Balachandran, A. P.; Queiroz, A. R.; Marques, A. M.; Teotonio-Sobrinho, P.
2007-07-01
Whenever the group {\\bb R}^n acts on an algebra {\\cal A} , there is a method to twist \\cal A to a new algebra {\\cal A}_\\theta which depends on an antisymmetric matrix θ (θμν = -θνμ = constant). The Groenewold-Moyal plane {\\cal A}_\\theta({\\bb R}^{d+1}) is an example of such a twisted algebra. We give a general construction to realize this twist in terms of {\\cal A} itself and certain 'charge' operators Qμ. For {\\cal A}_\\theta({\\bb R}^{d+1}), Q_\\mu are translation generators. This construction is then applied to twist the oscillators realizing the Kac-Moody (KM) algebra as well as the KM currents. They give different deformations of the KM algebra. From one of the deformations of the KM algebra, we construct, via the Sugawara construction, the Virasoro algebra. These deformations have an implication for statistics as well.
Algebraic complexities and algebraic curves over finite fields
Chudnovsky, D. V.; Chudnovsky, G. V.
1987-01-01
We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields. PMID:16593816
The Rogers-Ramanujan identities: Lie theoretic interpretation and proof
Lepowsky, James; Wilson, Robert Lee
1981-01-01
The two Rogers-Ramanujan identities, which equate certain infinite products with infinite sums, are among the most intriguing of the classical formal power series identitites. It has been found by Lepowsky and Milne that the product side of each of them differs by a certain factor from the principally specialized character of a certain standard module for the Euclidean Kac-Moody Lie algebra A1(1). On the other hand, the present authors have introduced an infinite-dimensional Heisenberg subalgebra [unk] of A1(1) which leads to a construction of A1(1) in terms of differential operators given by the homogeneous components of an “exponential generating function.” In the present announcement, we use [unk] to formulate a natural “abstract Rogers-Ramanujan identity” for an arbitrary standard A1(1)-module which turns out to coincide with the classical identities in the cases of the two corresponding standard modules. The abstract identity equates two expressions, one a product and the other a sum, for the principally specialized character of the space Ω of highest weight vectors or “vacuum states” for [unk] in the module. The construction of A1(1) leads to a concrete realization of Ω as the span of certain spaces of symmetric polynomials occurring as the homogeneous components of exponential generating functions. The summands in the Rogers-Ramanujan identities turn out to “count” the dimensions of these spaces. For general standard A1(1)-modules, we conjecture that the abstract identities agree with generalizations of the Rogers-Ramanujan identities due to Gordon, Andrews, and Bressoud. PMID:16592977
Preschoolers' Understanding of Lies and Innocent and Negligent Mistakes.
ERIC Educational Resources Information Center
Siegal, Michael; Peterson, Candida C.
1998-01-01
Examined preschoolers' ability to distinguish innocent and negligent mistakes from lies. Found that, when asked to identify a mistake or lie about a food's contact with contaminants and identify a bystander's reaction, children distinguished mistakes from lies; they could also discriminate between lies and both negligent mistakes that generate…
Spectacular science: the lie detector's ambivalent powers.
Bunn, Geoffrey C
2007-05-01
Spectacular science is a mode of scientific inquiry that is created and sustained by popular culture. In this article, I provide evidence for this claim by examining the history of the lie detector. Throughout the 20th century, the technology was nurtured by newspaper and magazine articles, movies, comic books, television shows, and advertisements. Analysis of this rich archive reveals the instrument to be, on the one hand, an automatically functioning machine, the epitome of science. But on the other hand, the lie detector is also a totemistic object that requires the skills of a charismatic magician to work at all. Nevertheless, the instrument was untroubled by such apparent contradictions, because it operated according to a spectacular mode of governance.
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.
Algebraic Methods to Design Signals
2015-08-27
group theory are employed to investigate the theory of their construction methods leading to new families of these arrays and some generalizations...sequences and arrays with desirable correlation properties. The methods used are very algebraic and number theoretic. Many new families of sequences...context of optical quantum computing, we prove that infinite families of anticirculant block weighing matrices can be obtained from generic weighing
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.
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.
Introduction to Image Algebra Ada
NASA Astrophysics Data System (ADS)
Wilson, Joseph N.
1991-07-01
Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.
