Integrable N dimensional systems on the Hopf algebra and q deformations

International Nuclear Information System (INIS)

Lisitsyn, Ya.V.; Shapovalov, A.V.

2000-01-01

The class of integrable classic and quantum systems on the Hopf algebra, describing the n of interacting particles, is plotted. The general structure of the integrable Hamiltonian system for the Hopf algebra A(g) of the Lee simple algebra g is obtained, wherefrom it follows, that motion integrals depend on the linear combinations k of the phase space coordinates. The q-deformation standard procedure is carried out and the corresponding integrable system is obtained. The general scheme is illustrated by the examples of the sl(2), sl(3) and o(3, 1) algebras. The exact solution is achieved for the N-dimensional Hamiltonian system quantum analog on the Hopf algebra A (sl(2)) through the method of noncommutative integration of linear differential equations [ru

Generalized exclusion and Hopf algebras

International Nuclear Information System (INIS)

Yildiz, A

2002-01-01

We propose a generalized oscillator algebra at the roots of unity with generalized exclusion and we investigate the braided Hopf structure. We find that there are two solutions: these are the generalized exclusions of the bosonic and fermionic types. We also discuss the covariance properties of these oscillators

A nonlinear deformed su(2) algebra with a two-color quasitriangular Hopf structure

International Nuclear Information System (INIS)

Bonatsos, D.; Daskaloyannis, C.; Kolokotronis, P.; Ludu, A.; Quesne, C.

1997-01-01

Nonlinear deformations of the enveloping algebra of su(2), involving two arbitrary functions of J 0 and generalizing the Witten algebra, were introduced some time ago by Delbecq and Quesne. In the present paper, the problem of endowing some of them with a Hopf algebraic structure is addressed by studying in detail a specific example, referred to as scr(A) q + (1). This algebra is shown to possess two series of (N+1)-dimensional unitary irreducible representations, where N=0,1,2,hor-ellipsis. To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed by proceeding in two steps. In the first one, a variant and extension of the deforming functional technique is introduced: variant because a map between two deformed algebras, su q (2) and scr(A) q + (1), is considered instead of a map between a Lie algebra and a deformed one, and extension because use is made of a two-valued functional, whose inverse is singular. As a result, the Hopf structure of su q (2) is carried over to scr(A) q + (1), thereby endowing the latter with a double Hopf structure. In the second step, the definition of the coproduct, counit, antipode, and scr(R)-matrix is extended so that the double Hopf algebra is enlarged into a new algebraic structure. The latter is referred to as a two-color quasitriangular Hopf algebra because the corresponding scr(R)-matrix is a solution of the colored Yang endash Baxter equation, where the open-quotes colorclose quotes parameters take two discrete values associated with the two series of finite-dimensional representations. copyright 1997 American Institute of Physics

From racks to pointed Hopf algebras

OpenAIRE

Andruskiewitsch, Nicolás; Graña, Matı́as

2003-01-01

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces (CX, c^q), where C is the field of complex numbers, X is a rack and q is a 2-cocycle on X with values in C^*. Racks and cohomology of racks appeared also in the work of topologists. This...

Feynman graphs and related Hopf algebras

International Nuclear Information System (INIS)

Duchamp, G H E; Blasiak, P; Horzela, A; Penson, K A; Solomon, A I

2006-01-01

In a recent series of communications we have shown that the reordering problem of bosons leads to certain combinatorial structures. These structures may be associated with a certain graphical description. In this paper, we show that there is a Hopf Algebra structure associated with this problem which is, in a certain sense, unique

On left Hopf algebras within the framework of inhomogeneous quantum groups for particle algebras

Energy Technology Data Exchange (ETDEWEB)

Rodriguez-Romo, Suemi [Facultad de Estudios Superiores Cuautitlan, Universidad Nacional Autonoma de Mexico (Mexico)

2012-10-15

We deal with some matters needed to construct concrete left Hopf algebras for inhomogeneous quantum groups produced as noncommutative symmetries of fermionic and bosonic creation/annihilation operators. We find a map for the bidimensional fermionic case, produced as in Manin's [Quantum Groups and Non-commutative Hopf Geometry (CRM Univ. de Montreal, 1988)] seminal work, named preantipode that fulfills all the necessary requirements to be left but not right on the generators of the algebra. Due to the complexity and importance of the full task, we consider our result as an important step that will be extended in the near future.

Twisted Acceleration-Enlarged Newton-Hooke Hopf Algebras

International Nuclear Information System (INIS)

Daszkiewicz, M.

2010-01-01

Ten Abelian twist deformations of acceleration-enlarged Newton-Hooke Hopf algebra are considered. The corresponding quantum space-times are derived as well. It is demonstrated that their contraction limit τ → ∞ leads to the new twisted acceleration-enlarged Galilei spaces. (author)

Self-dual Hopf quivers

International Nuclear Information System (INIS)

Huang Hualin; Li Libin; Ye Yu

2004-07-01

We study pointed graded self-dual Hopf algebras with a help of the dual Gabriel theorem for pointed Hopf algebras. Quivers of such Hopf algebras are said to be self-dual. An explicit classification of self-dual Hopf quivers is obtained. We also prove that finite dimensional coradically graded pointed self-dual Hopf algebras are generated by group-like and skew-primitive elements as associative algebras. This partially justifies a conjecture of Andruskiewitsch and Schneider and may help to classify finite dimensional self-dual pointed Hopf algebras

Domain wall solitons and Hopf algebraic translational symmetries in noncommutative field theories

International Nuclear Information System (INIS)

Sasai, Yuya; Sasakura, Naoki

2008-01-01

Domain wall solitons are the simplest topological objects in field theories. The conventional translational symmetry in a field theory is the generator of a one-parameter family of domain wall solutions, and induces a massless moduli field which propagates along a domain wall. We study similar issues in braided noncommutative field theories possessing Hopf algebraic translational symmetries. As a concrete example, we discuss a domain wall soliton in the scalar φ 4 braided noncommutative field theory in Lie-algebraic noncommutative space-time, [x i ,x j ]=2iκε ijk x k (i,j,k=1,2,3), which has a Hopf algebraic translational symmetry. We first discuss the existence of a domain wall soliton in view of Derrick's theorem, and construct explicitly a one-parameter family of solutions in perturbation of the noncommutativity parameter κ. We then find the massless moduli field which propagates on the domain wall soliton. We further extend our analysis to the general Hopf algebraic translational symmetry

Generating loop graphs via Hopf algebra in quantum field theory

International Nuclear Information System (INIS)

Mestre, Angela; Oeckl, Robert

2006-01-01

We use the Hopf algebra structure of the time-ordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected n-point functions. The recursion proceeds by loop order and vertex number

Hopf structure and Green ansatz of deformed parastatistics algebras

Energy Technology Data Exchange (ETDEWEB)

Aneva, Boyka [Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, bld. Tsarigradsko chaussee 72, BG-1784 Sofia (Bulgaria); Popov, Todor [Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, bld. Tsarigradsko chaussee 72, BG-1784 Sofia (Bulgaria)

2005-07-22

Deformed parabose and parafermi algebras are revised and endowed with Hopf structure in a natural way. The noncocommutative coproduct allows for construction of parastatistics Fock-like representations, built out of the simplest deformed Bose and Fermi representations. The construction gives rise to quadratic algebras of deformed anomalous commutation relations which define the generalized Green ansatz.

A generalization of the deformed algebra of quantum group SU(2)q for Hopf algebra

International Nuclear Information System (INIS)

Ludu, A.; Gupta, R.K.

1992-12-01

A generalization of the deformation of Lie algebra of SU(2) group is established for the Hopf algebra, by modifying the J 3 component in all of its defining commutators. The modification is carried out in terms of a polynomial f, of J 3 and the q-deformation parameter, which contains the known q-deformation functionals as its particular cases. (author). 20 refs

Twisting products in Hopf algebras and the construction of the quantum double

International Nuclear Information System (INIS)

Ferrer Santos, W.R.

1992-04-01

Let H be a finite dimensional Hopf algebra and B an (H, H*)-comodule algebra. The purpose of this note is to present a construction in which the product of B is twisted by the given actions. The constructions of the smash product and of the Quantum Double appear as special cases. (author). 7 refs

Twist deformations leading to κ-Poincaré Hopf algebra and their application to physics

International Nuclear Information System (INIS)

Jurić, Tajron; Meljanac, Stjepan; Samsarov, Andjelo

2016-01-01

We consider two twist operators that lead to kappa-Poincaré Hopf algebra, the first being an Abelian one and the second corresponding to a light-like kappa-deformation of Poincaré algebra. The adventage of the second one is that it is expressed solely in terms of Poincaré generators. In contrast to this, the Abelian twist goes out of the boundaries of Poincaré algebra and runs into envelope of the general linear algebra. Some of the physical applications of these two different twist operators are considered. In particular, we use the Abelian twist to construct the statistics flip operator compatible with the action of deformed symmetry group. Furthermore, we use the light-like twist operator to define a star product and subsequently to formulate a free scalar field theory compatible with kappa-Poincaré Hopf algebra and appropriate for considering the interacting ϕ 4 scalar field model on kappa-deformed space. (paper)

Weak C* Hopf Symmetry

OpenAIRE

Rehren, K. -H.

1996-01-01

Weak C* Hopf algebras can act as global symmetries in low-dimensional quantum field theories, when braid group statistics prevents group symmetries. Possibilities to construct field algebras with weak C* Hopf symmetry from a given theory of local observables are discussed.

International Workshop "Groups, Rings, Lie and Hopf Algebras"

CERN Document Server

2003-01-01

The volume is almost entirely composed of the research and expository papers by the participants of the International Workshop "Groups, Rings, Lie and Hopf Algebras", which was held at the Memorial University of Newfoundland, St. John's, NF, Canada. All four areas from the title of the workshop are covered. In addition, some chapters touch upon the topics, which belong to two or more areas at the same time. Audience: The readership targeted includes researchers, graduate and senior undergraduate students in mathematics and its applications.

Hopf-algebraic renormalization of QED in the linear covariant gauge

Energy Technology Data Exchange (ETDEWEB)

Kißler, Henry, E-mail: kissler@physik.hu-berlin.de

2016-09-15

In the context of massless quantum electrodynamics (QED) with a linear covariant gauge fixing, the connection between the counterterm and the Hopf-algebraic approach to renormalization is examined. The coproduct formula of Green’s functions contains two invariant charges, which give rise to different renormalization group functions. All formulas are tested by explicit computations to third loop order. The possibility of a finite electron self-energy by fixing a generalized linear covariant gauge is discussed. An analysis of subdivergences leads to the conclusion that such a gauge only exists in quenched QED.

q-deformed conformal superalgebra and its Hopf subalgebras

International Nuclear Information System (INIS)

Dobrev, V.K.; Lukierski, J.; Sobczyk, J.; Tolstoy, V.N.

1992-07-01

We present in detail a Hopf superalgebra U q (su(2,2/2)) which is a q-deformation of the conformal superalgebra su(2,2/1). The superalgebra U q (su(2,2/1)) contains as a subalgebra a q-deformed super-Poincare algebra and as Hopf subalgebras two conjugate 16-generator q-deformed super-Weyl algebras, which are q-deformation of parabolic subalgebras of su(2,2/1). We use several (anti-) involutions, including the standard Cartan involution and a *-antiinvolution under which the super-Weyl algebras are *-subalgebras of U q (su(2,2/1)). The q-deformed Lorentz algebra is Hopf subalgebra of both Weyl algebras and is preserved by all (anti-) involutions considered. (author). 26 refs

The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre

Energy Technology Data Exchange (ETDEWEB)

Farsad, Vanda

2017-06-14

This thesis is concerned with ''N pairs of symplectic fermions'' which are examples of logarithmic conformal field theories in two dimensions. The mathematical language of two-dimensional conformal field theories (on Riemannian surfaces of genus zero) are vertex operator algebras. The representation category of the even part of the symplectic fermion vertex operator super-algebra Rep V{sub ev} is conjecturally a factorisable finite ribbon tensor category. This determines an isomorphism of projective representations between two SL(2,Z)-actions associated to V{sub ev}. The first action is obtained by modular transformations on the space of so-called pseudo-trace functions of a vertex operator algebra. For V{sub ev} this was developed by A.M.Gaberdiel and I. Runkel. For the action one uses that Rep V{sub ev} is conjecturally a factorisable finite ribbon tensor category and thus carries a projective SL(2,Z)-action on a certain Hom-space [Ly1,Ly2,KL]. To do so we calculate the SL(2,Z)-action on the representation category of a general factorisable quasi-Hopf algebras. Then we show that Rep V{sub ev} is conjecturally ribbon equivalent to Rep Q, for Q a factorisable quasi-Hopf algebra, and calculate the SL(2,Z)-action explicitly on Rep Q. The result is that the two SL(2,Z)-action indeed agree. This poses the first example of such comparison for logarithmic conformal field theories.

The Epstein-Glaser approach to perturbative quantum field theory: graphs and Hopf algebras

International Nuclear Information System (INIS)

Lange, Alexander

2005-01-01

The paper aims at investigating perturbative quantum field theory in the approach of Epstein and Glaser (EG) and, in particular, its formulation in the language of graphs and Hopf algebras (HAs). Various HAs are encountered, each one associated with a special combination of physical concepts such as normalization, localization, pseudounitarity, causal regularization, and renormalization. The algebraic structures, representing the perturbative expansion of the S-matrix, are imposed on operator-valued distributions equipped with appropriate graph indices. Translation invariance ensures the algebras to be analytically well defined and graded total symmetry allows to formulate bialgebras. The algebraic results are given embedded in the corresponding physical framework, covering the two EG versions by Fredenhagen and Scharf that differ with respect to the concrete recursive implementation of causality. Besides, the ultraviolet divergences occurring in Feynman's representation are mathematically reasoned. As a final result, the change of the renormalization scheme in the context of EG is modeled via a HA and interpreted as the EG analog of Kreimer's HA

Quasi Hopf quantum symmetry in quantum theory

International Nuclear Information System (INIS)

Mack, G.; Schomerus, V.

1991-05-01

In quantum theory, internal symmetries more general than groups are possible. We show that quasitriangular quasi Hopf algebras G * as introduced by Drinfeld permit a consistent formulation of a transformation law of states in the physical Hilbert space H, of invariance of the ground state, and of a transformation law of field operators which is consistent with local braid relations of field operators as proposed by Froehlich. All this remains true when Drinfelds axioms are suitably weakened in order to build in truncated tensor products. Conversely, all the axioms of a weak quasitriangular quasi Hopf algebra are motivated from what physics demands of a symmetry. Unitarity requires in addition that G * admits a * -operation with certain properties. Invariance properties of Greens functions follow from invariance of the ground state and covariance of field operators as usual. Covariant adjoints and covariant products of field operators can be defined. The R-matrix elements in the local braid relations are in general operators in H. They are determined by the symmetry up to a phase factor. Quantum group algebras like U q (sl 2 ) with vertical strokeqvertical stroke=1 are examples of symmetries with special properties. We show that a weak quasitriangular quasi Hopf algebra G * is canonically associated with U q (sl 2 ) if q P =-1. We argue that these weak quasi Hopf algebras are the true symmetries of minimal conformal models. Their dual algebras G ('functions on the group') are neither commutative nor associative. (orig.)

Renormalization in quantum field theory and the Riemann-Hilbert problem. I. Hopf algebra structure of graphs and the main theorem

International Nuclear Information System (INIS)

Connes, A.; Kreimer, D.

2000-01-01

This paper gives a complete selfcontained proof of our result (1999) showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra H which is commutative asan algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra G whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of H. We show then that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop γ(z) element of G, z element of C, where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ + of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. (orig.)

Normal forms of Hopf-zero singularity

International Nuclear Information System (INIS)

Gazor, Majid; Mokhtari, Fahimeh

2015-01-01

The Lie algebra generated by Hopf-zero classical normal forms is decomposed into two versal Lie subalgebras. Some dynamical properties for each subalgebra are described; one is the set of all volume-preserving conservative systems while the other is the maximal Lie algebra of nonconservative systems. This introduces a unique conservative–nonconservative decomposition for the normal form systems. There exists a Lie-subalgebra that is Lie-isomorphic to a large family of vector fields with Bogdanov–Takens singularity. This gives rise to a conclusion that the local dynamics of formal Hopf-zero singularities is well-understood by the study of Bogdanov–Takens singularities. Despite this, the normal form computations of Bogdanov–Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic nonzero condition, complete results on the simplest Hopf-zero normal forms are obtained in terms of the conservative–nonconservative decomposition. Some practical formulas are derived and the results implemented using Maple. The method has been applied on the Rössler and Kuramoto–Sivashinsky equations to demonstrate the applicability of our results. (paper)

Normal forms of Hopf-zero singularity

Science.gov (United States)

Gazor, Majid; Mokhtari, Fahimeh

2015-01-01

The Lie algebra generated by Hopf-zero classical normal forms is decomposed into two versal Lie subalgebras. Some dynamical properties for each subalgebra are described; one is the set of all volume-preserving conservative systems while the other is the maximal Lie algebra of nonconservative systems. This introduces a unique conservative-nonconservative decomposition for the normal form systems. There exists a Lie-subalgebra that is Lie-isomorphic to a large family of vector fields with Bogdanov-Takens singularity. This gives rise to a conclusion that the local dynamics of formal Hopf-zero singularities is well-understood by the study of Bogdanov-Takens singularities. Despite this, the normal form computations of Bogdanov-Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic nonzero condition, complete results on the simplest Hopf-zero normal forms are obtained in terms of the conservative-nonconservative decomposition. Some practical formulas are derived and the results implemented using Maple. The method has been applied on the Rössler and Kuramoto-Sivashinsky equations to demonstrate the applicability of our results.

Spacetime from locality of interactions in deformations of special relativity: The example of κ -Poincaré Hopf algebra

Science.gov (United States)

Carmona, J. M.; Cortés, J. L.; Relancio, J. J.

2018-03-01

A new proposal for the notion of spacetime in a relativistic generalization of special relativity based on a modification of the composition law of momenta is presented. Locality of interactions is the principle which defines the spacetime structure for a system of particles. The formulation based on κ -Poincaré Hopf algebra is shown to be contained in this framework as a particular example.

Graded associative conformal algebras of finite type

OpenAIRE

Kolesnikov, Pavel

2011-01-01

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group $\\Gamma $ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group $G$ such that the identity component $G^0$ is the affine line and $G/G^0\\simeq \\Gamma $. A classification of simple...

On the algebraic structure of differential calculus on quantum groups

International Nuclear Information System (INIS)

Rad'ko, O.V.; Vladimirov, A.A.

1997-01-01

Intrinsic Hopf algebra structure of the Woronowicz differential complex is shown to generate quite naturally a bicovariant algebra of four basic objects within a differential calculus on quantum groups - coordinate functions, differential forms, Lie derivatives, and inner derivatives - as the cross-product algebra of two mutually dual graded Hopf algebras. This construction, properly taking into account Hopf-algebraic properties of Woronowicz's bicovariant calculus, provides a direct proof of the Cartan identity and of many other useful relations. A detailed comparison with other approaches is also given

Relative Hom-Hopf modules and total integrals

International Nuclear Information System (INIS)

Guo, Shuangjian; Zhang, Xiaohui; Wang, Shengxiang

2015-01-01

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) a right (H, α)-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of (A, β) in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral ϕ : (H, α) → (A, β) if and only if any representation of the pair (H, A) is injective in a functorial way, as a corepresentation of (H, α), which generalizes Doi’s result. Finally, we define a total quantum integral γ : H → Hom(H, A) and prove the following affineness criterion: if there exists a total quantum integral γ and the canonical map ψ : A⊗ B A → A ⊗ H, a⊗ B b ↦ β −1 (a) b [0] ⊗ α(b [1] ) is surjective, then the induction functor A⊗ B −:ℋ ~ (ℳ k ) B →ℋ ~ (ℳ k ) A H is an equivalence of categories

q-deformations of noncompact Lie (super-) algebras: The examples of q-deformed Lorentz, Weyl, Poincare' and (super-) conformal algebras

International Nuclear Information System (INIS)

Dobrev, V.K.

1992-01-01

We review and explain a canonical procedure for the q-deformation of the real forms G of complex Lie (super-) algebras associated with (generalized) Cartan matrices. Our procedure gives different q-deformations for the non-conjugate Cartan subalgebras of G. We give several in detail the q-deformed Lorentz and conformal (super-) algebras. The q-deformed conformal algebra contains as a subalgebra a q-deformed Poincare algebra and as Hopf subalgebras two conjugate 11-generator q-deformed Weyl algebras. The q-deformed Lorentz algebra in Hopf subalgebra of both Weyl algebras. (author). 24 refs

Recoupling Lie algebra and universal ω-algebra

International Nuclear Information System (INIS)

Joyce, William P.

2004-01-01

We formulate the algebraic version of recoupling theory suitable for commutation quantization over any gradation. This gives a generalization of graded Lie algebra. Underlying this is the new notion of an ω-algebra defined in this paper. ω-algebra is a generalization of algebra that goes beyond nonassociativity. We construct the universal enveloping ω-algebra of recoupling Lie algebras and prove a generalized Poincare-Birkhoff-Witt theorem. As an example we consider the algebras over an arbitrary recoupling of Z n graded Heisenberg Lie algebra. Finally we uncover the usual coalgebra structure of a universal envelope and substantiate its Hopf structure

BRST-operator for quantum Lie algebra and differential calculus on quantum groups

International Nuclear Information System (INIS)

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

2001-01-01

For A Hopf algebra one determined structure of differential complex in two dual external Hopf algebras: A external expansion and in A* dual algebra external expansion. The Heisenberg double of these two Hopf algebras governs the differential algebra for the Cartan differential calculus on A algebra. The forst differential complex is the analog of the de Rame complex. The second complex coincide with the standard complex. Differential is realized as (anti)commutator with Q BRST-operator. Paper contains recursion relation that determines unequivocally Q operator. For U q (gl(N)) Lie quantum algebra one constructed BRST- and anti-BRST-operators and formulated the theorem of the Hodge expansion [ru

Relative Hom-Hopf modules and total integrals

Energy Technology Data Exchange (ETDEWEB)

Guo, Shuangjian [School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025 (China); Zhang, Xiaohui [Department of Mathematics, Southeast University, Nanjing 210096 (China); Wang, Shengxiang, E-mail: wangsx-math@163.com [School of Mathematics and Finance, Chuzhou University, Chuzhou 239000 (China); Department of Mathematics, Nanjing University, Nanjing 210093 (China)

2015-02-15

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) a right (H, α)-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of (A, β) in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral ϕ : (H, α) → (A, β) if and only if any representation of the pair (H, A) is injective in a functorial way, as a corepresentation of (H, α), which generalizes Doi’s result. Finally, we define a total quantum integral γ : H → Hom(H, A) and prove the following affineness criterion: if there exists a total quantum integral γ and the canonical map ψ : A⊗{sub B}A → A ⊗ H, a⊗{sub B}b ↦ β{sup −1}(a) b{sub [0]} ⊗ α(b{sub [1]}) is surjective, then the induction functor A⊗{sub B}−:ℋ{sup ~}(ℳ{sub k}){sub B}→ℋ{sup ~}(ℳ{sub k}){sub A}{sup H} is an equivalence of categories.

The geometric Hopf invariant and surgery theory

CERN Document Server

Crabb, Michael

2017-01-01

Written by leading experts in the field, this monograph provides homotopy theoretic foundations for surgery theory on higher-dimensional manifolds. Presenting classical ideas in a modern framework, the authors carefully highlight how their results relate to (and generalize) existing results in the literature. The central result of the book expresses algebraic surgery theory in terms of the geometric Hopf invariant, a construction in stable homotopy theory which captures the double points of immersions. Many illustrative examples and applications of the abstract results are included in the book, making it of wide interest to topologists. Serving as a valuable reference, this work is aimed at graduate students and researchers interested in understanding how the algebraic and geometric topology fit together in the surgery theory of manifolds. It is the only book providing such a wide-ranging historical approach to the Hopf invariant, double points and surgery theory, with many results old and new. .

Quantum deformation of the affine transformation algebra

International Nuclear Information System (INIS)

Aizawa, N.; Sato, Haru-Tada

1994-01-01

We discuss a quantum deformation of the affine transformation algebra in one-dimensional space. It is shown that the quantum algebra has a non-cocommutative Hopf algebra structure, simple realizations and quantum tensor operators. (orig.)

Quantum Groupoids Acting on Semiprime Algebras

Directory of Open Access Journals (Sweden)

Inês Borges

2011-01-01

Full Text Available Following Linchenko and Montgomery's arguments we show that the smash product of an involutive weak Hopf algebra and a semiprime module algebra, satisfying a polynomial identity, is semiprime.

q-deformed Poincare algebra

International Nuclear Information System (INIS)

Ogievetsky, O.; Schmidke, W.B.; Wess, J.; Muenchen Univ.; Zumino, B.; Lawrence Berkeley Lab., CA

1992-01-01

The q-differential calculus for the q-Minkowski space is developed. The algebra of the q-derivatives with the q-Lorentz generators is found giving the q-deformation of the Poincare algebra. The reality structure of the q-Poincare algebra is given. The reality structure of the q-differentials is also found. The real Laplaacian is constructed. Finally the comultiplication, counit and antipode for the q-Poincare algebra are obtained making it a Hopf algebra. (orig.)

Bicrossproduct structure and graded contractions of deformed algebras

International Nuclear Information System (INIS)

Perez Bueno, J.C.

1997-01-01

It is shown that all members in the family of deformed Hopf algebras corresponding to the graded contractions of the inhomogeneous algebras iso(p,q), p + q = N, have a bicrossproduct structure. (author)

Bicovariant quantum algebras and quantum Lie algebras

International Nuclear Information System (INIS)

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

1993-01-01

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun(G q ) to U q g, given by elements of the pure braid group. These operators - the 'reflection matrix' Y= triple bond 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). (orig.)

Introduction to quantum algebras

International Nuclear Information System (INIS)

Kibler, M.R.

1992-09-01

The concept of a quantum algebra is made easy through the investigation of the prototype algebras u qp (2), su q (2) and u qp (1,1). The latter quantum algebras are introduced as deformations of the corresponding Lie algebras; this is achieved in a simple way by means of qp-bosons. The Hopf algebraic structure of u qp (2) is also discussed. The basic ingredients for the representation theory of u qp (2) are given. Finally, in connection with the quantum algebra u qp (2), the qp-analogues of the harmonic oscillator are discussed and of the (spherical and hyperbolical) angular momenta. (author) 50 refs

Twisted classical Poincare algebras

International Nuclear Information System (INIS)

Lukierski, J.; Ruegg, H.; Tolstoy, V.N.; Nowicki, A.

1993-11-01

We consider the twisting of Hopf structure for classical enveloping algebra U(g), where g is the inhomogeneous rotations algebra, with explicite formulae given for D=4 Poincare algebra (g=P 4 ). The comultiplications of twisted U F (P 4 ) are obtained by conjugating primitive classical coproducts by F element of U(c)xU(c), where c denotes any Abelian subalgebra of P 4 , and the universal R-matrices for U F (P 4 ) are triangular. As an example we show that the quantum deformation of Poincare algebra recently proposed by Chaichian and Demiczev is a twisted classical Poincare algebra. The interpretation of twisted Poincare algebra as describing relativistic symmetries with clustered 2-particle states is proposed. (orig.)

Some remarks about monoidal Hom-algebras | Alonso Álvarez ...

African Journals Online (AJOL)

Abstract. In this paper, for C a strict braided monoidal category with tensor product , we improve the denition of Hom-associative algebra by removing the multiplicativity condition of the automorphism . After that we state the close connection between the classical notions of (co)algebra, (co)module and Hopf algebra and the ...

Sugawara construction and the q-deformation of Virasoro (super)algebra

Energy Technology Data Exchange (ETDEWEB)

Chaichian, M. (Theory Div., CERN, Geneva (Switzerland)); Presnajder, P. (Dept. of Theoretical Physics, Comenius Univ., Bratislava (Czechoslovakia))

1992-02-27

The q-deformed Virasoro algebra is obtained using the bosonic annihilation and creation operators of the q-deformed infinite Heisenberg algebra H({infinity}){sub q}, which has the Hopf structure. The generators of the q-deformed Virasoro algebra are expressed as a Sugawara construction in terms of normal ordered binomials in these annihilation and creation operators and become double indexed as the reminiscence of a degeneracy removal. The obtained q-deformed Virasoro algebra with central extension reduces to the standard one in the non-deformed limit and in special representations (but not in general) possesses a simple (cocommutative) Hopf structure (not related to the one in H({infinity}){sub q}). The fermionic annihilation and creation operators corresponding to the q-deformed infinite Heisenberg superalgebra s-H({infinity}){sub q} necessary for a similar construction of the q-deformed Virasoro superalgebra are presented. (orig.).

Hopf Bifurcation Analysis for a Stochastic Discrete-Time Hyperchaotic System

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Jie Ran

2015-01-01

Full Text Available The dynamics of a discrete-time hyperchaotic system and the amplitude control of Hopf bifurcation for a stochastic discrete-time hyperchaotic system are investigated in this paper. Numerical simulations are presented to exhibit the complex dynamical behaviors in the discrete-time hyperchaotic system. Furthermore, the stochastic discrete-time hyperchaotic system with random parameters is transformed into its equivalent deterministic system with the orthogonal polynomial theory of discrete random function. In addition, the dynamical features of the discrete-time hyperchaotic system with random disturbances are obtained through its equivalent deterministic system. By using the Hopf bifurcation conditions of the deterministic discrete-time system, the specific conditions for the existence of Hopf bifurcation in the equivalent deterministic system are derived. And the amplitude control with random intensity is discussed in detail. Finally, the feasibility of the control method is demonstrated by numerical simulations.

Realizations of κ-Minkowski space, Drinfeld twists, and related symmetry algebras

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Juric, Tajron; Meljanac, Stjepan; Pikutic, Danijel [Ruder Boskovic Institute, Theoretical Physics Division, Zagreb (Croatia)

2015-11-15

Realizations of κ-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of the gl(n) generators. There are three one-parameter families of linear realizations for timelike and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between a deformed Heisenberg algebra, the star product, the coproduct of momenta, and the twist operator is presented. It is proved that for each linear realization there exists a Drinfeld twist satisfying normalization and cocycle conditions. κ-Deformed igl(n)-Hopf algebras are presented for all cases. The κ-Poincare-Weyl and κ-Poincare-Hopf algebras are discussed. The left-right dual κ-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All Drinfeld twists related to κ-Minkowski space are obtained from our construction. Finally, some physical applications are discussed. (orig.)

Realizations of κ-Minkowski space, Drinfeld twists, and related symmetry algebras

International Nuclear Information System (INIS)

Juric, Tajron; Meljanac, Stjepan; Pikutic, Danijel

2015-01-01

Realizations of κ-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of the gl(n) generators. There are three one-parameter families of linear realizations for timelike and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between a deformed Heisenberg algebra, the star product, the coproduct of momenta, and the twist operator is presented. It is proved that for each linear realization there exists a Drinfeld twist satisfying normalization and cocycle conditions. κ-Deformed igl(n)-Hopf algebras are presented for all cases. The κ-Poincare-Weyl and κ-Poincare-Hopf algebras are discussed. The left-right dual κ-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All Drinfeld twists related to κ-Minkowski space are obtained from our construction. Finally, some physical applications are discussed. (orig.)

Si'lnikov chaos and Hopf bifurcation analysis of Rucklidge system

International Nuclear Information System (INIS)

Wang Xia

2009-01-01

A three-dimensional autonomous system - the Rucklidge system is considered. By the analytical method, Hopf bifurcation of Rucklidge system may occur when choosing an appropriate bifurcation parameter. Using the undetermined coefficient method, the existence of heteroclinic and homoclinic orbits in the Rucklidge system is proved, and the explicit and uniformly convergent algebraic expressions of Si'lnikov type orbits are given. As a result, the Si'lnikov criterion guarantees that there exists the Smale horseshoe chaos motion for the Rucklidge system.

Fixed point algebras for easy quantum groups

DEFF Research Database (Denmark)

Gabriel, Olivier; Weber, Moritz

2016-01-01

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author,we prove...... that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S+ n,the free orthogonal quantum group O+ n and the quantum reflection groups Hs+ n. Our fixed point......-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups,which are related to Hopf-Galois extensions....

