The XI International Conference on Symmetry Methods in Physics (SYMPHYS-11) was held in Prague, the Czech Republic, from 21 to 24 June 2004. It was organized by the Bogoliubov Laboratory of Theoretical Physics, the Doppler Institute, and the Czech Technical University (Prague) in the framework of the Blokhintsev-Votruba programme. A total of about 100 scientists from all over the world took part in the conference.

The conference series was initiated by Professor Yakov A. Smorodinsky (1917-1992), an outstanding theoretical physicist. Professor Smorodinsky organized the first five conferences that were held at the Institute of Physics and Power Engineering in Obninsk from 1986 to 1991. The next three conferences of the series were held at the Joint Institute for Nuclear Research in Dubna in 1993, 1995 and 1997, while the last two took place at the Yerevan State University (Armenia) in 2001 and 2003.

The SYMPHYS-11 conference is devoted to general studies and applications of group theoretical methods in modern physics. It covers fields of research where symmetry-based methods play an important role. The programme included the following topics: symmetries of fundamental interactions; Lie groups, supergroups and nonlinear algebraic structures; symmetries of difference and differential equations; nonlinear systems and quantum chaos; quantum optics and coherent states; periodic and aperiodic structures.

BPS wall solutions in four-dimensional massive N = 2 nonlinear sigma models are studied in the off-shell harmonic superspace approach in which N = 2 supersymmetry is manifest. The general nonlinear sigma model can be described by an analytic harmonic potential which is the hyper-Kahler analog of the Kahler potential in N = 1 theory. We examine the massive nonlinear sigma model with multi-center four-dimensional target hyper-Kahler metrics and derive the corresponding BPS equation. We study in some detail two particular cases with the Taub-NUT and double Taub-NUT metrics. The latter embodies, as its two separate limits, both Taub-NUT and Eguchi-Hanson metrics. We find that domain wall solutions exist only in the double Taub-NUT case including its Eguchi-Hanson limit.

The anomaly of various currents in the noncommutative supersymmetric N = 1, U(1) gauge theory are calculated and the effective superpotential obtained.

We report on the Scherk-Schwarz reduction of D = 6 (2,0) supergravity coupled to matter and its interpretation as D = 5 gauged N = 2 supergravity of no-scale type.

A general construction of integrable hierarchies based on afine Lie algebras is presented. The models are specified according to some algebraic data and their time evolution is obtained from solutions of the zero curvature condition. Such framework provides an unified treatment of relativistic and non relativistic models. The extension to the construction of supersymmetric integrable hierarchies is proposed. An explicit example of N = 2 super mKdV and sinh-Gordon is presented.

We study open spin chains based on rational sl(N) and sl(M|N) R-matrices. We classify the solutions of the re ection equations, for both the soliton-preserving and soliton- non-preserving cases. We then write the Bethe equations for these open spin chains.

Integrable geometries were obtained by adding a total time derivative involving the components of the angular momentum to a given free Lagrangian. The motion on a sphere and its induced geometries are examined in details.

We study mathematical models of quasicrystalline materials/non-crystallographic solids with long range aperiodic order. A natural generalization of crystallographic lattices are the so-called Meyer sets. They are uniformly discrete, relatively dense point sets with the property of almost lattices. This property ensures that there is only a finite number of local configurations of atoms in the model of the material. The most commonly studied class of Meyer sets arises in the well known cut-and-project scheme. For cut-and-project sets with compact acceptance window we study a finite set of the Meyer property. This task can be transformed into the problem of covering of the difference set by open copies. The cardinality fof the minimal covering is called the Meyer number. We show that f is bounded on the space of convex compact sets. We give estimates on the universal upper bound of the Meyer number of dimension 2 and 3. We determine the values of f for some special types of dimension 2. We further show that f is not bounded if we relax the condition of convexity.

The general properties of the clock synchronization in Special Theory of Relativity with different "one way" velocities of light are discussed. It is argued that the customary irreducible element of conventionality of the synchronization problem may be eliminated.

