The Quantum Galilei Group

The quantum Galilei group $G_{\varkappa}$ is defined. The bicrossproduct structure of $G_{\varkappa}$ and the corresponding Lie algebra is revealed. The projective representations for the two-dimensional quantum Galilei group are constructed.Comment: AMSTe

The high energy semiclassical asymptotics of loci of roots of fundamental solutions for polynomial potentials

In the case of polynomial potentials all solutions to 1-D Schroedinger equation are entire functions totally determined by loci of their roots and their behaviour at infinity. In this paper a description of the first of the two properties is given for fundamental solutions for the high complex energy limit when the energy is quantized or not. In particular due to the fact that the limit considered is semiclassical it is shown that loci of roots of fundamental solutions are collected of selected Stokes lines (called exceptional) specific for the solution considered and are distributed along these lines in a specific way. A stable asymptotic limit of loci of zeros of fundamental solutions on their exceptional Stokes lines has island forms and there are infintely many of such roots islands on exceptional Stokes lines escaping to infinity and a finite number of them on exceptional Stokes lines which connect pairs of turning points. The results obtained for asymptotic roots distributions of fundamental solutions in the semiclassical high (complex) energy limit are of a general nature for polynomial potentials.Comment: 41 pages, 14 figure

Irreducible Representations of an Algebra underlying Hidden Symmetries of a class of Quasi Exactly Solvable Systems of Equations

The set of linear, differential operators preserving the vector space of couples of polynomials of degrees n and n-2 in one real variable leads to an abstract associative graded algebra A(2). The irreducible, finite dimensional representations of this algebra are classified into five infinite discrete sets and one exceptional case. Their matrix elements are given explicitely. The results are related to the theory of quasi exactly solvable equations.Comment: 38 pages, late

Quadratic integrals of motion for the systems of identical particles

The dynamical systems of identical particles admitting quadratic integrals of motion are classified. The relevant integrals are explicitly constructed and their relation to separation of variables in H-J equation is clarified.Comment: 6 pages, no figure

Global Symmetries of Noncommutative Space-time

The global counterpart of infinitesimal symmetries of noncommutative space-time is discussed.Comment: 7 pages, no figures; minor changes in the bibliography; final version accepted for publication in Phys. Rev.

Forestry and the aquatic environment: studies in an Irish context

International audienceResearch on the interaction between plantation forestry and aquatic environments is essential to develop environmentally compatible and sustainable management further. Given, in Ireland, the generally low levels of atmospheric pollution, its geology and maritime climate, and the unique fauna and flora due to its island history, such studies are important not only in the regional context, but also internationally, as they provide an opportunity to examine the effect of forestry and forest management practices on aquatic systems per se, without the complications of acidification. Here, some of the major findings of forestry and water research in Ireland have been reviewed and compared with those from the UK and elsewhere. Plantation forests do not exacerbate acidification in the south of Ireland (Munster) as a whole so that the influence of forestry on water chemistry is far less important than in other parts of the country (such as Wicklow and Mayo). The main forestry influence on streams in Munster is more likely through physical factors, but their nature is unclear. In a few catchments some negative effects are evident, but in many others apparently positive forest effects occur. In this context, smaller scale catchment-level effects appear to be more important in explaining the various relationships between plantation forests and stream ecology than larger scale regional factors. The management of riparian zones, particularly in forested catchments, is of major importance for the structure and functioning of aquatic communities and further work is needed on best management practices. It is suggested that it is unreasonable to base forest management on national Forest-Fisheries guidelines since regions vary too much and the signal from local conditions is too strong. The approach for environmentally benign, scientifically sound forestry management has to be at the catchment scale. Trees in the right places may be beneficial ecologically but further work is needed to identify these locations. The introduction of new forest management practices such as adoption of new species mixes and continuous forest cover are at an early stage in Ireland and their influence on aquatic systems is unknown. Keywords: forest-stream interactions, Irish plantation forestry, hydrochemistry, macroinvertebrates, salmonids, forest managemen

κ\kappa-Deformed Statistics and Classical Fourmomentum Addition Law

We consider $\kappa$-deformed relativistic symmetries described algebraically by modified Majid-Ruegg bicrossproduct basis and investigate the quantization of field oscillators for the $\kappa$-deformed free scalar fields on $\kappa$-Minkowski space. By modification of standard multiplication rule, we postulate the $\kappa$-deformed algebra of bosonic creation and annihilation operators. Our algebra permits to define the n-particle states with classical addition law for the fourmomenta in a way which is not in contradiction with the nonsymmetric quantum fourmomentum coproduct. We introduce $\kappa$-deformed Fock space generated by our $\kappa$-deformed oscillators which satisfy the standard algebraic relations with modified $\kappa$-multiplication rule. We show that such a $\kappa$-deformed bosonic Fock space is endowed with the conventional bosonic symmetry properties. Finally we discuss the role of $\kappa$-deformed algebra of oscillators in field-theoretic noncommutative framework.Comment: LaTeX, 12 pages. V2: second part of chapter 4 changed, new references and comments added. V3: formula (14) corrected. Some additional explanations added. V4: further comments about algebraic structure are adde