Volume-preserving normal forms of Hopf-zero singularity

International Nuclear Information System (INIS)

Gazor, Majid; Mokhtari, Fahimeh

2013-01-01

A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a nonzero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any nondegenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified Rössler and generalized Kuramoto–Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple. (paper)

Volume-preserving normal forms of Hopf-zero singularity

Science.gov (United States)

Gazor, Majid; Mokhtari, Fahimeh

2013-10-01

A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a nonzero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any nondegenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified Rössler and generalized Kuramoto-Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple.

Algebra, Geometry and Mathematical Physics Conference

CERN Document Server

Paal, Eugen; Silvestrov, Sergei; Stolin, Alexander

2014-01-01

This book collects the proceedings of the Algebra, Geometry and Mathematical Physics Conference, held at the University of Haute Alsace, France, October 2011. Organized in the four areas of algebra, geometry, dynamical symmetries and conservation laws and mathematical physics and applications, the book covers deformation theory and quantization; Hom-algebras and n-ary algebraic structures; Hopf algebra, integrable systems and related math structures; jet theory and Weil bundles; Lie theory and applications; non-commutative and Lie algebra and more. The papers explore the interplay between research in contemporary mathematics and physics concerned with generalizations of the main structures of Lie theory aimed at quantization, and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, non-commutative geometry and applications in physics and beyond. The book benefits a broad audience of researchers a...

Hopf Bifurcation of Compound Stochastic van der Pol System

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Shaojuan Ma

2016-01-01

Full Text Available Hopf bifurcation analysis for compound stochastic van der Pol system with a bound random parameter and Gaussian white noise is investigated in this paper. By the Karhunen-Loeve (K-L expansion and the orthogonal polynomial approximation, the equivalent deterministic van der Pol system can be deduced. Based on the bifurcation theory of nonlinear deterministic system, the critical value of bifurcation parameter is obtained and the influence of random strength δ and noise intensity σ on stochastic Hopf bifurcation in compound stochastic system is discussed. At last we found that increased δ can relocate the critical value of bifurcation parameter forward while increased σ makes it backward and the influence of δ is more sensitive than σ. The results are verified by numerical simulations.

On the Hopf structure of Up,q(gl(1/1)) and the universal Τ-matrix of Funp,q(GL(1/1))

International Nuclear Information System (INIS)

Chakrabarti, R.; Jagannathan, R.

1994-08-01

Using the technique developed by Fronsdal and Galindo (Lett. Math. Phys, 27 (1993) 57) for studying the Hopf duality between the quantum algebras Fun p,q (GL(2)) and U p,q (gl(2)), the Hopf structure of U p,q (gl(1/1)), dual to Fun p,q (GL(1/1)), is derived and the corresponding universal Τ-matrix of Fun p,q (GL(1/1)), embodying the suitably modified exponential relationship U p,q (gl(1/1)) → Fun p,q (GL(1/1)), is obtained. (author). 10 refs

On the Computation of Degenerate Hopf Bifurcations for n-Dimensional Multiparameter Vector Fields

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Michail P. Markakis

2016-01-01

Full Text Available The restriction of an n-dimensional nonlinear parametric system on the center manifold is treated via a new proper symbolic form and analytical expressions of the involved quantities are obtained as functions of the parameters by lengthy algebraic manipulations combined with computer assisted calculations. Normal forms regarding degenerate Hopf bifurcations up to codimension 3, as well as the corresponding Lyapunov coefficients and bifurcation portraits, can be easily computed for any system under consideration.

Hopf solitons in the AFZ model

International Nuclear Information System (INIS)

Gillard, Mike

2011-01-01

The Aratyn–Ferreira–Zimerman (AFZ) model is a conformal field theory in three-dimensional space. It has solutions that are topological solitons classified by an integer-valued Hopf index. There exist infinitely many axial solutions which have been found analytically. Static axial, knot and linked solitons are found numerically using a modified volume preserving flow for Hopf index one to eight, allowing for comparison with other Hopf soliton models. Solutions include a static trefoil knot at Hopf index five. A one-parameter family of conformal Skyrme–Faddeev models, consisting of linear combinations of the Nicole and AFZ models, are also investigated numerically. The transition of solutions for Hopf index four is mapped across these models. A topological change between linked and axial solutions occurs, with fewer models (or a limited range of parameter values) permitting axial solitons than linked solitons at Hopf index four

From Quantum Mechanics to Quantum Field Theory: The Hopf route

Energy Technology Data Exchange (ETDEWEB)

Solomon, A I [Physics and Astronomy Department, Open University, Milton Keynes MK7 6AA (United Kingdom); Duchamp, G H E [Institut Galilee, LIPN, CNRS UMR 7030 99 Av. J.-B. Clement, F-93430 Villetaneuse (France); Blasiak, P; Horzela, A [H. Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Division of Theoretical Physics, ul. Eliasza-Radzikowskiego 152, PL 31-342 Krakow (Poland); Penson, K A, E-mail: a.i.solomon@open.ac.uk, E-mail: gduchamp2@free.fr, E-mail: pawel.blasiak@ifj.edu.pl, E-mail: andrzej.horzela@ifj.edu.pl, E-mail: penson@lptl.jussieu.fr [Lab.de Phys.Theor. de la Matiere Condensee, University of Paris VI (France)

2011-03-01

We show that the combinatorial numbers known as Bell numbers are generic in quantum physics. This is because they arise in the procedure known as Normal ordering of bosons, a procedure which is involved in the evaluation of quantum functions such as the canonical partition function of quantum statistical physics, inter alia. In fact, we shall show that an evaluation of the non-interacting partition function for a single boson system is identical to integrating the exponential generating function of the Bell numbers, which is a device for encapsulating a combinatorial sequence in a single function. We then introduce a remarkable equality, the Dobinski relation, and use it to indicate why renormalisation is necessary in even the simplest of perturbation expansions for a partition function. Finally we introduce a global algebraic description of this simple model, giving a Hopf algebra, which provides a starting point for extensions to more complex physical systems.

Twisted vertex algebras, bicharacter construction and boson-fermion correspondences

International Nuclear Information System (INIS)

Anguelova, Iana I.

2013-01-01

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

Wiener-Hopf factorization of piecewise meromorphic matrix-valued functions

International Nuclear Information System (INIS)

Adukov, Victor M

2009-01-01

Let D + be a multiply connected domain bounded by a contour Γ, let D - be the complement of D + union Γ in C-bar=C union {∞}, and a(t) be a continuous invertible matrix-valued function on Γ which can be meromorphically extended into the open disconnected set D - (as a piecewise meromorphic matrix-valued function). An explicit solution of the Wiener-Hopf factorization problem for a(t) is obtained and the partial factorization indices of a(t) are calculated. Here an explicit solution of a factorization problem is meant in the sense of reducing it to the investigation of finitely many systems of linear algebraic equations with matrices expressed in closed form, that is, in quadratures. Bibliography: 15 titles.

NUMERICAL HOPF BIFURCATION OF DELAY-DIFFERENTIAL EQUATIONS

Institute of Scientific and Technical Information of China (English)

无

2006-01-01

In this paper we consider the numerical solution of some delay differential equations undergoing a Hopf bifurcation. We prove that if the delay differential equations have a Hopf bifurcation point atλ=λ*, then the numerical solution of the equation also has a Hopf bifurcation point atλh =λ* + O(h).

Hopf Structures on Standard Young Tableaux

International Nuclear Information System (INIS)

Loday, Jean-Louis; Popov, Todor

2010-01-01

We review the Poirier-Reutenauer Hopf structure on Standard Young Tableaux and show that it is a distinguished member of a family of Hopf structures. The family in question is related to deformed parastatistics.

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

Directory of Open Access Journals (Sweden)

Ion C. Baianu

2009-04-01

Full Text Available A novel algebraic topology approach to supersymmetry (SUSY and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin models with the extended quantum symmetry of entangled, 'string-net condensed' (ground states.

WIENER-HOPF SOLVER WITH SMOOTH PROBABILITY DISTRIBUTIONS OF ITS COMPONENTS

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Mr. Vladimir A. Smagin

2016-12-01

Full Text Available The Wiener – Hopf solver with smooth probability distributions of its component is presented. The method is based on hyper delta approximations of initial distributions. The use of Fourier series transformation and characteristic function allows working with the random variable method concentrated in transversal axis of absc.

Strong result for real zeros of random algebraic polynomials

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T. Uno

2001-01-01

Full Text Available An estimate is given for the lower bound of real zeros of random algebraic polynomials whose coefficients are non-identically distributed dependent Gaussian random variables. Moreover, our estimated measure of the exceptional set, which is independent of the degree of the polynomials, tends to zero as the degree of the polynomial tends to infinity.

Hopf Algebroids and Their Cyclic Theory

NARCIS (Netherlands)

Kowalzig, N.

2009-01-01

The main objective of this thesis is to clarify concepts of generalised symmetries in noncommutative geometry (i.e., the noncommutative analogue of groupoids and Lie algebroids) and their associated (co)homologies. These ideas are incorporated by the notion of Hopf algebroids and Hopf-cyclic

Application of laser speckle to randomized numerical linear algebra

Science.gov (United States)

Valley, George C.; Shaw, Thomas J.; Stapleton, Andrew D.; Scofield, Adam C.; Sefler, George A.; Johannson, Leif

2018-02-01

We propose and simulate integrated optical devices for accelerating numerical linear algebra (NLA) calculations. Data is modulated on chirped optical pulses and these propagate through a multimode waveguide where speckle provides the random projections needed for NLA dimensionality reduction.

Hopf solitons in the Nicole model

International Nuclear Information System (INIS)

Gillard, Mike; Sutcliffe, Paul

2010-01-01

The Nicole model is a conformal field theory in a three-dimensional space. It has topological soliton solutions classified by the integer-valued Hopf charge, and all currently known solitons are axially symmetric. A volume-preserving flow is used to construct soliton solutions numerically for all Hopf charges from 1 to 8. It is found that the known axially symmetric solutions are unstable for Hopf charges greater than 2 and new lower energy solutions are obtained that include knots and links. A comparison with the Skyrme-Faddeev model suggests many universal features, though there are some differences in the link types obtained in the two theories.

κ-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems

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Andrzej Borowiec

2010-10-01

Full Text Available Some classes of Deformed Special Relativity (DSR theories are reconsidered within the Hopf algebraic formulation. For this purpose we shall explore a minimal framework of deformed Weyl-Heisenberg algebras provided by a smash product construction of DSR algebra. It is proved that this DSR algebra, which uniquely unifies κ-Minkowski spacetime coordinates with Poincaré generators, can be obtained by nonlinear change of generators from undeformed one. Its various realizations in terms of the standard (undeformed Weyl-Heisenberg algebra opens the way for quantum mechanical interpretation of DSR theories in terms of relativistic (Stückelberg version Quantum Mechanics. On this basis we review some recent results concerning twist realization of κ-Minkowski spacetime described as a quantum covariant algebra determining a deformation quantization of the corresponding linear Poisson structure. Formal and conceptual issues concerning quantum κ-Poincaré and κ-Minkowski algebras as well as DSR theories are discussed. Particularly, the so-called ''q-analog'' version of DSR algebra is introduced. Is deformed special relativity quantization of doubly special relativity remains an open question. Finally, possible physical applications of DSR algebra to description of some aspects of Planck scale physics are shortly recalled.

The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders

International Nuclear Information System (INIS)

Gurau, Razvan

2012-01-01

Random tensor models for a generic complex tensor generalize matrix models in arbitrary dimensions and yield a theory of random geometries. They support a 1/N expansion dominated by graphs of spherical topology. Their Schwinger Dyson equations, generalizing the loop equations of matrix models, translate into constraints satisfied by the partition function. The constraints have been shown, in the large N limit, to close a Lie algebra indexed by colored rooted D-ary trees yielding a first generalization of the Virasoro algebra in arbitrary dimensions. In this paper we complete the Schwinger Dyson equations and the associated algebra at all orders in 1/N. The full algebra of constraints is indexed by D-colored graphs, and the leading order D-ary tree algebra is a Lie subalgebra of the full constraints algebra.

Linking the Gauss-Bonnet-Chern theorem, essential HOPF maps and membrane solitons with exotic spin and statistics

International Nuclear Information System (INIS)

Tze, Chia-Hsiung

1989-01-01

By way of the Gauss-Bonnet-Chern theorem, we present a higher dimensional extension of Polyakov's regularization of Wilson loops of point solitons. Spacetime paths of extended objects become hyper-ribbons with self-linking, twisting and writhing numbers. specifically we discuss the exotic spin and statistical phase entanglements of geometric n-membrane solitons of D-dimensional KP 1 σ-models with an added Hopf-Chern-Simons term where (n, D, K) = (0, 3, C), (2, 7, H), (6, 15, Ω). They are uniquely linked to the complex and quaternion and octonion division algebras. 22 refs

Generalized semilocal theories and higher Hopf maps

International Nuclear Information System (INIS)

Hindmarsh, M.; Holman, R.; Kephart, T.W.; Vachaspati, T.

1993-01-01

In semilocal theories, the vacuum manifold is fibered in a non-trivial way by the action of the gauge group. Here we generalize the original semilocal theory (which was based on the Hopf bundle S 3 → S1 S 2 ) to realize the next Hopf bundle S 7 →S 3 S 4 , and its extensions S 2n+1 → S3 HP n . The semilocal defects in this class of theories are classified by π 3 (S 3 ), and are interpreted as constrained instantons or generalized sphaleron configurations. We fail to find a field theoretic realization of the final Hopf bundle S 15 →S 7 S 8 , but are able to construct other semilocal spaces realizing Stiefel bundles over grassmannian spaces. (orig.)

Hopf bifurcation in a delayed reaction-diffusion-advection population model

Science.gov (United States)

Chen, Shanshan; Lou, Yuan; Wei, Junjie

2018-04-01

In this paper, we investigate a reaction-diffusion-advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction-diffusion-advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.

Rank of quantized universal enveloping algebras and modular functions

International Nuclear Information System (INIS)

Majid, S.; Soibelman, Ya.S.

1991-01-01

We compute an intrinsic rank invariant for quasitriangular Hopf algebras in the case of general quantum groups U q (g). As a function of q the rank has remarkable number theoretic properties connected with modular covariance and Galois theory. A number of examples are treated in detail, including rank (U q (su(3)) and rank (U q (e 8 )). We briefly indicate a physical interpretation as relating Chern-Simons theory with the theory of a quantum particle confined to an alcove of g. (orig.)

Radford's biproducts and Yetter-Drinfeld modules for monoidal Hom-Hopf algebras

Energy Technology Data Exchange (ETDEWEB)

Liu, Ling, E-mail: ntliuling@gmail.com [College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004 (China); Shen, Bingliang [Shanghai University of Finance and Economics Zhejiang College, Jinhua 321013 (China)

2014-03-15

Let (H, α) be a monoidal Hom-bialgebra and (B, β) be a left (H, α)-Hom-module algebra and also a left (H, α)-Hom-comodule coalgebra. Then in this paper, we first introduce the notion of a Hom-smash coproduct, which is a monoidal Hom-coalgebra. Second, we find sufficient and necessary conditions for the Hom-smash product algebra structure and the Hom-smash coproduct coalgebra structure on B ⊗ H to afford B ⊗ H a monoidal Hom-bialgebra structure, generalizing the well-known Radford's biproduct, where the conditions are equivalent to that (B, β) is a bialgebra in the category of Hom-Yetter-Drinfeld modules H/H HYD. Finally, we illustrate the category of Hom-Yetter-Drinfeld modules H/H HYD and prove that the category H/H HYD is a braided monoidal category.

Global Hopf Bifurcation for a Predator-Prey System with Three Delays

Science.gov (United States)

Jiang, Zhichao; Wang, Lin

2017-06-01

In this paper, a delayed predator-prey model is considered. The existence and stability of the positive equilibrium are investigated by choosing the delay τ = τ1 + τ2 as a bifurcation parameter. We see that Hopf bifurcation can occur as τ crosses some critical values. The direction of the Hopf bifurcations and the stability of the bifurcation periodic solutions are also determined by using the center manifold and normal form theory. Furthermore, based on the global Hopf bifurcation theorem for general function differential equations, which was established by J. Wu using fixed point theorem and degree theory methods, the existence of global Hopf bifurcation is investigated. Finally, numerical simulations to support the analytical conclusions are carried out.

Proceedings – Mathematical Sciences | Indian Academy of Sciences

Indian Academy of Sciences (India)

Volume 116 Issue 4 November 2006 pp 443-458 Operator Theory/Operator Algebras/Quantum Invariants. The Planar Algebra of a Semisimple and Cosemisimple Hopf Algebra · Vijay Kodiyalam V S Sunder · More Details Abstract Fulltext PDF. To a semisimple and cosemisimple Hopf algebra over an algebraically closed ...

Double Hopf bifurcation in delay differential equations

Directory of Open Access Journals (Sweden)

Redouane Qesmi

2014-07-01

Full Text Available The paper addresses the computation of elements of double Hopf bifurcation for retarded functional differential equations (FDEs with parameters. We present an efficient method for computing, simultaneously, the coefficients of center manifolds and normal forms, in terms of the original FDEs, associated with the double Hopf singularity up to an arbitrary order. Finally, we apply our results to a nonlinear model with periodic delay. This shows the applicability of the methodology in the study of delay models arising in either natural or technological problems.

Three lectures on quantum groups: Representations, duality, real forms

International Nuclear Information System (INIS)

Dobrev, V.K.

1992-07-01

Quantum groups appeared first as quantum algebra, i.e. as one parameter deformations of the numerical enveloping algebras of complex Lie algebras, in the study of the algebraic aspects of quantum integrable systems. Then quantum algebras related to triparametric solutions of the quantum Yang-Baxter equation were axiomatically introduced as (pseudo) quasi-triangular Hopf algebras. Later, a theory of formal deformations has been developed and the notion of quasi-Hopf algebra has been introduced. In other approaches to quantum groups the objects are called quantum matrix groups and are Hopf algebras in chirality to the quantum algebras. The representations of U q (G), the chirality and the real forms associated to these approaches are discussed here. Refs

Analysis of stability and Hopf bifurcation for a delayed logistic equation

International Nuclear Information System (INIS)

Sun Chengjun; Han Maoan; Lin Yiping

2007-01-01

The dynamics of a logistic equation with discrete delay are investigated, together with the local and global stability of the equilibria. In particular, the conditions under which a sequence of Hopf bifurcations occur at the positive equilibrium are obtained. Explicit algorithm for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.]. Global existence of periodic solutions is also established by using a global Hopf bifurcation result of Wu [Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 350:1998;4799-38.

Subcritical Hopf Bifurcation and Stochastic Resonance of Electrical Activities in Neuron under Electromagnetic Induction

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Yu-Xuan Fu

2018-02-01

Full Text Available The FitzHugh–Nagumo model is improved to consider the effect of the electromagnetic induction on single neuron. On the basis of investigating the Hopf bifurcation behavior of the improved model, stochastic resonance in the stochastic version is captured near the bifurcation point. It is revealed that a weak harmonic oscillation in the electromagnetic disturbance can be amplified through stochastic resonance, and it is the cooperative effect of random transition between the resting state and the large amplitude oscillating state that results in the resonant phenomenon. Using the noise dependence of the mean of interburst intervals, we essentially suggest a biologically feasible clue for detecting weak signal by means of neuron model with subcritical Hopf bifurcation. These observations should be helpful in understanding the influence of the magnetic field to neural electrical activity.

Hopf bifurcation and uncontrolled stochastic traffic-induced chaos in an RED-AQM congestion control system

International Nuclear Information System (INIS)

Wang Jun-Song; Yuan Rui-Xi; Gao Zhi-Wei; Wang De-Jin

2011-01-01

We study the Hopf bifurcation and the chaos phenomena in a random early detection-based active queue management (RED-AQM) congestion control system with a communication delay. We prove that there is a critical value of the communication delay for the stability of the RED-AQM control system. Furthermore, we show that the system will lose its stability and Hopf bifurcations will occur when the delay exceeds the critical value. When the delay is close to its critical value, we demonstrate that typical chaos patterns may be induced by the uncontrolled stochastic traffic in the RED-AQM control system even if the system is still stable, which reveals a new route to the chaos besides the bifurcation in the network congestion control system. Numerical simulations are given to illustrate the theoretical results. (general)

Adaptive Control of Electromagnetic Suspension System by HOPF Bifurcation

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Aming Hao

2013-01-01

Full Text Available EMS-type maglev system is essentially nonlinear and unstable. It is complicated to design a stable controller for maglev system which is under large-scale disturbance and parameter variance. Theory analysis expresses that this phenomenon corresponds to a HOPF bifurcation in mathematical model. An adaptive control law which adjusts the PID control parameters is given in this paper according to HOPF bifurcation theory. Through identification of the levitated mass, the controller adjusts the feedback coefficient to make the system far from the HOPF bifurcation point and maintain the stability of the maglev system. Simulation result indicates that adjusting proportion gain parameter using this method can extend the state stability range of maglev system and avoid the self-excited vibration efficiently.

Probe Knots and Hopf Insulators with Ultracold Atoms

Science.gov (United States)

Deng, Dong-Ling; Wang, Sheng-Tao; Sun, Kai; Duan, L.-M.

2018-01-01

Knots and links are fascinating and intricate topological objects. Their influence spans from DNA and molecular chemistry to vortices in superfluid helium, defects in liquid crystals and cosmic strings in the early universe. Here we find that knotted structures also exist in a peculiar class of three-dimensional topological insulators—the Hopf insulators. In particular, we demonstrate that the momentum-space spin textures of Hopf insulators are twisted in a nontrivial way, which implies the presence of various knot and link structures. We further illustrate that the knots and nontrivial spin textures can be probed via standard time-of-flight images in cold atoms as preimage contours of spin orientations in stereographic coordinates. The extracted Hopf invariants, knots, and links are validated to be robust to typical experimental imperfections. Our work establishes the existence of knotted structures in Hopf insulators, which may have potential applications in spintronics and quantum information processing. D.L.D., S.T.W. and L.M.D. are supported by the ARL, the IARPA LogiQ program, and the AFOSR MURI program, and supported by Tsinghua University for their visits. K.S. acknowledges the support from NSF under Grant No. PHY1402971. D.L.D. is also supported by JQI-NSF-PFC and LPS-MPO-CMTC at the final stage of this paper.

Quantum algebras as quantizations of dual Poisson–Lie groups

International Nuclear Information System (INIS)

Ballesteros, Ángel; Musso, Fabio

2013-01-01

A systematic computational approach for the explicit construction of any quantum Hopf algebra (U z (g), Δ z ) starting from the Lie bialgebra (g, δ) that gives the first-order deformation of the coproduct map Δ z is presented. The procedure is based on the well-known ‘quantum duality principle’, namely the fact that any quantum algebra can be viewed as the quantization of the unique Poisson–Lie structure (G*, Λ g ) on the dual group G*, which is obtained by exponentiating the Lie algebra g* defined by the dual map δ*. From this perspective, the coproduct for U z (g) is just the pull-back of the group law for G*, and the Poisson analogues of the quantum commutation rules for U z (g) are given by the unique Poisson–Lie structure Λ g on G* whose linearization is the Poisson analogue of the initial Lie algebra g. This approach is shown to be a very useful technical tool in order to solve the Lie bialgebra quantization problem explicitly since, once a Lie bialgebra (g, δ) is given, the full dual Poisson–Lie group (G*, Λ) can be obtained either by applying standard Poisson–Lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs. As a consequence, the quantization of (G*, Λ) will give rise to the full U z (g) quantum algebra, provided that ordering problems are appropriately fixed through the choice of certain local coordinates on G* whose coproduct fulfils a precise ‘quantum symmetry’ property. The applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant Lie algebras such as sl(2,R), the (2+1) anti-de Sitter algebra so(2, 2) and the Poincaré algebra in (3+1) dimensions. (paper)

Hopf bifurcation analysis of Chen circuit with direct time delay feedback

International Nuclear Information System (INIS)

Hai-Peng, Ren; Wen-Chao, Li; Ding, Liu

2010-01-01

Direct time delay feedback can make non-chaotic Chen circuit chaotic. The chaotic Chen circuit with direct time delay feedback possesses rich and complex dynamical behaviours. To reach a deep and clear understanding of the dynamics of such circuits described by delay differential equations, Hopf bifurcation in the circuit is analysed using the Hopf bifurcation theory and the central manifold theorem in this paper. Bifurcation points and bifurcation directions are derived in detail, which prove to be consistent with the previous bifurcation diagram. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodical orbit (oscillation) in Chen circuit with direct time delay feedback. Bifurcation boundaries are derived using the Hopf bifurcation analysis, which will be helpful for determining the parameters in the stabilisation of the originally chaotic circuit

Hopf bifurcation in an Internet congestion control model

International Nuclear Information System (INIS)

Li Chunguang; Chen Guanrong; Liao Xiaofeng; Yu Juebang

2004-01-01

We consider an Internet model with a single link accessed by a single source, which responds to congestion signals from the network, and study bifurcation of such a system. By choosing the gain parameter as a bifurcation parameter, we prove that Hopf bifurcation occurs. The stability of bifurcating periodic solutions and the direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical example is given to verify the theoretical analysis

Introduction to quantized LIE groups and algebras

International Nuclear Information System (INIS)

Tjin, T.

1992-01-01

In this paper, the authors give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups the authors study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then the authors explain in detail the concept of quantization for them. As an example the quantization of sl 2 is explicitly carried out. Next, the authors show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction, the authors explicitly construct the universal R matrix for the quantum sl 2 algebra. In the last section, the authors deduce all finite-dimensional irreducible representations for q a root of unity. The authors also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory

Non-commutative algebra of functions of 4-dimensional quantum Hall droplet

International Nuclear Information System (INIS)

Chen Yixin; Hou Boyu; Hou Boyuan

2002-01-01

We develop the description of non-commutative geometry of the 4-dimensional quantum Hall fluid's theory proposed recently by Zhang and Hu. The non-commutative structure of fuzzy S 4 , which is the base of the bundle S 7 obtained by the second Hopf fibration, i.e., S 7 /S 3 =S 4 , appears naturally in this theory. The fuzzy monopole harmonics, which are the essential elements in the non-commutative algebra of functions on S 4 , are explicitly constructed and their obeying the matrix algebra is obtained. This matrix algebra is associative. We also propose a fusion scheme of the fuzzy monopole harmonics of the coupling system from those of the subsystems, and determine the fusion rule in such fusion scheme. By products, we provide some essential ingredients of the theory of SO(5) angular momentum. In particular, the explicit expression of the coupling coefficients, in the theory of SO(5) angular momentum, are given. We also discuss some possible applications of our results to the 4-dimensional quantum Hall system and the matrix brane construction in M-theory

Twisted quantum doubles

Directory of Open Access Journals (Sweden)

Daijiro Fukuda

2004-01-01

Full Text Available Using diagrammatic pictures of tensor contractions, we consider a Hopf algebra (Aop⊗ℛλA** twisted by an element ℛλ∈A*⊗Aop corresponding to a Hopf algebra morphism λ:A→A. We show that this Hopf algebra is quasitriangular with the universal R-matrix coming from ℛλ when λ2=idA, generalizing the quantum double construction which corresponds to the case λ=idA.

Polarization of light and Hopf fibration

International Nuclear Information System (INIS)

Jurco, B.

1987-01-01

A set of polarization states of quasi-monochromatic light is described geometrically in terms of the Hopf fibration. Several associated alternative polarization parametrizations are given explicitly, including the Stokes parameters. (author). 8 refs

Nonresonant Double Hopf Bifurcation in Toxic Phytoplankton-Zooplankton Model with Delay

Science.gov (United States)

Yuan, Rui; Jiang, Weihua; Wang, Yong

This paper investigates a toxic phytoplankton-zooplankton model with Michaelis-Menten type phytoplankton harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, fold, Hopf, fold-Hopf and double Hopf bifurcation, when the parameters change and go through some of the critical values, the dynamical properties of the system will change also, such as the stability, equilibrium points and the periodic orbit. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations, and the completion bifurcation set by calculating the universal unfoldings near the double Hopf bifurcation point are given by the normal form theory and center manifold theorem. We obtained that the stable coexistent equilibrium point and stable periodic orbit alternate regularly when the digestion time delay is within some finite value. That is, we derived the pattern for the occurrence, and disappearance of a stable periodic orbit. Furthermore, we calculated the approximation expression of the critical bifurcation curve using the digestion time delay and the harvesting rate as parameters, and determined a large range in terms of the harvesting rate for the phytoplankton and zooplankton to coexist in a long term.

Noether analysis of the twisted Hopf symmetries of canonical noncommutative spacetimes

International Nuclear Information System (INIS)

Amelino-Camelia, Giovanni; Gubitosi, Giulia; Marciano, Antonino; Martinetti, Pierre; Mercati, Flavio; Briscese, Fabio

2008-01-01

We study the twisted Hopf-algebra symmetries of observer-independent canonical spacetime noncommutativity, for which the commutators of the spacetime coordinates take the form [x^ μ ,x^ ν ]=iθ μν with observer-independent (and coordinate-independent) θ μν . We find that it is necessary to introduce nontrivial commutators between transformation parameters and spacetime coordinates, and that the form of these commutators implies that all symmetry transformations must include a translation component. We show that with our noncommutative transformation parameters the Noether analysis of the symmetries is straightforward, and we compare our canonical-noncommutativity results with the structure of the conserved charges and the ''no-pure-boost'' requirement derived in a previous study of κ-Minkowski noncommutativity. We also verify that, while at intermediate stages of the analysis we do find terms that depend on the ordering convention adopted in setting up the Weyl map, the final result for the conserved charges is reassuringly independent of the choice of Weyl map and (the corresponding choice of) star product.

New examples of frobenius extensions

CERN Document Server

Kadison, Lars

1999-01-01

This volume is based on the author's lecture courses to algebraists at Munich and at G�teborg. He presents, for the first time in book form, a unified approach from the point of view of Frobenius algebras/extensions to diverse topics, such as Jones' subfactor theory, Hopf algebras and Hopf subalgebras, the Yang-Baxter Equation and 2-dimensional topological quantum field theories. Other Features: Initial steps toward a theory of noncommutative ring extensions. Self-contained sections on Azumaya algebras and strongly separable algebras. Applications and generalizations of Morita theory and Azumaya algebra due to Hirata and Sugano. Understanding the text requires no prior background in Frobenius algebras or Hopf algebras. An index and a thorough list of further references are included. There is an appendix giving a brief historical guide to the literature.

Hopf bifurcation of the stochastic model on business cycle

International Nuclear Information System (INIS)

Xu, J; Wang, H; Ge, G

2008-01-01

A stochastic model on business cycle was presented in thas paper. Simplifying the model through the quasi Hamiltonian theory, the Ito diffusion process was obtained. According to Oseledec multiplicative ergodic theory and singular boundary theory, the conditions of local and global stability were acquired. Solving the stationary FPK equation and analyzing the stationary probability density, the stochastic Hopf bifurcation was explained. The result indicated that the change of parameter awas the key factor to the appearance of the stochastic Hopf bifurcation

The magnon kinematics of the AdS/CFT correspondence

International Nuclear Information System (INIS)

Gomez, Cesar; Hernandez, Rafael

2006-01-01

The planar dilatation operator of N = 4 supersymmetric Yang-Mills is the Hamiltonian of an integrable spin chain whose length is allowed to fluctuate. We will identify the dynamics of length fluctuations of planar N = 4 Yang-Mills with the existence of an abelian Hopf algebra Z symmetry with non-trivial co-multiplication and antipode. The intertwiner conditions for this Hopf algebra will restrict the allowed magnon irreps to those leading to the magnon dispersion relation. We will discuss magnon kinematics and crossing symmetry on the spectrum of Z. We also consider general features of the underlying Hopf algebra with Z as central Hopf subalgebra, and discuss the giant magnon semiclassical regime

Competition of Spatial and Temporal Instabilities under Time Delay near Codimension-Two Turing-Hopf Bifurcations

International Nuclear Information System (INIS)

Wang Huijuan; Ren Zhi

2011-01-01

Competition of spatial and temporal instabilities under time delay near the codimension-two Turing-Hopf bifurcations is studied in a reaction-diffusion equation. The time delay changes remarkably the oscillation frequency, the intrinsic wave vector, and the intensities of both Turing and Hopf modes. The application of appropriate time delay can control the competition between the Turing and Hopf modes. Analysis shows that individual or both feedbacks can realize the control of the transformation between the Turing and Hopf patterns. Two-dimensional numerical simulations validate the analytical results. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)

Hopf bifurcation for tumor-immune competition systems with delay

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Ping Bi

2014-01-01

Full Text Available In this article, a immune response system with delay is considered, which consists of two-dimensional nonlinear differential equations. The main purpose of this paper is to explore the Hopf bifurcation of a immune response system with delay. The general formula of the direction, the estimation formula of period and stability of bifurcated periodic solution are also given. Especially, the conditions of the global existence of periodic solutions bifurcating from Hopf bifurcations are given. Numerical simulations are carried out to illustrate the the theoretical analysis and the obtained results.

