Taekyun KIM, A S HEGAZI, M MANSOUR

Pages: 315 - 322

Recently, B. A. Kupershmidt have constructed a reflection symmetries of q-Bernoulli polynomials (see [9]). In this paper we give another construction of a q-Bernoulli polynomials, which form Barnes' multiple Bernoulli polynomials at q = 1, cf. [1, 13,14]. By using q-Volkenborn integration, we can also...

Joseph BERNSTEIN, Ossip SCHWARZMAN

Pages: 323 - 351

We prove that, for the irreducible complex crystallographic Coxeter group W, the
following conditions are equivalent: a) W is generated by reflections;
b) the analytic variety X/W is isomorphic to a weighted projective space.
The result is of interest, for example, in application to topological conformal...

Paolo CASATI, Giovanni ORTENZI

Pages: 352 - 364

We present and study bihamiltonian equations of Euler type which include a n

Rafael DIAZ, Eddy PARIGUAN

Pages: 365 - 376

We introduce perturbative Feynman integrals in the context of q-calculus generalizing the Gaussian q-integrals introduced by Diaz and Teruel. We provide analytic as well as combinatorial interpretations for the Feynman-Jackson integrals.

S.M. MYENI, P.G.L. LEACH

Pages: 377 - 392

The complete symmetry group of a 1 + 1 linear evolution equation has been demon-
strated to be represented by the six-dimensional Lie algebra of point symmetries
sl(2, R)?s W , where W is the three-dimensional Heisenberg-Weyl algebra. The infinite number of solution symmetries does not play a role...

A.M. GRUNDLAND, B. HUARD

Pages: 393 - 419

In this paper we employ a “direct method” to construct rank-k solutions, express-
ible in Riemann invariants, to hyperbolic system of first order quasilinear differential
equations in many dimensions. The most important feature of our approach is the
analysis of group invariance properties of these...

Daniel CONDURACHE, Vladimir MARTINUSI

Pages: 420 - 440

We study the well-known Kepler’s problem by introducing a new vectorial regularization. This helps deduce Kepler’s equations by a simple and unified method. Some integral temporal means are also obtained by means of this regularization. The vectorial eccentricity plays a fundamental part in this approach.

Hailiang LIU

Pages: 441 - 466

The Korteweg de Vries (KdV) equation is well known as an approximation model
for small amplitude and long waves in different physical contexts, but wave breaking phenomena related to short wavelengths are not captured in. In this work we consider a class of nonlocal dispersive wave equations which...

Viktor ABRAMOV

Pages: 1 - 8

Given an associative unital ZN -graded algebra over the complex numbers we construct
the graded q-differential algebra by means of a graded q-commutator, where q is a
primitive N-th root of unity. The N-differential d of the graded q-differential algebra
is a homogeneous endomorphism of degree...

V ABRAMOV, O LIIVAPUU

Pages: 9 - 20

We propose a geometric approach to the BRST-symmetries of the Lagrangian of a
topological quantum field theory on a four dimensional manifold based on the formalism of superconnections. Making use of a graded q-differential algebra, where q is a
primitive N-th root of unity, we also propose a notion...

Nadezda BAZUNOVA

Pages: 21 - 26

In this article, we generalize a construction of graded q-differential algebra with ternary
differential satisfying the property d3
= 0 and q-Leibniz rule on the non-coordinate
case, that is on the case where the differentials of generators of underlying algebra do
not coincide with generators...

C. BURDIK, S. POSTA, O NAVRATIL

Pages: 27 - 36

We study the exact solvable 3 × 3 matrix model of the type G2. We apply the construction similar to that one, which give the 2 × 2 matrix model. But in the studied
case the construction does not give symmetric matrix potential. We conceive that the
exact solvable 3 × 3 matrix potential model of...

Cestmir BURDIK, Eugen PAAL, Juri VIRKEPU

Pages: 37 - 43

Canonical formalism for plane rotations is developed. This group can be seen as
a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and
Hamiltonian are explicitly constructed and their physical interpretation are given. The
Euler-Lagrange and Hamiltonian canonical equations...

Jurgen FUCHS

Pages: 44 - 54

The graphical description of morphisms in rigid monoidal categories, in particular in
ribbon categories, is summarized. It is illustrated with various examples of algebraic
structures in such categories, like algebras, (weak) bi-algebras, Frobenius algebras,
and modules and bimodules. Nakayama...

Tatjana GRAMUSHNJAK, Peeter PUUSEMP

Pages: 55 - 65

Let n be an integer such that n 3 and Cm denote a cyclic group of order m . It
is proved that there exist exactly 17 non-isomorphic groups of order 22n+1
which can
be represented as a semidirect product (C2n × C2n ) C2. These groups are given by
generators and defining relations.