Little groups are the subgroups of the Poincare group whose transformations leave the four-momentum of a relativistic particle invariant. Massless particle representations can be obtained from their massive counterparts through a contraction procedure. The two by two matrix representations of little groups as well as of other kinematical effects of special relativity, like Wigner rotations are observed to coincide with the matrix formulations of some interesting classical ray optics phenomena. Examples include beam cycles in laser cavities, image focusing in a one-lens-camera, multilens optics and interferometers. Thus, it is argued that optical implementations can be exploited as analogue processors for special relativity.

We paraquantize the bosonic (resp. the Neuveu Shwarz spinning) string theory. Unlike the Ardalan and Mansouri work, the paraquantum system is so that both the center of mass variables and the excitation modes of the string verify paracommutation relations. We find existence possibilities of parabosonic (resp. paraspinning) strings defined in a noncommutative space-time at space-time dimensions other than D = 26 (resp. D = 10). We investigate then the existence possibilities of the. D = 3, 4, 6 parasuperstring. The two cases, parabose-parafermi (resp. bose-parafermi) superstrings are considered. In the first one, the spectrum is discussed through the partition functions for D = 3, 4, 6. Despite of the parastatistical algebraic structure of the dynamical variables, the combined set of the generators of the symmetries forms the algebra of the Super Symmetric Quantum Mechanic (resp. the ParaSSQM in the sense of Beckers and Debergh).

We review the spin bit model describing anomalous dimensions of the operators of Super Yang-Mills theory. We concentrate here on the scalar sector. In the limit of large N this model coincides with integrable spin chain while at finite N it has nontrivial chain splitting and joining interaction.

We demonstrate that two-dimensional N = 8 supersymmetric quantum mechanics which inherits the most interesting properties of N = 2; d = 4 SYM can be constructed if the reduction to one dimension is performed in terms of the basic object - N = 2; d = 4 vector multiplet. In such a reduction only complex scalar fields from the N = 2; d = 4 vector multiplet become physical bosons in d = 1, while the rest of the bosonic components are reduced to auxiliary fields thus giving rise to (2, 8, 6) supermultiplet in d = 1. We construct the most general action for this supermultiplet with all possible FI terms included and explicitly demonstrate that the action possesses duality symmetry extended to the fermionic sector of theory. To deal with the second-class constraints presented in the system, we introduce the Dirac brackets for the canonical variables and find supercharges and Hamiltonian which form the N = 8 super Poincare algebra with central charges. Finally, we explicitly present the generalization of the two-dimensional N = 8 SQM to the 2k-dimensional case with a special Kahler geometry in the target space.

In this paper we present some recent and new developments in the theory of p- mechanics. p-Mechanics is a consistent physical theory which contains both classical and quantum mechanics. The Heisenberg group and its representation theory is the basis of p-mechanics. We give a summary of recent results on p-mechanical observables, states and canonical transformations. In doing so we exhibit relations between the quantum and classical image of these objects. We also present some new work on the Kepler/Coulomb problem. This involves constructing a new Hilbert space which represents the dynamics of the Kepler/Coulomb problem in a simple form.

Lie algebraic methods, which have been used widely for stationary states of quantum mechanical systems are extended here to treat time–dependent problems. Difficulties may arise at points where the potential is discontinuous or has discontinuous derivatives and from certain imposed boundary conditions. The simplicity and elegance of the usual algebraic methods can be retained for such problems by redefining the domain of the operators using techniques developed by Lighthill to introduce generalized functions. We treat a model double–well subject to time–varying external fields as well as prob­lems with time–dependent boundary conditions.

Exact isotropic and anisotropic plasma equilibria are constructed as solutions to nonlinear 3D Magnetohydrodynamic (MHD) and anisotropic Chew-Goldberger-Low (CGL) plasma equilibrium equations, using the representation of equilibrium equations in coordinates connected with magnetic surfaces. Infinite-dimensional symmetries of MHD and CGL equilibrium equations used in this construction are discussed from the prospective of Lie group analysis. The infinite-parameter set of transformations between MHD and CGL equilibrium systems is employed to produce families of anisotropic (CGL) equilibria from particular isotropic (MHD) ones. Solutions produced with the presented method are generally fully 3D solutions with no geometrical symmetries; they have different topologies and physical properties, and can serve as models of astrophysical phenomena.