Using Max-Plus Algebra for the Evaluation of Stochastic Process Algebra Prefixes

NARCIS (Netherlands)

Cloth, L.; de Alfaro, L.; Gilmore, S.; Bohnenkamp, H.C.; Haverkort, Boudewijn R.H.M.

2001-01-01

In this paper, the concept of complete finite prefixes for process algebra expressions is extended to stochastic models. Events are supposed to happen after a delay that is determined by random variables assigned to the preceding conditions. Max-plus algebra expressions are shown to provide an

Algebras of Random Operators Associated to Delone Dynamical Systems

International Nuclear Information System (INIS)

Lenz, Daniel; Stollmann, Peter

2003-01-01

We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C*-subalgebra to discuss a Shubin trace formula

An Approach to Robust Control of the Hopf Bifurcation

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Giacomo Innocenti

2011-01-01

Full Text Available The paper illustrates a novel approach to modify the Hopf bifurcation nature via a nonlinear state feedback control, which leaves the equilibrium properties unchanged. This result is achieved by recurring to linear and nonlinear transformations, which lead the system to locally assume the ordinary differential equation representation. Third-order models are considered, since they can be seen as proper representatives of a larger class of systems. The explicit relationship between the control input and the Hopf bifurcation nature is obtained via a frequency approach, that does not need the computation of the center manifold.

Stability and Hopf bifurcation analysis of a prey-predator system with two delays

International Nuclear Information System (INIS)

Li Kai; Wei Junjie

2009-01-01

In this paper, we have considered a prey-predator model with Beddington-DeAngelis functional response and selective harvesting of predator species. Two delays appear in this model to describe the time that juveniles take to mature. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By analyzing the associated characteristic equation, its linear stability is investigated and Hopf bifurcations are demonstrated. The stability and direction of the Hopf bifurcation are determined by applying the normal form method and the center manifold theory. Numerical simulation results are given to support the theoretical predictions.

DINÁMICA DE LA BIFURCACIÓN DE HOPF EN UNA CLASE DE MODELOS DE COMPETENCIA QUE EXHIBEN LA BIFURCACIÓN ZIP Hopf Bifurcation Dynamic in a Class of Competence Model Exhibiting Zip Bifurcation

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Carlos Mario Escobar Callejas

2011-12-01

Full Text Available En el presente artículo de investigación se caracteriza el tipo de bifurcación de Hopf que se presenta en el fenómeno de la bifurcación de zip para un sistema tridimensional no lineal de ecuaciones diferenciales que satisface las condiciones planteadas por Butler y Farkas, las cuales modelan la competición de dos especies predadoras por una presa singular que se regenera. Se demuestra que en todas las variedades bidimensionales invariantes del sistema considerado se desarrolla una bifurcación de Hopf supercrítica lo cual es una extensión de algunos resultados sobre el tipo de bifurcación de Hopf que se forma en el fenómeno de la bifurcación de zip en sistema con respuesta funcional del predador del tipo Holling II, [1].This research article characterizes the type of Hopf bifurcation occurring in the Zip bifurcation phenomenon for a non-linear 3D system of differential equations which meets the conditions stated by Butler and Farkas to model competition of two predators struggling for a prey. It is shown that a supercritical Hopf bifurcation is developed in all invariant two-dimensional varieties of the system considered, which is an extension of some results about the kind of Hopf bifurcation which is formed in the Zip bifurcation phenomenon in a system with functional response of the Holling-type predator.

Analysis of stability and Hopf bifurcation for a viral infectious model with delay

International Nuclear Information System (INIS)

Sun Chengjun; Cao Zhijie; Lin Yiping

2007-01-01

In this paper, a four-dimensional viral infectious model with delay is considered. The stability of the two equilibria and the existence of Hopf bifurcation are investigated. It is found that there are stability switches and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. Using the normal form theory and center manifold argument [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981], the explicit formulaes which determine the stability, the direction and the period of bifurcating periodic solutions are derived. Numerical simulations are carried out to illustrate the validity of the main results

Macdonald operators and homological invariants of the colored Hopf link

International Nuclear Information System (INIS)

Awata, Hidetoshi; Kanno, Hiroaki

2011-01-01

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov-Rozansky homology as a special case. We prove the polynomiality of the invariants obtained by GIKV's proposal for arbitrary representations. We derive a closed formula of the invariants of the colored Hopf link for antisymmetric representations. We argue that a little amendment of GIKV's proposal is required to make all the coefficients of the polynomial non-negative integers. (paper)

Quantum torsors

OpenAIRE

Grunspan, C.

2003-01-01

This text gives some results about quantum torsors. Our starting point is an old reformulation of torsors recalled recently by Kontsevich. We propose an unification of the definitions of torsors in algebraic geometry and in Poisson geometry. Any quantum torsor is equipped with two comodule-algebra structures over Hopf algebras and these structures commute with each other. In the finite dimensional case, these two Hopf algebras share the same finite dimension. We show that any Galois extension...

Stability and Hopf Bifurcation Analysis on a Nonlinear Business Cycle Model

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Liming Zhao

2016-01-01

Full Text Available This study begins with the establishment of a three-dimension business cycle model based on the condition of a fixed exchange rate. Using the established model, the reported study proceeds to describe and discuss the existence of the equilibrium and stability of the economic system near the equilibrium point as a function of the speed of market regulation and the degree of capital liquidity and a stable region is defined. In addition, the condition of Hopf bifurcation is discussed and the stability of a periodic solution, which is generated by the Hopf bifurcation and the direction of the Hopf bifurcation, is provided. Finally, a numerical simulation is provided to confirm the theoretical results. This study plays an important role in theoretical understanding of business cycle models and it is crucial for decision makers in formulating macroeconomic policies as detailed in the conclusions of this report.

Hopf bifurcation and chaos in a third-order phase-locked loop

Science.gov (United States)

Piqueira, José Roberto C.

2017-01-01

Phase-locked loops (PLLs) are devices able to recover time signals in several engineering applications. The literature regarding their dynamical behavior is vast, specifically considering that the process of synchronization between the input signal, coming from a remote source, and the PLL local oscillation is robust. For high-frequency applications it is usual to increase the PLL order by increasing the order of the internal filter, for guarantying good transient responses; however local parameter variations imply structural instability, thus provoking a Hopf bifurcation and a route to chaos for the phase error. Here, one usual architecture for a third-order PLL is studied and a range of permitted parameters is derived, providing a rule of thumb for designers. Out of this range, a Hopf bifurcation appears and, by increasing parameters, the periodic solution originated by the Hopf bifurcation degenerates into a chaotic attractor, therefore, preventing synchronization.

General distributions in process algebra

NARCIS (Netherlands)

Katoen, Joost P.; d' Argenio, P.R.; Brinksma, Hendrik; Hermanns, H.; Katoen, Joost P.

2001-01-01

This paper is an informal tutorial on stochastic process algebras, i.e., process calculi where action occurrences may be subject to a delay that is governed by a (mostly continuous) random variable. Whereas most stochastic process algebras consider delays determined by negative exponential

Numerical Hopf bifurcation of Runge-Kutta methods for a class of delay differential equations

International Nuclear Information System (INIS)

Wang Qiubao; Li Dongsong; Liu, M.Z.

2009-01-01

In this paper, we consider the discretization of parameter-dependent delay differential equation of the form y ' (t)=f(y(t),y(t-1),τ),τ≥0,y element of R d . It is shown that if the delay differential equation undergoes a Hopf bifurcation at τ=τ * , then the discrete scheme undergoes a Hopf bifurcation at τ(h)=τ * +O(h p ) for sufficiently small step size h, where p≥1 is the order of the Runge-Kutta method applied. The direction of numerical Hopf bifurcation and stability of bifurcating invariant curve are the same as that of delay differential equation.

Notes on quantum groups

International Nuclear Information System (INIS)

Pressley, A.; Chari, V.; Tata Inst. of Fundamental Research, Bombay

1990-01-01

The authors presents an introduction to quantum groups defined as a deformation of the universal enveloping algebra of a Lie algebra. After the description of Hopf algebras with some examples the approach of Drinfel'd and Jimbo is described, where the quantization of a Lie algebra represents a Hopf algebra, defined over the algebra of formal power series in an indetermined h. The authors show that this approach arises from a r-matrix, which satisfies the classical Yang-Baxter equation. As example quantum sl 2 is considered. Furthermore the approaches of Manin and Woroniwicz and the R-matrix approach are described. (HSI)

Generalized Cole–Hopf transformations for generalized Burgers ...

Indian Academy of Sciences (India)

2015-10-15

Oct 15, 2015 ... Cole–Hopf transformations; Burgers equation; invariance analysis. ... was to generate nonlinear parabolic equations from a linear parabolic equation via a ..... BMV acknowledges the financial support to attend the NMI Workshop ... [16] P J Olver, Applications of Lie Groups to differential equations, Graduate ...

Hopf bifurcation in a environmental defensive expenditures model with time delay

International Nuclear Information System (INIS)

Russu, Paolo

2009-01-01

In this paper a three-dimensional environmental defensive expenditures model with delay is considered. The model is based on the interactions among visitors V, quality of ecosystem goods E, and capital K, intended as accommodation and entertainment facilities, in Protected Areas (PAs). The tourism user fees (TUFs) are used partly as a defensive expenditure and partly to increase the capital stock. The stability and existence of Hopf bifurcation are investigated. It is that stability switches and Hopf bifurcation occurs when the delay t passes through a sequence of critical values, τ 0 . It has been that the introduction of a delay is a destabilizing process, in the sense that increasing the delay could cause the bio-economics to fluctuate. Formulas about the stability of bifurcating periodic solution and the direction of Hopf bifurcation are exhibited by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the results.

Stability and Hopf bifurcation for a delayed SLBRS computer virus model.

Science.gov (United States)

Zhang, Zizhen; Yang, Huizhong

2014-01-01

By incorporating the time delay due to the period that computers use antivirus software to clean the virus into the SLBRS model a delayed SLBRS computer virus model is proposed in this paper. The dynamical behaviors which include local stability and Hopf bifurcation are investigated by regarding the delay as bifurcating parameter. Specially, direction and stability of the Hopf bifurcation are derived by applying the normal form method and center manifold theory. Finally, an illustrative example is also presented to testify our analytical results.

Stability and Hopf Bifurcation for a Delayed SLBRS Computer Virus Model

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Zizhen Zhang

2014-01-01

Full Text Available By incorporating the time delay due to the period that computers use antivirus software to clean the virus into the SLBRS model a delayed SLBRS computer virus model is proposed in this paper. The dynamical behaviors which include local stability and Hopf bifurcation are investigated by regarding the delay as bifurcating parameter. Specially, direction and stability of the Hopf bifurcation are derived by applying the normal form method and center manifold theory. Finally, an illustrative example is also presented to testify our analytical results.

Braided quantum field theories and their symmetries

International Nuclear Information System (INIS)

Sasai, Yuya; Sasakura, Naoki

2007-01-01

Braided quantum field theories, proposed by Oeckl, can provide a framework for quantum field theories that possess Hopf algebra symmetries. In quantum field theories, symmetries lead to non-perturbative relations among correlation functions. We study Hopf algebra symmetries and such relations in the context of braided quantum field theories. We give the four algebraic conditions among Hopf algebra symmetries and braided quantum field theories that are required for the relations to hold. As concrete examples, we apply our analysis to the Poincare symmetries of two examples of noncommutative field theories. One is the effective quantum field theory of three-dimensional quantum gravity coupled to spinless particles formulated by Freidel and Livine, and the other is noncommutative field theory on the Moyal plane. We also comment on quantum field theory in κ-Minkowski spacetime. (author)

Hopf bifurcation in love dynamical models with nonlinear couples and time delays

International Nuclear Information System (INIS)

Liao Xiaofeng; Ran Jiouhong

2007-01-01

A love dynamical models with nonlinear couples and two delays is considered. Local stability of this model is studied by analyzing the associated characteristic transcendental equation. We find that the Hopf bifurcation occurs when the sum of the two delays varies and passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Numerical example is given to illustrate our results

Necessary and sufficient conditions for Hopf bifurcation in tri-neuron equation with a delay

International Nuclear Information System (INIS)

Liu Xiaoming; Liao Xiaofeng

2009-01-01

In this paper, we consider the delayed differential equations modeling three-neuron equations with only a time delay. Using the time delay as a bifurcation parameter, necessary and sufficient conditions for Hopf bifurcation to occur are derived. Numerical results indicate that for this model, Hopf bifurcation is likely to occur at suitable delay parameter values.

Stability and Hopf Bifurcation of Fractional-Order Complex-Valued Single Neuron Model with Time Delay

Science.gov (United States)

Wang, Zhen; Wang, Xiaohong; Li, Yuxia; Huang, Xia

2017-12-01

In this paper, the problems of stability and Hopf bifurcation in a class of fractional-order complex-valued single neuron model with time delay are addressed. With the help of the stability theory of fractional-order differential equations and Laplace transforms, several new sufficient conditions, which ensure the stability of the system are derived. Taking the time delay as the bifurcation parameter, Hopf bifurcation is investigated and the critical value of the time delay for the occurrence of Hopf bifurcation is determined. Finally, two representative numerical examples are given to show the effectiveness of the theoretical results.

Delay Induced Hopf Bifurcation of an Epidemic Model with Graded Infection Rates for Internet Worms

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Tao Zhao

2017-01-01

Full Text Available A delayed SEIQRS worm propagation model with different infection rates for the exposed computers and the infectious computers is investigated in this paper. The results are given in terms of the local stability and Hopf bifurcation. Sufficient conditions for the local stability and the existence of Hopf bifurcation are obtained by using eigenvalue method and choosing the delay as the bifurcation parameter. In particular, the direction and the stability of the Hopf bifurcation are investigated by means of the normal form theory and center manifold theorem. Finally, a numerical example is also presented to support the obtained theoretical results.

Hopf bifurcation and chaos from torus breakdown in voltage-mode controlled DC drive systems

International Nuclear Information System (INIS)

Dai Dong; Ma Xikui; Zhang Bo; Tse, Chi K.

2009-01-01

Period-doubling bifurcation and its route to chaos have been thoroughly investigated in voltage-mode and current-mode controlled DC motor drives under simple proportional control. In this paper, the phenomena of Hopf bifurcation and chaos from torus breakdown in a voltage-mode controlled DC drive system is reported. It has been shown that Hopf bifurcation may occur when the DC drive system adopts a more practical proportional-integral control. The phenomena of period-adding and phase-locking are also observed after the Hopf bifurcation. Furthermore, it is shown that the stable torus can breakdown and chaos emerges afterwards. The work presented in this paper provides more complete information about the dynamical behaviors of DC drive systems.

The Boundary-Hopf-Fold Bifurcation in Filippov Systems

NARCIS (Netherlands)

Efstathiou, Konstantinos; Liu, Xia; Broer, Henk W.

2015-01-01

This paper studies the codimension-3 boundary-Hopf-fold (BHF) bifurcation of planar Filippov systems. Filippov systems consist of at least one discontinuity boundary locally separating the phase space to disjoint components with different dynamics. Such systems find applications in several fields,

Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay

International Nuclear Information System (INIS)

Ding Yuting; Jiang Weihua; Wang Hongbin

2012-01-01

Highlights: ► We derive the unfolding of a financial system with Hopf-pitchfork bifurcation. ► We show the coexistence of a pair of stable small amplitudes periodic solutions. ► At the same time, also there is a pair of stable large amplitudes periodic solutions. ► Chaos can appear by period-doubling bifurcation far away from Hopf-pitchfork value. ► The study will be useful for interpreting economics phenomena in theory. - Abstract: In this paper, we identify the critical point for a Hopf-pitchfork bifurcation in a nonlinear financial system with delay, and derive the normal form up to third order with their unfolding in original system parameters near the bifurcation point by normal form method and center manifold theory. Furthermore, we analyze its local dynamical behaviors, and show the coexistence of a pair of stable periodic solutions. We also show that there coexist a pair of stable small-amplitude periodic solutions and a pair of stable large-amplitude periodic solutions for different initial values. Finally, we give the bifurcation diagram with numerical illustration, showing that the pair of stable small-amplitude periodic solutions can also exist in a large region of unfolding parameters, and the financial system with delay can exhibit chaos via period-doubling bifurcations as the unfolding parameter values are far away from the critical point of the Hopf-pitchfork bifurcation.

Real zeros of classes of random algebraic polynomials

Directory of Open Access Journals (Sweden)

K. Farahmand

2003-01-01

Full Text Available There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0+a1x+a2x2+⋯+an−1xn−1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients {aj}j=0n−1 it is shown that the above expected number is asymptotic to O(logn. This order for the expected number of zeros remains valid for the case when the coefficients are grouped into two, each group with a different variance. However, it was recently shown that if the coefficients are non-identically distributed such that the variance of the jth term is (nj the expected number of zeros of the polynomial increases to O(n. The present paper provides the value for this asymptotic formula for the polynomials with the latter variances when they are grouped into three with different patterns for their variances.

Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays

International Nuclear Information System (INIS)

Karaoglu, Esra; Merdan, Huseyin

2014-01-01

Highlights: • A ratio-dependent predator–prey system involving two discrete maturation time delays is studied. • Hopf bifurcations are analyzed by choosing delay parameters as bifurcation parameters. • When a delay parameter passes through a critical value, Hopf bifurcations occur. • The direction of bifurcation, the period and the stability of periodic solution are also obtained. - Abstract: In this paper we give a detailed Hopf bifurcation analysis of a ratio-dependent predator–prey system involving two different discrete delays. By analyzing the characteristic equation associated with the model, its linear stability is investigated. Choosing delay terms as bifurcation parameters the existence of Hopf bifurcations is demonstrated. Stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. Furthermore, some of the bifurcation properties including direction, stability and period are given. Finally, theoretical results are supported by some numerical simulations

Forced phase-locked response of a nonlinear system with time delay after Hopf bifurcation

International Nuclear Information System (INIS)

Ji, J.C.; Hansen, Colin H.

2005-01-01

The trivial equilibrium of a nonlinear autonomous system with time delay may become unstable via a Hopf bifurcation of multiplicity two, as the time delay reaches a critical value. This loss of stability of the equilibrium is associated with two coincident pairs of complex conjugate eigenvalues crossing the imaginary axis. The resultant dynamic behaviour of the corresponding nonlinear non-autonomous system in the neighbourhood of the Hopf bifurcation is investigated based on the reduction of the infinite-dimensional problem to a four-dimensional centre manifold. As a result of the interaction between the Hopf bifurcating periodic solutions and the external periodic excitation, a primary resonance can occur in the forced response of the system when the forcing frequency is close to the Hopf bifurcating periodic frequency. The method of multiple scales is used to obtain four first-order ordinary differential equations that determine the amplitudes and phases of the phase-locked periodic solutions. The first-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration of the delay-differential equation. It is also found that the steady state solutions of the nonlinear non-autonomous system may lose their stability via either a pitchfork or Hopf bifurcation. It is shown that the primary resonance response may exhibit symmetric and asymmetric phase-locked periodic motions, quasi-periodic motions, chaotic motions, and coexistence of two stable motions

Nonfamilial acrokeratosis verruciformis of Hopf

Directory of Open Access Journals (Sweden)

Nidhi Patel

2015-01-01

Full Text Available Acrokeratosis verruciformis (AKV of Hopf is an autosomal dominant genodermatosis with unknown etiology. It is characterized by multiple flat-topped keratotic papules resembling planar warts located mainly on the dorsum of hands and feet. Superficial ablation is the treatment of choice. A 41-year-old female presented with multiple hyperpigmented, hyperkeratotic papules and plaques over flexor aspect of both forearms, extensors of both legs and dorsum of the feet. Histopathology showed changes of AKV. Patient was treated with a combination of topical corticosteroids and cryotherapy with no visible improvement.

Deformations of spacetime and internal symmetries

Directory of Open Access Journals (Sweden)

Gresnigt Niels G.

2017-01-01

Full Text Available Algebraic deformations provide a systematic approach to generalizing the symmetries of a physical theory through the introduction of new fundamental constants. The applications of deformations of Lie algebras and Hopf algebras to both spacetime and internal symmetries are discussed. As a specific example we demonstrate how deforming the classical flavor group S U(3 to the quantum group S Uq(3 ≡ U q (su(3 (a Hopf algebra and taking into account electromagnetic mass splitting within isospin multiplets leads to new and exceptionally accurate baryon mass sum rules that agree perfectly with experimental data.

Hopf bifurcation formula for first order differential-delay equations

Science.gov (United States)

Rand, Richard; Verdugo, Anael

2007-09-01

This work presents an explicit formula for determining the radius of a limit cycle which is born in a Hopf bifurcation in a class of first order constant coefficient differential-delay equations. The derivation is accomplished using Lindstedt's perturbation method.

Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

Directory of Open Access Journals (Sweden)

Zizhen Zhang

2013-01-01

Full Text Available A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

Satake diagrams of affine Kac-Moody algebras

Energy Technology Data Exchange (ETDEWEB)

Tripathy, L K [S B R Government Womens' College, Berhampur, Orissa 760 001 (India); Pati, K C [Department of Physics, Khallikote College, Berhampur, Orissa 760 001 (India)

2006-02-10

Satake diagrams of affine Kac-Moody algebras (untwisted and twisted) are obtained from their Dynkin diagrams. These diagrams give a classification of restricted root systems associated with these algebras. In the case of simple Lie algebras, these root systems and Satake diagrams correspond to symmetric spaces which have recently found many physical applications in quantum integrable systems, quantum transport problems, random matrix theories etc. We hope these types of root systems may have similar applications in theoretical physics in future and may correspond to symmetric spaces analogue of affine Kac-Moody algebras if they exist.

Handbook of algebra Vol. 1

CERN Document Server

1996-01-01

Handbook of Algebra defines algebra as consisting of many different ideas, concepts and results. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. Each chapter of the book combines some of the features of both a graduate-level textbook and a research-level survey. This book is divided into eight sections. Section 1A focuses on linear algebra and discusses such concepts as matrix functions and equations and random matrices. Section 1B cover linear d

Model Reduction of Nonlinear Aeroelastic Systems Experiencing Hopf Bifurcation

KAUST Repository

Abdelkefi, Abdessattar

2013-06-18

In this paper, we employ the normal form to derive a reduced - order model that reproduces nonlinear dynamical behavior of aeroelastic systems that undergo Hopf bifurcation. As an example, we consider a rigid two - dimensional airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. We apply the center manifold theorem on the governing equations to derive its normal form that constitutes a simplified representation of the aeroelastic sys tem near flutter onset (manifestation of Hopf bifurcation). Then, we use the normal form to identify a self - excited oscillator governed by a time - delay ordinary differential equation that approximates the dynamical behavior while reducing the dimension of the original system. Results obtained from this oscillator show a great capability to predict properly limit cycle oscillations that take place beyond and above flutter as compared with the original aeroelastic system.

Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays

Directory of Open Access Journals (Sweden)

Huitao Zhao

2013-01-01

Full Text Available A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998 for functional differential equations, the global existence of the periodic solutions is obtained. Finally, an example for numerical simulations is also included.

Wiener-Hopf operators on spaces of functions on R+ with values in a Hilbert space

OpenAIRE

Petkova, Violeta

2006-01-01

A Wiener-Hopf operator on a Banach space of functions on R+ is a bounded operator T such that P^+S_{-a}TS_a=T, for every positive a, where S_a is the operator of translation by a. We obtain a representation theorem for the Wiener-Hopf operators on a large class of functions on R+ with values in a separable Hilbert space.

La factorización de una transformada de Fourier en el método de Wiener-Hopf

OpenAIRE

José Rosales-Ortega; Carlos Márquez Rivera

2009-01-01

Using the Wiener-Hopf method, we factorize the Fourier Transform of the kernel of a singular integral equation as the product of two functions: one holomorphic in the upper semiplan and the other holomophic in the lower semiplan. Keywords: function product, Fourier transform, Wiener-Hopf method.

Vertex algebras and algebraic curves

CERN Document Server

Frenkel, Edward

2004-01-01

Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book co...

Algebraic partial Boolean algebras

International Nuclear Information System (INIS)

Smith, Derek

2003-01-01

Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell-Kochen-Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are 'algebraic' in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell-Kochen-Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B(T) is finite when the underlying space H is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B(T) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B(T) that are generated by a given set T. We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B(T) is infinite just if that root lattice has an A 5 sublattice. Finally, we characterize the rays of B(T) when T consists of the rays spanning the minimal vectors of the root lattice E 8

La factorización de una transformada de Fourier en el método de Wiener-Hopf

Directory of Open Access Journals (Sweden)

José Rosales-Ortega

2009-02-01

Full Text Available Using the Wiener-Hopf method, we factorize the Fourier Transform of the kernel of a singular integral equation as the product of two functions: one holomorphic in the upper semiplan and the other holomophic in the lower semiplan. Keywords: function product, Fourier transform, Wiener-Hopf method.

Homotopy of operads and Grothendieck–Teichmüller groups part 1 the algebraic theory and its topological background

CERN Document Server

Fresse, Benoit

2017-01-01

The Grothendieck-Teichmüller group was defined by Drinfeld in quantum group theory with insights coming from the Grothendieck program in Galois theory. The ultimate goal of this book is to explain that this group has a topological interpretation as a group of homotopy automorphisms associated to the operad of little 2-discs, which is an object used to model commutative homotopy structures in topology. This volume gives a comprehensive survey on the algebraic aspects of this subject. The book explains the definition of an operad in a general context, reviews the definition of the little discs operads, and explains the definition of the Grothendieck-Teichmüller group from the viewpoint of the theory of operads. In the course of this study, the relationship between the little discs operads and the definition of universal operations associated to braided monoidal category structures is explained. Also provided is a comprehensive and self-contained survey of the applications of Hopf algebras to the definition of...

Local stability and Hopf bifurcation in small-world delayed networks

International Nuclear Information System (INIS)

Li Chunguang; Chen Guanrong

2004-01-01

The notion of small-world networks, recently introduced by Watts and Strogatz, has attracted increasing interest in studying the interesting properties of complex networks. Notice that, a signal or influence travelling on a small-world network often is associated with time-delay features, which are very common in biological and physical networks. Also, the interactions within nodes in a small-world network are often nonlinear. In this paper, we consider a small-world networks model with nonlinear interactions and time delays, which was recently considered by Yang. By choosing the nonlinear interaction strength as a bifurcation parameter, we prove that Hopf bifurcation occurs. We determine the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation by applying the normal form theory and the center manifold theorem. Finally, we show a numerical example to verify the theoretical analysis

Local stability and Hopf bifurcation in small-world delayed networks

Energy Technology Data Exchange (ETDEWEB)

Li Chunguang E-mail: cgli@uestc.edu.cn; Chen Guanrong E-mail: gchen@ee.cityu.edu.hk

2004-04-01

The notion of small-world networks, recently introduced by Watts and Strogatz, has attracted increasing interest in studying the interesting properties of complex networks. Notice that, a signal or influence travelling on a small-world network often is associated with time-delay features, which are very common in biological and physical networks. Also, the interactions within nodes in a small-world network are often nonlinear. In this paper, we consider a small-world networks model with nonlinear interactions and time delays, which was recently considered by Yang. By choosing the nonlinear interaction strength as a bifurcation parameter, we prove that Hopf bifurcation occurs. We determine the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation by applying the normal form theory and the center manifold theorem. Finally, we show a numerical example to verify the theoretical analysis.

Topology, ergodic theory, real algebraic geometry Rokhlin's memorial

CERN Document Server

Turaev, V

2001-01-01

This book is dedicated to the memory of the outstanding Russian mathematician, V. A. Rokhlin (1919-1984). It is a collection of research papers written by his former students and followers, who are now experts in their fields. The topics in this volume include topology (the Morse-Novikov theory, spin bordisms in dimension 6, and skein modules of links), real algebraic geometry (real algebraic curves, plane algebraic surfaces, algebraic links, and complex orientations), dynamics (ergodicity, amenability, and random bundle transformations), geometry of Riemannian manifolds, theory of Teichmüller

Cup products in Hopf cyclic cohomology with coefficients in contramodules

OpenAIRE

Rangipour, Bahram

2010-01-01

We use stable anti Yetter-Drinfeld contramodules to improve the cup products in Hopf cyclic cohomology. The improvement fixes the lack of functoriality of the cup products previously defined and show that the cup products are sensitive to the coefficients.

Linear Algebra and Smarandache Linear Algebra

OpenAIRE

Vasantha, Kandasamy

2003-01-01

The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and vector spaces over finite p...

Degenerate Hopf bifurcation in a self-exciting Faraday disc dynamo

Indian Academy of Sciences (India)

Weiquan Pan

2017-05-31

May 31, 2017 ... Recently, self-exciting Faraday disk dynamo is also a topic of con- cern [16–20]. ..... Hopf bifurcation. (a) Projected on the x–z plane and (b) pro- ... Key Lab of Com- plex System Optimization and Big Data Processing. (No.

Enlarged symmetry algebras of spin chains, loop models, and S-matrices

International Nuclear Information System (INIS)

Read, N.; Saleur, H.

2007-01-01

The symmetry algebras of certain families of quantum spin chains are considered in detail. The simplest examples possess m states per site (m>=2), with nearest-neighbor interactions with U(m) symmetry, under which the sites transform alternately along the chain in the fundamental m and its conjugate representation m-bar. We find that these spin chains, even with arbitrary coefficients of these interactions, have a symmetry algebra A m much larger than U(m), which implies that the energy eigenstates fall into sectors that for open chains (i.e., free boundary conditions) can be labeled by j=0,1,...,L, for the 2L-site chain such that the degeneracies of all eigenvalues in the jth sector are generically the same and increase rapidly with j. For large j, these degeneracies are much larger than those that would be expected from the U(m) symmetry alone. The enlarged symmetry algebra A m (2L) consists of operators that commute in this space of states with the Temperley-Lieb algebra that is generated by the set of nearest-neighbor interaction terms; A m (2L) is not a Yangian. There are similar results for supersymmetric chains with gl(m+n|n) symmetry of nearest-neighbor interactions, and a richer representation structure for closed chains (i.e., periodic boundary conditions). The symmetries also apply to the loop models that can be obtained from the spin chains in a spacetime or transfer matrix picture. In the loop language, the symmetries arise because the loops cannot cross. We further define tensor products of representations (for the open chains) by joining chains end to end. The fusion rules for decomposing the tensor product of representations labeled j 1 and j 2 take the same form as the Clebsch-Gordan series for SU(2). This and other structures turn the symmetry algebra A m into a ribbon Hopf algebra, and we show that this is 'Morita equivalent' to the quantum group U q (sl 2 ) for m=q+q -1 . The open-chain results are extended to the cases vertical bar m vertical

Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebras

International Nuclear Information System (INIS)

Gebert, R.W.

1993-09-01

The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory. In this context Borcherds algebras arise as certain ''physical'' subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction into this rapidly-developing area of mathematics. Based on the machinery of formal calculus we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake Monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analysed from the point of view of symmetry in quantum theory and the construction of the Monster Lie algebra is sketched. (orig.)

Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus

Directory of Open Access Journals (Sweden)

Tao Dong

2012-01-01

Full Text Available By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

Stability and Hopf bifurcation in a simplified BAM neural network with two time delays.