Lars HELLSTROM

Pages: 66 - 75

This paper is an informal collection of observations on how established rewriting techniques can be applied to or need to be adapted for use in non-associative algebras,
operads, and PROPs.

Daniel LARSSON, Gunnar SIGURDSSON, Sergei D SILVESTROV

Pages: 76 - 87

This paper explores the quasi-deformation scheme devised in [1, 3] as applied to the
simple Lie algebra sl2(F) for specific choices of the involved parameters and underlying
algebras. One of the main points of this method is that the quasi-deformed algebra
comes endowed with a canonical twisted...

Eugen PAAL

Pages: 87 - 92

Deformation equation of a non-associative deformation in operad is proposed. Its
integrability condition (the Bianchi identity) is considered. Algebraic meaning of the
latter is explained.

Peeter PUUSEMP

Pages: 93 - 101

It is proved that among the finite groups of order less than 32 only the tetrahedral
group and the binary tetrahedral group are not determined by their endomorphism
semigroups in the class of all groups.

Maido RAHULA, Vitali RETSNOI

Pages: 102 - 109

Total differentiation operators as linear vector fields, their flows, invariants and symmetries form the geometry of jet space. In the jet space the dragging of tensor fields
obeys the exponential law.
The composition of smooth maps induces a composition of jets in corresponding
jet spaces....

Gunnar SIGURDSSON, Sergei D SILVESTROV

Pages: 110 - 128

We describe realizations of the colour analogue of the Heisenberg Lie algebra by power
series in non-commuting indeterminates satisfying Heisenberg's canonical commutation relations of quantum mechanics. The obtained formulas are used to construct new
operator representations of the colour Heisenberg...

Julia Y.-MAGNUSSON, Maxim SAMSONOV, Alexander STOLIN

Pages: 129 - 136

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter
equation. The method is based on so-called affinization of certain seaweed algebras
and their quantum analogues.

Enrique LOUBET

Pages: 158 - 162

We clarify and extend some remarks raised in [5] [Constantin A, J. Math. Phys. 46
(2005), 023506] about the evolution of compactly supported initial data under the
Camassa-Holm flow.

Rodica CIMPOIASU, Radu CONSTANTINESCU

Pages: 285 - 292

The paper investigates some special Lie type symmetries and associated invariant
quantities which appear in the case of the 2D Ricci flow equation in conformal gauge.
Starting from the invariants some simple classes of solutions will be determined.

Fethi BOUZEFFOUR

Pages: 293 - 301

We solve the problem of constructing entire functions where ln M(r; f) grows like ln2
r
from their values at q-n
, for 0 < q < 1. As application we give a product formula for
the basic sine function.

Joseph BERNSTEIN, Ossip SCHWARZMAN

Pages: 163 - 182

We classify (up to an isomorphism in the category of affine groups) the complex
crystallographic groups generated by reflections and such that d, its linear part, is
a Coxeter group, i.e., d is generated by "real" reflections of order 2.

Ting XIAO, Yunbo ZENG

Pages: 193 - 204

For the first time we show that the quasiclassical limit of the symmetry constraint of
the Sato operator for the KP hierarchy leads to the generalized Zakharov reduction of
the Sato function for the dispersionless KP (dKP) hierarchy which has been proved to
be result of symmetry constraint of the...

Nian-Ning HUANG, Hao CAI, Tian YAN, Fan-Rong XU

Pages: 302 - 314

The Hamiltonian formalism of the Landau-Lifschitz equation for a spin chain with
full anisotropy is formulated completely, which constructs a stable base for further
investigations.

GEGENHASI, Xing-Biao HU

Pages: 183 - 192

S E KORENBLIT, V V SEMENOV

Pages: 271 - 284

The formal Heisenberg equations of the Federbush model are linearized and then are
directly integrated applying the method of dynamical mappings. The fundamental
role of two-dimensional free massless pseudo-scalar fields is revealed for this procedure
together with their locality condition taken into...

M BRUSCHI

Pages: 205 - 210

New Cellular Automata associated with the Schroedinger discrete spectral problem
are derived. These Cellular Automata possess an infinite (countable) set of constants
of motion.

V ROSENHAUS

Pages: 255 - 270

We study local conservation laws and corresponding boundary conditions for the ptential Zabolotskaya-Khokhlov equation in (3+1)-dimensional case. We analyze an
infinite Lie point symmetry group of the equation, and generate a finite number of
conserved quantities corresponding to infinite symmetries...

F CALOGERO, J-P FRANCOISE

Pages: 231 - 254

We revisit an integrable (indeed, superintegrable and solvable) many-body model itroduced almost two decades ago by Gibbons and Hermsen and by Wojciechowski,
and we modify it so that its generic solutions are all isochronous (namely, completely
periodic with fixed period). We then show how this model...