In many recent astrophysical applications of the theory of dense matter it is necessary to investigate the properties of rapidly rotating compact objects within general relativity theory. The reason of this development is the hope that changes in the internal structure of the dense matter, e.g. during phase transitions, could have observable consequences for the dynamics of the rotational behavior of these objects. Particular examples are the observations of glitches and postglitch relaxation in pulsars, which are discussed as signals for superfluidity in nuclear matter and the suggestion that the braking index is remarkably enhanced when a quark matter core occurs in the center of a pulsar during its spin–down evolution. Further constraints for the nuclear equation of state come from the observation of quasi–periodic brightness oscillations (QPO’s) in low–mass–X–ray binaries, which entail mass and radius limits for rapidly rotating neutron stars.

We prove that the original Penrose tilings of the plane admit an infinite number of independent scaling factors and an infinite number of in ation centers. Our results are based on the definition of these tilings in terms of strip projection method proposed by Katz and Duneau shortly after the discovery of quasicrystals.

Moyal–deformed hierarchies of soliton equations can be extended to larger hierarchies by including additional evolution equations with respect to the deformation parameters. A general framework is presented in which the extension is universally determined and which applies to several deformed hierarchies, including the noncommutative KP, modified KP, and Toda lattice hierarchy. We prove a Birkhoff factorization relation for the extended ncKP and ncmKP hierarchies. Also reductions of the latter hierarchies are briefly discussed. Furthermore, some results concerning the extended ncKP hierarchy are recalled from previous work.

We give character formulae for the positive energy unitary irreducible representations of the N-extended D = 4 conformal superalgebras su(2; 2=N). Using these we also derive decompositions of long superfields as they descend to the unitarity threshold. These results are also applicable to irreps of the complex Lie superalgebras sl(4=N).

We review non-Hermitian Hamiltonians following Mostafazadeh, while expanding on the underlying mathematical details. To conclude, we shortly summarize pseudo-supersymmetric quantum mechanics.

We consider solutions of a generalization of the Camassa-Holm hierarchy to (2+1) dimensions that include, in particular, the well-known multipeakons solutions for the celebrated Camassa-Holm equation.

We present some facts related to the charged nontopological solutions of nonlinear field equations known as Q-balls. Using simplified field equations from the bosonic sector of the supersymmetric model we discuss an approximate solution with the spherical symmetry.

We report on the existence of symmetry plane-groups for quasiperiodic point-sets named beta-lattices. Like lattices are vector superpositions of integers, beta-lattices are vector superpositions of beta-integers. When beta > 1 is a quadratic Pisot-Vijayaraghavan (PV) algebraic unit, the set of beta-integers can be equipped with an abelian group structure and an internal multiplicative law. When beta is equal to special values, we show that these arithmetic and algebraic structures lead to freely generated symmetry plane- groups for beta-lattices. These plane-groups are based on repetitions of discrete adapted rotations and translations we shall refer to as beta-rotations and beta-translations. Hence beta-lattices, endowed with beta-rotations and beta-translations, can be viewed like lattices. We also show that, at large distances, beta-lattices and their symmetries behave asymptotically like lattices and lattice symmetries respectively.

We review our recent results concerning two-dimensional smooth orientable surfaces immersed in su(N) Lie algebras. These are derived from the sigma model defined on Minkowski space. The structural equations of such surfaces expressed in terms of any regular solution of the model are found. This is carried out using a moving frame adapted to the surface. A procedure for construction of such surfaces is proposed and illustrated by several examples obtained from the model in dim = 1.