Science.gov (United States)

Cao, Jinde; Xiao, Min

2007-03-01

Various local periodic solutions may represent different classes of storage patterns or memory patterns, and arise from the different equilibrium points of neural networks (NNs) by applying Hopf bifurcation technique. In this paper, a bidirectional associative memory NN with four neurons and multiple delays is considered. By applying the normal form theory and the center manifold theorem, analysis of its linear stability and Hopf bifurcation is performed. An algorithm is worked out for determining the direction and stability of the bifurcated periodic solutions. Numerical simulation results supporting the theoretical analysis are also given.

Hopf and Bautin Bifurcation in a Tritrophic Food Chain Model with Holling Functional Response Types III and IV

Science.gov (United States)

Castellanos, Víctor; Castillo-Santos, Francisco Eduardo; Dela-Rosa, Miguel Angel; Loreto-Hernández, Iván

In this paper, we analyze the Hopf and Bautin bifurcation of a given system of differential equations, corresponding to a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. We distinguish two cases, when the prey has linear or logistic growth. In both cases we guarantee the existence of a limit cycle bifurcating from an equilibrium point in the positive octant of ℝ3. In order to do so, for the Hopf bifurcation we compute explicitly the first Lyapunov coefficient, the transversality Hopf condition, and for the Bautin bifurcation we also compute the second Lyapunov coefficient and verify the regularity conditions.

Nonintegrability of the unfolding of the fold-Hopf bifurcation

Science.gov (United States)

Yagasaki, Kazuyuki

2018-02-01

We consider the unfolding of the codimension-two fold-Hopf bifurcation and prove its meromorphic nonintegrability in the meaning of Bogoyavlenskij for almost all parameter values. Our proof is based on a generalized version of the Morales-Ramis-Simó theory for non-Hamiltonian systems and related variational equations up to second order are used.

CP1 model with Hopf interaction: the quantum theory

International Nuclear Information System (INIS)

Chakraborty, B.; Ghosh, Subir; Malik, R.P.

2001-01-01

The CP 1 model with Hopf interaction is quantised following the Batalin-Tyutin (BT) prescription. In this scheme, extra BT fields are introduced which allow for the existence of only commuting first-class constraints. Explicit expression for the quantum correction to the expectation value of the energy density and angular momentum in the physical sector of this model is derived. The result shows, in the particular operator ordering prescription we have chosen to work with, that the quantum effect has the usual divergent contribution of O(ℎ 2 ) in the energy expectation value. But, interestingly the Hopf term, though topological in nature, can have a finite O(ℎ) contribution to energy density in the homotopically nontrivial topological sector. The angular momentum operator, however, is found to have no quantum correction at O(ℎ), indicating the absence of any fractional spin even at this quantum level. Finally, the extended Lagrangian incorporating the BT auxiliary fields is computed in the conventional framework of BRST formalism exploiting Faddeev-Popov technique of path integral method

Toward the classification of differential calculi on κ-Minkowski space and related field theories

Energy Technology Data Exchange (ETDEWEB)

Jurić, Tajron; Meljanac, Stjepan; Pikutić, Danijel [Ruđer Bošković Institute, Theoretical Physics Division,Bijenička c.54, HR-10002 Zagreb (Croatia); Štrajn, Rina [Dipartimento di Matematica e Informatica, Università di Cagliari,viale Merello 92, I-09123 Cagliari (Italy); INFN, Sezione di Cagliari,Cagliari (Italy)

2015-07-13

Classification of differential forms on κ-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the κ-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to κ-Poincaré Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications.

Hopf bifurcations in a fractional reaction–diffusion model for the ...

African Journals Online (AJOL)

The phenomenon of hopf bifurcation has been well-studied and applied to many physical situations to explain behaviour of solutions resulting from differential and partial differential equations. This phenomenon is applied to a fractional reaction diffusion model for tumor invasion and development. The result suggests that ...

Stability and Hopf bifurcation analysis of a new system

International Nuclear Information System (INIS)

Huang Kuifei; Yang Qigui

2009-01-01

In this paper, a new chaotic system is introduced. The system contains special cases as the modified Lorenz system and conjugate Chen system. Some subtle characteristics of stability and Hopf bifurcation of the new chaotic system are thoroughly investigated by rigorous mathematical analysis and symbolic computations. Meanwhile, some numerical simulations for justifying the theoretical analysis are also presented.

Quantum W-algebras and elliptic algebras

International Nuclear Information System (INIS)

Feigin, B.; Kyoto Univ.; Frenkel, E.

1996-01-01

We define a quantum W-algebra associated to sl N as an associative algebra depending on two parameters. For special values of the parameters, this algebra becomes the ordinary W-algebra of sl N , or the q-deformed classical W-algebra of sl N . We construct free field realizations of the quantum W-algebras and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in U q (n). (orig.)

The Kustaanheimo-Stiefel transformation, the hydrogen-oscillator connection and orthogonal polynomials

International Nuclear Information System (INIS)

Campigotto, C.

1991-03-01

The purpose of this paper is to investigate some special cases of nonbijective canonical transformations known as Hurwitz transformations. They are connected with the Hopf fiber bundles and contain the Kustaanheimo-Stiefel (KS) transformation as a particular case. The latter transformation corresponds to the map S 3 x R + → S 2 x R + , i.e., the S 3 /S 1 = S 2 Hopf fibration. A generalization of the KS transformation is given by the map S 7 x R + → S 4 x R + , i.e., the S 7 /S 3 = S 4 Hopf fibration. The general algebraic framework for studying these transformations is given by the Cayley-Dickson algebras in eight dimensions

Stability of small-amplitude periodic solutions near Hopf bifurcations in time-delayed fully-connected PLL networks

Science.gov (United States)

Ferruzzo Correa, Diego P.; Bueno, Átila M.; Castilho Piqueira, José R.

2017-04-01

In this paper we investigate stability conditions for small-amplitude periodic solutions emerging near symmetry-preserving Hopf bifurcations in a time-delayed fully-connected N-node PLL network. The study of this type of systems which includes the time delay between connections has attracted much attention among researchers mainly because the delayed coupling between nodes emerges almost naturally in mathematical modeling in many areas of science such as neurobiology, population dynamics, physiology and engineering. In a previous work it has been shown that symmetry breaking and symmetry preserving Hopf bifurcations can emerge in the parameter space. We analyze the stability along branches of periodic solutions near fully-synchronized Hopf bifurcations in the fixed-point space, based on the reduction of the infinite-dimensional space onto a two-dimensional center manifold in normal form. Numerical results are also presented in order to confirm our analytical results.

The Yoneda algebra of a K2 algebra need not be another K2 algebra

OpenAIRE

Cassidy, T.; Phan, C.; Shelton, B.

2010-01-01

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.

On the classical origins of yangian symmetry in integrable field theory

International Nuclear Information System (INIS)

MacKay, N.J.

1992-01-01

We show that Drinfeld's yangian algebra, studied recently as the algebra of conserved charges in certain two-dimensional integrable quantum field theories, is also present in the classical theory as a Poisson-Hopf algebra, and exhibit explicitly the Serre relations, coproduct and antipode. (orig.)

Hopf Bifurcation of a Delayed Epidemic Model with Information Variable and Limited Medical Resources

Directory of Open Access Journals (Sweden)

Caijuan Yan

2014-01-01

Full Text Available We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratio ℛ01, we obtain sufficient conditions under which the endemic equilibrium E* of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.

Cylindric-like algebras and algebraic logic

CERN Document Server

Ferenczi, Miklós; Németi, István

2013-01-01

Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways:  as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.

The Hopf fibration over S8 admits no S1-subfibration

International Nuclear Information System (INIS)

Loo, B.; Verjovsky, A.

1990-10-01

It is shown that there does not exist a PL-bundle over S 8 with fibre and total space PL-manifolds homotopy equivalent to CP 3 and CP 7 respectively. Consequently, the Hopf fibration over S 8 admits no subfibration by PL-circles. (author). 27 refs

The Hopf fibration over S8 admits no S1-subfibration

International Nuclear Information System (INIS)

Loo, B.; Verjovsky, A.

1990-05-01

It is shown that there does not exist a PL-bundle over S 8 with fibre and total space PL-manifolds homotopy equivalent to CP 3 and CP 7 respectively. Consequently, the Hopf fibration over S 8 admits no subfibration by PL-circles. (author). 27 refs

Boundary maps for $C^*$-crossed products with R with an application to the quantum Hall effect

OpenAIRE

Kellendonk, Johannes; Schulz-Baldes, Hermann

2004-01-01

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with R is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t.Connes' pairing of cyclic cohomology with K-theory. As an application, we prove...

Quaestiones Mathematicae - Vol 38, No 5 (2015)

African Journals Online (AJOL)

The formal theory of Hopf algebras part I: Hopf monoids in a monoidal category · EMAIL FULL TEXT EMAIL FULL TEXT DOWNLOAD ... Analytical construction of peaked solutions for the nonlinear evolution of an electromagnetic pulse propagating through a plasma · EMAIL FULL TEXT EMAIL FULL TEXT DOWNLOAD ...

Introduction to relation algebras relation algebras

CERN Document Server

Givant, Steven

2017-01-01

The first volume of a pair that charts relation algebras from novice to expert level, this text offers a comprehensive grounding for readers new to the topic. Upon completing this introduction, mathematics students may delve into areas of active research by progressing to the second volume, Advanced Topics in Relation Algebras; computer scientists, philosophers, and beyond will be equipped to apply these tools in their own field. The careful presentation establishes first the arithmetic of relation algebras, providing ample motivation and examples, then proceeds primarily on the basis of algebraic constructions: subalgebras, homomorphisms, quotient algebras, and direct products. Each chapter ends with a historical section and a substantial number of exercises. The only formal prerequisite is a background in abstract algebra and some mathematical maturity, though the reader will also benefit from familiarity with Boolean algebra and naïve set theory. The measured pace and outstanding clarity are particularly ...

White noise on bialgebras

CERN Document Server

Schürmann, Michael

1993-01-01

Stochastic processes with independent increments on a group are generalized to the concept of "white noise" on a Hopf algebra or bialgebra. The main purpose of the book is the characterization of these processes as solutions of quantum stochastic differential equations in the sense of R.L. Hudsonand K.R. Parthasarathy. The notes are a contribution to quantum probability but they are also related to classical probability, quantum groups, and operator algebras. The Az ma martingales appear as examples of white noise on a Hopf algebra which is a deformation of the Heisenberg group. The book will be of interest to probabilists and quantum probabilists. Specialists in algebraic structures who are curious about the role of their concepts in probablility theory as well as quantum theory may find the book interesting. The reader should havesome knowledge of functional analysis, operator algebras, and probability theory.

From Rota-Baxter algebras to pre-Lie algebras

International Nuclear Information System (INIS)

An Huihui; Ba, Chengming

2008-01-01

Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension ≤3 and their corresponding pre-Lie algebras

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

International Nuclear Information System (INIS)

Marquette, Ian

2013-01-01

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

Yoneda algebras of almost Koszul algebras

Indian Academy of Sciences (India)

Abstract. Let k be an algebraically closed field, A a finite dimensional connected. (p,q)-Koszul self-injective algebra with p, q ≥ 2. In this paper, we prove that the. Yoneda algebra of A is isomorphic to a twisted polynomial algebra A![t; β] in one inde- terminate t of degree q +1 in which A! is the quadratic dual of A, β is an ...

Quantum algebra Uqp(u2) and application to the rotational collective dynamics of the nuclei

International Nuclear Information System (INIS)

Barbier, R.

1995-01-01

This thesis concerns some aspects of new symmetries in Nuclear Physics. It comprises three parts. The first one is devoted to the study of the quantum algebra U qp (u 2 ). More precisely, we develop its Hopf algebraic structure and we study its co-product structure. The bases of the representation theory of U qp (u 2 ) are introduced. On one hand, we construct the finite-dimensional irreducible representations of U qp (u 2 ). On the other hand, we calculate the Clebsch-Gordan coefficients with the projection operator method. To complete our study, we construct some deformed boson mappings of the quantum algebras U qp (u 2 ), U q 2 (su 2 ) and U qp (u 1,1 ). The second part deals with the construction of a new phenomenological model of the non rigid rotator. This model is based on the quantum algebra U qp (u 2 ). The rotational energy and the E2 reduced transition probabilities are obtained. They depend on the two deformation parameters q and p of the quantum algebra. We show how the use of the two-parameter deformation of the algebra U qp (u 2 ) leads to a generalization of the U q (su 2 )-rotator model. We also introduce a new model of the anharmonic oscillator on the basis of the quantum algebra U qp (u 2 ). We show that the system of the U q (su 2 )-rotator and of the anharmonic oscillator can be coupled with the use of the deformation parameters of U qp (u 2 ). A ro-vibration energy formula and expansion 'a la' Dunham are obtained. The aim of the lest part is to apply our non rigid rotator model to the rotational collective dynamics of the superdeformed nuclei of the A∼130 - 150 and A∼190 mass regions and deformed nuclei of the actinide and rare earth series. We adjust the free parameters of our model and compare our results with those arising from four other models of the non rigid rotator. A comparative analysis is given in terms of transition energies. We calculate the dynamical moments of inertia with the fitted parameters. A comparison between the

Complexity and Hopf Bifurcation Analysis on a Kind of Fractional-Order IS-LM Macroeconomic System

Science.gov (United States)

Ma, Junhai; Ren, Wenbo

On the basis of our previous research, we deepen and complete a kind of macroeconomics IS-LM model with fractional-order calculus theory, which is a good reflection on the memory characteristics of economic variables, we also focus on the influence of the variables on the real system, and improve the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. The conditions of Hopf bifurcation in fractional-order system models are briefly demonstrated, and the fractional order when Hopf bifurcation occurs is calculated, showing the inherent complex dynamic characteristics of the system. With numerical simulation, bifurcation, strange attractor, limit cycle, waveform and other complex dynamic characteristics are given; and the order condition is obtained with respect to time. We find that the system order has an important influence on the running state of the system. The system has a periodic motion when the order meets the conditions of Hopf bifurcation; the fractional-order system gradually stabilizes with the change of the order and parameters while the corresponding integer-order system diverges. This study has certain significance to policy-making about macroeconomic regulation and control.

Local and global Hopf bifurcation analysis in a neutral-type neuron system with two delays

Science.gov (United States)

Lv, Qiuyu; Liao, Xiaofeng

2018-03-01

In recent years, neutral-type differential-difference equations have been applied extensively in the field of engineering, and their dynamical behaviors are more complex than that of the delay differential-difference equations. In this paper, the equations used to describe a neutral-type neural network system of differential difference equation with two delays are studied (i.e. neutral-type differential equations). Firstly, by selecting τ1, τ2 respectively as a parameter, we provide an analysis about the local stability of the zero equilibrium point of the equations, and sufficient conditions of asymptotic stability for the system are derived. Secondly, by using the theory of normal form and applying the theorem of center manifold introduced by Hassard et al., the Hopf bifurcation is found and some formulas for deciding the stability of periodic solutions and the direction of Hopf bifurcation are given. Moreover, by applying the theorem of global Hopf bifurcation, the existence of global periodic solution of the system is studied. Finally, an example is given, and some computer numerical simulations are taken to demonstrate and certify the correctness of the presented results.

Quantum cluster algebras and quantum nilpotent algebras

Science.gov (United States)

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

An introduction to algebraic geometry and algebraic groups

CERN Document Server

Geck, Meinolf

2003-01-01

An accessible text introducing algebraic geometries and algebraic groups at advanced undergraduate and early graduate level, this book develops the language of algebraic geometry from scratch and uses it to set up the theory of affine algebraic groups from first principles.Building on the background material from algebraic geometry and algebraic groups, the text provides an introduction to more advanced and specialised material. An example is the representation theory of finite groups of Lie type.The text covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups

On the quantum symmetry of the chiral Ising model

Science.gov (United States)

Vecsernyés, Peter

1994-03-01

We introduce the notion of rational Hopf algebras that we think are able to describe the superselection symmetries of rational quantum field theories. As an example we show that a six-dimensional rational Hopf algebra H can reproduce the fusion rules, the conformal weights, the quantum dimensions and the representation of the modular group of the chiral Ising model. H plays the role of the global symmetry algebra of the chiral Ising model in the following sense: (1) a simple field algebra F and a representation π on Hπ of it is given, which contains the c = {1}/{2} unitary representations of the Virasoro algebra as subrepresentations; (2) the embedding U: H → B( Hπ) is such that the observable algebra π( A) - is the invariant subalgebra of B( Hπ) with respect to the left adjoint action of H and U(H) is the commutant of π( A); (3) there exist H-covariant primary fields in B( Hπ), which obey generalized Cuntz algebra properties and intertwine between the inequivalent sectors of the observables.

Associated quantum vector bundles and symplectic structure on a quantum space

International Nuclear Information System (INIS)

Coquereaux, R.; Garcia, A.O.; Trinchero, R.

2000-01-01

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bundles without requiring the use of a (bicovariant) differential structure over H. Moreover, if H is coquasitriangular, it coacts naturally on the associated bundle, and the differential structure is covariant. We apply this construction to the case of the finite quotient of the SL q (2) function Hopf algebra at a root of unity (q 3 = 1) as the structure group, and a reduced 2-dimensional quantum plane as both the 'base manifold' and fibre, getting an algebra which generalizes the notion of classical phase space for this quantum space. We also build explicitly a differential complex for this phase space algebra, and find that levels 0 and 2 support a (co)representation of the quantum symplectic group. On this phase space we define vector fields, and with the help of the Sp q structure we introduce a symplectic form relating 1-forms to vector fields. This leads naturally to the introduction of Poisson brackets, a necessary step to do 'classical' mechanics on a quantum space, the quantum plane. (author)

Stability and Hopf Bifurcation of a Reaction-Diffusion Neutral Neuron System with Time Delay

Science.gov (United States)

Dong, Tao; Xia, Linmao

2017-12-01

In this paper, a type of reaction-diffusion neutral neuron system with time delay under homogeneous Neumann boundary conditions is considered. By constructing a basis of phase space based on the eigenvectors of the corresponding Laplace operator, the characteristic equation of this system is obtained. Then, by selecting time delay and self-feedback strength as the bifurcating parameters respectively, the dynamic behaviors including local stability and Hopf bifurcation near the zero equilibrium point are investigated when the time delay and self-feedback strength vary. Furthermore, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form and the center manifold theorem for the corresponding partial differential equation. Finally, two simulation examples are given to verify the theory.

Hopf bifurcation in a partial dependent predator-prey system with delay

International Nuclear Information System (INIS)

Zhao Huitao; Lin Yiping

2009-01-01

In this paper, a partial dependent predator-prey model with time delay is studied by using the theory of functional differential equation and Hassard's method, the condition on which positive equilibrium exists and Hopf bifurcation occurs are given. Finally, numerical simulations are performed to support the analytical results, and the chaotic behaviors are observed.

Views on the Hopf bifurcation with respect to voltage instabilities

Energy Technology Data Exchange (ETDEWEB)

Roa-Sepulveda, C A [Universidad de Concepcion, Concepcion (Chile). Dept. de Ingenieria Electrica; Knight, U G [Imperial Coll. of Science and Technology, London (United Kingdom). Dept. of Electrical and Electronic Engineering

1994-12-31

This paper presents a sensitivity study of the Hopf bifurcation phenomenon which can in theory appear in power systems, with reference to the dynamics of the process and the impact of demand characteristics. Conclusions are drawn regarding power levels at which these bifurcations could appear and concern the concept of the imaginary axis as a `hard` limit eigenvalue analyses. (author) 20 refs., 31 figs.

The relation between quantum W algebras and Lie algebras

International Nuclear Information System (INIS)

Boer, J. de; Tjin, T.

1994-01-01

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary sl 2 embeddings we show that a large set W of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set W contains many known W algebras such as W N and W 3 (2) . Our formalism yields a completely algorithmic method for calculating the W algebra generators and their operator product expansions, replacing the cumbersome construction of W algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that any W algebra in W can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Therefore any realization of this semisimple affine Lie algebra leads to a realization of the W algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolutions for all algebras in W. Some examples are explicitly worked out. (orig.)

Macular variant of acrokeratosis verruciformis of Hopf

Directory of Open Access Journals (Sweden)

Rita Vipul Vora

2017-01-01

Full Text Available Acrokeratosis verruciformis (AKV of Hopf is an autosomal dominant condition characterized by multiple flesh-colored or lightly pigmented flat or convex warty papules over dorsa of hands, feet, knees, elbows, and forearms. It affects both sexes and is usually present at birth or appears in early childhood. Two forms of the disease have been described, namely, classical AKV and sporadic AKV. Histological examination differentiates it from other similar conditions. Superficial ablation is the treatment of choice. We represent a case of a young female with extensive lesions over contralateral limbs, of classical AKV interspersed with multiple hypopigmented macular lesions of AKV.

2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras

International Nuclear Information System (INIS)

Ayupov, Shavkat; Kudaybergenov, Karimbergen

2016-01-01

The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2 n (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation. (paper)

Global Hopf bifurcation analysis on a BAM neural network with delays

Science.gov (United States)

Sun, Chengjun; Han, Maoan; Pang, Xiaoming

2007-01-01

A delayed differential equation that models a bidirectional associative memory (BAM) neural network with four neurons is considered. By using a global Hopf bifurcation theorem for FDE and a Bendixon's criterion for high-dimensional ODE, a group of sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large.

Global Hopf bifurcation analysis on a BAM neural network with delays

International Nuclear Information System (INIS)

Sun Chengjun; Han Maoan; Pang Xiaoming

2007-01-01

A delayed differential equation that models a bidirectional associative memory (BAM) neural network with four neurons is considered. By using a global Hopf bifurcation theorem for FDE and a Bendixon's criterion for high-dimensional ODE, a group of sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large

Quantum cluster algebra structures on quantum nilpotent algebras

CERN Document Server

Goodearl, K R

2017-01-01

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein-Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts.

Concurrence of Quantum States: Algebraic Dynamical Method Study XXX Models in a Time-Depending Random External Field

International Nuclear Information System (INIS)

Fu Chuanji; Zhu Qinsheng; Wu Shaoyi

2010-01-01

Based on algebraic dynamics and the concept of the concurrence of the entanglement, we investigate the evolutive properties of the two-qubit entanglement that formed by Heisenberg XXX models under a time-depending external held. For this system, the property of the concurrence that is only dependent on the coupling constant J and total values of the external field is proved. Furthermore, we found that the thermal concurrence of the system under a static random external field is a function of the coupling constant J, temperature T, and the magnitude of external held. (general)

Quadratic algebras

CERN Document Server

Polishchuk, Alexander

2005-01-01

Quadratic algebras, i.e., algebras defined by quadratic relations, often occur in various areas of mathematics. One of the main problems in the study of these (and similarly defined) algebras is how to control their size. A central notion in solving this problem is the notion of a Koszul algebra, which was introduced in 1970 by S. Priddy and then appeared in many areas of mathematics, such as algebraic geometry, representation theory, noncommutative geometry, K-theory, number theory, and noncommutative linear algebra. The book offers a coherent exposition of the theory of quadratic and Koszul algebras, including various definitions of Koszulness, duality theory, Poincar�-Birkhoff-Witt-type theorems for Koszul algebras, and the Koszul deformation principle. In the concluding chapter of the book, they explain a surprising connection between Koszul algebras and one-dependent discrete-time stochastic processes.

Regularity of C*-algebras and central sequence algebras

DEFF Research Database (Denmark)

Christensen, Martin S.

The main topic of this thesis is regularity properties of C*-algebras and how these regularity properties are re ected in their associated central sequence algebras. The thesis consists of an introduction followed by four papers [A], [B], [C], [D]. In [A], we show that for the class of simple...... Villadsen algebra of either the rst type with seed space a nite dimensional CW complex, or the second type, tensorial absorption of the Jiang-Su algebra is characterized by the absence of characters on the central sequence algebra. Additionally, in a joint appendix with Joan Bosa, we show that the Villadsen...... algebra of the second type with innite stable rank fails the corona factorization property. In [B], we consider the class of separable C*-algebras which do not admit characters on their central sequence algebra, and show that it has nice permanence properties. We also introduce a new divisibility property...

Real division algebras and other algebras motivated by physics

International Nuclear Information System (INIS)

Benkart, G.; Osborn, J.M.

1981-01-01

In this survey we discuss several general techniques which have been productive in the study of real division algebras, flexible Lie-admissible algebras, and other nonassociative algebras, and we summarize results obtained using these methods. The principal method involved in this work is to view an algebra A as a module for a semisimple Lie algebra of derivations of A and to use representation theory to study products in A. In the case of real division algebras, we also discuss the use of isotopy and the use of a generalized Peirce decomposition. Most of the work summarized here has appeared in more detail in various other papers. The exceptions are results on a class of algebras of dimension 15, motivated by physics, which admit the Lie algebra sl(3) as an algebra of derivations

Second Hopf map and supersymmetric mechanics with Yang monopole

International Nuclear Information System (INIS)

Gonzales, M.; Toppan, F.; Kuznetsova, Z.; Nersessian, F.; Yeghikyan, V.

2009-01-01

We propose to use the second Hopf map for the reduction (via SU(2) group action) of the eight-dimensional supersymmetric mechanics to five-dimensional supersymmetric systems specified by the presence of an SU(2) Yang monopole. For our purpose we develop the relevant Lagrangian reduction procedure. The reduced system is characterized by its invariance under the N = 5 or N = 4 supersymmetry generators (with or without an additional conserved BRST charge operator) which commute with the su(2) generators. (author)

Hom-Novikov algebras

International Nuclear Information System (INIS)

Yau, Donald

2011-01-01

We study a twisted generalization of Novikov algebras, called Hom-Novikov algebras, in which the two defining identities are twisted by a linear map. It is shown that Hom-Novikov algebras can be obtained from Novikov algebras by twisting along any algebra endomorphism. All algebra endomorphisms on complex Novikov algebras of dimensions 2 or 3 are computed, and their associated Hom-Novikov algebras are described explicitly. Another class of Hom-Novikov algebras is constructed from Hom-commutative algebras together with a derivation, generalizing a construction due to Dorfman and Gel'fand. Two other classes of Hom-Novikov algebras are constructed from Hom-Lie algebras together with a suitable linear endomorphism, generalizing a construction due to Bai and Meng.

Boundary maps for C*-crossed products with R with an application to the quantum Hall effect

CERN Document Server

Kellendonk, J

2003-01-01

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with $\\RR$ is the Connes-Thom isomorphism. In this article, the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map in cyclic cohomology. It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t. Connes' pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schrödinger operators.

Linear algebra meets Lie algebra: the Kostant-Wallach theory

OpenAIRE

Shomron, Noam; Parlett, Beresford N.

2008-01-01

In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.

Introduction to quantum groups

International Nuclear Information System (INIS)

Sudbery, A.

1996-01-01

These pedagogical lectures contain some motivation for the study of quantum groups; a definition of ''quasi triangular Hopf algebra'' with explanations of all the concepts required to build it up; descriptions of quantised universal enveloping algebras and the quantum double; and an account of quantised function algebras and the action of quantum groups on quantum spaces. (author)

Extended Virasoro algebra and algebra of area preserving diffeomorphisms

International Nuclear Information System (INIS)

Arakelyan, T.A.

1990-01-01

The algebra of area preserving diffeomorphism plays an important role in the theory of relativistic membranes. It is pointed out that the relation between this algebra and the extended Virasoro algebra associated with the generalized Kac-Moody algebras G(T 2 ). The highest weight representation of these infinite-dimensional algebras as well as of their subalgebras is studied. 5 refs

Quantum groups, non-commutative differential geometry and applications

International Nuclear Information System (INIS)

Schupp, P.; California Univ., Berkeley, CA

1993-01-01

The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on non-commutative or quantum spaces, providing powerful but easy-to-use mathematical tools for applications in physics and related sciences. A generalization of unitary time evolution is proposed and studied for a simple 2-level system, leading to non-conservation of microscopic entropy, a phenomenon new to quantum mechanics. A Cartan calculus that combines functions, forms, Lie derivatives and inner derivations along general vector fields into one big algebra is constructed for quantum groups and then extended to quantum planes. The construction of a tangent bundle on a quantum group manifold and an BRST type approach to quantum group gauge theory are given as further examples of applications. The material is organized in two parts: Part I studies vector fields on quantum groups, emphasizing Hopf algebraic structures, but also introducing a ''quantum geometric'' construction. Using a generalized semi-direct product construction we combine the dual Hopf algebras A of functions and U of left-invariant vector fields into one fully bicovariant algebra of differential operators. The pure braid group is introduced as the commutant of Δ(U). It provides invariant maps A → U and thereby bicovariant vector fields, casimirs and metrics. This construction allows the translation of undeformed matrix expressions into their less obvious quantum algebraic counter parts. We study this in detail for quasitriangular Hopf algebras, giving the determinant and orthogonality relation for the ''reflection'' matrix. Part II considers the additional structures of differential forms and finitely generated quantum Lie algebras -- it is devoted to the construction of the Cartan calculus, based on an undeformed Cartan identity

An application of the division algebras, Jordan algebras and split composition algebras

International Nuclear Information System (INIS)

Foot, R.; Joshi, G.C.

1992-01-01

It has been established that the covering group of the Lorentz group in D = 3, 4, 6, 10 can be expressed in a unified way, based on the four composition division algebras R, C, Q and O. In this paper, the authors discuss, in this framework, the role of the complex numbers of quantum mechanics. A unified treatment of quantum-mechanical spinors is given. The authors provide an explicit demonstration that the vector and spinor transformations recently constructed from a subgroup of the reduced structure group of the Jordan algebras M n 3 are indeed the Lorentz transformations. The authors also show that if the division algebras in the construction of the covering groups of the Lorentz groups in D = 3, 4, 6, 10 are replaced by the split composition algebras, then the sequence of groups SO(2, 2), SO(3, 3) and SO(5, 5) result. The analysis is presumed to be self-contained as the relevant aspects of the division algebras and Jordan algebras are reviewed. Some applications to physical theory are indicated

Interactions Between Representation Ttheory, Algebraic Topology and Commutative Algebra

CERN Document Server

Pitsch, Wolfgang; Zarzuela, Santiago

2016-01-01

This book includes 33 expanded abstracts of selected talks given at the two workshops "Homological Bonds Between Commutative Algebra and Representation Theory" and "Brave New Algebra: Opening Perspectives," and the conference "Opening Perspectives in Algebra, Representations, and Topology," held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program "Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA)." Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest dev...

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

Science.gov (United States)

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.

Monomial algebras

CERN Document Server

Villarreal, Rafael

2015-01-01

The book stresses the interplay between several areas of pure and applied mathematics, emphasizing the central role of monomial algebras. It unifies the classical results of commutative algebra with central results and notions from graph theory, combinatorics, linear algebra, integer programming, and combinatorial optimization. The book introduces various methods to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings and blowup algebra-emphasizing square free quadratics, hypergraph clutters, and effective computational methods.

Algebra

CERN Document Server

Tabak, John

2004-01-01

Looking closely at algebra, its historical development, and its many useful applications, Algebra examines in detail the question of why this type of math is so important that it arose in different cultures at different times. The book also discusses the relationship between algebra and geometry, shows the progress of thought throughout the centuries, and offers biographical data on the key figures. Concise and comprehensive text accompanied by many illustrations presents the ideas and historical development of algebra, showcasing the relevance and evolution of this branch of mathematics.

Iwahori-Hecke algebras and Schur algebras of the symmetric group

CERN Document Server

Mathas, Andrew

1999-01-01

This volume presents a fully self-contained introduction to the modular representation theory of the Iwahori-Hecke algebras of the symmetric groups and of the q-Schur algebras. The study of these algebras was pioneered by Dipper and James in a series of landmark papers. The primary goal of the book is to classify the blocks and the simple modules of both algebras. The final chapter contains a survey of recent advances and open problems. The main results are proved by showing that the Iwahori-Hecke algebras and q-Schur algebras are cellular algebras (in the sense of Graham and Lehrer). This is proved by exhibiting natural bases of both algebras which are indexed by pairs of standard and semistandard tableaux respectively. Using the machinery of cellular algebras, which is developed in Chapter 2, this results in a clean and elegant classification of the irreducible representations of both algebras. The block theory is approached by first proving an analogue of the Jantzen sum formula for the q-Schur algebras. T...

Algebra of pseudo-differential operators over C*-algebra

International Nuclear Information System (INIS)

Mohammad, N.