Algebra: A Challenge at the Crossroads of Policy and Practice
ERIC Educational Resources Information Center
Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.
2011-01-01
The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…
NASA Astrophysics Data System (ADS)
Cai, Jinfa
2014-12-01
Drawing on evidence from the Longitudinal Investigation of the Effect of Curriculum on Algebra Learning (LieCal) Project, issues related to mathematics curriculum reform and student learning are discussed. The LieCal Project was designed to longitudinally investigate the impact of a reform mathematics curriculum called the Connected Mathematics Project (CMP) in the USA on teachers' teaching and students' learning. Using a three-level conceptualization of curriculum (intended, implemented, and attained), a variety of evidence from the LieCal Project is presented to show the impact of mathematics curriculum reform on teachers' teaching and students' learning. This paper synthesizes findings from the two longitudinal studies spanning 7 years of the LieCal Project both to show the kind of impact curriculum has on teachers' teaching and students' learning and to suggest powerful but feasible ways researchers can investigate curriculum effect on both teaching and learning.
The role of nonmetricity in metric-affine theories of gravity
NASA Astrophysics Data System (ADS)
Vitagliano, Vincenzo
2014-02-01
The intriguing choice to treat alternative theories of gravity by means of the Palatini approach, namely elevating the affine connection to the role of independent variable, contains the seed of some interesting (usually under-explored) generalizations of General Relativity, the metric-affine theories of gravity. The peculiar aspect of these theories is to provide a natural way for matter fields to be coupled to the independent connection through the covariant derivative built from the connection itself. Adopting a procedure borrowed from the effective field theory prescriptions, we study the dynamics of metric-affine theories of increasing order, that in the complete version include invariants built from curvature, nonmetricity and torsion. We show that even including terms obtained from nonmetricity and torsion to the second order density Lagrangian, the connection lacks dynamics and acts as an auxiliary field that can be algebraically eliminated, resulting in some extra interactions between metric and matter fields. Dedicated to the memory of Francesco Caracciolo
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
NASA Astrophysics Data System (ADS)
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
Chemical binding affinity estimation using MSB
NASA Astrophysics Data System (ADS)
Weaver, John B.; Rauwerdink, Adam M.
2011-03-01
Binding affinity can be estimated in several ways in the laboratory but there is no viable way to estimate binding affinity in vivo without assumptions on the number of binding sites. Magnetic spectroscopy of nanoparticle Brownian motion, MSB, measures the rotational Brownian motion. The MSB signal is affected by nanoparticle binding affinity so it provides a mechanism to measure the chemical binding affinity. We present a possible mechanism to quantify the binding affinity and test that mechanism using viscous solutions.
NASA Astrophysics Data System (ADS)
Abedi-Fardad, J.; Rezaei-Aghdam, A.; Haghighatdoost, Gh.
2017-01-01
We classify all four-dimensional real Lie bialgebras of symplectic type and obtain the classical r-matrices for these Lie bialgebras and Poisson structures on all the associated four-dimensional Poisson-Lie groups. We obtain some new integrable models where a Poisson-Lie group plays the role of the phase space and its dual Lie group plays the role of the symmetry group of the system.
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.
Affine Contractions on the Plane
ERIC Educational Resources Information Center
Celik, D.; Ozdemir, Y.; Ureyen, M.
2007-01-01
Contractions play a considerable role in the theory of fractals. However, it is not easy to find contractions which are not similitudes. In this study, it is shown by counter examples that an affine transformation of the plane carrying a given triangle onto another triangle may not be a contraction even if it contracts edges, heights or medians.…
Affinity-aware checkpoint restart
Saini, Ajay; Rezaei, Arash; Mueller, Frank; Hargrove, Paul; Roman, Eric
2014-12-08
Current checkpointing techniques employed to overcome faults for HPC applications result in inferior application performance after restart from a checkpoint for a number of applications. This is due to a lack of page and core affinity awareness of the checkpoint/restart (C/R) mechanism, i.e., application tasks originally pinned to cores may be restarted on different cores, and in case of non-uniform memory architectures (NUMA), quite common today, memory pages associated with tasks on a NUMA node may be associated with a different NUMA node after restart. Here, this work contributes a novel design technique for C/R mechanisms to preserve task-to-core maps and NUMA node specific page affinities across restarts. Experimental results with BLCR, a C/R mechanism, enhanced with affinity awareness demonstrate significant performance benefits of 37%-73% for the NAS Parallel Benchmark codes and 6-12% for NAMD with negligible overheads instead of up to nearly four times longer an execution times without affinity-aware restarts on 16 cores.