We present two different quantum deformations for the (anti)de Sitter algebras and groups. The former is a non-standard (triangular) deformation of SO(4; 2) realized as the conformal group of the (3+1)D Minkowskian spacetime, while the latter is a standard (quasitriangular) deformation of both SO(2; 2) and SO(3; 1) expressed as the kinematical groups of the (2+1)D anti-de Sitter and de Sitter spacetimes, respectively. The Hopf structure of the quantum algebra and a study of the dual quantum group are presented for each deformation. These results enable us to propose new non-commutative spacetimes that can be interpreted as generalizations of the kappa-Minkowski space, either by considering a variable deformation parameter (depending on the boost coordinates) in the conformal deformation, or by introducing an explicit curvature/cosmological constant in the kinematical one; kappa-Minkowski turns out to be the common first-order structure for all of these quantum spaces. Some properties provided by these deformations, such as dimensions of the deformation parameter (related with the Planck length), space isotropy, deformed boost transformations, etc., are also commented.

We present a general procedure for determining quasi–exact solvability of the Dirac and the Pauli equation with an underlying sl(2) symmetry. This procedure makes full use of the close connection between quasi–exactly solvable systems and supersymmetry.

We treat the problem of normally ordering expressions involving the standard boson operators a, a+ where [a, a+] = 1. We show that a simple product formula for formal power series - essentially an extension of the Taylor expansion - leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions - in essence, a combinatorial field theory. We apply these techniques to some examples related to specific physical Hamiltonians.

Presented results were achieved in collaboration with Miloslav Havlicek, Jiri Patera and Jiri Tolar. We consider the Pauli grading of the Lie algebra sl(3, C) and use a concept of graded contractions to construct non-isomorphic Lie algebras of dimension 8, while preserving the Pauli grading. We show how the symmetry group of a grading simplifies the solution of contraction equations and identification of results. We give examples of resulting Lie algebras. Complete results will be published elsewhere.

In this paper we sketch the foundations of recoupling theory. Introduction of an indistinguishability principle leads to Pauli Exclusion and confinement. We discuss its application to SU(3) colour.

This paper is concerned with the application of the group SO(4; 2) SU(2) to the periodic table of chemical elements. It is shown how the Madelung rule of the atomic shell model can be used for setting up a periodic table that can be further rationalized via the group SO(4; 2) SU(2) and some of its subgroups. Qualitative results are obtained from the table and the general lines of a programme for a quantitative approach to the properties of chemical elements are developed on the basis of the group SO(4; 2) SU(2).

Definition of maximum entanglement in terms of a novel variational principle for quantum uctuations and its corollaries are discussed.

A one-dimensional model with interacting families of Calogero-type particles is studied. It includes harmonic, two-body and three-body interactions among particles. We find the exact eigenenergies corresponding to a class of the exact eigenstates of the model. We emphasize the universal SU(1; 1) structure of the model. We show how SU(1; 1) generators for the whole system are composed of SU(1; 1) generators of arbitrary subsystems. By imposing the conditions for the absence of the three-body interaction, we find certain relations between the coupling constants.

Commutativity of the position operator components is one of the Newton-Wigner postulates for the localized states tacitly included in the list. Omitting it gives additional possibilities to use the Poincare group representations for the analysis of the concept of the relativistic quantum localized states.

The method of categorical extension of the Cayley-Klein groups is developed. The method uses the Cayley-Klein spaces, as objects of the Cayley-Klein category, endowed with all possible linear relations or bilinear forms as morphisms.

We investigate a non-trivial extension of the D-dimensional Poincare algebra. Matrix representations are obtained. The bosonic multiplets contain antisymmetric tensor fields. It turns out that this symmetry acts in a natural geometric way on these p-forms. Some field theoretical aspects of this symmetry are studied and invariant Lagrangians are explicitly given.

We study local conservation laws for the potential Zabolotskaya-Khokhlov equation in three-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 through appropriate boundary conditions.

We explain some features about toric self dual structures and toric quaternionic Kahler manifolds in four dimensions. Applications are outlined.