1982-08-01

Algebras of pseudo-differential operators over C*-algebras are studied for the special case when in Hormander class Ssub(rho,delta)sup(m)(Ω) Ω = Rsup(n); rho = 1, delta = 0, m any real number, and the C*-algebra is infinite dimensional non-commutative. The space B, i.e. the set of A-valued C*-functions in Rsup(n) (or Rsup(n) x Rsup(n)) whose derivatives are all bounded, plays an important role. A denotes C*-algebra. First the operator class Ssub(phi,0)sup(m) is defined, and through it, the class Lsub(1,0)sup(m) of pseudo-differential operators. Then the basic asymptotic expansion theorems concerning adjoint and product of operators of class Ssub(1,0)sup(m) are stated. Finally, proofs are given of L 2 -continuity theorem and the main theorem, which states that algebra of all pseudo-differential operators over C*-algebras is itself C*-algebra

Commutative algebra with a view toward algebraic geometry

CERN Document Server

Eisenbud, David

1995-01-01

Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algeb...

Jordan algebras versus C*- algebras

International Nuclear Information System (INIS)

Stormer, E.

1976-01-01

The axiomatic formulation of quantum mechanics and the problem of whether the observables form self-adjoint operators on a Hilbert space, are discussed. The relation between C*- algebras and Jordan algebras is studied using spectral theory. (P.D.)

Implicative Algebras

African Journals Online (AJOL)

Tadesse

In this paper we introduce the concept of implicative algebras which is an equivalent definition of lattice implication algebra of Xu (1993) and further we prove that it is a regular Autometrized. Algebra. Further we remark that the binary operation → on lattice implicative algebra can never be associative. Key words: Implicative ...

Open algebraic surfaces

CERN Document Server

Miyanishi, Masayoshi

2000-01-01

Open algebraic surfaces are a synonym for algebraic surfaces that are not necessarily complete. An open algebraic surface is understood as a Zariski open set of a projective algebraic surface. There is a long history of research on projective algebraic surfaces, and there exists a beautiful Enriques-Kodaira classification of such surfaces. The research accumulated by Ramanujan, Abhyankar, Moh, and Nagata and others has established a classification theory of open algebraic surfaces comparable to the Enriques-Kodaira theory. This research provides powerful methods to study the geometry and topology of open algebraic surfaces. The theory of open algebraic surfaces is applicable not only to algebraic geometry, but also to other fields, such as commutative algebra, invariant theory, and singularities. This book contains a comprehensive account of the theory of open algebraic surfaces, as well as several applications, in particular to the study of affine surfaces. Prerequisite to understanding the text is a basic b...

Separable algebras

CERN Document Server

Ford, Timothy J

2017-01-01

This book presents a comprehensive introduction to the theory of separable algebras over commutative rings. After a thorough introduction to the general theory, the fundamental roles played by separable algebras are explored. For example, Azumaya algebras, the henselization of local rings, and Galois theory are rigorously introduced and treated. Interwoven throughout these applications is the important notion of étale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups. The text is accessible to graduate students who have finished a first course in algebra, and it includes necessary foundational material, useful exercises, and many nontrivial examples.

About the differential calculus on the quantum groups

International Nuclear Information System (INIS)

Bernard, D.

1992-01-01

Given a solution R of the Yang-Baxter equation admitting a quasi-triangular decomposition we define a quasi-triangular quantum Lie algebra. We describe how to any quasi-triangular quantum Lie algebra U(G R ) is associated a Hopf algebra F(G R ) with a differential calculus on it such that the algebra of the quantum Lie derivatives is the algebra U(G R ). This allows us to make the connection between the differential calculus on quantum groups and the exchange algebras of the algebraic Bethe ansatz. (orig.)

Twisted supersymmetry in a deformed Wess-Zumino model in (2+1) dimensions

Energy Technology Data Exchange (ETDEWEB)

Palechor, C.; Ferrari, A.F.; Quinto, A.G. [Universidade Federal do ABC - UFABC,Avenida dos Estados 5001, 0910-580, Santo André, SP (Brazil)

2017-01-12

Non-anticommutative deformations have been studied in the context of supersymmetry (SUSY) in three and four space-time dimensions, and the general picture is that highly nontrivial to deform supersymmetry in a way that still preserves some of its important properties, both at the formal algebraic level (e.g., preserving the associativity of the deformed theory) as well as at the physical level (e.g., maintaining renormalizability). The Hopf algebra formalism allows the definition of algebraically consistent deformations of SUSY, but this algebraic consistency does not guarantee that physical models build upon these structures will be consistent from the physical point of view. We will investigate a deformation induced by a Drinfel’d twist of the N=1 SUSY algebra in three space-time dimensions. The use of the Hopf algebra formalism allows the construction of deformed N=1 SUSY algebras that should still preserve a deformed version of supersymmetry. We will construct the simplest deformed version of the Wess-Zumino model in this context, but we will show that despite the consistent algebraic structure, the model in question is not invariant under SUSY transformation and is not renormalizable. We will comment on the relation of these results with previous ones discussed in the literature regarding similar four-dimensional constructions.

Generalized EMV-Effect Algebras

Science.gov (United States)

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

2018-04-01

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

Special set linear algebra and special set fuzzy linear algebra

OpenAIRE

Kandasamy, W. B. Vasantha; Smarandache, Florentin; Ilanthenral, K.

2009-01-01

The authors in this book introduce the notion of special set linear algebra and special set fuzzy Linear algebra, which is an extension of the notion set linear algebra and set fuzzy linear algebra. These concepts are best suited in the application of multi expert models and cryptology. This book has five chapters. In chapter one the basic concepts about set linear algebra is given in order to make this book a self contained one. The notion of special set linear algebra and their fuzzy analog...

Banach Synaptic Algebras

Science.gov (United States)

Foulis, David J.; Pulmannov, Sylvia

2018-04-01

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

Hopf bifurcation and eigenvalue sensitivity analysis of doubly fed induction generator wind turbine system

DEFF Research Database (Denmark)

Yang, Li Hui; Xu, Zhao; Østergaard, Jacob

2010-01-01

This paper first presents the Hopf bifurcation analysis for a vector-controlled doubly fed induction generator (DFIG) which is widely used in wind power conversion systems. Using three-phase back-to-back pulse-width-modulated (PWM) converters, DFIG can keep stator frequency constant under variabl...

Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays

International Nuclear Information System (INIS)

Song Yongli; Han Maoan; Peng Yahong

2004-01-01

We consider a Lotka-Volterra competition system with two delays. We first investigate the stability of the positive equilibrium and the existence of Hopf bifurcations, and then using the normal form theory and center manifold argument, derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions

Grassmann algebras

International Nuclear Information System (INIS)

Garcia, R.L.

1983-11-01

The Grassmann algebra is presented briefly. Exponential and logarithm of matrices functions, whose elements belong to this algebra, are studied with the help of the SCHOONSCHIP and REDUCE 2 algebraic manipulators. (Author) [pt

Diagnosing students' misconceptions in algebra: results from an experimental pilot study.

Science.gov (United States)

Russell, Michael; O'Dwyer, Laura M; Miranda, Helena

2009-05-01

Computer-based diagnostic assessment systems hold potential to help teachers identify sources of poor performance and to connect teachers and students to learning activities designed to help advance students' conceptual understandings. The present article presents findings from a study that examined how students' performance in algebra and their overcoming of common algebraic misconceptions were affected by the use of a diagnostic assessment system that focused on important algebra concepts. This study used a four-group randomized cluster trial design in which teachers were assigned randomly to one of four groups: a "business as usual" control group, a partial intervention group that was provided with access to diagnostic tests results, a partial intervention group that was provided with access to the learning activities, and a full intervention group that was given access to the test results and learning activities. Data were collected from 905 students (6th-12th grade) nested within 44 teachers. We used hierarchical linear modeling techniques to compare the effects of full, partial, and no (control) intervention on students' algebraic ability and misconceptions. The analyses indicate that full intervention had a net positive effect on ability and misconception measures.

Hopf bifurcation of an (n + 1) -neuron bidirectional associative memory neural network model with delays.

Science.gov (United States)

Xiao, Min; Zheng, Wei Xing; Cao, Jinde

2013-01-01

Recent studies on Hopf bifurcations of neural networks with delays are confined to simplified neural network models consisting of only two, three, four, five, or six neurons. It is well known that neural networks are complex and large-scale nonlinear dynamical systems, so the dynamics of the delayed neural networks are very rich and complicated. Although discussing the dynamics of networks with a few neurons may help us to understand large-scale networks, there are inevitably some complicated problems that may be overlooked if simplified networks are carried over to large-scale networks. In this paper, a general delayed bidirectional associative memory neural network model with n + 1 neurons is considered. By analyzing the associated characteristic equation, the local stability of the trivial steady state is examined, and then the existence of the Hopf bifurcation at the trivial steady state is established. By applying the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction and stability of the bifurcating periodic solution. Furthermore, the paper highlights situations where the Hopf bifurcations are particularly critical, in the sense that the amplitude and the period of oscillations are very sensitive to errors due to tolerances in the implementation of neuron interconnections. It is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons. Numerical simulations are carried out to illustrate the main results.

Algebraic geometry

CERN Document Server

Lefschetz, Solomon

2005-01-01

An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.

Converting nested algebra expressions into flat algebra expressions

NARCIS (Netherlands)

Paredaens, J.; Van Gucht, D.

1992-01-01

Nested relations generalize ordinary flat relations by allowing tuple values to be either atomic or set valued. The nested algebra is a generalization of the flat relational algebra to manipulate nested relations. In this paper we study the expressive power of the nested algebra relative to its

Fibered F-Algebra

OpenAIRE

Kleyn, Aleks

2007-01-01

The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.

The bubble algebra: structure of a two-colour Temperley-Lieb Algebra

International Nuclear Information System (INIS)

Grimm, Uwe; Martin, Paul P

2003-01-01

We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley-Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang-Baxter equations

Conformal algebras of two-dimensional disordered systems

International Nuclear Information System (INIS)

Gurarie, Victor; Ludwig, Andreas W.W.

2002-01-01

We discuss the structure of two-dimensional conformal field theories at a central charge c=0 describing critical disordered systems, polymers and percolation. We construct a novel extension of the c=0 Virasoro algebra, characterized by a number b measuring the effective number of massless degrees of freedom, and by a logarithmic partner of the stress tensor. It is argued to be present at a generic random critical point, lacking super Kac-Moody, or other higher symmetries, and is a tool to describe and classify such theories. Interestingly, this algebra is not only consistent with, but indeed naturally accommodates in general an underlying global supersymmetry. Polymers and percolation realize this algebra. Unexpectedly, we find that the c=0 Kac table of the degenerate fields contains two distinct theories with b=5/6 and b=-5/8 which we conjecture to correspond to percolation and polymers, respectively. A given Kac-table field can be degenerate only in one of them. Remarkably, we also find this algebra, and thereby an ensuing hidden supersymmetry, realized at general replica-averaged critical points, for which we derive an explicit formula for b. (author). Letter-to-the-editor

Algebraic monoids, group embeddings, and algebraic combinatorics

CERN Document Server

Li, Zhenheng; Steinberg, Benjamin; Wang, Qiang

2014-01-01

This book contains a collection of fifteen articles and is dedicated to the sixtieth birthdays of Lex Renner and Mohan Putcha, the pioneers of the field of algebraic monoids.   Topics presented include:   v  structure and representation theory of reductive algebraic monoids v  monoid schemes and applications of monoids v  monoids related to Lie theory v  equivariant embeddings of algebraic groups v  constructions and properties of monoids from algebraic combinatorics v  endomorphism monoids induced from vector bundles v  Hodge–Newton decompositions of reductive monoids   A portion of these articles are designed to serve as a self-contained introduction to these topics, while the remaining contributions are research articles containing previously unpublished results, which are sure to become very influential for future work. Among these, for example, the important recent work of Michel Brion and Lex Renner showing that the algebraic semigroups are strongly π-regular.   Graduate students as well a...

Leavitt path algebras

CERN Document Server

Abrams, Gene; Siles Molina, Mercedes

2017-01-01

This book offers a comprehensive introduction by three of the leading experts in the field, collecting fundamental results and open problems in a single volume. Since Leavitt path algebras were first defined in 2005, interest in these algebras has grown substantially, with ring theorists as well as researchers working in graph C*-algebras, group theory and symbolic dynamics attracted to the topic. Providing a historical perspective on the subject, the authors review existing arguments, establish new results, and outline the major themes and ring-theoretic concepts, such as the ideal structure, Z-grading and the close link between Leavitt path algebras and graph C*-algebras. The book also presents key lines of current research, including the Algebraic Kirchberg Phillips Question, various additional classification questions, and connections to noncommutative algebraic geometry. Leavitt Path Algebras will appeal to graduate students and researchers working in the field and related areas, such as C*-algebras and...

Algebraic methods in random matrices and enumerative geometry

CERN Document Server

Eynard, Bertrand

2008-01-01

We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defined a sequence of differential forms, and a sequence of complex numbers Fg . We recall the definition of the invariants Fg, and we explain their main properties, in particular symplectic invariance, integrability, modularity,... Then, we give several example of applications, in particular matrix models, enumeration of discrete surfaces (maps), algebraic geometry and topological strings, non-intersecting brownian motions,...

The planar algebra of a semisimple and cosemisimple Hopf algebra

Indian Academy of Sciences (India)

M. Senthilkumar (Newgen Imaging) 1461 1996 Oct 15 13:05:22

The C(ounit) and T(race) relations. *. *. *. *. *. *. = = a1 a. 2 b b a a. Sa. Figure 4. The E(xchange) and A(ntipode) relations. and R being ... To prove the reverse inequality, we construct a functional λ±: P (L)0± → k which is non-trivial and show that it descends to the quotient P0± of P (L)0± by I (R)0± . The motivation for the ...

Hopf bifurcation of a ratio-dependent predator-prey system with time delay

International Nuclear Information System (INIS)

Celik, Canan

2009-01-01

In this paper, we consider a ratio dependent predator-prey system with time delay where the dynamics is logistic with the carrying capacity proportional to prey population. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the system based on the normal form approach and the center manifold theory. Finally, we illustrate our theoretical results by numerical simulations.

Approximation of complex algebraic numbers by algebraic numbers of bounded degree

OpenAIRE

Bugeaud, Yann; Evertse, Jan-Hendrik

2007-01-01

We investigate how well complex algebraic numbers can be approximated by algebraic numbers of degree at most n. We also investigate how well complex algebraic numbers can be approximated by algebraic integers of degree at most n+1. It follows from our investigations that for every positive integer n there are complex algebraic numbers of degree larger than n that are better approximable by algebraic numbers of degree at most n than almost all complex numbers. As it turns out, these numbers ar...

Nonlinear Characteristics of Randomly Excited Transonic Flutter

DEFF Research Database (Denmark)

Christiansen, Lasse Engbo; Lehn-Schiøler, Tue; Mosekilde, Erik

2002-01-01

. When this model is extended by the introduction of nonlinear terms, it can reproduce the subcritical Hopf bifurcation. We hereafter consider the effects of subjecting simplified versions of the model to random external excitations representing the fluctuations present in the airflow. These models can......The paper describes the effects of random external excitations on the onset and dynamical characteristics of transonic flutter (i.e. large-amplitude, self-sustained oscillations) for a high aspect ratio wing. Wind tunnel experiments performed at the National Aerospace Laboratory (NAL) in Japan have...

Operadic formulation of topological vertex algebras and gerstenhaber or Batalin-Vilkovisky algebras

International Nuclear Information System (INIS)

Huang Yizhi

1994-01-01

We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak topological vertex algebra) by combining this operadic formulation with a theorem of Getzler (or of Cohen) which formulates Batalin-Vilkovisky algebras (or Gerstenhaber algebras) in terms of the homology of the framed little disk operad (or of the little disk operad). (orig.)

Wn(2) algebras

International Nuclear Information System (INIS)

Feigin, B.L.; Semikhatov, A.M.

2004-01-01

We construct W-algebra generalizations of the sl-circumflex(2) algebra-W algebras W n (2) generated by two currents E and F with the highest pole of order n in their OPE. The n=3 term in this series is the Bershadsky-Polyakov W 3 (2) algebra. We define these algebras as a centralizer (commutant) of the Uqs-bar (n vertical bar 1) quantum supergroup and explicitly find the generators in a factored, 'Miura-like' form. Another construction of the W n (2) algebras is in terms of the coset sl-circumflex(n vertical bar 1)/sl-circumflex(n). The relation between the two constructions involves the 'duality' (k+n-1)(k'+n-1)=1 between levels k and k' of two sl-circumflex(n) algebras

Uniform in Time Description for Weak Solutions of the Hopf Equation with Nonconvex Nonlinearity

Directory of Open Access Journals (Sweden)

Antonio Olivas Martinez

2009-01-01

Full Text Available We consider the Riemann problem for the Hopf equation with concave-convex flux functions. Applying the weak asymptotics method we construct a uniform in time description for the Cauchy data evolution and show that the use of this method implies automatically the appearance of the Oleinik E-condition.

The vacuum preserving Lie algebra of a classical W-algebra

International Nuclear Information System (INIS)

Feher, L.; Tsutsui, I.

1993-07-01

We simplify and generalize an argument due to Bowcock and Watts showing that one can associate a finite Lie algebra (the 'classical vacuum preserving algebra') containing the Moebius sl(2) subalgebra to any classical W-algebra. Our construction is based on a kinematical analysis of the Poisson brackets of quasi-fields. In the case of the W S G -subalgebra S of a simple Lie algebra G, we exhibit a natural isomorphism between this finite Lie algebra and G whereby the Moebius sl(2) is identified with S. (orig.)

Nonflexible Lie-admissible algebras

International Nuclear Information System (INIS)

Myung, H.C.

1978-01-01

We discuss the structure of Lie-admissible algebras which are defined by nonflexible identities. These algebras largely arise from the antiflexible algebras, 2-varieties and associator dependent algebras. The nonflexible Lie-admissible algebras in our discussion are in essence byproducts of the study of nonassociative algebras defined by identities of degree 3. The main purpose is to discuss the classification of simple Lie-admissible algebras of nonflexible type

On 2-Banach algebras

International Nuclear Information System (INIS)

Mohammad, N.; Siddiqui, A.H.

1987-11-01

The notion of a 2-Banach algebra is introduced and its structure is studied. After a short discussion of some fundamental properties of bivectors and tensor product, several classical results of Banach algebras are extended to the 2-Banach algebra case. A condition under which a 2-Banach algebra becomes a Banach algebra is obtained and the relation between algebra of bivectors and 2-normed algebra is discussed. 11 refs

Algebra

CERN Document Server

Flanders, Harley

1975-01-01

Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a

Lukasiewicz-Moisil algebras

CERN Document Server

Boicescu, V; Georgescu, G; Rudeanu, S

1991-01-01

The Lukasiewicz-Moisil algebras were created by Moisil as an algebraic counterpart for the many-valued logics of Lukasiewicz. The theory of LM-algebras has developed to a considerable extent both as an algebraic theory of intrinsic interest and in view of its applications to logic and switching theory.This book gives an overview of the theory, comprising both classical results and recent contributions, including those of the authors. N-valued and &THgr;-valued algebras are presented, as well as &THgr;-algebras with negation.Mathematicians interested in lattice theory or symbolic logic, and computer scientists, will find in this monograph stimulating material for further research.

Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction

Science.gov (United States)

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…

Probabilistic Signal Recovery and Random Matrices

Science.gov (United States)

2016-12-08

that classical methods for linear regression (such as Lasso) are applicable for non- linear data. This surprising finding has already found several...we studied the complexity of convex sets. In numerical linear algebra , we analyzed the fastest known randomized approximation algorithm for...and perfect matchings In numerical linear algebra , we studied the fastest known randomized approximation algorithm for computing the permanents of

A Geometric Problem and the Hopf Lemma. Ⅱ

Institute of Scientific and Technical Information of China (English)

YanYan LI; Louis NIRENBERG

2006-01-01

A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in Rn+1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn+1 =constant in case M satisfies: for any two points (X′, Xn+1), (X′, ^Xn+1)on M, with Xn+1 ＞ ^Hn+1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part Ⅰ dealt with corresponding one dimensional problems.

Stability and Hopf bifurcation in a delayed competitive web sites model

International Nuclear Information System (INIS)

Xiao Min; Cao Jinde

2006-01-01

The delayed differential equations modeling competitive web sites, based on the Lotka-Volterra competition equations, are considered. Firstly, the linear stability is investigated. It is found that there is a stability switch for time delay, and Hopf bifurcation occurs when time delay crosses through a critical value. Then the direction and stability of the bifurcated periodic solutions are determined, using the normal form theory and the center manifold reduction. Finally, some numerical simulations are carried out to illustrate the results found

The Boolean algebra and central Galois algebras

Directory of Open Access Journals (Sweden)

George Szeto

2001-01-01

Full Text Available Let B be a Galois algebra with Galois group G, Jg={b∈B∣bx=g(xb   for all   x∈B} for g∈G, and BJg=Beg for a central idempotent eg. Then a relation is given between the set of elements in the Boolean algebra (Ba,≤ generated by {0,eg∣g∈G} and a set of subgroups of G, and a central Galois algebra Be with a Galois subgroup of G is characterized for an e∈Ba.

The global quantum duality principle: theory, examples, and applications

OpenAIRE

Gavarini, Fabio

2003-01-01

Let R be an integral domain, h non-zero in R such that R/hR is a field, and HA the category of torsionless (or flat) Hopf algebras over R. We call any H in HA "quantized function algebra" (=QFA), resp. "quantized (restricted) universal enveloping algebra" (=QrUEA), at h if H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. We establish an "inner" Galois' correspond...

Wavelets and quantum algebras

International Nuclear Information System (INIS)

Ludu, A.; Greiner, M.

1995-09-01

A non-linear associative algebra is realized in terms of translation and dilation operators, and a wavelet structure generating algebra is obtained. We show that this algebra is a q-deformation of the Fourier series generating algebra, and reduces to this for certain value of the deformation parameter. This algebra is also homeomorphic with the q-deformed su q (2) algebra and some of its extensions. Through this algebraic approach new methods for obtaining the wavelets are introduced. (author). 20 refs

Novikov-Jordan algebras

OpenAIRE

Dzhumadil'daev, A. S.

2002-01-01

Algebras with identity $(a\\star b)\\star (c\\star d) -(a\\star d)\\star(c\\star b)$ $=(a,b,c)\\star d-(a,d,c)\\star b$ are studied. Novikov algebras under Jordan multiplication and Leibniz dual algebras satisfy this identity. If algebra with such identity has unit, then it is associative and commutative.

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

Science.gov (United States)

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)

Higher order differential calculus on SLq(N)

International Nuclear Information System (INIS)

Heckenberger, I.; Schueler, A.

1997-01-01

Let Γ be a bicovariant first order differential calculus on a Hopf algebra A. There are three possibilities to construct a differential N 0 -graded Hopf algebra Γcirconflex which contains Γ as its first order part. In all cases Γcirconflex is a quotient Γcirconflex = Γ x /J of the tensor algebra by some suitable ideal. We distinguish three possible choices u J, s J, and w J, where the first one generates the universal differential calculus (over Γ) and the last one is Woronowicz' external algebra. Let q be a transcendental complex number and let Γ be one of the N 2 -dimensional bicovariant first order differential calculi on the quantum group SL q (N). Then for N ≥ 3 the three ideals coincide. For Woronowicz' external algebra we calculate the dimensions of the spaces of left-invariant and bi-invariant k-forms. In this case each bi-invariant form is closed. In case of 4D ± calculi on SL q (2) the universal calculus is strictly larger than the other two calculi. In particular, the bi-invariant 1-form is not closed. (author)

Noise-induced transitions at a Hopf bifurcation in a first-order delay-differential equation

International Nuclear Information System (INIS)

Longtin, A.

1991-01-01

The influence of colored noise on the Hopf bifurcation in a first-order delay-differential equation (DDE), a model paradigm for nonlinear delayed feedback systems, is considered. First, it is shown, using a stability analysis, how the properties of the DDE depend on the ratio R of system delay to response time. When this ratio is small, the DDE behaves more like a low-dimensional system of ordinary differential equations (ODE's); when R is large, one obtains a singular perturbation limit in which the behavior of the DDE approaches that of a discrete time map. The relative magnitude of the additive and multiplicative noise-induced postponements of the Hopf bifurcation are numerically shown to depend on the ratio R. Although both types of postponements are minute in the large-R limit, they are almost equal due to an equivalence of additive and parametric noise for discrete time maps. When R is small, the multiplicative shift is larger than the additive one at large correlation times, but the shifts are equal for small correlation times. In fact, at constant noise power, the postponement is only slightly affected by the correlation time of the noise, except when the noise becomes white, in which case the postponement drastically decreases. This is a numerical study of the stochastic Hopf bifurcation, in ODE's or DDE's, that looks at the effect of noise correlation time at constant power. Further, it is found that the slope at the fixed point averaged over the stochastic-parameter motion acts, under certain conditions, as a quantitative indicator of oscillation onset in the presence of noise. The problem of how properties of the DDE carry over to ODE's and to maps is discussed, along with the proper theoretical framework in which to study nonequilibrium phase transitions in this class of functional differential equations

(Quasi-)Poisson enveloping algebras

OpenAIRE

Yang, Yan-Hong; Yao, Yuan; Ye, Yu

2010-01-01

We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.

Iterated Leavitt Path Algebras

International Nuclear Information System (INIS)

Hazrat, R.

2009-11-01

Leavitt path algebras associate to directed graphs a Z-graded algebra and in their simplest form recover the Leavitt algebras L(1,k). In this note, we introduce iterated Leavitt path algebras associated to directed weighted graphs which have natural ± Z grading and in their simplest form recover the Leavitt algebras L(n,k). We also characterize Leavitt path algebras which are strongly graded. (author)

Mixed-Mode Oscillations Due to a Singular Hopf Bifurcation in a Forest Pest Model

DEFF Research Database (Denmark)

Brøns, Morten; Desroches, Mathieu; Krupa, Martin

2015-01-01

In a forest pest model, young trees are distinguished from old trees. The pest feeds on old trees. The pest grows on a fast scale, the young trees on an intermediate scale, and the old trees on a slow scale. A combination of a singular Hopf bifurcation and a “weak return” mechanism, characterized...

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

Science.gov (United States)

Lawrence, Lettie Carol

1997-08-01

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

Free probability and random matrices

CERN Document Server

Mingo, James A

2017-01-01

This volume opens the world of free probability to a wide variety of readers. From its roots in the theory of operator algebras, free probability has intertwined with non-crossing partitions, random matrices, applications in wireless communications, representation theory of large groups, quantum groups, the invariant subspace problem, large deviations, subfactors, and beyond. This book puts a special emphasis on the relation of free probability to random matrices, but also touches upon the operator algebraic, combinatorial, and analytic aspects of the theory. The book serves as a combination textbook/research monograph, with self-contained chapters, exercises scattered throughout the text, and coverage of important ongoing progress of the theory. It will appeal to graduate students and all mathematicians interested in random matrices and free probability from the point of view of operator algebras, combinatorics, analytic functions, or applications in engineering and statistical physics.

Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks.

Science.gov (United States)

Wang, Zhen; Campbell, Sue Ann

2017-11-01

We consider the networks of N identical oscillators with time delayed, global circulant coupling, modeled by a system of delay differential equations with Z N symmetry. We first study the existence of Hopf bifurcations induced by the coupling time delay and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to a case study: a network of FitzHugh-Nagumo neurons with diffusive coupling. For this model, we derive the asymptotic stability, global asymptotic stability, absolute instability, and stability switches of the equilibrium point in the plane of coupling time delay (τ) and excitability parameter (a). We investigate the patterns of cluster oscillations induced by the time delay and determine the direction and stability of the bifurcating periodic orbits by employing the multiple timescales method and normal form theory. We find that in the region where stability switching occurs, the dynamics of the system can be switched from the equilibrium point to any symmetric cluster oscillation, and back to equilibrium point as the time delay is increased.

Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks

Science.gov (United States)

Wang, Zhen; Campbell, Sue Ann

2017-11-01

We consider the networks of N identical oscillators with time delayed, global circulant coupling, modeled by a system of delay differential equations with ZN symmetry. We first study the existence of Hopf bifurcations induced by the coupling time delay and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to a case study: a network of FitzHugh-Nagumo neurons with diffusive coupling. For this model, we derive the asymptotic stability, global asymptotic stability, absolute instability, and stability switches of the equilibrium point in the plane of coupling time delay (τ) and excitability parameter (a). We investigate the patterns of cluster oscillations induced by the time delay and determine the direction and stability of the bifurcating periodic orbits by employing the multiple timescales method and normal form theory. We find that in the region where stability switching occurs, the dynamics of the system can be switched from the equilibrium point to any symmetric cluster oscillation, and back to equilibrium point as the time delay is increased.

Algebraic topological entropy

International Nuclear Information System (INIS)

Hudetz, T.

1989-01-01

As a 'by-product' of the Connes-Narnhofer-Thirring theory of dynamical entropy for (originally non-Abelian) nuclear C * -algebras, the well-known variational principle for topological entropy is eqivalently reformulated in purly algebraically defined terms for (separable) Abelian C * -algebras. This 'algebraic variational principle' should not only nicely illustrate the 'feed-back' of methods developed for quantum dynamical systems to the classical theory, but it could also be proved directly by 'algebraic' methods and could thus further simplify the original proof of the variational principle (at least 'in principle'). 23 refs. (Author)

Linearizing W-algebras

International Nuclear Information System (INIS)

Krivonos, S.O.; Sorin, A.S.

1994-06-01

We show that the Zamolodchikov's and Polyakov-Bershadsky nonlinear algebras W 3 and W (2) 3 can be embedded as subalgebras into some linear algebras with finite set of currents. Using these linear algebras we find new field realizations of W (2) 3 and W 3 which could be a starting point for constructing new versions of W-string theories. We also reveal a number of hidden relationships between W 3 and W (2) 3 . We conjecture that similar linear algebras can exist for other W-algebra as well. (author). 10 refs

Quantum deformed su(mvertical stroke n) algebra and superconformal algebra on quantum superspace

International Nuclear Information System (INIS)

Kobayashi, Tatsuo

1993-01-01

We study a deformed su(mvertical stroke n) algebra on a quantum superspace. Some interesting aspects of the deformed algebra are shown. As an application of the deformed algebra we construct a deformed superconformal algebra. From the deformed su(1vertical stroke 4) algebra, we derive deformed Lorentz, translation of Minkowski space, iso(2,2) and its supersymmetric algebras as closed subalgebras with consistent automorphisms. (orig.)

Linear algebraic groups

CERN Document Server

Springer, T A

1998-01-01

"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style… [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature…appear for the first time in a text." –Mathematical Reviews (Review of the Second Edition) "This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of t...

Extended conformal algebras

International Nuclear Information System (INIS)

Goddard, Peter

1990-01-01

The algebra of the group of conformal transformations in two dimensions consists of two commuting copies of the Virasoro algebra. In many mathematical and physical contexts, the representations of ν which are relevant satisfy two conditions: they are unitary and they have the ''positive energy'' property that L o is bounded below. In an irreducible unitary representation the central element c takes a fixed real value. In physical contexts, the value of c is a characteristic of a theory. If c < 1, it turns out that the conformal algebra is sufficient to ''solve'' the theory, in the sense of relating the calculation of the infinite set of physically interesting quantities to a finite subset which can be handled in principle. For c ≥ 1, this is no longer the case for the algebra alone and one needs some sort of extended conformal algebra, such as the superconformal algebra. It is these algebras that this paper aims at addressing. (author)

Generalized symmetry algebras

International Nuclear Information System (INIS)

Dragon, N.

1979-01-01

The possible use of trilinear algebras as symmetry algebras for para-Fermi fields is investigated. The shortcomings of the examples are argued to be a general feature of such generalized algebras. (author)

Directed Abelian algebras and their application to stochastic models.

