Invariants of Triangular Lie Algebras

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.Comment: LaTeX2e, 16 pages; misprints are corrected, some proofs are extende

New results on group classification of nonlinear diffusion-convection equations

Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the general form $f(x)u_t=(D(u)u_x)_x+K(u)u_x.$ We obtain new interesting cases of such equations with the density $f$ localized in space, which have large invariance algebra. Exact solutions of these equations are constructed. We also consider the problem of investigation of the possible local trasformations for an arbitrary pair of equations from the class under consideration, i.e. of describing all the possible partial equivalence transformations in this class.Comment: LaTeX2e, 19 page

Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations

A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations $f(x)u_{tt}=(H(u)u_x)_x+K(u)u_x$, is given, by using a compatibility method and additional equivalence transformations. A number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Furthermore, the possible additional equivalence transformations between equations from the class under consideration are investigated. Exact solutions of special forms of these equations are also constructed via classical Lie method and generalized conditional transformations. Local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations.Comment: 23 page

Realizations of Real Low-Dimensional Lie Algebras

Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject. Known results on classification of low-dimensional real Lie algebras, their automorphisms, differentiations, ideals, subalgebras and realizations are reviewed.Comment: LaTeX2e, 39 pages. Essentially exetended version. Misprints in Appendix are correcte

Anomalously interacting new extra vector bosons and their first LHC constraints

In this review phenomenological consequences of the Standard Model extension by means of new spin-1 chiral fields with the internal quantum numbers of the electroweak Higgs doublets are summarized. The prospects for resonance production and detection of the chiral vector $Z^*$ and $W^{*\pm}$ bosons at the LHC energies are considered. The $Z^*$ boson can be observed as a Breit-Wigner resonance peak in the invariant dilepton mass distributions in the same way as the well-known extra gauge $Z'$ bosons. However, the $Z^*$ bosons have unique signatures in transverse momentum, angular and pseudorapidity distributions of the final leptons, which allow one to distinguish them from other heavy neutral resonances. In 2010, with 40 pb$^{-1}$ of the LHC proton-proton data at the energy 7 TeV, the ATLAS detector was used to search for narrow resonances in the invariant mass spectrum of $e^+e^-$ and $\mu^+\mu^-$ final states and high-mass charged states decaying to a charged lepton and a neutrino. No statistically significant excess above the Standard Model expectation was observed. The exclusion mass limits of 1.15 TeV$/c^2$ and 1.35 TeV$/c^2$ were obtained for the chiral neutral $Z^*$ and charged $W^*$ bosons, respectively. These are the first direct limits on the $W^*$ and $Z^*$ boson production. For almost all currently considered exotic models the relevant signal is expected in the central dijet rapidity region. On the contrary, the chiral bosons do not contribute to this region but produce an excess of dijet events far away from it. For these bosons the appropriate kinematic restrictions lead to a dip in the centrality ratio distribution over the dijet invariant mass instead of a bump expected in the most exotic models.Comment: 24 pages, 34 figure, based on talk given by V.A.Bednyakov at 15th Lomonosov conference, 22.08.201

Fluctuating selection models and Mcdonald-Kreitman type analyses

It is likely that the strength of selection acting upon a mutation varies through time due to changes in the environment. However, most population genetic theory assumes that the strength of selection remains constant. Here we investigate the consequences of fluctuating selection pressures on the quantification of adaptive evolution using McDonald-Kreitman (MK) style approaches. In agreement with previous work, we show that fluctuating selection can generate evidence of adaptive evolution even when the expected strength of selection on a mutation is zero. However, we also find that the mutations, which contribute to both polymorphism and divergence tend, on average, to be positively selected during their lifetime, under fluctuating selection models. This is because mutations that fluctuate, by chance, to positive selected values, tend to reach higher frequencies in the population than those that fluctuate towards negative values. Hence the evidence of positive adaptive evolution detected under a fluctuating selection model by MK type approaches is genuine since fixed mutations tend to be advantageous on average during their lifetime. Never-the-less we show that methods tend to underestimate the rate of adaptive evolution when selection fluctuates

Two-Fermion Production in Electron-Positron Collisions

This report summarizes the results of the two-fermion working group of the LEP2-MC workshop, held at CERN from 1999 to 2000. Recent developments in the theoretical calculations of the two fermion production process in the electron-positron collision at LEP2 center of the mass energies are reported. The Bhabha process and the production of muon, tau, neutrino and quark pairs is covered. On the basis of comparison of various calculations, theoretical uncertainties are estimated and compared with those needed for the final LEP2 data analysis. The subjects for the further studies are identified.Comment: 2-fermion working group report of the LEP2 Monte Carlo Workshop 1999/2000, 113 pages, 24 figures, 35 table