Affinity-aware checkpoint restart
Saini, Ajay; Rezaei, Arash; Mueller, Frank; ...
2014-12-08
Current checkpointing techniques employed to overcome faults for HPC applications result in inferior application performance after restart from a checkpoint for a number of applications. This is due to a lack of page and core affinity awareness of the checkpoint/restart (C/R) mechanism, i.e., application tasks originally pinned to cores may be restarted on different cores, and in case of non-uniform memory architectures (NUMA), quite common today, memory pages associated with tasks on a NUMA node may be associated with a different NUMA node after restart. Here, this work contributes a novel design technique for C/R mechanisms to preserve task-to-core mapsmore » and NUMA node specific page affinities across restarts. Experimental results with BLCR, a C/R mechanism, enhanced with affinity awareness demonstrate significant performance benefits of 37%-73% for the NAS Parallel Benchmark codes and 6-12% for NAMD with negligible overheads instead of up to nearly four times longer an execution times without affinity-aware restarts on 16 cores.« less
ELECTRON AFFINITIES OF INORGANIC RADICALS.
energy in the latter compound is 110 kcals/mole, distinctly higher than in ammonia. Cyanogen (CN)2 and hydrocyanic acid (HCN) yield values for the...ions very readily, and the electron affinity is 49 kcals/mole. A comparison with the results from thiocyanic acid (HNCS) indicates that the H-N bond
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.
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.
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…
ERIC Educational Resources Information Center
Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos
2016-01-01
This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…
Semiempirical Theories of the Affinities of Negative Atomic Ions
NASA Technical Reports Server (NTRS)
Edie, John W.
1961-01-01
The determination of the electron affinities of negative atomic ions by means of direct experimental investigation is limited. To supplement the meager experimental results, several semiempirical theories have been advanced. One commonly used technique involves extrapolating the electron affinities along the isoelectronic sequences, The most recent of these extrapolations Is studied by extending the method to Include one more member of the isoelectronic sequence, When the results show that this extension does not increase the accuracy of the calculations, several possible explanations for this situation are explored. A different approach to the problem is suggested by the regularities appearing in the electron affinities. Noting that the regular linear pattern that exists for the ionization potentials of the p electrons as a function of Z, repeats itself for different degrees of ionization q, the slopes and intercepts of these curves are extrapolated to the case of the negative Ion. The method is placed on a theoretical basis by calculating the Slater parameters as functions of q and n, the number of equivalent p-electrons. These functions are no more than quadratic in q and n. The electron affinities are calculated by extending the linear relations that exist for the neutral atoms and positive ions to the negative ions. The extrapolated. slopes are apparently correct, but the intercepts must be slightly altered to agree with experiment. For this purpose one or two experimental affinities (depending on the extrapolation method) are used in each of the two short periods. The two extrapolation methods used are: (A) an isoelectronic sequence extrapolation of the linear pattern as such; (B) the same extrapolation of a linearization of this pattern (configuration centers) combined with an extrapolation of the other terms of the ground configurations. The latter method Is preferable, since it requires only experimental point for each period. The results agree within
Theoretical proton affinity and fluoride affinity of nerve agent VX.
Bera, Narayan C; Maeda, Satoshi; Morokuma, Keiji; Viggiano, Al A
2010-12-23
Proton affinity and fluoride affinity of nerve agent VX at all of its possible sites were calculated at the RI-MP2/cc-pVTZ//B3LYP/6-31G* and RI-MP2/aug-cc-pVTZ//B3LYP/6-31+G* levels, respectively. The protonation leads to various unique structures, with H(+) attached to oxygen, nitrogen, and sulfur atoms; among which the nitrogen site possesses the highest proton affinity of -ΔE ∼ 251 kcal/mol, suggesting that this is likely to be the major product. In addition some H(2), CH(4) dissociation as well as destruction channels have been found, among which the CH(4) + [Et-O-P(═O)(Me)-S-(CH(2))(2)-N(+)(iPr)═CHMe] product and the destruction product forming Et-O-P(═O)(Me)-SMe + CH(2)═N(+)(iPr)(2) are only 9 kcal/mol less stable than the most stable N-protonated product. For fluoridization, the S-P destruction channel to give Et-O-P(═O)(Me)(F) + [S-(CH(2))(2)-N-(iPr)(2)](-) is energetically the most favorable, with a fluoride affinity of -ΔE ∼ 44 kcal. Various F(-) ion-molecule complexes are also found, with the one having F(-) interacting with two hydrogen atoms in different alkyl groups to be only 9 kcal/mol higher than the above destruction product. These results suggest VX behaves quite differently from surrogate systems.