Science.gov (United States)

Alcaraz, F C; Rittenberg, V

2008-10-01

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

Galilean contractions of W-algebras

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Jørgen Rasmussen

2017-09-01

Full Text Available Infinite-dimensional Galilean conformal algebras can be constructed by contracting pairs of symmetry algebras in conformal field theory, such as W-algebras. Known examples include contractions of pairs of the Virasoro algebra, its N=1 superconformal extension, or the W3 algebra. Here, we introduce a contraction prescription of the corresponding operator-product algebras, or equivalently, a prescription for contracting tensor products of vertex algebras. With this, we work out the Galilean conformal algebras arising from contractions of N=2 and N=4 superconformal algebras as well as of the W-algebras W(2,4, W(2,6, W4, and W5. The latter results provide evidence for the existence of a whole new class of W-algebras which we call Galilean W-algebras. We also apply the contraction prescription to affine Lie algebras and find that the ensuing Galilean affine algebras admit a Sugawara construction. The corresponding central charge is level-independent and given by twice the dimension of the underlying finite-dimensional Lie algebra. Finally, applications of our results to the characterisation of structure constants in W-algebras are proposed.

Quantum affine algebras and deformations of the virasoro and W-algebras

International Nuclear Information System (INIS)

Frenkel, E.; Reshetikhin, N.

1996-01-01

Using the Wakimoto realization of quantum affine algebras we define new Poisson algebras, which are q-deformations of the classical W-algebras. We also define their free field realizations, i.e. homomorphisms into some Heisenberg-Poisson algebras. The formulas for these homomorphisms coincide with formulas for spectra of transfer-matrices in the corresponding quantum integrable models derived by the Bethe-Ansatz method. (orig.)

Algebraic entropy for algebraic maps

International Nuclear Information System (INIS)

Hone, A N W; Ragnisco, Orlando; Zullo, Federico

2016-01-01

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Bäcklund transformations. (letter)

On hyper BCC-algebras

OpenAIRE

Borzooei, R. A.; Dudek, W. A.; Koohestani, N.

2006-01-01

We study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.

On hyper BCC-algebras

Directory of Open Access Journals (Sweden)

R. A. Borzooei

2006-01-01

Full Text Available We study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.

Applications of differential algebra to single-particle dynamics in storage rings

International Nuclear Information System (INIS)

Yan, Y.

1991-09-01

Recent developments in the use of differential algebra to study single-particle beam dynamics in charged-particle storage rings are the subject of this paper. Chapter 2 gives a brief review of storage rings. The concepts of betatron motion and synchrotron motion, and their associated resonances, are introduced. Also introduced are the concepts of imperfections, such as off-momentum, misalignment, and random and systematic errors, and their associated corrections. The chapter concludes with a discussion of numerical simulation principles and the concept of one-turn periodic maps. In Chapter 3, the discussion becomes more focused with the introduction of differential algebras. The most critical test for differential algebraic mapping techniques -- their application to long-term stability studies -- is discussed in Chapter 4. Chapter 5 presents a discussion of differential algebraic treatment of dispersed betatron motion. The paper concludes in Chapter 6 with a discussion of parameterization of high-order maps

Algebraic theory of numbers

CERN Document Server

Samuel, Pierre

2008-01-01

Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Gal

Topology-preserving quantum deformation with non-numerical parameter

Science.gov (United States)

Aukhadiev, Marat; Grigoryan, Suren; Lipacheva, Ekaterina

2013-11-01

We introduce a class of compact quantum semigroups, that we call semigroup deformations of compact Abelian qroups. These objects arise from reduced semigroup -algebras, the generalization of the Toeplitz algebra. We study quantum subgroups, quantum projective spaces and quantum quotient groups for such objects, and show that the group is contained as a compact quantum subgroup in the deformation of itself. The connection with the weak Hopf algebra notion is described. We give a grading on the -algebra of the compact quantum semigroups constructed.

Construction of Nijenhuis operators and dendriform trialgebras

Directory of Open Access Journals (Sweden)

Philippe Leroux

2004-01-01

Full Text Available We construct Nijenhuis operators from particular bialgebras called dendriform-Nijenhuis bialgebras. It turns out that such Nijenhuis operators commute with TD-operators, a kind of Baxter-Rota operators, and are therefore closely related to dendriform trialgebras. This allows the construction of associative algebras, called dendriform-Nijenhuis algebras, made out of nine operations and presenting an exotic combinatorial property. We also show that the augmented free dendriform-Nijenhuis algebra and its commutative version have a structure of connected Hopf algebras. Examples are given.

The BRS algebra of a free differential algebra

International Nuclear Information System (INIS)

Boukraa, S.

1987-04-01

We construct in this work, the Weil and the universal BRS algebras of theories that can have as a gauge symmetry a free differential (Sullivan) algebra, the natural extension of Lie algebras allowing the definition of p-form gauge potentials (p>1). The finite gauge transformations of these potentials are deduced from the infinitesimal ones and the group structure is shown. The geometrical meaning of these p-form gauge potentials is given by the notion of a Quillen superconnection. (author). 19 refs

Pseudo-Riemannian Novikov algebras

Energy Technology Data Exchange (ETDEWEB)

Chen Zhiqi; Zhu Fuhai [School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 (China)], E-mail: chenzhiqi@nankai.edu.cn, E-mail: zhufuhai@nankai.edu.cn

2008-08-08

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.

On the PR-algebras

International Nuclear Information System (INIS)

Lebedenko, V.M.

1978-01-01

The PR-algebras, i.e. the Lie algebras with commutation relations of [Hsub(i),Hsub(j)]=rsub(ij)Hsub(i)(i< j) type are investigated. On the basis of former results a criterion for the membership of 2-solvable Lie algebras to the PR-algebra class is given. The conditions imposed by the criterion are formulated in the linear algebra language

Introduction to W-algebras

International Nuclear Information System (INIS)

Takao, Masaru

1989-01-01

We review W-algebras which are generated by stress tensor and primary fields. Associativity plays an important role in determining the extended algebra and further implies the algebras to exist for special values of central charges. Explicitly constructing the algebras including primary fields of spin less than 4, we investigate the closure structure of the Jacobi identity of the extended algebras. (author)

(Modular Effect Algebras are Equivalent to (Frobenius Antispecial Algebras

Directory of Open Access Journals (Sweden)

Dusko Pavlovic

2017-01-01

Full Text Available Effect algebras are one of the generalizations of Boolean algebras proposed in the quest for a quantum logic. Frobenius algebras are a tool of categorical quantum mechanics, used to present various families of observables in abstract, often nonstandard frameworks. Both effect algebras and Frobenius algebras capture their respective fragments of quantum mechanics by elegant and succinct axioms; and both come with their conceptual mysteries. A particularly elegant and mysterious constraint, imposed on Frobenius algebras to characterize a class of tripartite entangled states, is the antispecial law. A particularly contentious issue on the quantum logic side is the modularity law, proposed by von Neumann to mitigate the failure of distributivity of quantum logical connectives. We show that, if quantum logic and categorical quantum mechanics are formalized in the same framework, then the antispecial law of categorical quantum mechanics corresponds to the natural requirement of effect algebras that the units are each other's unique complements; and that the modularity law corresponds to the Frobenius condition. These correspondences lead to the equivalence announced in the title. Aligning the two formalisms, at the very least, sheds new light on the concepts that are more clearly displayed on one side than on the other (such as e.g. the orthogonality. Beyond that, it may also open up new approaches to deep and important problems of quantum mechanics (such as the classification of complementary observables.

An algorithm to construct the basic algebra of a skew group algebra

NARCIS (Netherlands)

Horobeţ, E.

2016-01-01

We give an algorithm for the computation of the basic algebra Morita equivalent to a skew group algebra of a path algebra by obtaining formulas for the number of vertices and arrows of the new quiver Qb. We apply this algorithm to compute the basic algebra corresponding to all simple quaternion

Intervals in Generalized Effect Algebras and their Sub-generalized Effect Algebras

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Zdenka Riečanová

2013-01-01

Full Text Available We consider subsets G of a generalized effect algebra E with 0∈G and such that every interval [0, q]G = [0, q]E ∩ G of G (q ∈ G , q ≠ 0 is a sub-effect algebra of the effect algebra [0, q]E. We give a condition on E and G under which every such G is a sub-generalized effect algebra of E.

The C*-algebra of a vector bundle and fields of Cuntz algebras

OpenAIRE

Vasselli, Ezio

2004-01-01

We study the Pimsner algebra associated with the module of continuous sections of a Hilbert bundle, and prove that it is a continuous bundle of Cuntz algebras. We discuss the role of such Pimsner algebras w.r.t. the notion of inner endomorphism. Furthermore, we study bundles of Cuntz algebras carrying a global circle action, and assign to them a class in the representable KK-group of the zero-grade bundle. We compute such class for the Pimsner algebra of a vector bundle.

Simple relation algebras

CERN Document Server

Givant, Steven

2017-01-01

This monograph details several different methods for constructing simple relation algebras, many of which are new with this book. By drawing these seemingly different methods together, all are shown to be aspects of one general approach, for which several applications are given. These tools for constructing and analyzing relation algebras are of particular interest to mathematicians working in logic, algebraic logic, or universal algebra, but will also appeal to philosophers and theoretical computer scientists working in fields that use mathematics. The book is written with a broad audience in mind and features a careful, pedagogical approach; an appendix contains the requisite background material in relation algebras. Over 400 exercises provide ample opportunities to engage with the material, making this a monograph equally appropriate for use in a special topics course or for independent study. Readers interested in pursuing an extended background study of relation algebras will find a comprehensive treatme...

Algebraic polynomials with random coefficients

Directory of Open Access Journals (Sweden)

K. Farahmand

2002-01-01

Full Text Available This paper provides an asymptotic value for the mathematical expected number of points of inflections of a random polynomial of the form a0(ω+a1(ω(n11/2x+a2(ω(n21/2x2+…an(ω(nn1/2xn when n is large. The coefficients {aj(w}j=0n, w∈Ω are assumed to be a sequence of independent normally distributed random variables with means zero and variance one, each defined on a fixed probability space (A,Ω,Pr. A special case of dependent coefficients is also studied.

Boolean algebra

CERN Document Server

Goodstein, R L

2007-01-01

This elementary treatment by a distinguished mathematician employs Boolean algebra as a simple medium for introducing important concepts of modern algebra. Numerous examples appear throughout the text, plus full solutions.

Combinatorial commutative algebra

CERN Document Server

Miller, Ezra

2005-01-01

Offers an introduction to combinatorial commutative algebra, focusing on combinatorial techniques for multigraded polynomial rings, semigroup algebras, and determined rings. The chapters in this work cover topics ranging from homological invariants of monomial ideals and their polyhedral resolutions, to tools for studying algebraic varieties.

Topological أ-algebras with Cأ-enveloping algebras II

Indian Academy of Sciences (India)

necessarily complete) pro-Cأ-topology which coincides with the relative uniform .... problems in Cأ-algebras, Phillips introduced more general weakly Cأ- .... Banach أ-algebra obtained by completing A=Np in the norm jjxpjjp ¼ pًxق where.

C*-algebras by example

CERN Document Server

Davidson, Kenneth R

1996-01-01

The subject of C*-algebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of Brown, Douglas, and Fillmore on extensions of C*-algebras and Elliott's use of K-theory to provide a useful classification of AF algebras. These results were the beginning of a marvelous new set of tools for analyzing concrete C*-algebras. This book is an introductory graduate level text which presents the basics of the subject through a detailed analysis of several important classes of C*-algebras. The development of operator algebras in the last twenty yea

Noncommutative geometry and twisted conformal symmetry

International Nuclear Information System (INIS)

Matlock, Peter

2005-01-01

The twist-deformed conformal algebra is constructed as a Hopf algebra with twisted coproduct. This allows for the definition of conformal symmetry in a noncommutative background geometry. The twisted coproduct is reviewed for the Poincare algebra and the construction is then extended to the full conformal algebra. The case of Moyal-type noncommutativity of the coordinates is considered. It is demonstrated that conformal invariance need not be viewed as incompatible with noncommutative geometry; the noncommutativity of the coordinates appears as a consequence of the twisting, as has been shown in the literature in the case of the twisted Poincare algebra

Non-freely generated W-algebras and construction of N=2 super W-algebras

International Nuclear Information System (INIS)

Blumenhagen, R.

1994-07-01

Firstly, we investigate the origin of the bosonic W-algebras W(2, 3, 4, 5), W(2, 4, 6) and W(2, 4, 6) found earlier by direct construction. We present a coset construction for all three examples leading to a new type of finitely, non-freely generated quantum W-algebras, which we call unifying W-algebras. Secondly, we develop a manifest covariant formalism to construct N = 2 super W-algebras explicitly on a computer. Applying this algorithm enables us to construct the first four examples of N = 2 super W-algebras with two generators and the N = 2 super W 4 algebra involving three generators. The representation theory of the former ones shows that all examples could be divided into four classes, the largest one containing the N = 2 special type of spectral flow algebras. Besides the W-algebra of the CP(3) Kazama-Suzuki coset model, the latter example with three generators discloses a second solution which could also be explained as a unifying W-algebra for the CP(n) models. (orig.)

Q-creation and annihilation tensors for the two parameters deformation of U(SU(2))

International Nuclear Information System (INIS)

Wehrhahn, R.F.; Vraceanu, D.

1993-03-01

The Jordan-Schwinger construction for the Hopf algebra U qp (su(2)) is realized. The creation and annihilation tensor operators together with their tensor products including the Casimir operators are calculated. (orig.)

Quantum field theory and coalgebraic logic in theoretical computer science.

Science.gov (United States)

Basti, Gianfranco; Capolupo, Antonio; Vitiello, Giuseppe

2017-11-01

We suggest that in the framework of the Category Theory it is possible to demonstrate the mathematical and logical dual equivalence between the category of the q-deformed Hopf Coalgebras and the category of the q-deformed Hopf Algebras in quantum field theory (QFT), interpreted as a thermal field theory. Each pair algebra-coalgebra characterizes a QFT system and its mirroring thermal bath, respectively, so to model dissipative quantum systems in far-from-equilibrium conditions, with an evident significance also for biological sciences. Our study is in fact inspired by applications to neuroscience where the brain memory capacity, for instance, has been modeled by using the QFT unitarily inequivalent representations. The q-deformed Hopf Coalgebras and the q-deformed Hopf Algebras constitute two dual categories because characterized by the same functor T, related with the Bogoliubov transform, and by its contravariant application T op , respectively. The q-deformation parameter is related to the Bogoliubov angle, and it is effectively a thermal parameter. Therefore, the different values of q identify univocally, and label the vacua appearing in the foliation process of the quantum vacuum. This means that, in the framework of Universal Coalgebra, as general theory of dynamic and computing systems ("labelled state-transition systems"), the so labelled infinitely many quantum vacua can be interpreted as the Final Coalgebra of an "Infinite State Black-Box Machine". All this opens the way to the possibility of designing a new class of universal quantum computing architectures based on this coalgebraic QFT formulation, as its ability of naturally generating a Fibonacci progression demonstrates. Copyright © 2017 Elsevier Ltd. All rights reserved.

Algebraic conformal field theory

International Nuclear Information System (INIS)

Fuchs, J.; Nationaal Inst. voor Kernfysica en Hoge-Energiefysica

1991-11-01

Many conformal field theory features are special versions of structures which are present in arbitrary 2-dimensional quantum field theories. So it makes sense to describe 2-dimensional conformal field theories in context of algebraic theory of superselection sectors. While most of the results of the algebraic theory are rather abstract, conformal field theories offer the possibility to work out many formulae explicitly. In particular, one can construct the full algebra A-bar of global observables and the endomorphisms of A-bar which represent the superselection sectors. Some explicit results are presented for the level 1 so(N) WZW theories; the algebra A-bar is found to be the enveloping algebra of a Lie algebra L-bar which is an extension of the chiral symmetry algebra of the WZW theory. (author). 21 refs., 6 figs

Boolean algebra essentials

CERN Document Server

Solomon, Alan D

2012-01-01

REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Boolean Algebra includes set theory, sentential calculus, fundamental ideas of Boolean algebras, lattices, rings and Boolean algebras, the structure of a Boolean algebra, and Boolean

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

Science.gov (United States)

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

2018-03-01

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

On the exterior structure of graphs

International Nuclear Information System (INIS)

Kastler, Daniel

2004-01-01

After a detailed ab initio description of the exterior structure of graphs as handled by Connes and Kreimer in their work on renormalization (illustrated by the example of the φ 3 model in six dimensions) we spell out in detail their study of the Lie algebra of infinitesimal characters and of the group of characters of the Hopf algebra of Feynman graphs

Hopf bifurcation in a reaction-diffusive two-species model with nonlocal delay effect and general functional response

International Nuclear Information System (INIS)

Han, Renji; Dai, Binxiang

2017-01-01

Highlights: • We model general two-dimensional reaction-diffusion with nonlocal delay. • The existence of unique positive steady state is studied. • The bilinear form for the proposed system is given. • The existence, direction of Hopf bifurcation are given by symmetry method. - Abstract: A nonlocal delayed reaction-diffusive two-species model with Dirichlet boundary condition and general functional response is investigated in this paper. Based on the Lyapunov–Schmidt reduction, the existence, bifurcation direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady-state solution are obtained, where the time delay is taken as the bifurcation parameter. Moreover, the general results are applied to a diffusive Lotka–Volterra type food-limited population model with nonlocal delay effect, and it is found that diffusion and nonlocal delay can also affect the other dynamic behavior of the system by numerical experiments.

Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain

International Nuclear Information System (INIS)

Wang Fengyan; Pang Guoping

2008-01-01

In this paper, we propose and study a model of a hybrid ratio-dependent three species food chain, which is constituted by a hybrid type subsystem of prey and middle-predator and a middle-top predators' subsystem with Holling type-II functional response. We investigate the persistence and Hopf bifurcation of the system. Computer simulations are carried out to explain the mathematical conclusions. The chaotic attractor is obtained for suitable choice of parametric values

Continuum analogues of contragredient Lie algebras

International Nuclear Information System (INIS)

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

1989-03-01

We present an axiomatic formulation of a new class of infinite-dimensional Lie algebras - the generalizations of Z-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras ''continuum Lie algebras''. The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential Cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered. (author). 9 refs

Hopf bifurcation of a free boundary problem modeling tumor growth with two time delays

International Nuclear Information System (INIS)

Xu Shihe

2009-01-01

In this paper, a free boundary problem modeling tumor growth with two discrete delays is studied. The delays respectively represents the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis. We show the influence of time delays on the Hopf bifurcation when one of delays as a bifurcation parameter.

Lectures on algebraic statistics

CERN Document Server

Drton, Mathias; Sullivant, Seth

2009-01-01

How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.

Quiver W-algebras

Science.gov (United States)

Kimura, Taro; Pestun, Vasily

2018-06-01

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

Abstract algebra

CERN Document Server

Garrett, Paul B

2007-01-01

Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal

College algebra

CERN Document Server

Kolman, Bernard

1985-01-01

College Algebra, Second Edition is a comprehensive presentation of the fundamental concepts and techniques of algebra. The book incorporates some improvements from the previous edition to provide a better learning experience. It provides sufficient materials for use in the study of college algebra. It contains chapters that are devoted to various mathematical concepts, such as the real number system, the theory of polynomial equations, exponential and logarithmic functions, and the geometric definition of each conic section. Progress checks, warnings, and features are inserted. Every chapter c

Pre-Algebra Essentials For Dummies

CERN Document Server

Zegarelli, Mark

2010-01-01

Many students worry about starting algebra. Pre-Algebra Essentials For Dummies provides an overview of critical pre-algebra concepts to help new algebra students (and their parents) take the next step without fear. Free of ramp-up material, Pre-Algebra Essentials For Dummies contains content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical pre-algebra course, from fractions, decimals, and percents to scientific notation and simple variable equations. This guide is also a perfect reference for parents who need to review critical pre-algebra

Representations of quantum bicrossproduct algebras

International Nuclear Information System (INIS)

Arratia, Oscar; Olmo, Mariano A del

2002-01-01

We present a method to construct induced representations of quantum algebras which have a bicrossproduct structure. We apply this procedure to some quantum kinematical algebras in (1+1) dimensions with this kind of structure: null-plane quantum Poincare algebra, non-standard quantum Galilei algebra and quantum κ-Galilei algebra

On control of Hopf bifurcation in time-delayed neural network system

International Nuclear Information System (INIS)

Zhou Shangbo; Liao Xiaofeng; Yu Juebang; Wong Kwokwo

2005-01-01

The control of Hopf bifurcations in neural network systems is studied in this Letter. The asymptotic stability theorem and the relevant corollary for linearized nonlinear dynamical systems are proven. In particular, a novel method for analyzing the local stability of a dynamical system with time-delay is suggested. For the time-delayed system consisting of one or two neurons, a washout filter based control model is proposed and analyzed. By employing the stability theorems derived, we investigate the stability of a control system and state the relevant theorems for choosing the parameters of the stabilized control system

Algorithms in Algebraic Geometry

CERN Document Server

Dickenstein, Alicia; Sommese, Andrew J

2008-01-01

In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its

Computer algebra and operators

Science.gov (United States)

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.

Abstract Algebra to Secondary School Algebra: Building Bridges

Science.gov (United States)

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…

Infinite dimension algebra and conformal symmetry

International Nuclear Information System (INIS)

Ragoucy-Aubezon, E.

1991-04-01

A generalisation of Kac-Moody algebras (current algebras defined on a circle) to algebras defined on a compact supermanifold of any dimension and with any number of supersymmetries is presented. For such a purpose, we compute all the central extensions of loop algebras defined on this supermanifold, i.e. all the cohomology classes of these loop algebras. Then, we try to extend the relation (i.e. semi-direct sum) that exists between the two dimensional conformal algebras (called Virasoro algebra) and the usual Kac-Moody algebras, by considering the derivation algebra of our extended Kac-Moody algebras. The case of superconformal algebras (used in superstrings theories) is treated, as well as the cases of area-preserving diffeomorphisms (used in membranes theories), and Krichever-Novikov algebras (used for interacting strings). Finally, we present some generalizations of the Sugawara construction to the cases of extended Kac-Moody algebras, and Kac-Moody of superalgebras. These constructions allow us to get new realizations of the Virasoro, and Ramond, Neveu-Schwarz algebras

Equivalency of two-dimensional algebras

International Nuclear Information System (INIS)

Santos, Gildemar Carneiro dos; Pomponet Filho, Balbino Jose S.

2011-01-01

Full text: Let us consider a vector z = xi + yj over the field of real numbers, whose basis (i,j) satisfy a given algebra. Any property of this algebra will be reflected in any function of z, so we can state that the knowledge of the properties of an algebra leads to more general conclusions than the knowledge of the properties of a function. However structural properties of an algebra do not change when this algebra suffers a linear transformation, though the structural constants defining this algebra do change. We say that two algebras are equivalent to each other whenever they are related by a linear transformation. In this case, we have found that some relations between the structural constants are sufficient to recognize whether or not an algebra is equivalent to another. In spite that the basis transform linearly, the structural constants change like a third order tensor, but some combinations of these tensors result in a linear transformation, allowing to write the entries of the transformation matrix as function of the structural constants. Eventually, a systematic way to find the transformation matrix between these equivalent algebras is obtained. In this sense, we have performed the thorough classification of associative commutative two-dimensional algebras, and find that even non-division algebra may be helpful in solving non-linear dynamic systems. The Mandelbrot set was used to have a pictorial view of each algebra, since equivalent algebras result in the same pattern. Presently we have succeeded in classifying some non-associative two-dimensional algebras, a task more difficult than for associative one. (author)

Complex quantum groups

International Nuclear Information System (INIS)

Drabant, B.; Schlieker, M.

1993-01-01

The complex quantum groups are constructed. They are q-deformations of the real Lie groups which are obtained as the complex groups corresponding to the Lie algebras of type A n-1 , B n , C n . Following the ideas of Faddeev, Reshetikhin and Takhtajan Hopf algebras of regular functionals U R for these complexified quantum groups are constructed. One has thus in particular found a construction scheme for the q-Lorentz algebra to be identified as U(sl q (2,C). (orig.)

Axis Problem of Rough 3-Valued Algebras

Institute of Scientific and Technical Information of China (English)

Jianhua Dai; Weidong Chen; Yunhe Pan

2006-01-01

The collection of all the rough sets of an approximation space has been given several algebraic interpretations, including Stone algebras, regular double Stone algebras, semi-simple Nelson algebras, pre-rough algebras and 3-valued Lukasiewicz algebras. A 3-valued Lukasiewicz algebra is a Stone algebra, a regular double Stone algebra, a semi-simple Nelson algebra, a pre-rough algebra. Thus, we call the algebra constructed by the collection of rough sets of an approximation space a rough 3-valued Lukasiewicz algebra. In this paper,the rough 3-valued Lukasiewicz algebras, which are a special kind of 3-valued Lukasiewicz algebras, are studied. Whether the rough 3-valued Lukasiewicz algebra is a axled 3-valued Lukasiewicz algebra is examined.

Enveloping σ-C C C-algebra of a smooth Frechet algebra crossed ...

Indian Academy of Sciences (India)

Home; Journals; Proceedings – Mathematical Sciences; Volume 116; Issue 2. Enveloping -*-Algebra of a Smooth Frechet Algebra Crossed Product by R R , K -Theory and Differential Structure in *-Algebras. Subhash J Bhatt. Regular Articles Volume 116 Issue 2 May 2006 pp 161-173 ...

Hecke algebras with unequal parameters

CERN Document Server

Lusztig, G

2003-01-01

Hecke algebras arise in representation theory as endomorphism algebras of induced representations. One of the most important classes of Hecke algebras is related to representations of reductive algebraic groups over p-adic or finite fields. In 1979, in the simplest (equal parameter) case of such Hecke algebras, Kazhdan and Lusztig discovered a particular basis (the KL-basis) in a Hecke algebra, which is very important in studying relations between representation theory and geometry of the corresponding flag varieties. It turned out that the elements of the KL-basis also possess very interesting combinatorial properties. In the present book, the author extends the theory of the KL-basis to a more general class of Hecke algebras, the so-called algebras with unequal parameters. In particular, he formulates conjectures describing the properties of Hecke algebras with unequal parameters and presents examples verifying these conjectures in particular cases. Written in the author's precise style, the book gives rese...

Divergence of Scientific Heuristic Method and Direct Algebraic Instruction

Science.gov (United States)

Calucag, Lina S.

2016-01-01

This is an experimental study, made used of the non-randomized experimental and control groups, pretest-posttest designs. The experimental and control groups were two separate intact classes in Algebra. For a period of twelve sessions, the experimental group was subjected to the scientific heuristic method, but the control group instead was given…

25 Years of Quantum Groups: from Definition to Classification

Directory of Open Access Journals (Sweden)

A. Stolin

2008-01-01

Full Text Available In mathematics and theoretical physics, quantum groups are certain non-commutative, non-cocommutative Hopf algebras, which first appeared in the theory of quantum integrable models and later they were formalized by Drinfeld and Jimbo. In this paper we present a classification scheme for quantum groups, whose classical limit is a polynomial Lie algebra. As a consequence we obtain deformed XXX and XXZ Hamiltonians. 

Stability Switches, Hopf Bifurcations, and Spatio-temporal Patterns in a Delayed Neural Model with Bidirectional Coupling

Science.gov (United States)

Song, Yongli; Zhang, Tonghua; Tadé, Moses O.

2009-12-01

The dynamical behavior of a delayed neural network with bi-directional coupling is investigated by taking the delay as the bifurcating parameter. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. In particular, we obtain that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural activities. Numerical simulations are given to illustrate the obtained results and show the existence of bursts in some interval of the time for large enough delay.

Categories and Commutative Algebra

CERN Document Server

Salmon, P

2011-01-01

L. Badescu: Sur certaines singularites des varietes algebriques.- D.A. Buchsbaum: Homological and commutative algebra.- S. Greco: Anelli Henseliani.- C. Lair: Morphismes et structures algebriques.- B.A. Mitchell: Introduction to category theory and homological algebra.- R. Rivet: Anneaux de series formelles et anneaux henseliens.- P. Salmon: Applicazioni della K-teoria all'algebra commutativa.- M. Tierney: Axiomatic sheaf theory: some constructions and applications.- C.B. Winters: An elementary lecture on algebraic spaces.

Particle-like structure of Lie algebras

Science.gov (United States)

Vinogradov, A. M.

2017-07-01

If a Lie algebra structure 𝔤 on a vector space is the sum of a family of mutually compatible Lie algebra structures 𝔤i's, we say that 𝔤 is simply assembled from the 𝔤i's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the 𝔤i's, one obtains a Lie algebra assembled in two steps from 𝔤i's, and so on. We describe the process of modular disassembling of a Lie algebra into a unimodular and a non-unimodular part. We then study two inverse questions: which Lie algebras can be assembled from a given family of Lie algebras, and from which Lie algebras can a given Lie algebra be assembled. We develop some basic assembling and disassembling techniques that constitute the elements of a new approach to the general theory of Lie algebras. The main result of our theory is that any finite-dimensional Lie algebra over an algebraically closed field of characteristic zero or over R can be assembled in a finite number of steps from two elementary constituents, which we call dyons and triadons. Up to an abelian summand, a dyon is a Lie algebra structure isomorphic to the non-abelian 2-dimensional Lie algebra, while a triadon is isomorphic to the 3-dimensional Heisenberg Lie algebra. As an example, we describe constructions of classical Lie algebras from triadons.

Dynamical entropy of C* algebras and Von Neumann algebras

International Nuclear Information System (INIS)

Connes, A.; Narnhofer, H.; Thirring, W.

1986-01-01

The definition of the dynamical entropy is extended for automorphism groups of C * algebras. As example the dynamical entropy of the shift of a lattice algebra is studied and it is shown that in some cases it coincides with the entropy density. (Author)

Numerical Exploration of Kaldorian Macrodynamics: Hopf-Neimark Bifurcations and Business Cycles with Fixed Exchange Rates

Directory of Open Access Journals (Sweden)

Toichiro Asada

2007-01-01

Full Text Available We explore numerically a three-dimensional discrete-time Kaldorian macrodynamic model in an open economy with fixed exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods market α, and the degree of capital mobility β on the stability of equilibrium and on the existence of business cycles. We determine the stability region in the parameter space and find that increase of α destabilizes the equilibrium more quickly than increase of β. We determine the Hopf-Neimark bifurcation curve along which business cycles are generated, and discuss briefly the occurrence of Arnold tongues. Bifurcation and Lyapunov exponent diagrams are computed providing information on the emergence, persistence, and amplitude of the cycles and illustrating the complex dynamics involved. Examples of cycles and other attractors are presented. Finally, we discuss a two-dimensional variation of the model related to a “wealth effect,” called model 2, and show that in this case, α does not destabilize the equilibrium more quickly than β, and that a Hopf-Neimark bifurcation curve does not exist in the parameter space, therefore model 2 does not produce cycles.

Gradings on simple Lie algebras

CERN Document Server

Elduque, Alberto

2013-01-01

Gradings are ubiquitous in the theory of Lie algebras, from the root space decomposition of a complex semisimple Lie algebra relative to a Cartan subalgebra to the beautiful Dempwolff decomposition of E_8 as a direct sum of thirty-one Cartan subalgebras. This monograph is a self-contained exposition of the classification of gradings by arbitrary groups on classical simple Lie algebras over algebraically closed fields of characteristic not equal to 2 as well as on some nonclassical simple Lie algebras in positive characteristic. Other important algebras also enter the stage: matrix algebras, the octonions, and the Albert algebra. Most of the presented results are recent and have not yet appeared in book form. This work can be used as a textbook for graduate students or as a reference for researchers in Lie theory and neighboring areas.

The Effects of the Elevate Math Summer Program on Math Achievement and Algebra Readiness. REL 2015-096

Science.gov (United States)

Snipes, Jason; Huang, Chun-Wei; Jaquet, Karina; Finkelstein, Neal

2015-01-01

The Effects of the Elevate Math summer program on math achievement and algebra readiness: This randomized trial examined the effects of the Elevate Math summer program on math achievement and algebra readiness, as well as math interest and self-efficacy, among rising 8th grade students in California's Silicon Valley. The Elevate Math summer math…

Classical trajectories and quantum field theory

International Nuclear Information System (INIS)

Vitiello, Giuseppe; Istituto Nazionale di Fisica Nucleare, Salerno

2005-01-01

The density matrix and the Wigner function formalism requires the doubling of the degrees of freedom in quantum mechanics (QM) and quantum field theory (QFT). The doubled degrees of freedom play the role of the thermal bath or environment degrees of freedom and are entangled with the system degrees of freedom. They also account for quantum noise in the fluctuating random forces in the system-environment coupling. The algebraic structure of QFT turns out to be the one of the deformed Hopf algebra. In such a frame, the trajectories in the space of the unitarily inequivalent representations of the canonical commutation relations turn out to be classical trajectories and, under convenient conditions, they may exhibit properties typical of classical chaotic trajectories in nonlinear dynamics. The quantum Brownian motion and the two-slit experiment in QM are discussed in connection with the doubling of the degrees of freedom. (author)

Topological conformal algebra and BRST algebra in non-critical string theories

International Nuclear Information System (INIS)

Fujikawa, Kazuo; Suzuki, Hiroshi.

