Short Proofs for Cut-and-Paste Sorting of Permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of $[n]$ using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of $[n]$ can be transformed to the identity in at most \flr{2n/3} such moves and that some permutations require at least \flr{n/2} moves.Comment: 7 pages, 2 figure

On Holomorphic Effective Actions of Hypermultiplets Coupled to External Gauge Superfields

We study the structure of holomorphic effective action for hypermultiplet models interacting with background super Yang-Mills fields. A general form of holomorphic effective action is found for hypermultiplet belonging to arbitrary representation of any semisimple compact Lie group spontaneously broken to its maximal abelian subgroup. The applications of obtained results to hypermultiplets in fundamental and adjoint representations of the SU(n), SO(n), Sp(n) groups are considered.Comment: 8 pages, no figure

A class of Lorentzian Kac-Moody algebras

We consider a natural generalisation of the class of hyperbolic Kac-Moody algebras. We describe in detail the conditions under which these algebras are Lorentzian. We also construct their fundamental weights, and analyse whether they possess a real principal so(1,2) subalgebra. Our class of algebras include the Lorentzian Kac-Moody algebras that have recently been proposed as symmetries of M-theory and the closed bosonic string.Comment: 40 pages TeX, 5 eps-figure

Duality Symmetries and G^{+++} Theories

We show that the non-linear realisations of all the very extended algebras G^{+++}, except the B and C series which we do not consider, contain fields corresponding to all possible duality symmetries of the on-shell degrees of freedom of these theories. This result also holds for G_2^{+++} and we argue that the non-linear realisation of this algebra accounts precisely for the form fields present in the corresponding supersymmetric theory. We also find a simple necessary condition for the roots to belong to a G^{+++} algebra.Comment: 35 pages. v2: 2 appendices added, other minor corrections. v3: tables corrected, other minor changes, one appendix added, refs. added. Version published in Class. Quant. Gra

Generalised geometry, eleven dimensions and E11

We construct the non-linear realisation of E11 and its first fundamental representation in eleven dimensions at low levels. The fields depend on the usual coordinates of space-time as well as two form and five form coordinates. We derive the terms in the dynamics that contain the three form and six form fields and show that when we restricted their field dependence to be only on the usual space-time we recover the correct self-duality relation. Should this result generalise to the gravity fields then the non-linear realisation is an extension of the maximal supergravity theory, as previously conjectured. We also comment on the connections between the different approaches to generalised geometry.Comment: 17 pages, Trivial typos corrected in version one and a substantial note added which gives the equation of motion relating the gravity field to its dua

Generalised Space-time and Gauge Transformations

We consider the generalised space-time introduced by the author in 2003 in the context of the non-linear realisation of the semi-direct product of E11 and its first fundamental representation. For all the fields we propose gauge transformations which are compatible with the underlying E11 structure. A crucial role is played by the generalised vielbein that the generalised space-time possess. We work out the explicit form of the gauge transformations, at low levels, in four, five and eleven dimensions.Comment: 33 page

Kac-Moody Symmetries of Ten-dimensional Non-maximal Supergravity Theories

A description of the bosonic sector of ten-dimensional N=1 supergravity as a non-linear realisation is given. We show that if a suitable extension of this theory were invariant under a Kac-Moody algebra, then this algebra would have to contain a rank eleven Kac-Moody algebra, that can be identified to be a particular real form of very-extended D_8. We also describe the extension of N=1 supergravity coupled to an abelian vector gauge field as a non-linear realisation, and find the Kac-Moody algebra governing the symmetries of this theory to be very-extended B_8. Finally, we discuss the related points for the N=1 supergravity coupled to an arbitrary number of abelian vector gauge fields

An eco-physiological study of sesleria caerulea (L.) Ard. subsp. calcarea

Not availabl

Guest Editor\u27s Introduction

Guest editor\u27s introduction for 54.3/