We construct non-relativistic Lagrangian field models by enforcing Galilean covariance with a (4; 1) Minkowski manifold followed by a projection onto the (3; 1) Newtonian space- time. We discuss scalar, Fermi and gauge fields, as well as interactions between some of these fields. The Galilean covariant formalism provides an elegant construction of the Lagrangians which describe the electric and magnetic limits of Galilean electromagnetism. As further examples of scalar fields, we discuss various models of fluids and super fluids. Then, we turn to linear wave equations, and consider the Dirac Lagrangian which allows one to retrieve the Levy-Leblond wave equations. We examine the situation where the Fermi field interacts with an abelian gauge field. Finally, we study the Bhabha equations for spins 0 and 1.

Through a direct implementation of the saturation regime resulting from the unitarity limit in the impact parameter representation, we explore various possibilities for the energy dependence of hadronic scattering. We show that it is possible to obtain a good description of the scattering amplitude from a hard pomeron provided one includes non-linear effects

It's well known, that the symmetries of a system of differential equations allow transforming its solutions to solutions of this system. Using this property, from two known solutions of the theory of plasticity: the solution of Nadai for circular cavity stressed by normal and shear pressure, and Prandtl's solution for a block compressed between perfectly rough plates, there were constructed new analytical exact solutions of the system of two-dimensional ideal plasticity equations.

Although symmetry methods and analysis are a necessary ingredient in every physicist's toolkit, rather less use has been made of combinatorial methods. One exception is in the realm of Statistical Physics, where the calculation of the partition function, for example, is essentially a combinatorial problem. In this talk we shall show that one approach is via the normal ordering of the second quantized operators appearing in the partition function. This in turn leads to a combinatorial graphical description, giving essentially Feynman-type graphs associated with the theory. We illustrate this methodology by the explicit calculation of two model examples, the free boson gas and a super uid boson model. We show how the calculation of partition functions can be facilitated by knowledge of the combinatorics of the boson normal ordering problem; this naturally gives rise to the Bell numbers of combinatorics. The associated graphical representation of these numbers gives a perturbation expansion in terms of a sequence of graphs analogous to zero-dimensional Feynman diagrams.

We consider systems of diffusion equations that have considerable interest in Soil Science and Mathematical Biology. We construct non-local symmetries, known as potential symmetries. Furthermore we present linearizing mappings.

In this work, we present a general procedure, which is able to generate new exact solitonic models in 1+1 dimensions, from a known one, consisting of two coupled scalar fields. An interesting consequence of the method, is that of the appearing of nontrivial extensions, where the deformed systems presents other BPS solitons than that appearing in the original model. Finally we take a particular example, in order to check the above mentioned features.

Generalized supersymmetries admitting bosonic tensorial central charges are classi- fied in accordance with their division algebra properties. Division algebra consistent constraints lead (in the complex and quaternionic cases) to the classes of hermitian and holomorphic generalized supersymmetries. Applications to the analytic continuation of the M–algebra to the Euclidean and the systematic investigation of certain classes of models in generic space–times are briefly mentioned.

We study the perturbation theory for the example of a topological Batalin-Vilkovisky theory and show that it is free from the UV divergences. In fact, the property of finiteness can be generalized for some class of theories, which we describe here. 3-dimensional Chern-Simons and 2-dimensional topological Yang-Mills theories belong to this class.

This paper focus's upon the derivation of the similarity solutions of a free boundary problem arising in glaciology. With reference to shallow ice sheet ow we present a potential symmetry analysis of the second order non-linear degenerate parabolic equation that describe non-Newtonian ice sheet dynamics in the isothermal case. A full classical and also a non-classical symmetry analysis is presented. A particular example of a similarity solution to a problem formulated with Cauchy boundary conditions is described. This demonstrates the existence of a free moving boundary and also an accumulation-ablation function with realistic physical properties.

Zero-total momentum two-electron wavefunction in crystal space-groups are constructed making use of induced representation method and projection operator technique. Theory is applied to analysis of nodal structure of Cooper pairs in unconventional supercondectors. Theoretical results are compared with experimental data.

Parallels between the concepts of symmetry, supersymmetry and (recently introduced) PT -symmetry are reviewed and discussed, with particular emphasis on the new insight in quantum mechanics which is rendered possible by their combined use.