1991-03-01

The operator algebra in non-critical string theories is studied by treating the cosmological term as a perturbation. The algebra of covariantly regularized BRST and related currents contains a twisted N = 2 superconformal algebra only at d = -2 in bosonic strings, and a twisted N = 3 superconformal algebra only at d = ±∞ in spinning strings. The bosonic string at d = -2 is examined by replacing the string coordinate by a fermionic matter with c = -2. The resulting bc-βγ system accommodates various forms of BRST cohomology, and the ghost number assignment and BRST cohomology are different in the c = -2 string theory and two-dimensional topological gravity. (author)

Algebraic K-theory

CERN Document Server

Srinivas, V

1996-01-01

Algebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and graduate students in mathematics. The book is based on lectures given at the author's home institution, the Tata Institute in Bombay, and elsewhere. A detailed appendix on topology was provided in the first edition to make the treatment accessible to readers with a limited background in topology. The second edition also includes an appendix on algebraic geometry that contains the required definitions and results needed to understand the core of the book; this makes the book accessible to a wider audience. A central part of the book is a detailed exposition of the ideas of Quillen as contained in his classic papers "Higher Algebraic K-Theory, I, II." A more elementary proof of the theorem of Merkujev--Suslin is given in this edition; this makes the treatment of this topic self-contained. An application ...

Head First Algebra A Learner's Guide to Algebra I

CERN Document Server

Pilone, Tracey

2008-01-01

Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have great news: Head First Algebra is designed for you. Full of engaging stories and practical, real-world explanations, this book will help you learn everything from natural numbers and exponents to solving systems of equations and graphing polynomials. Along the way, you'll go beyond solving hundreds of repetitive problems, and actually use what you learn to make real-life decisions. Does it make sense to buy two years of insurance on a car that depreciates as soon as you drive i

Novikov algebras with associative bilinear forms

Energy Technology Data Exchange (ETDEWEB)

Zhu Fuhai; Chen Zhiqi [School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 (China)

2007-11-23

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.

Lie algebra in quantum physics by means of computer algebra

OpenAIRE

Kikuchi, Ichio; Kikuchi, Akihito

2017-01-01

This article explains how to apply the computer algebra package GAP (www.gap-system.org) in the computation of the problems in quantum physics, in which the application of Lie algebra is necessary. The article contains several exemplary computations which readers would follow in the desktop PC: such as, the brief review of elementary ideas of Lie algebra, the angular momentum in quantum mechanics, the quark eight-fold way model, and the usage of Weyl character formula (in order to construct w...

Bacterial effector HopF2 interacts with AvrPto and suppresses Arabidopsis innate immunity at the plasma membrane

Science.gov (United States)

Plant pathogenic bacteria inject a cocktail of effector proteins into host plant cells to modulate the host immune response, thereby promoting pathogenicity. How or whether these effectors work cooperatively is largely unknown. The Pseudomonas syringae DC3000 effector HopF2 suppresses the host plan...

Tensor spaces and exterior algebra

CERN Document Server

Yokonuma, Takeo

1992-01-01

This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. To facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. In particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.

Profinite algebras and affine boundedness

OpenAIRE

Schneider, Friedrich Martin; Zumbrägel, Jens

2015-01-01

We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a topological algebra, whereas for topological groups, rings, semigroups, and distributive lattices, profiniteness turns out to be a purely topological property as it is is equivalent to the underlying topological space being a Stone space. Condensing the core...

A comparison of equality in computer algebra and correctness in mathematical pedagogy (II)

OpenAIRE

Bradford, Russell; Davenport, James H; Sangwin, C

2010-01-01

A perennial problem in computer-aided assessment is that “a right answer”, pedagogically speaking, is not the same thing as “a mathematically correct expression”, as verified by a computer algebra system, or indeed other techniques such as random evaluation. Paper I in this series considered the difference in cases where there was “the right answer”, typically calculus questions. Here we look at some other cases, notably in linear algebra, where there can be many “right answers”, but still th...

Double-partition Quantum Cluster Algebras

DEFF Research Database (Denmark)

Jakobsen, Hans Plesner; Zhang, Hechun

2012-01-01

A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but sub-classes have been studied previously by other authors. The algebras are indexed by double parti- tions or double flag varieties. Equivalently, they are indexed by broken lines L. By grouping...... together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of one broken line to another. Compatible pairs can be written down. The algebras are equal to their upper cluster algebras. The variables of the quantum seeds are given by elements of the dual canonical basis....

Algebra II workbook for dummies

CERN Document Server

Sterling, Mary Jane

2014-01-01

To succeed in Algebra II, start practicing now Algebra II builds on your Algebra I skills to prepare you for trigonometry, calculus, and a of myriad STEM topics. Working through practice problems helps students better ingest and retain lesson content, creating a solid foundation to build on for future success. Algebra II Workbook For Dummies, 2nd Edition helps you learn Algebra II by doing Algebra II. Author and math professor Mary Jane Sterling walks you through the entire course, showing you how to approach and solve the problems you encounter in class. You'll begin by refreshing your Algebr

Very true operators on MTL-algebras

Directory of Open Access Journals (Sweden)

Wang Jun Tao

2016-01-01

Full Text Available The main goal of this paper is to investigate very true MTL-algebras and prove the completeness of the very true MTL-logic. In this paper, the concept of very true operators on MTL-algebras is introduced and some related properties are investigated. Also, conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra are given via this operator. Moreover, very true filters on very true MTL-algebras are studied. In particular, subdirectly irreducible very true MTL-algebras are characterized and an analogous of representation theorem for very true MTL-algebras is proved. Then, the left and right stabilizers of very true MTL-algebras are introduced and some related properties are given. As applications of stabilizer of very true MTL-algebras, we produce a basis for a topology on very true MTL-algebras and show that the generated topology by this basis is Baire, connected, locally connected and separable. Finally, the corresponding logic very true MTL-logic is constructed and the soundness and completeness of this logic are proved based on very true MTL-algebras.

W-realization of Lie algebras. Application to so(4,2) and Poincare algebras

International Nuclear Information System (INIS)

Barbarin, F.; Ragoucy, E.; Sorba, P.

1996-05-01

The property of some finite W-algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra G is exploited for the obtention of new G-realizations from a 'canonical' differential one. The method is applied to the conformal algebra so(4,2) and therefore yields also results for its Poincare subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated to - the induced representation technique. (author)

Extended Kac-Moody algebras and applications

International Nuclear Information System (INIS)

Ragoucy, E.; Sorba, P.

1991-04-01

The notion of a Kac-Moody algebra defined on the S 1 circle is extended to super Kac-Moody algebras defined on MxG N , M being a smooth closed compact manifold of dimension greater than one, and G N the Grassman algebra with N generators. All the central extensions of these algebras are computed. Then, for each such algebra the derivation algebra constructed from the MxG N diffeomorphism is determined. The twists of such super Kac-Moody algebras as well as the generalization to non-compact surfaces are partially studied. Finally, the general construction is applied to the study of conformal and superconformal algebras, as well as area-preserving diffeomorphisms algebra and its supersymmetric extension. (author) 65 refs

Two-Level Adaptive Algebraic Multigrid for a Sequence of Problems with Slowly Varying Random Coefficients [Adaptive Algebraic Multigrid for Sequence of Problems with Slowly Varying Random Coefficients

Energy Technology Data Exchange (ETDEWEB)

Kalchev, D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Ketelsen, C. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Vassilevski, P. S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2013-11-07

Our paper proposes an adaptive strategy for reusing a previously constructed coarse space by algebraic multigrid to construct a two-level solver for a problem with nearby characteristics. Furthermore, a main target application is the solution of the linear problems that appear throughout a sequence of Markov chain Monte Carlo simulations of subsurface flow with uncertain permeability field. We demonstrate the efficacy of the method with extensive set of numerical experiments.

Lie algebras and applications

CERN Document Server

Iachello, Francesco

2015-01-01

This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators...

A heterogenous Cournot duopoly with delay dynamics: Hopf bifurcations and stability switching curves

Science.gov (United States)

Pecora, Nicolò; Sodini, Mauro

2018-05-01

This article considers a Cournot duopoly model in a continuous-time framework and analyze its dynamic behavior when the competitors are heterogeneous in determining their output decision. Specifically the model is expressed in the form of differential equations with discrete delays. The stability conditions of the unique Nash equilibrium of the system are determined and the emergence of Hopf bifurcations is shown. Applying some recent mathematical techniques (stability switching curves) and performing numerical simulations, the paper confirms how different time delays affect the stability of the economy.

(Fuzzy) Ideals of BN-Algebras

Science.gov (United States)

Walendziak, Andrzej

2015-01-01

The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained. PMID:26125050

Non-relativistic Bondi-Metzner-Sachs algebra

Science.gov (United States)

Batlle, Carles; Delmastro, Diego; Gomis, Joaquim

2017-09-01

We construct two possible candidates for non-relativistic bms4 algebra in four space-time dimensions by contracting the original relativistic bms4 algebra. bms4 algebra is infinite-dimensional and it contains the generators of the Poincaré algebra, together with the so-called super-translations. Similarly, the proposed nrbms4 algebras can be regarded as two infinite-dimensional extensions of the Bargmann algebra. We also study a canonical realization of one of these algebras in terms of the Fourier modes of a free Schrödinger field, mimicking the canonical realization of relativistic bms4 algebra using a free Klein-Gordon field.

The Unitality of Quantum B-algebras

Science.gov (United States)

Han, Shengwei; Xu, Xiaoting; Qin, Feng

2018-02-01

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

On Weak-BCC-Algebras

Science.gov (United States)

Thomys, Janus; Zhang, Xiaohong

2013-01-01

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

G-identities of non-associative algebras

International Nuclear Information System (INIS)

Bakhturin, Yu A; Zaitsev, M V; Sehgal, S K

1999-01-01

The main class of algebras considered in this paper is the class of algebras of Lie type. This class includes, in particular, associative algebras, Lie algebras and superalgebras, Leibniz algebras, quantum Lie algebras, and many others. We prove that if a finite group G acts on such an algebra A by automorphisms and anti-automorphisms and A satisfies an essential G-identity, then A satisfies an ordinary identity of degree bounded by a function that depends on the degree of the original identity and the order of G. We show in the case of ordinary Lie algebras that if L is a Lie algebra, a finite group G acts on L by automorphisms and anti-automorphisms, and the order of G is coprime to the characteristic of the field, then the existence of an identity on skew-symmetric elements implies the existence of an identity on the whole of L, with the same kind of dependence between the degrees of the identities. Finally, we generalize Amitsur's theorem on polynomial identities in associative algebras with involution to the case of alternative algebras with involution

W-realization of Lie algebras. Application to so(4,2) and Poincare algebras

Energy Technology Data Exchange (ETDEWEB)

Barbarin, F.; Ragoucy, E.; Sorba, P.

1996-05-01

The property of some finite W-algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra G is exploited for the obtention of new G-realizations from a `canonical` differential one. The method is applied to the conformal algebra so(4,2) and therefore yields also results for its Poincare subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated to - the induced representation technique. (author). 12 refs.

On Dunkl angular momenta algebra

Energy Technology Data Exchange (ETDEWEB)

Feigin, Misha [School of Mathematics and Statistics, University of Glasgow,15 University Gardens, Glasgow G12 8QW (United Kingdom); Hakobyan, Tigran [Yerevan State University,1 Alex Manoogian, 0025 Yerevan (Armenia); Tomsk Polytechnic University,Lenin Ave. 30, 634050 Tomsk (Russian Federation)

2015-11-17

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 subalgebra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.

Algebraic computing

International Nuclear Information System (INIS)

MacCallum, M.A.H.

1990-01-01

The implementation of a new computer algebra system is time consuming: designers of general purpose algebra systems usually say it takes about 50 man-years to create a mature and fully functional system. Hence the range of available systems and their capabilities changes little between one general relativity meeting and the next, despite which there have been significant changes in the period since the last report. The introductory remarks aim to give a brief survey of capabilities of the principal available systems and highlight one or two trends. The reference to the most recent full survey of computer algebra in relativity and brief descriptions of the Maple, REDUCE and SHEEP and other applications are given. (author)

Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra

CERN Document Server

Cox, David A; O'Shea, Donal

2015-01-01

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem, and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D). The book may serve as a first or second course in undergraduate abstract algebra and, with some supplementation perhaps, for beginning graduate level courses in algebraic geom...

Connections between algebra, combinatorics, and geometry

CERN Document Server

Sather-Wagstaff, Sean

2014-01-01

Commutative algebra, combinatorics, and algebraic geometry are thriving areas of mathematical research with a rich history of interaction. Connections Between Algebra, Combinatorics, and Geometry contains lecture notes, along with exercises and solutions, from the Workshop on Connections Between Algebra and Geometry held at the University of Regina from May 29-June 1, 2012. It also contains research and survey papers from academics invited to participate in the companion Special Session on Interactions Between Algebraic Geometry and Commutative Algebra, which was part of the CMS Summer Meeting at the University of Regina held June 2–3, 2012, and the meeting Further Connections Between Algebra and Geometry, which was held at the North Dakota State University, February 23, 2013. This volume highlights three mini-courses in the areas of commutative algebra and algebraic geometry: differential graded commutative algebra, secant varieties, and fat points and symbolic powers. It will serve as a useful resou...

Abstract algebra for physicists

International Nuclear Information System (INIS)

Zeman, J.

1975-06-01

Certain recent models of composite hadrons involve concepts and theorems from abstract algebra which are unfamiliar to most theoretical physicists. The algebraic apparatus needed for an understanding of these models is summarized here. Particular emphasis is given to algebraic structures which are not assumed to be associative. (2 figures) (auth)

Basic notions of algebra

CERN Document Server

Shafarevich, Igor Rostislavovich

2005-01-01

This book is wholeheartedly recommended to every student or user of mathematics. Although the author modestly describes his book as 'merely an attempt to talk about' algebra, he succeeds in writing an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields, commutative rings and groups studied in every university math course, through Lie groups and algebras to cohomology and category theory, the author shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches

Characterizations of locally C*-algebras

International Nuclear Information System (INIS)

Mohammad, N.; Somasundaram, S.

1991-08-01

We seek the generalization of the Gelfand-Naimark theorems for locally C*-algebras. Precisely, if A is a unital commutative locally C*-algebra, then it is shown that A is *-isomorphic (topologically and algebraically) to C(Δ). Further, if A is any locally C*-algebra, then it is realized as a closed *-subalgebra of some L(H) up to a topological algebraic *-isomorphism. Also, a brief exposition of the Gelfand-Naimark-Segal construction is given and some of its consequences are discussed. (author). 16 refs

Galois Theory of Differential Equations, Algebraic Groups and Lie Algebras

NARCIS (Netherlands)

Put, Marius van der

1999-01-01

The Galois theory of linear differential equations is presented, including full proofs. The connection with algebraic groups and their Lie algebras is given. As an application the inverse problem of differential Galois theory is discussed. There are many exercises in the text.

Bifurcation of cubic nonlinear parallel plate-type structure in axial flow

International Nuclear Information System (INIS)

Lu Li; Yang Yiren

2005-01-01

The Hopf bifurcation of plate-type beams with cubic nonlinear stiffness in axial flow was studied. By assuming that all the plates have the same deflections at any instant, the nonlinear model of plate-type beam in axial flow was established. The partial differential equation was turned into an ordinary differential equation by using Galerkin method. A new algebraic criterion of Hopf bifurcation was utilized to in our analysis. The results show that there's no Hopf bifurcation for simply supported plate-type beams while the cantilevered plate-type beams has. At last, the analytic expression of critical flow velocity of cantilevered plate-type beams in axial flow and the purely imaginary eigenvalues of the corresponding linear system were gotten. (authors)

Brauer algebras of type B

NARCIS (Netherlands)

Cohen, A.M.; Liu, S.

2011-01-01

For each n>0, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type Dn+1 and point out a cellular

A course in BE-algebras

CERN Document Server

Mukkamala, Sambasiva Rao

2018-01-01

This book presents a unified course in BE-algebras with a comprehensive introduction, general theoretical basis and several examples. It introduces the general theoretical basis of BE-algebras, adopting a credible style to offer students a conceptual understanding of the subject. BE-algebras are important tools for certain investigations in algebraic logic, because they can be considered as fragments of any propositional logic containing a logical connective implication and the constant "1", which is considered as the logical value “true”.  Primarily aimed at graduate and postgraduate students of mathematics, it also helps researchers and mathematicians to build a strong foundation in applied abstract algebra. Presenting insights into some of the abstract thinking that constitutes modern abstract algebra, it provides a transition from elementary topics to advanced topics in BE-algebras. With abundant examples and exercises arranged after each section, it offers readers a comprehensive, easy-to-follow int...

Markovian representations of current algebras

International Nuclear Information System (INIS)

Streater, R.F.

1977-01-01

Hegerfeldt's concept (Commun. Math. Phys.; 35:155 (1974)) of T-positivity in Euclidean random fields is generalized to non-commutative probability theory, that is, to Euclidean Fermi fields and to current algebra with possible Schwinger terms. The axioms used imply the Wightman axioms. A non-Abelian form of Markovicity is introduced, and is shown to imply T-positivity if a reflection property holds. The investigation suggests a generalization of Nelson-Symanzik positivity, which might be valid in cases when the extension of the Schwinger functions to coinciding arguments is not expected to maintain both commutativity and positivity (or anti-commutativity and positivity). (author)

On Associative Conformal Algebras of Linear Growth

OpenAIRE

Retakh, Alexander

2000-01-01

Lie conformal algebras appear in the theory of vertex algebras. Their relation is similar to that of Lie algebras and their universal enveloping algebras. Associative conformal algebras play a role in conformal representation theory. We introduce the notions of conformal identity and unital associative conformal algebras and classify finitely generated simple unital associative conformal algebras of linear growth. These are precisely the complete algebras of conformal endomorphisms of finite ...

The Linear Span of Projections in AH Algebras and for Inclusions of C*-Algebras

Directory of Open Access Journals (Sweden)

Dinh Trung Hoa

2013-01-01

Full Text Available In the first part of this paper, we show that an AH algebra A=lim→(Ai,ϕi has the LP property if and only if every element of the centre of Ai belongs to the closure of the linear span of projections in A. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation in the sense of Bratteli and Elliott. The second contribution of this paper is that for an inclusion of unital C*-algebras P⊂A with a finite Watatani index, if a faithful conditional expectation E:A→P has the Rokhlin property in the sense of Kodaka et al., then P has the LP property under the condition thatA has the LP property. As an application, let A be a simple unital C*-algebra with the LP property, α an action of a finite group G onto Aut(A. If α has the Rokhlin property in the sense of Izumi, then the fixed point algebra AG and the crossed product algebra A ⋊α G have the LP property. We also point out that there is a symmetry on the CAR algebra such that its fixed point algebra does not have the LP property.

Non-commutative multiple-valued logic algebras

CERN Document Server

Ciungu, Lavinia Corina

2014-01-01

This monograph provides a self-contained and easy-to-read introduction to non-commutative multiple-valued logic algebras; a subject which has attracted much interest in the past few years because of its impact on information science, artificial intelligence and other subjects.   A study of the newest results in the field, the monograph includes treatment of pseudo-BCK algebras, pseudo-hoops, residuated lattices, bounded divisible residuated lattices, pseudo-MTL algebras, pseudo-BL algebras and pseudo-MV algebras. It provides a fresh perspective on new trends in logic and algebras in that algebraic structures can be developed into fuzzy logics which connect quantum mechanics, mathematical logic, probability theory, algebra and soft computing.   Written in a clear, concise and direct manner, Non-Commutative Multiple-Valued Logic Algebras will be of interest to masters and PhD students, as well as researchers in mathematical logic and theoretical computer science.

Automorphic Lie algebras with dihedral symmetry

International Nuclear Information System (INIS)

Knibbeler, V; Lombardo, S; A Sanders, J

2014-01-01

The concept of automorphic Lie algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. automorphic Lie algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever–Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is sl 2 (C) and the poles of the automorphic Lie algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie algebra structure. In this research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base Lie algebra. We find a uniform description of automorphic Lie algebras with dihedral symmetry, valid for poles at exceptional and generic orbits. (paper)

Certain number-theoretic episodes in algebra

CERN Document Server

Sivaramakrishnan, R

2006-01-01

Many basic ideas of algebra and number theory intertwine, making it ideal to explore both at the same time. Certain Number-Theoretic Episodes in Algebra focuses on some important aspects of interconnections between number theory and commutative algebra. Using a pedagogical approach, the author presents the conceptual foundations of commutative algebra arising from number theory. Self-contained, the book examines situations where explicit algebraic analogues of theorems of number theory are available. Coverage is divided into four parts, beginning with elements of number theory and algebra such as theorems of Euler, Fermat, and Lagrange, Euclidean domains, and finite groups. In the second part, the book details ordered fields, fields with valuation, and other algebraic structures. This is followed by a review of fundamentals of algebraic number theory in the third part. The final part explores links with ring theory, finite dimensional algebras, and the Goldbach problem.

Fusion rules of chiral algebras

International Nuclear Information System (INIS)

Gaberdiel, M.

1994-01-01

Recently we showed that for the case of the WZW and the minimal models fusion can be understood as a certain ring-like tensor product of the symmetry algebra. In this paper we generalize this analysis to arbitrary chiral algebras. We define the tensor product of conformal field theory in the general case and prove that it is associative and symmetric up to equivalence. We also determine explicitly the action of the chiral algebra on this tensor product. In the second part of the paper we demonstrate that this framework provides a powerful tool for calculating restrictions for the fusion rules of chiral algebras. We exhibit this for the case of the W 3 algebra and the N=1 and N=2 NS superconformal algebras. (orig.)

Computations in finite-dimensional Lie algebras

Directory of Open Access Journals (Sweden)

A. M. Cohen

1997-12-01

Full Text Available This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System, within the computer algebra package GAP. A first sketch of the package can be found in Cohen and de Graaf[1]. Since then, in a collaborative effort with G. Ivanyos, the authors have continued to develop algorithms which were implemented in ELIAS by the second author. These activities are part of a bigger project, called ACELA and financed by STW, the Dutch Technology Foundation, which aims at an interactive book on Lie algebras (cf. Cohen and Meertens [2]. This paper gives a global description of the main ways in which to present Lie algebras on a computer. We focus on the transition from a Lie algebra abstractly given by an array of structure constants to a Lie algebra presented as a subalgebra of the Lie algebra of n×n matrices. We describe an algorithm typical of the structure analysis of a finite-dimensional Lie algebra: finding a Levi subalgebra of a Lie algebra.

Coset realization of unifying W-algebras

International Nuclear Information System (INIS)

Blumenhagen, R.; Huebel, R.

1994-06-01

We construct several quantum coset W-algebras, e.g. sl(2,R)/U(1) and sl(2,R)+sl(2,R)/sl(2,R), and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying W-algebras of Casimir W-algebras. We show that it is possible to give coset realizations of various types of unifying W-algebras, e.g. the diagonal cosets based on the symplectic Lie algebras sp(2n) realize the unifying W-algebras which have previously been introduced as 'WD -n '. In addition, minimal models of WD -n are studied. The coset realizations provide a generalization of level-rank-duality of dual coset pairs. As further examples of finitely nonfreely generated quantum W-algebras we discuss orbifolding of W-algebras which on the quantum level has different properties than in the classical case. We demonstrate in some examples that the classical limit according to Bowcock and Watts of these nonfreely finitely generated quantum W-algebras probably yields infinitely nonfreely generated classical W-algebras. (orig.)

Algebra

CERN Document Server

Sepanski, Mark R

2010-01-01

Mark Sepanski's Algebra is a readable introduction to the delightful world of modern algebra. Beginning with concrete examples from the study of integers and modular arithmetic, the text steadily familiarizes the reader with greater levels of abstraction as it moves through the study of groups, rings, and fields. The book is equipped with over 750 exercises suitable for many levels of student ability. There are standard problems, as well as challenging exercises, that introduce students to topics not normally covered in a first course. Difficult problems are broken into manageable subproblems

Identities and derivations for Jacobian algebras

International Nuclear Information System (INIS)

Dzhumadil'daev, A.S.

2001-09-01

Constructions of n-Lie algebras by strong n-Lie-Poisson algebras are given. First cohomology groups of adjoint module of Jacobian algebras are calculated. Minimal identities of 3-Jacobian algebra are found. (author)

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

Science.gov (United States)

Sialaros, Michalis; Christianidis, Jean

2016-06-01

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

Cohomology of Effect Algebras

Directory of Open Access Journals (Sweden)

Frank Roumen

2017-01-01

Full Text Available We will define two ways to assign cohomology groups to effect algebras, which occur in the algebraic study of quantum logic. The first way is based on Connes' cyclic cohomology. The resulting cohomology groups are related to the state space of the effect algebra, and can be computed using variations on the Kunneth and Mayer-Vietoris sequences. The second way involves a chain complex of ordered abelian groups, and gives rise to a cohomological characterization of state extensions on effect algebras. This has applications to no-go theorems in quantum foundations, such as Bell's theorem.

Classroom Connectivity and Algebra 1 Achievement: A Three-Year Longitudinal Study

Science.gov (United States)

Irving, Karen E.; Pape, Stephen J.; Owens, Douglas T.; Abrahamson, Louis; Silver, David; Sanalan, Vehbi A.

2016-01-01

Findings from three years of a longitudinal randomized control trial involving a national U.S. sample of Algebra 1 teachers and students are reported. The study examines the effects of a connected classroom technology (CCT) professional development and classroom intervention on student achievement when compared to classroom instruction with…

Topology general & algebraic

CERN Document Server

Chatterjee, D

2007-01-01

About the Book: This book provides exposition of the subject both in its general and algebraic aspects. It deals with the notions of topological spaces, compactness, connectedness, completeness including metrizability and compactification, algebraic aspects of topological spaces through homotopy groups and homology groups. It begins with the basic notions of topological spaces but soon going beyond them reaches the domain of algebra through the notions of homotopy, homology and cohomology. How these approaches work in harmony is the subject matter of this book. The book finally arrives at the

Principles of linear algebra with Mathematica

CERN Document Server

Shiskowski, Kenneth M

2013-01-01

A hands-on introduction to the theoretical and computational aspects of linear algebra using Mathematica® Many topics in linear algebra are simple, yet computationally intensive, and computer algebra systems such as Mathematica® are essential not only for learning to apply the concepts to computationally challenging problems, but also for visualizing many of the geometric aspects within this field of study. Principles of Linear Algebra with Mathematica uniquely bridges the gap between beginning linear algebra and computational linear algebra that is often encountered in applied settings,

Einstein algebras and general relativity

International Nuclear Information System (INIS)

Heller, M.

1992-01-01

A purely algebraic structure called an Einstein algebra is defined in such a way that every spacetime satisfying Einstein's equations is an Einstein algebra but not vice versa. The Gelfand representation of Einstein algebras is defined, and two of its subrepresentations are discussed. One of them is equivalent to the global formulation of the standard theory of general relativity; the other one leads to a more general theory of gravitation which, in particular, includes so-called regular singularities. In order to include other types of singularities one must change to sheaves of Einstein algebras. They are defined and briefly discussed. As a test of the proposed method, the sheaf of Einstein algebras corresponding to the space-time of a straight cosmic string with quasiregular singularity is constructed. 22 refs

Analytic real algebras.

Science.gov (United States)

Seo, Young Joo; Kim, Young Hee

2016-01-01

In this paper we construct some real algebras by using elementary functions, and discuss some relations between several axioms and its related conditions for such functions. We obtain some conditions for real-valued functions to be a (edge) d -algebra.

Biderivations of W-algebra $W(2,2)$ and Virasoro algebra without skewsymmetric condition and their applications

OpenAIRE

Tang, Xiaomin

2016-01-01

In this paper, we characterize the biderivations of W-algebra $W(2,2)$ and Virasoro algebra $Vir$ without skewsymmetric condition. We get two classes of non-inner biderivations. As applications, we also get the forms of linear commuting maps on W-algebra $W(2,2)$ and Virasoro algebra $Vir$.

Evolution algebras generated by Gibbs measures

International Nuclear Information System (INIS)

Rozikov, Utkir A.; Tian, Jianjun Paul

2009-03-01

In this article we study algebraic structures of function spaces defined by graphs and state spaces equipped with Gibbs measures by associating evolution algebras. We give a constructive description of associating evolution algebras to the function spaces (cell spaces) defined by graphs and state spaces and Gibbs measure μ. For finite graphs we find some evolution subalgebras and other useful properties of the algebras. We obtain a structure theorem for evolution algebras when graphs are finite and connected. We prove that for a fixed finite graph, the function spaces have a unique algebraic structure since all evolution algebras are isomorphic to each other for whichever Gibbs measures are assigned. When graphs are infinite graphs then our construction allows a natural introduction of thermodynamics in studying of several systems of biology, physics and mathematics by theory of evolution algebras. (author)

Complexity dynamics and Hopf bifurcation analysis based on the first Lyapunov coefficient about 3D IS-LM macroeconomics system

Science.gov (United States)

Ma, Junhai; Ren, Wenbo; Zhan, Xueli

2017-04-01

Based on the study of scholars at home and abroad, this paper improves the three-dimensional IS-LM model in macroeconomics, analyzes the equilibrium point of the system and stability conditions, focuses on the parameters and complex dynamic characteristics when Hopf bifurcation occurs in the three-dimensional IS-LM macroeconomics system. In order to analyze the stability of limit cycles when Hopf bifurcation occurs, this paper further introduces the first Lyapunov coefficient to judge the limit cycles, i.e. from a practical view of the business cycle. Numerical simulation results show that within the range of most of the parameters, the limit cycle of 3D IS-LM macroeconomics is stable, that is, the business cycle is stable; with the increase of the parameters, limit cycles becomes unstable, and the value range of the parameters in this situation is small. The research results of this paper have good guide significance for the analysis of macroeconomics system.

Classical algebraic chromodynamics

International Nuclear Information System (INIS)

Adler, S.L.

1978-01-01

I develop an extension of the usual equations of SU(n) chromodynamics which permits the consistent introduction of classical, noncommuting quark source charges. The extension involves adding a singlet gluon, giving a U(n) -based theory with outer product P/sup a/(u,v) = (1/2)(d/sup a/bc + if/sup a/bc)(u/sup b/v/sup c/ - v/sup b/u/sup c/) which obeys the Jacobi identity, inner product S (u,v) = (1/2)(u/sup a/v/sup a/ + v/sup a/u/sup a/), and with the n 2 gluon fields elevated to algebraic fields over the quark color charge C* algebra. I show that provided the color charge algebra satisfies the condition S (P (u,v),w) = S (u,P (v,w)) for all elements u,v,w of the algebra, all the standard derivations of Lagrangian chromodynamics continue to hold in the algebraic chromodynamics case. I analyze in detail the color charge algebra in the two-particle (qq, qq-bar, q-barq-bar) case and show that the above consistency condition is satisfied for the following unique (and, interestingly, asymmetric) choice of quark and antiquark charges: Q/sup a//sub q/ = xi/sup a/, Q/sup a//sub q/ = xi-bar/sup a/ + delta/sup a/0(n/2)/sup 3/2/1, with xi/sup a/xi/sup b/ = (1/2)(d/sup a/bc + if/sup a/bc) xi/sup c/, xi-bar/sup a/xi-bar/sup b/ = -(1/2)(d/sup a/bc - if/sup a/bc) xi-bar/sup c/. The algebraic structure of the two-particle U(n) force problem, when expressed on an appropriately diagonalized basis, leads for all n to a classical dynamics problem involving an ordinary SU(2) Yang-Mills field with uniquely specified classical source charges which are nonparallel in the color-singlet state. An explicit calculation shows that local algebraic U(n) gauge transformations lead only to a rigid global rotation of axes in the overlying classical SU(2) problem, which implies that the relative orientations of the classical source charges have physical significance

Investigating the use of interval algebra to schedule mechanically steered multistatic radars

CSIR Research Space (South Africa)

Focke, RW

2015-02-01

Full Text Available to test random target locations and motion. A novel IA algorithm that makes use of reduced point algebra was generated that allowed execution time to be reduced. Another, simpler novel contribution was an IA algorithm that ensured that radar tasks are only...

