On algebraic equations satisfied by correlators in Wess-Zumino-Witten models

Apart from the Knizhnik-Zamolodchikov differential equations, correlation functions in Wess-Zumino-Witten models of conformal field theory satisfy a certain system of algebraic equations. We show that hypergeometric solutions of KZ equations constructed in [3] do satisfy these algebraic equations.     

Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations

In this Letter, homotopy perturbation method (HPM) is directly applied to modified Camassa-Holm and Degasperis-Procesi equations to obtain their solitary wave solutions. The results show the applicability, accuracy and efficiency of HPM in solving nonlinear differential equations with fully nonlinear dispersion term. It is predicted that HPM can be widely applied in engineering problems.     

An explicit one-soliton solution of the (2+1)-dimensional generalized sine-gordon equations

The (n+1)-dimensional differential geometric generalization of the sine-Gordon equation (SGE) given by Tenenblat and Terng is solved explicitly in the casen=2 to obtain a one-soliton solution. The solution yields the soliton solution of the (1+1)-dimensional SGE in the limit as one of the three independent variables approaches infinity. However, more than one variable plays the role of time in these limits.     

Lie-admissible structures on Witt type algebras

In this paper we study Lie-admissible structures on Witt type algebras. Witt type algebras are Γ$\Gamma$-graded Lie algebras (where Γ$\Gamma$ is an abelian group) which generalize the Witt algebra. We give all third power-associative and flexible Lie-admissible structures on these algebras. In particular we generalize some results on the Witt algebra. After describing the second scalar cohomology group of Witt type algebras, we investigate third power-associative and flexible Lie-admissible structures on the central extension of some Witt type algebras. Finally we study a left-symmetric structure induced by a symplectic form for some Witt type algebras.     

Symmetric functions in the Kontsevich-Witten intersection theory of the moduli space of curves

We show that symmetric functions introduced by Di Francesco and co-workers, to prove Kontsevich's formula and Witten's conjecture in the intersection theory of the moduli space of curves are, up to sign, SchurQ-functions.On leave from Institute of Mathematics, Polish Academy of Sciences, Toru, Poland     

Stability conditions for Cohen-Grossberg neural networks with time-varying delays

Global asymptotic stability for Cohen-Grossberg neural networks (CGNNs) with time-varying delays is investigated. Criteria are proposed to guarantee the stability and uniqueness of equilibrium point of CGNNs via LMI approach. A numerical example is illustrated to show the effectiveness of our results.     

Deformations of Lie algebras of vector fields arising from families of schemes

Fialowski and Schlichenmaier constructed global deformations of the infinitesimally and formally rigid Lie algebra of polynomial vector fields on the circle from families of projective curves with marked points. In the present article, we show how to obtain these examples in a conceptual way. For this, we define a stack of deformations of Lie algebras and study the morphism from the moduli stack to this deformation stack which associates to a family of marked curves the Lie algebra of vertical vector fields on the punctured family.     

Generalization of the Toda chain system to the elliptic curve case

We propose a system of two equations which, when some of its parameters vanish, separates into two equations describing independent one-dimensional Toda chains. The system has its foundation in the discrete transformations of the Landau-Lifshitz equation which is closely connected with elliptic curves. Nontrivial solutions of the system are found in an explicit form.     

The Lie algebra of the sl(2, ℂ)-valued automorphic functions on a torus

It is shown that the Lie algebra of the automorphic, meromorphic sl(2, )-valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2, )-valued loop algebra, while the latter goes into the Lie algebra (A ₁ ⁽¹⁾ )/(centre).     