M EDWARDS, M C NUCCI

Pages: 211 - 230

Lie group analysis is applied to a core group model for sexually transmitted disease
formulated by Hadeler and Castillo-Chavez [Hadeler K P and Castillo-Chavez C, A
core group model for disease transmission, Math. Biosci. 128 (1995), 4155]. Several
instances of integrability even linearity are found...

Mats EHRNSTROM

Pages: 1 - 8

We consider a nontrivial symmetric periodic gravity wave on a current with nondcreasing vorticity. It is shown that if the surface profile is monotone between trough
and crest, it is in fact strictly monotone. The result is valid for both finite and infinite
depth.

Junyi ZHU, Xianguo GENG

Pages: 81 - 89

The generalized dressing method is extended to variable-coefficient AKNS equations,
including a variable-coefficient coupled nonlinear Schr¨odinger equation and a variablcoefficient coupled mKdV equation. A general variable-coefficient KP equation is
proposed and decomposed into the two 1+1 dimensional...

Betti HARTMANN, Wojtek J ZAKRZEWSKI

Pages: 111 - 116

We study solitons arising in a system describing the interaction of a two-dimensional
discrete hexagonal lattice with an additional electron field (or, in general, an exciton
field). We assume that this interaction is electron-phonon-like. In our previous paper
[4] we have studied the existence of...

Boris A KUPERSHMIDT

Pages: 145 - 157

1-dimensional polytropic gas dynamics is integrable for trivial reasons, having 2 < 3
components. It is realized as a subsystem of two different integrable systems: an
infinite-component hydrodynamic chain of Lax type, and a 3-component system not
of Lax type.

Sergey Yu VERNOV

Pages: 50 - 63

The Painlev´e test is very useful to construct not only the Laurent series solutions of
systems of nonlinear ordinary differential equations but also the elliptic and trigonmetric ones. The standard methods for constructing the elliptic solutions consist of
two independent steps: transformation of...

E A KUZNETSOV

Pages: 64 - 80

In this paper we give a brief review of the recent results obtained by the author and
his co-authors for description of three-dimensional vortical incompressible flows in the
hydrodynamic type systems. For such flows we introduce a new mixed LagrangiaEulerian description - the so called vortex line...

Douglas BALDWIN, Willy HEREMAN

Pages: 90 - 110

The automation of the traditional Painlev´e test in Mathematica is discussed. The
package PainleveTest.m allows for the testing of polynomial systems of nonlinear
ordinary and partial differential equations which may be parameterized by arbitrary
functions (or constants). Except where limited by memory,...

Blazej M SZABLIKOWSKI

Pages: 117 - 128

Appropriate restrictions of Lax operators which allows to construction of (2+1dimensional integrable field systems, coming from centrally extended algebra of pseuddifferential operators, are reviewed. The gauge transformation and the reciprocal link
between three classes of Lax hierarchies are established.

Uma N IYER

Pages: 34 - 49

N BEN FRAJ, S OMRI

Pages: 19 - 33

We classify nontrivial deformations of the standard embedding of the Lie superalgebra
K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie supealgebra of superpseudodifferential operators on the supercircle. This approach leads
to the deformations of the central charge induced...

A MAHARAJ, P G L LEACH

Pages: 129 - 144

We study the dominant terms of systems of Lotka-Volterra-type which arise in the
the mathematical modelling of the evolution of many divers natural systems from the
viewpoint of both symmetry and singularity analyses. The connections between an
increase in the amount of symmetry possessed by the system...

A S HEGAZI, M MANSOUR

Pages: 9 - 18

In this paper, we define a new q-analogy of the Bernoulli polynomials and the
Bernoulli numbers and we deduced some important relations of them. Also, we dduced a q-analogy of the Euler-Maclaurin formulas. Finally, we present a relation
between the q-gamma function and the q-Bernoulli polynomials.

Alexei ZHEDANOV

Pages: 333 - 356

We consider the generalized eigenvalue problem A = B for two operators A, B.
Self-similar closure of this problem under a simplest Darboux transformation gives rise
to two possible types of regular algebras of dimension 2 with generators A, B. Realiztion of the operators A, B by tri-diagonal operators...

E V GUDKOVA

Pages: 197 - 205

The problem of the classification of integrable truncations of the Toda chain is dicussed. A new example of the cutting off constraint is found.

Harry W BRADEN, Victor Z ENOLSKII, Andrew N W HONE

Pages: 46 - 62

The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear
recurrence relation. In recent work, one of us has proved that the general term in
such sequences can be expressed in terms of the Weierstrass sigma function for an
associated elliptic curve. Here we derive the analogous...

Christopher ATHORNE

Pages: 15 - 27

By associating polynomials and power series expansions with sln(C) modules we dscribe the theory of Padé approximants in terms of tensor products of representations
and interpret their recurrence relations algebraically. The treatment links with the
theory of Hirota derivatives and discrete integrable...