On the classification of quantum W-algebras

International Nuclear Information System (INIS)

Bowcock, P.; Watts, G.T.M.

1992-01-01

In this paper we consider the structure of general quantum W-algebras. We introduce the notions of deformability, positive-definiteness, and reductivity of a W-algebra. We show that one can associate a reductive finite Lie algebra to each reductive W-algebra. The finite Lie algebra is also endowed with a preferred sl(2) subalgebra, which gives the conformal weights of the W-algebra. We extend this to cover W-algebras containing both bosonic and fermionic fields, and illustrate our ideas with the Poisson bracket algebras of generalised Drinfeld-Sokolov hamiltonian systems. We then discuss the possibilities of classifying deformable W-algebras which fall outside this class in the context of automorphisms of Lie algebras. In conclusion we list the cases in which the W-algebra has no weight-one fields, and further, those in which it has only one weight-two field. (orig.)

Lie-Algebras. Pt. 1

International Nuclear Information System (INIS)

Baeuerle, G.G.A.; Kerf, E.A. de

1990-01-01

The structure of the laws in physics is largely based on symmetries. This book is on Lie algebras, the mathematics of symmetry. It gives a thorough mathematical treatment of finite dimensional Lie algebras and Kac-Moody algebras. Concepts such as Cartan matrix, root system, Serre's construction are carefully introduced. Although the book can be read by an undergraduate with only an elementary knowledge of linear algebra, the book will also be of use to the experienced researcher. Experience has shown that students who followed the lectures are well-prepared to take on research in the realms of string-theory, conformal field-theory and integrable systems. 48 refs.; 66 figs.; 3 tabs

Semiprojectivity of universal -algebras generated by algebraic elements

DEFF Research Database (Denmark)

Shulman, Tatiana

2012-01-01

Let be a polynomial in one variable whose roots all have multiplicity more than 1. It is shown that the universal -algebra of a relation , , is semiprojective and residually finite-dimensional. Applications to polynomially compact operators are given.......Let be a polynomial in one variable whose roots all have multiplicity more than 1. It is shown that the universal -algebra of a relation , , is semiprojective and residually finite-dimensional. Applications to polynomially compact operators are given....

Macdonald index and chiral algebra

Science.gov (United States)

Song, Jaewon

2017-08-01

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

Vertex algebras and mirror symmetry

International Nuclear Information System (INIS)

Borisov, L.A.

2001-01-01

Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi-Yau manifolds. This should eventually allow us to rewrite the whole story of toric mirror symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties.This paper could also be of interest to physicists, because it contains explicit description of holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in terms of free bosons and fermions. (orig.)

From starproducts to Drinfeld-twists. Present and future applications

International Nuclear Information System (INIS)

Koch, Florian

2008-01-01

Physics comes up with models that invoke noncommutative structures in configuration space. Such structures are dual to the deformed coalgebra sector of a represented symmetry algebra. In the mean time such deformations are performed in terms of the symmetry algebra itself via twists or quasitriangular structures. One might thus find oneself in the bad situation that the symmetry algebra is not large enough to provide the required twist that dually matches the noncommutative structure found. It thus has to remain in the unpleasant state of being without any notion of symmetry. We show how starproducts can be pushed to twists by introducing a larger algebra that accommodates any finite dimensional representation of a Lie-algebra. This new algebra is similar to a Heisenberg-algebra but in contrast to the latter can be enhanced to a Hopf-algebra. Some Examples are given. (author)

Advanced modern algebra part 2

CERN Document Server

Rotman, Joseph J

2017-01-01

This book is the second part of the new edition of Advanced Modern Algebra (the first part published as Graduate Studies in Mathematics, Volume 165). Compared to the previous edition, the material has been significantly reorganized and many sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra, respectively. The book can be used as a text for a second abstract algebra graduate course, as a source of additional material to a first abstract algebra graduate course, or for self-study.

A q-deformed Lorentz algebra

International Nuclear Information System (INIS)

Schmidke, W.B.; Wess, J.; Muenchen Univ.; Zumino, B.; Lawrence Berkeley Lab., CA

1991-01-01

We derive a q-deformed version of the Lorentz algebra by deformating the algebra SL(2, C). The method is based on linear representations of the algebra on the complex quantum spinor space. We find that the generators usually identified with SL q (2, C) generate SU q (2) only. Four additional generators are added which generate Lorentz boosts. The full algebra of all seven generators and their coproduct is presented. We show that in the limit q→1 the generators are those of the classical Lorentz algebra plus an additional U(1). Thus we have a deformation of SL(2, C)xU(1). (orig.)

Introduction to algebraic quantum field theory

International Nuclear Information System (INIS)

Horuzhy, S.S.

1990-01-01

This volume presents a systematic introduction to the algebraic approach to quantum field theory. The structure of the contents corresponds to the way the subject has advanced. It is shown how the algebraic approach has developed from the purely axiomatic theory of observables via superselection rules into the dynamical formalism of fields and observables. Chapter one discusses axioms and their consequences -many of which are now classical theorems- and deals, in general, with the axiomatic theory of local observable algebras. The absence of field concepts makes this theory incomplete and, in chapter two, superselection rules are shown to be the key to the reconstruction of fields from observables. Chapter three deals with the algebras of Wightman fields, first unbounded operator algebras, then Von Neumann field algebras (with a special section on wedge region algebras) and finally local algebras of free and generalised free fields. (author). 447 refs.; 4 figs

Comments on N=4 superconformal algebras

International Nuclear Information System (INIS)

Rasmussen, J.

2001-01-01

We present a new and asymmetric N=4 superconformal algebra for arbitrary central charge, thus completing our recent work on its classical analogue with vanishing central charge. Besides the Virasoro generator and 4 supercurrents, the algebra consists of an internal SU(2)xU(1) Kac-Moody algebra in addition to two spin 1/2 fermions and a bosonic scalar. The algebra is shown to be invariant under a linear twist of the generators, except for a unique value of the continuous twist parameter. At this value, the invariance is broken and the algebra collapses to the small N=4 superconformal algebra. The asymmetric N=4 superconformal algebra may be seen as induced by an affine SL(2 vertical bar 2) current superalgebra. Replacing SL(2 vertical bar 2) with the coset SL(2 vertical bar 2)/U(1), results directly in the small N=4 superconformal algebra

The theory of algebraic numbers

CERN Document Server

Pollard, Harry

1998-01-01

An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.

Quantum Heisenberg groups and Sklyanin algebras

International Nuclear Information System (INIS)

Andruskiewitsch, N.; Devoto, J.; Tiraboschi, A.

1993-05-01

We define new quantizations of the Heisenberg group by introducing new quantizations in the universal enveloping algebra of its Lie algebra. Matrix coefficients of the Stone-von Neumann representation are preserved by these new multiplications on the algebra of functions on the Heisenberg group. Some of the new quantizations provide also a new multiplication in the algebra of theta functions; we obtain in this way Sklyanin algebras. (author). 23 refs

Classification of simple flexible Lie-admissible algebras

International Nuclear Information System (INIS)

Okubo, S.; Myung, H.C.

1979-01-01

Let A be a finite-dimensional flexible Lie-admissible algebra over the complex field such that A - is a simple Lie algebra. It is shown that either A is itself a Lie algebra isomorphic to A - or A - is a Lie algebra of type A/sub n/ (n greater than or equal to 2). In the latter case, A is isomorphic to the algebra defined on the space of (n + 1) x (n + 1) traceless matrices with multiplication given by x * y = μxy + (1 - μ)yx - (1/(n + 100 Tr (xy) E where μ is a fixed scalar, xy denotes the matrix operators in Lie algebras which has been studied in theoretical physics. We also discuss a broader class of Lie algebras over arbitrary field of characteristic not equal to 2, called quasi-classical, which includes semisimple as well as reductive Lie algebras. For this class of Lie algebras, we can introduce a multiplication which makes the adjoint operator space into an associative algebra. When L is a Lie algebra with nondegenerate killing form, it is shown that the adjoint operator algebra of L in the adjoint representation becomes a commutative associative algebra with unit element and its dimension is 1 or 2 if L is simple over the complex field. This is related to the known result that a Lie algebra of type A/sub n/ (n greater than or equal to 2) alone has a nonzero completely symmetric adjoint operator in the adjoint representation while all other algebras have none. Finally, Lie-admissible algebras associated with bilinear form are investigated

Quantitative Algebraic Reasoning

DEFF Research Database (Denmark)

Mardare, Radu Iulian; Panangaden, Prakash; Plotkin, Gordon

2016-01-01

We develop a quantitative analogue of equational reasoning which we call quantitative algebra. We define an equality relation indexed by rationals: a =ε b which we think of as saying that “a is approximately equal to b up to an error of ε”. We have 4 interesting examples where we have a quantitative...... equational theory whose free algebras correspond to well known structures. In each case we have finitary and continuous versions. The four cases are: Hausdorff metrics from quantitive semilattices; pWasserstein metrics (hence also the Kantorovich metric) from barycentric algebras and also from pointed...

Anyons, deformed oscillator algebras and projectors

International Nuclear Information System (INIS)

Engquist, Johan

2009-01-01

We initiate an algebraic approach to the many-anyon problem based on deformed oscillator algebras. The formalism utilizes a generalization of the deformed Heisenberg algebras underlying the operator solution of the Calogero problem. We define a many-body Hamiltonian and an angular momentum operator which are relevant for a linearized analysis in the statistical parameter ν. There exists a unique ground state and, in spite of the presence of defect lines, the anyonic weight lattices are completely connected by the application of the oscillators of the algebra. This is achieved by supplementing the oscillator algebra with a certain projector algebra.

Alternative algebraic approaches in quantum chemistry

International Nuclear Information System (INIS)

Mezey, Paul G.

2015-01-01

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

Alternative algebraic approaches in quantum chemistry

Energy Technology Data Exchange (ETDEWEB)

Mezey, Paul G., E-mail: paul.mezey@gmail.com [Canada Research Chair in Scientific Modeling and Simulation, Department of Chemistry and Department of Physics and Physical Oceanography, Memorial University of Newfoundland, 283 Prince Philip Drive, St. John' s, NL A1B 3X7 (Canada)

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.

On graded algebras of global dimension 3

International Nuclear Information System (INIS)

Piontkovskii, D I

2001-01-01

Assume that a graded associative algebra A over a field k is minimally presented as the quotient algebra of a free algebra F by the ideal I generated by a set f of homogeneous elements. We study the following two extensions of A: the algebra F-bar=F/I oplus I/I 2 oplus ... associated with F with respect to the I-adic filtration, and the homology algebra H of the Shafarevich complex Sh(f,F) (which is a non-commutative version of the Koszul complex). We obtain several characterizations of algebras of global dimension 3. In particular, the A-algebra H in this case is free, and the algebra F-bar is isomorphic to the quotient algebra of a free A-algebra by the ideal generated by a so-called strongly free (or inert) set

Unipotent and nilpotent classes in simple algebraic groups and lie algebras

CERN Document Server

Liebeck, Martin W

2012-01-01

This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of...

Hopf Bifurcation Analysis of a Gene Regulatory Network Mediated by Small Noncoding RNA with Time Delays and Diffusion

Science.gov (United States)

Li, Chengxian; Liu, Haihong; Zhang, Tonghua; Yan, Fang

2017-12-01

In this paper, a gene regulatory network mediated by small noncoding RNA involving two time delays and diffusion under the Neumann boundary conditions is studied. Choosing the sum of delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the corresponding characteristic equation. It is shown that the sum of delays can induce Hopf bifurcation and the diffusion incorporated into the system can effect the amplitude of periodic solutions. Furthermore, the spatially homogeneous periodic solution always exists and the spatially inhomogeneous periodic solution will arise when the diffusion coefficients of protein and mRNA are suitably small. Particularly, the small RNA diffusion coefficient is more robust and its effect on model is much less than protein and mRNA. Finally, the explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, numerical simulations are carried out to illustrate our theoretical analysis.

Elements of mathematics algebra

CERN Document Server

Bourbaki, Nicolas

2003-01-01

This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's, Algèbre, Chapters 4 to 7 (1981). This completes Algebra, 1 to 3, by establishing the theories of commutative fields and modules over a principal ideal domain. Chapter 4 deals with polynomials, rational fractions and power series. A section on symmetric tensors and polynomial mappings between modules, and a final one on symmetric functions, have been added. Chapter 5 was entirely rewritten. After the basic theory of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving way to a section on Galois theory. Galois theory is in turn applied to finite fields and abelian extensions. The chapter then proceeds to the study of general non-algebraic extensions which cannot usually be found in textbooks: p-bases, transcendental extensions, separability criterions, regular extensions. Chapter 6 treats ordered groups and fields and...

Brauer algebra of type F4

NARCIS (Netherlands)

Liu, S.

2012-01-01

We present an algebra related to the Coxeter group of type F4 which can be viewed as the Brauer algebra of type F4 and is obtained as a subalgebra of the Brauer algebra of type E6. We also describe some properties of this algebra.

Brauer algebras of type F4

NARCIS (Netherlands)

Liu, S.

2013-01-01

We present an algebra related to the Coxeter group of type F4 which can be viewed as the Brauer algebra of type F4 and is obtained as a subalgebra of the Brauer algebra of type E6. We also describe some properties of this algebra.

New examples of continuum graded Lie algebras

International Nuclear Information System (INIS)

Savel'ev, M.V.

1989-01-01

Several new examples of continuum graded Lie algebras which provide an additional elucidation of these algebras are given. Here, in particular, the Kac-Moody algebras, the algebra S 0 Diff T 2 of infinitesimal area-preserving diffeomorphisms of the torus T 2 , the Fairlie, Fletcher and Zachos sine-algebras, etc., are described as special cases of the cross product Lie algebras. 8 refs

Generalized Galilean algebras and Newtonian gravity

Science.gov (United States)

González, N.; Rubio, G.; Salgado, P.; Salgado, S.

2016-04-01

The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.

Circle Maps and C*-algebras

DEFF Research Database (Denmark)

Schmidt, Thomas Lundsgaard

such a map, generalising the transformation groupoid of a local homeomorphism first introduced by Renault in \\cite{re}. We conduct a detailed study of the relationship between the dynamics of $\\phi$, the properties of these groupoids, the structure of their corresponding reduced groupoid $C^*$-algebras, and......, for certain classes of maps, the K-theory of these algebras. When the map $\\phi$ is transitive, we show that the algebras $C^*_r(\\Gamma_\\phi)$ and $C^*_r(\\Gamma_\\phi^+)$ are purely infinite and satisfy the Universal Coefficient Theorem. Furthermore, we find necessary and sufficient conditions for simplicity...... of these algebras in terms of dynamical properties of $\\phi$. We proceed to consider the situation when the algebras are non-simple, and describe the primitive ideal spectrum in this case. We prove that any irreducible representation factors through the $C^*$-algebra of the reduction of the groupoid to the orbit...

The Boolean algebra of Galois algebras

Directory of Open Access Journals (Sweden)

Lianyong Xue

2003-02-01

Full Text Available Let B be a Galois algebra with Galois group G, Jg={b∈B|bx=g(xb for all x∈B} for each g∈G, and BJg=Beg for a central idempotent eg, Ba the Boolean algebra generated by {0,eg|g∈G}, e a nonzero element in Ba, and He={g∈G|eeg=e}. Then, a monomial e is characterized, and the Galois extension Be, generated by e with Galois group He, is investigated.

Geometry of Spin: Clifford Algebraic Approach

Indian Academy of Sciences (India)

Then the various algebraic properties of Pauli matricesare studied as properties of matrix algebra. What has beenshown in this article is that Pauli matrices are a representationof Clifford algebra of spin and hence all the propertiesof Pauli matrices follow from the underlying algebra. Cliffordalgebraic approach provides a ...

Differential operators and W-algebra

International Nuclear Information System (INIS)

Vaysburd, I.; Radul, A.

1992-01-01

The connection between W-algebras and the algebra of differential operators is conjectured. The bosonized representation of the differential operator algebra with c=-2n and all the subalgebras are examined. The degenerate representations and null-state classifications for c=-2 are presented. (orig.)

Donaldson invariants in algebraic geometry

International Nuclear Information System (INIS)

Goettsche, L.

2000-01-01

In these lectures I want to give an introduction to the relation of Donaldson invariants with algebraic geometry: Donaldson invariants are differentiable invariants of smooth compact 4-manifolds X, defined via moduli spaces of anti-self-dual connections. If X is an algebraic surface, then these moduli spaces can for a suitable choice of the metric be identified with moduli spaces of stable vector bundles on X. This can be used to compute Donaldson invariants via methods of algebraic geometry and has led to a lot of activity on moduli spaces of vector bundles and coherent sheaves on algebraic surfaces. We will first recall the definition of the Donaldson invariants via gauge theory. Then we will show the relation between moduli spaces of anti-self-dual connections and moduli spaces of vector bundles on algebraic surfaces, and how this makes it possible to compute Donaldson invariants via algebraic geometry methods. Finally we concentrate on the case that the number b + of positive eigenvalues of the intersection form on the second homology of the 4-manifold is 1. In this case the Donaldson invariants depend on the metric (or in the algebraic geometric case on the polarization) via a system of walls and chambers. We will study the change of the invariants under wall-crossing, and use this in particular to compute the Donaldson invariants of rational algebraic surfaces. (author)

Fractional supersymmetry and infinite dimensional lie algebras

International Nuclear Information System (INIS)

Rausch de Traubenberg, M.

2001-01-01

In an earlier work extensions of supersymmetry and super Lie algebras were constructed consistently starting from any representation D of any Lie algebra g. Here it is shown how infinite dimensional Lie algebras appear naturally within the framework of fractional supersymmetry. Using a differential realization of g this infinite dimensional Lie algebra, containing the Lie algebra g as a sub-algebra, is explicitly constructed

Quantum algebra of N superspace

International Nuclear Information System (INIS)

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

2007-01-01

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 κ-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

On criteria for algebraic independence of collections of functions satisfying algebraic difference relations

Directory of Open Access Journals (Sweden)

Hiroshi Ogawara

2017-01-01

Full Text Available This paper gives conditions for algebraic independence of a collection of functions satisfying a certain kind of algebraic difference relations. As applications, we show algebraic independence of two collections of special functions: (1 Vignéras' multiple gamma functions and derivatives of the gamma function, (2 the logarithmic function, \\(q\\-exponential functions and \\(q\\-polylogarithm functions. In a similar way, we give a generalization of Ostrowski's theorem.

An algebra of reversible computation.

Science.gov (United States)

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.

On an extension of the Weil algebra

International Nuclear Information System (INIS)

Palev, Ch.

An extension of the Weil algebra Wsub(n), generated by an appropriate topology is considered. The topology is introduced in such a way that algebraic operations in Wsub(n) to be continuous. The algebraic operations in Wsub(n) are extended by a natural way to a complement, which is noted as an extended Weil algebra. It turns out that the last algebra contains isomorphically the Heisenberg group. By the same way an arbitrary enveloping algebra of a Lie group may be extended. The extended algebra will contain the initial Lie group. (S.P.)

Exponentiation and deformations of Lie-admissible algebras

International Nuclear Information System (INIS)

Myung, H.C.

1982-01-01

The exponential function is defined for a finite-dimensional real power-associative algebra with unit element. The application of the exponential function is focused on the power-associative (p,q)-mutation of a real or complex associative algebra. Explicit formulas are computed for the (p,q)-mutation of the real envelope of the spin 1 algebra and the Lie algebra so(3) of the rotation group, in light of earlier investigations of the spin 1/2. A slight variant of the mutated exponential is interpreted as a continuous function of the Lie algebra into some isotope of the corresponding linear Lie group. The second part of this paper is concerned with the representation and deformation of a Lie-admissible algebra. The second cohomology group of a Lie-admissible algebra is introduced as a generalization of those of associative and Lie algebras in the Hochschild and Chevalley-Eilenberg theory. Some elementary theory of algebraic deformation of Lie-admissible algebras is discussed in view of generalization of that of associative and Lie algebras. Lie-admissible deformations are also suggested by the representation of Lie-admissible algebras. Some explicit examples of Lie-admissible deformation are given in terms of the (p,q)-mutation of associative deformation of an associative algebra. Finally, we discuss Lie-admissible deformations of order one

Clifford algebras and the minimal representations of the 1D N-extended supersymmetry algebra

International Nuclear Information System (INIS)

Toppan, Francesco

2008-01-01

The Atiyah-Bott-Shapiro classification of the irreducible Clifford algebra is used to derive general properties of the minimal representations of the 1D N-Extended Supersymmetry algebra (the Z 2 -graded symmetry algebra of the Supersymmetric Quantum Mechanics) linearly realized on a finite number of fields depending on a real parameter t, the time. (author)

Contraction of graded su(2) algebra

International Nuclear Information System (INIS)

Patra, M.K.; Tripathy, K.C.

1989-01-01

The Inoenu-Wigner contraction scheme is extended to Lie superalgebras. The structure and representations of extended BRS algebra are obtained from contraction of the graded su(2) algebra. From cohomological consideration, we demonstrate that the graded su(2) algebra is the only superalgebra which, on contraction, yields the full BRS algebra. (orig.)

Atomic effect algebras with compression bases

International Nuclear Information System (INIS)

Caragheorgheopol, Dan; Tkadlec, Josef

2011-01-01

Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.

Operator theory, operator algebras and applications

CERN Document Server

Lebre, Amarino; Samko, Stefan; Spitkovsky, Ilya

2014-01-01

This book consists of research papers that cover the scientific areas of the International Workshop on Operator Theory, Operator Algebras and Applications, held in Lisbon in September 2012. The volume particularly focuses on (i) operator theory and harmonic analysis (singular integral operators with shifts; pseudodifferential operators, factorization of almost periodic matrix functions; inequalities; Cauchy type integrals; maximal and singular operators on generalized Orlicz-Morrey spaces; the Riesz potential operator; modification of Hadamard fractional integro-differentiation), (ii) operator algebras (invertibility in groupoid C*-algebras; inner endomorphisms of some semi group, crossed products; C*-algebras generated by mappings which have finite orbits; Folner sequences in operator algebras; arithmetic aspect of C*_r SL(2); C*-algebras of singular integral operators; algebras of operator sequences) and (iii) mathematical physics (operator approach to diffraction from polygonal-conical screens; Poisson geo...

A twisted generalization of Novikov-Poisson algebras

OpenAIRE

Yau, Donald

2010-01-01

Hom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras, are studied. Hom-Novikov-Poisson algebras are shown to be closed under tensor products and several kinds of twistings. Necessary and sufficient conditions are given under which Hom-Novikov-Poisson algebras give rise to Hom-Poisson algebras.

Process Algebra and Markov Chains

NARCIS (Netherlands)

Brinksma, Hendrik; Hermanns, H.; Brinksma, Hendrik; Hermanns, H.; Katoen, Joost P.

This paper surveys and relates the basic concepts of process algebra and the modelling of continuous time Markov chains. It provides basic introductions to both fields, where we also study the Markov chains from an algebraic perspective, viz. that of Markov chain algebra. We then proceed to study

Process algebra and Markov chains

NARCIS (Netherlands)

Brinksma, E.; Hermanns, H.; Brinksma, E.; Hermanns, H.; Katoen, J.P.

2001-01-01

This paper surveys and relates the basic concepts of process algebra and the modelling of continuous time Markov chains. It provides basic introductions to both fields, where we also study the Markov chains from an algebraic perspective, viz. that of Markov chain algebra. We then proceed to study

Vertex ring-indexed Lie algebras

International Nuclear Information System (INIS)

Fairlie, David; Zachos, Cosmas

2005-01-01

Infinite-dimensional Lie algebras are introduced, which are only partially graded, and are specified by indices lying on cyclotomic rings. They may be thought of as generalizations of the Onsager algebra, but unlike it, or its sl(n) generalizations, they are not subalgebras of the loop algebras associated with sl(n). In a particular interesting case associated with sl(3), their indices lie on the Eisenstein integer triangular lattice, and these algebras are expected to underlie vertex operator combinations in CFT, brane physics, and graphite monolayers

Invariants of triangular Lie algebras

International Nuclear Information System (INIS)

Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman

2007-01-01

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated

Q-systems as cluster algebras

International Nuclear Information System (INIS)

Kedem, Rinat

2008-01-01

Q-systems first appeared in the analysis of the Bethe equations for the XXX model and generalized Heisenberg spin chains (Kirillov and Reshetikhin 1987 Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov. 160 211-21, 301). Such systems are known to exist for any simple Lie algebra and many other Kac-Moody algebras. We formulate the Q-system associated with any simple, simply-laced Lie algebras g in the language of cluster algebras (Fomin and Zelevinsky 2002 J. Am. Math. Soc. 15 497-529), and discuss the relation of the polynomiality property of the solutions of the Q-system in the initial variables, which follows from the representation-theoretical interpretation, to the Laurent phenomenon in cluster algebras (Fomin and Zelevinsky 2002 Adv. Appl. Math. 28 119-44)

Waterloo Workshop on Computer Algebra

CERN Document Server

Zima, Eugene; WWCA-2016; Advances in computer algebra : in honour of Sergei Abramov's' 70th birthday

2018-01-01

This book discusses the latest advances in algorithms for symbolic summation, factorization, symbolic-numeric linear algebra and linear functional equations. It presents a collection of papers on original research topics from the Waterloo Workshop on Computer Algebra (WWCA-2016), a satellite workshop of the International Symposium on Symbolic and Algebraic Computation (ISSAC’2016), which was held at Wilfrid Laurier University (Waterloo, Ontario, Canada) on July 23–24, 2016.   This workshop and the resulting book celebrate the 70th birthday of Sergei Abramov (Dorodnicyn Computing Centre of the Russian Academy of Sciences, Moscow), whose highly regarded and inspirational contributions to symbolic methods have become a crucial benchmark of computer algebra and have been broadly adopted by many Computer Algebra systems.

Construction of harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces

International Nuclear Information System (INIS)

Konderak, J.

1988-09-01

Defined here is an orthogonal multiplication for vector spaces with indefinite nondegenerate scalar product. This is then used, via the Hopf construction, to obtain harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces. Examples of harmonic maps are constructed using Clifford algebras. (author). 6 refs

The Cuntz algebra Q_N and C*-algebras of product systems

DEFF Research Database (Denmark)

Hong, Jeong Hee; Larsen, Nadia S.; Szymanski, Wojciech

2011-01-01

We consider a product system over the multiplicative group semigroup N^x of Hilbert bimodules which is implicit in work of S. Yamashita and of the second named author. We prove directly, using universal properties, that the associated Nica-Toeplitz algebra is an extension of the C*-algebra Q...

Lectures on algebraic quantum field theory and operator algebras

International Nuclear Information System (INIS)

Schroer, Bert

2001-04-01

In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as why mathematicians are/should be interested in algebraic quantum field theory would be equally fitting. besides a presentation of the framework and the main results of local quantum physics these notes may serve as a guide to frontier research problems in mathematical. (author)

Assessing Algebraic Solving Ability: A Theoretical Framework

Science.gov (United States)

Lian, Lim Hooi; Yew, Wun Thiam

2012-01-01

Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…

Associative and Lie deformations of Poisson algebras

OpenAIRE

Remm, Elisabeth

2011-01-01

Considering a Poisson algebra as a non associative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this non associative algebra. This gives a natural interpretation of deformations which preserves the underlying associative structure and we study deformations which preserve the underlying Lie algebra.

Paragrassmann analysis and covariant quantum algebras

International Nuclear Information System (INIS)

Filippov, A.T.; Isaev, A.P.; Kurdikov, A.B.; Pyatov, P.N.

1993-01-01

This report is devoted to the consideration from the algebraic point of view the paragrassmann algebras with one and many paragrassmann generators Θ i , Θ p+1 i = 0. We construct the paragrassmann versions of the Heisenberg algebra. For the special case, this algebra is nothing but the algebra for coordinates and derivatives considered in the context of covariant differential calculus on quantum hyperplane. The parameter of deformation q in our case is (p+1)-root of unity. Our construction is nondegenerate only for even p. Taking bilinear combinations of paragrassmann derivatives and coordinates we realize generators for the covariant quantum algebras as tensor products of (p+1) x (p+1) matrices. (orig./HSI)

CLASSIFICATION OF 4-DIMENSIONAL GRADED ALGEBRAS

OpenAIRE

Armour, Aaron; Chen, Hui-Xiang; ZHANG, Yinhuo

2009-01-01

Let k be an algebraically closed field. The algebraic and geometric classification of finite dimensional algebras over k with ch(k) not equal 2 was initiated by Gabriel in [6], where a complete list of nonisomorphic 4-dimensional k-algebras was given and the number of irreducible components of the variety Alg(4) was discovered to be 5. The classification of 5-dimensional k-algebras was done by Mazzola in [10]. The number of irreducible components of the variety Alg(5) is 10. With the dimensio...

Banana Algebra: Compositional syntactic language extension

DEFF Research Database (Denmark)

Andersen, Jacob; Brabrand, Claus; Christiansen, David Raymond

2013-01-01

We propose an algebra of languages and transformations as a means of compositional syntactic language extension. The algebra provides a layer of high-level abstractions built on top of languages (captured by context-free grammars) and transformations (captured by constructive catamorphisms...... algebra as presented in the paper is implemented as the Banana Algebra Tool which may be used to syntactically extend languages in an incremental and modular fashion via algebraic composition of previously defined languages and transformations. We demonstrate and evaluate the tool via several kinds...

Linear algebra

CERN Document Server

Liesen, Jörg

2015-01-01

This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exerc...

JB*-Algebras of Topological Stable Rank 1

Directory of Open Access Journals (Sweden)

Akhlaq A. Siddiqui

2007-01-01

Full Text Available In 1976, Kaplansky introduced the class JB*-algebras which includes all C*-algebras as a proper subclass. The notion of topological stable rank 1 for C*-algebras was originally introduced by M. A. Rieffel and was extensively studied by various authors. In this paper, we extend this notion to general JB*-algebras. We show that the complex spin factors are of tsr 1 providing an example of special JBW*-algebras for which the enveloping von Neumann algebras may not be of tsr 1. In the sequel, we prove that every invertible element of a JB*-algebra is positive in certain isotope of ; if the algebra is finite-dimensional, then it is of tsr 1 and every element of is positive in some unitary isotope of . Further, it is established that extreme points of the unit ball sufficiently close to invertible elements in a JB*-algebra must be unitaries and that in any JB*-algebras of tsr 1, all extreme points of the unit ball are unitaries. In the end, we prove the coincidence between the λ-function and λu-function on invertibles in a JB*-algebra.

(Super)conformal algebra on the (super)torus

International Nuclear Information System (INIS)

Mezincescu, L.; Nepomechie, R.I.; Zachos, C.K.

1989-01-01

A generalization of the Virasoro algebra has recently been introduced by Krichever and Novikov (KN). The KN algebra describes the algebra of general conformal transformations in a basis appropriate to a genus-g Riemann surface. We examine in detail the genus-one KN algebra, and find explicit expressions for the central extension. We, further, construct explicitly the superconformal algebra of the supertorus, which yields supersymmetric generalizations of the genus-one KN algebra. A novel feature of the odd-spin-structure case is that the algebra includes a central element which is anticommuting. We comment on possible applications to string theory. (orig.)

Spin-4 extended conformal algebras

International Nuclear Information System (INIS)

Kakas, A.C.

1988-01-01

We construct spin-4 extended conformal algebras using the second hamiltonian structure of the KdV hierarchy. In the presence of a U(1) current a family of spin-4 algebras exists but the additional requirement that the spin-1 and spin-4 currents commute fixes the algebra uniquely. (orig.)

Computer algebra applications

International Nuclear Information System (INIS)

Calmet, J.

1982-01-01