Compatible poisson-Lie structures on the loop group of SL2

We define a 1-parameter family ofr-matrices on the loop algebra of sl₂, defining compatible Poisson structures on the associated loop group, which degenerate into the rational and trigonometric structures, and study the Manin triples associated with them.To our friend Dimitri Gurevich, on his 50th birthday     

Synchronization of delayed fuzzy cellular neural networks based on adaptive control

This Letter studies synchronization of delayed fuzzy cellular neural networks with all the parameters unknown. To enhance the coupled strength dynamically and be more suitable for the reality, we add fuzzy theory to the traditional cellular neural networks. By the Lyapunov-Lasall principle of functional differential equations, some new stability criteria are obtained via adaptive control. To the best of our knowledge, there has few work studying fuzzy cellular neural networks. Moreover, the approaches developed here extend the ideas and techniques derived in recent literatures. In the end, an example and its simulation were given to illustrate the simpleness and effectiveness of our main results.     

Suppression of Chaos and Phase Locking in Two Coupled Nonidentical Neurons under Periodic Input

Dynamical behaviour of two coupled neurons with at least one of them being chaotic is presented. Bifurcation diagrams and Lyapunov exponents are calculated to diagnose the dynamical behaviour of the coupled neurons with the increasing coupling strength. It is found that when the coupling strength increases, a chaotic neuron can be controlled by the coupling between neurons. At the same time, phase locking is studied by the maxima of the differences of instantaneous phases and average frequencies between two coupled neurons, and the inherent connection of phase locking and the suppression of chaos is formulated. It is observed that the onset of phase locking is closely related to the suppression of chaos. Finally, a way for suppression of chaos in two coupled nonidentical neurons under periodic input is suggested.     

On pseudo-continuities in the eigenvalue continuum of one-speed neutron transport theory

We report on a new closed-form approximant to the singular eigenfunction transient solution of the one-speed, one-dimensional neutron transport equation with an anisotropic scattering kernel. It is proved that the associated eigenvalue continuum must be pointwise perforated.     

On The Structure and Representations of the Insertion–Elimination Lie Algebra

We examine the structure of the insertion–elimination Lie algebra on rooted trees introduced in Connes and Kreimer (Ann. Henri Poincar 3(3):411–433, 2002). It possesses a triangular structure , like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a “lowest weight” . We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.      

Hypergeometric solutions of Knizhnik-Zamolodchikov equations

Explicit integral formulas are presented for the solutions of Knizhnik-Zamolodchikov equations associated with an arbitrary Kac-Moody Lie algebra.     

Induction for weak symplectic Banach manifolds

The symplectic induction procedure is extended to the case of weak symplectic Banach manifolds. Using this procedure, one constructs hierarchies of integrable Hamiltonian systems related to the Banach Lie–Poisson spaces of k$k$-diagonal trace class operators.     

Mirror supersymmetry in the space of (super) extended Poincaré algebras p(p,q)

We classify extended Poincaré Lie superalgebras and Lie algebras of any signature (p, q), i.e. Lie superalgebras and ₂-graded Lie algebras g = g₀ + g₁, where g₀ = s₀(V) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = ^{p, q} of signature (p, q) and g₁ is a spin 1/2 s₀(V)-module extended to a s₀-module with kernel V.As a result of the classification, we obtain, if g₁ = S is the spinor module, the numbers L ⁺(n, s) (resp. L ^–(n, s)) of independent such Lie super algebras (resp. Lie algebras), which are periodic functions of the dimension n=p+q (mod 8) and the signature s=p–q (mod 8) and satisfy: L ⁺(–n, s)=L ^– (n, s).Supported by Max-Planck-Institut für Mathematik (Bonn).Supported by the Alexander von Humboldt Foundation, MSRI (Berkeley) and SFB 256 (Bonn University).     

Bounds for the chaotic region in the Lorenz model

In a previous paper, the authors made an extensive numerical study of the Lorenz model, changing all three parameters of the system. We conjectured that the region of parameters where the Lorenz model is chaotic is bounded for fixed r. In this paper, we give a theoretical proof of the conjecture by obtaining theoretical bounds for the chaotic region and by using Fenichel theory. The theoretical bounds are complemented with numerical studies performed using the Maximum Lyapunov Exponent and OFLI2 techniques, and a comparison of both sets of results is shown. Finally, we provide a complete three-dimensional model of the chaotic regime depending on the three parameters.