Warren, Kelly L; Dodd, Elyse; Raynor, Graham; Peterson, Carole
2012-01-01
In this investigation, 514 university students judged whether children were telling the truth about highly emotional events. Eight children (half female, half 8-9 and the remainder 12-14 years old) had been injured seriously enough to require emergency room treatment and were interviewed a few days later. Each was yoked to three other children matched in age and gender who fabricated accounts under one of three conditions: lies that were unprepared, prepared (24 hours to prepare), and coached by parents. Participants were at chance when judging true accounts as well as unprepared and prepared lies. However, 74% of the coached lies were judged as true. Participants' confidence in their judgments, age, experience with children, and relevant coursework/training did not improve judgments.
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.
Trell, Erik
2014-12-10
Santilli’s revolutionary iso-, geno- and hypermathematics have provided the original straight line Lie groups and algebras with a span and coherence in all dimensions, and thus already at the infinitesimal level an extension in the Cartesian sense, allowing a continuous self-similar cyclical realization of matter from the elementary particle threshold level via the atomic to molecular and visible scale where it meets and marries with modern nanotechnology in the form of an isotropic vector matrix of space-filling octahedron-tetrahedron composition. This is distributed as an electron transition matrix with Bohr shell model stratified signature and is here directly outlining a new, centrally coordinated organic composition and chart of the periodic system as specifically exemplified by the noble gases.
Lie symmetry analysis and exact solutions of the quasigeostrophic two-layer problem
NASA Astrophysics Data System (ADS)
Bihlo, Alexander; Popovych, Roman O.
2011-03-01
The quasigeostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras, we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Wherever possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.
Lie symmetry analysis, conservation laws, solitary and periodic waves for a coupled Burger equation
NASA Astrophysics Data System (ADS)
Xu, Mei-Juan; Tian, Shou-Fu; Tu, Jian-Min; Zhang, Tian-Tian
2017-01-01
Under investigation in this paper is a generalized (2 + 1)-dimensional coupled Burger equation with variable coefficients, which describes lots of nonlinear physical phenomena in geophysical fluid dynamics, condense matter physics and lattice dynamics. By employing the Lie group method, the symmetry reductions and exact explicit solutions are obtained, respectively. Based on a direct method, the conservations laws of the equation are also derived. Furthermore, by virtue of the Painlevé analysis, we successfully obtain the integrable condition on the variable coefficients, which plays an important role in further studying the integrability of the equation. Finally, its auto-Bäcklund transformation as well as some new analytic solutions including solitary and periodic waves are also presented via algebraic and differential manipulation.
NASA Astrophysics Data System (ADS)
Trell, Erik
2014-12-01
Santilli's revolutionary iso-, geno- and hypermathematics have provided the original straight line Lie groups and algebras with a span and coherence in all dimensions, and thus already at the infinitesimal level an extension in the Cartesian sense, allowing a continuous self-similar cyclical realization of matter from the elementary particle threshold level via the atomic to molecular and visible scale where it meets and marries with modern nanotechnology in the form of an isotropic vector matrix of space-filling octahedron-tetrahedron composition. This is distributed as an electron transition matrix with Bohr shell model stratified signature and is here directly outlining a new, centrally coordinated organic composition and chart of the periodic system as specifically exemplified by the noble gases.
Ribbon tableaux, Hall{endash}Littlewood functions, quantum affine algebras, and unipotent varieties
Lascoux, A.; Leclerc, B.; Thibon, J.
