Quantum integers and cyclotomy

A sequence of functions {f_n(q)}_{n=1}^{\infty} satisfies the functional equation for multiplication of quantum integers if f_{mn}(q) = f_m(q)f_n(q^m) for all positive integers m and n. This paper describes the structure of all sequences of rational functions with rational coefficients that satisfy this functional equation.Comment: 11 pages; LaTe

Majorana Neutrinos and Same-Sign Dilepton Production at LHC and in Rare Meson Decays

We discuss same-sign dilepton production mediated by Majorana neutrinos in high-energy proton-proton collisions pp\ra \ell^+ \ell^{\prime +}X for $\ell,~ \ell^\prime = e,~ \mu,~ \tau$ at the LHC energy $\sqrt{s}=14$ TeV, and in the rare decays of $K$, $D$, $D_s$, and $B$ mesons of the type M^{+}\ra M^{\prime -}\ell ^{+}\ell ^{\prime+}. For the $pp$ reaction, assuming one heavy Majorana neutrino of mass $m_N$, we present discovery limits in the $(m_{N},|U_{\ell N}U_{\ell^\prime N}|)$ plane where $U_{\ell N}$ are the mixing parameters. Taking into account the present limits from low energy experiments, we show that at LHC for the nominal luminosity L=100 fb$^{-1}$ there is no room for observable same-sign dilepton signals. However, increasing the integrated luminosity by a factor 30, one will have sensitivity to heavy Majorana neutrinos up to a mass $m_N\leq 1.5$ TeV only in the dilepton channels $\mu\mu$ and $\mu \tau$, but other dilepton states will not be detectable due to the already existing strong constraints. We work out a large number of rare meson decays, both for the light and heavy Majorana neutrino scenarios, and argue that the present experimental bounds on the branching ratios are too weak to set reasonable limits on the effective Majorana masses.Comment: 18 pages, 4 figures (requires graphicx), a coefficient in Eq. (4) corrected leading to drastic reduction in the Majorana-induced same-sign dilepton cross-section at LHC; revised Figs. 2 and 3; references adde

Scientific provocation as a method for stimulating the participation of distance learning students

The paper reviews provocation as a possible motivating factor to stimulate the interest of distance learning students in the subject studied, to catalyze their research potential and make them active participants in the learning process. It is mostly applicable in the event-oriented learning model. The distance learning model may include a great variety of teaching methods, which, if properly selected, multiply its positive effects and make it a preferable alternative to the conventional learning model. The introduction of a provocative element in the learning process of case studies based on real or hypothetical cases and situations and requiring field research and collection of information as well as the role-playing model relates the educational process to the real-life business and provides the students with the necessary attitude and skills to conduct independent research as well as to gather and process the collected information

Asymptotic behaviour of the spectrum of a waveguide with distant perturbations

We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in a certain sense, and the distance between their ''supports'' tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schroedinger operator, curved and deformed waveguide, delta interaction, and integral operator

Homogenization of the planar waveguide with frequently alternating boundary conditions

We consider Laplacian in a planar strip with Dirichlet boundary condition on the upper boundary and with frequent alternation boundary condition on the lower boundary. The alternation is introduced by the periodic partition of the boundary into small segments on which Dirichlet and Neumann conditions are imposed in turns. We show that under the certain condition the homogenized operator is the Dirichlet Laplacian and prove the uniform resolvent convergence. The spectrum of the perturbed operator consists of its essential part only and has a band structure. We construct the leading terms of the asymptotic expansions for the first band functions. We also construct the complete asymptotic expansion for the bottom of the spectrum