1997-02-01
We introduce a new family of symmetric functions, which are q analogs of products of Schur functions, defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation scr(F){sub q} of U{sub q}(sl{sub n}), and are related to Hall{endash}Littlewood functions via the geometry of flag varieties. We present a series of conjectures, and prove them in special cases. The essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of scr(F){sub q} using ribbon tableaux. {copyright} {ital 1997 American Institute of Physics.}
Pants on fire: the electrophysiological signature of telling a lie.
Pfister, Roland; Foerster, Anna; Kunde, Wilfried
2014-01-01
Even though electroencephalography has played a prominent role for lie detection via personally relevant information, the electrophysiological signature of active lying is still elusive. We addressed this signature with two experiments in which participants helped a virtual police officer to locate a knife. Crucially, before this response, they announced whether they would lie or tell the truth about the knife's location. This design allowed us to study the signature of lie-telling in the absence of rare and personally significant oddball stimuli that are typically used for lie detection via electrophysiological markers, especially the P300 component. Our results indicate that active lying attenuated P300 amplitudes as well as N200 amplitudes for such non-oddball stimuli. These results support accounts that stress the high cognitive demand of lie-telling, including the need to suppress the truthful response and to generate a lie.
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…
Learning from Student Approaches to Algebraic Proofs
ERIC Educational Resources Information Center
D'Ambrosio, Beatriz S.; Kastberg, Signe E.; Viola dos Santos, Joao Ricardo
2010-01-01
Many mathematics teachers struggle to support their students' developing understanding of proof as an essential element in investigations of mathematics. The area of mathematics where the development of an understanding of proof is most challenging is algebra. In the case of algebraic proof, analysis of student written work on tasks that demand…
From operator algebras to superconformal field theory
Kawahigashi, Yasuyuki
2010-01-15
We survey operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory of Jones, and certain aspects of noncommutative geometry of Connes.
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)
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…
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)
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…
Algebraic Thinking through Koch Snowflake Constructions
ERIC Educational Resources Information Center
Ghosh, Jonaki B.
2016-01-01
Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…
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…
Using Students' Interests as Algebraic Models
ERIC Educational Resources Information Center
Whaley, Kenneth A.
2012-01-01
Fostering algebraic thinking is an important goal for middle-grades mathematics teachers. Developing mathematical reasoning requires that teachers cultivate students' habits of mind. Teachers develop students' understanding of algebra by engaging them in tasks that involve modeling and representation. This study was designed to investigate how…
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.
THE RADICAL OF A JORDAN ALGEBRA
McCrimmon, Kevin
1969-01-01
In this paper we define a Jacobson radical for Jordan algebras analogous to that for associative algebras and show that it enjoys many of the properties of the associative radical. We then relate the corresponding notion of “semisimplicity” to the previously defined notion of “nondegeneracy” (Jacobson, N., these Proceedings, 55, 243-251 (1966)). PMID:16591736
The operator algebra approach to quantum groups
Kustermans, Johan; Vaes, Stefaan
2000-01-01
A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory. PMID:10639116
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…
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…
Gary M. Klingler Algebra Teacher Assistance Packages
ERIC Educational Resources Information Center
Klingler, Gary
2005-01-01
Several packages designed by Elizabeth Marquez for mathematics teachers of grades 6-12, officially entitled the Teacher Assistance Package in Support of Better Algebra Assessment, is a series of resources developed to accompany ET's End-of-Course Algebra Assessment. It is designed to enhance teachers classroom assessment by providing examples of…
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…
Symbolic Notations and Students' Achievements in Algebra
ERIC Educational Resources Information Center
Peter, Ebiendele E.; Olaoye, Adetunji A.
2013-01-01
This study focuses on symbolic notations and its impact on students' achievement in Algebra. The main reason for this study rests on the observation from personal and professional experiences on students' increasing hatred for Algebra. One hundred and fifty (150) Senior Secondary School Students (SSS) from Ojo Local Education District, Ojo, Lagos,…
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)
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…
Practicing Algebraic Skills: A Conceptual Approach
ERIC Educational Resources Information Center
Friedlander, Alex; Arcavi, Abraham
2012-01-01
Traditionally, a considerable part of teaching and learning algebra has focused on routine practice and the application of rules, procedures, and techniques. Although today's computerized environments may have decreased the need to master algebraic skills, procedural competence is still a central component in any mathematical activity. However,…
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…
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…