Central factorials under the Kontorovich-Lebedev transform of polynomials

We show that slight modifications of the Kontorovich-Lebedev transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL-transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.Comment: also available at http://cmup.fc.up.pt/cmup/ since the 2nd August 201

Surface-plasmon-polariton wave propagation guided by a metal slab in a sculptured nematic thin film

Surface-plasmon-polariton~(SPP) wave propagation guided by a metal slab in a periodically nonhomogeneous sculptured nematic thin film~(SNTF) was studied theoretically. The morphologically significant planes of the SNTF on both sides of the metal slab could either be aligned or twisted with respect to each other. The canonical boundary-value problem was formulated, solved for SPP-wave propagation, and examined to determine the effect of slab thickness on the multiplicity and the spatial profiles of SPP waves. Decrease in slab thickness was found to result in more intense coupling of two metal/SNTF interfaces. But when the metal slab becomes thicker, the coupling between the two interfaces reduces and SPP waves localize to one of the two interfaces. The greater the coupling between the two metal/SNTF interfaces, the smaller is the phase speed.Comment: 17 page

Phase Splitting for Periodic Lie Systems

In the context of the Floquet theory, using a variation of parameter argument, we show that the logarithm of the monodromy of a real periodic Lie system with appropriate properties admits a splitting into two parts, called dynamic and geometric phases. The dynamic phase is intrinsic and linked to the Hamiltonian of a periodic linear Euler system on the co-algebra. The geometric phase is represented as a surface integral of the symplectic form of a co-adjoint orbit.Comment: (v1) 15 pages. (v2) 16 pages. Some typos corrected. References and further comments added. Final version to appear in J. Phys. A

Breaking of chalk fallen onto the floor

This problem was presented at the IPT 2022. The main goal was to find out What is the maximum height a piece of chalk might be dropped without breaking for a given surface, estimate parameters on which the height depends and suggest throwing techniques which minimize the breakage probabilit

Scar functions in the Bunimovich Stadium billiard

In the context of the semiclassical theory of short periodic orbits, scar functions play a crucial role. These wavefunctions live in the neighbourhood of the trajectories, resembling the hyperbolic structure of the phase space in their immediate vicinity. This property makes them extremely suitable for investigating chaotic eigenfunctions. On the other hand, for all practical purposes reductions to Poincare sections become essential. Here we give a detailed explanation of resonances and scar functions construction in the Bunimovich stadium billiard and the corresponding reduction to the boundary. Moreover, we develop a method that takes into account the departure of the unstable and stable manifolds from the linear regime. This new feature extends the validity of the expressions.Comment: 21 pages, 10 figure

Dynamic Phase Transition in a Time-Dependent Ginzburg-Landau Model in an Oscillating Field

The Ginzburg-Landau model below its critical temperature in a temporally oscillating external field is studied both theoretically and numerically. As the frequency or the amplitude of the external force is changed, a nonequilibrium phase transition is observed. This transition separates spatially uniform, symmetry-restoring oscillations from symmetry-breaking oscillations. Near the transition a perturbation theory is developed, and a switching phenomenon is found in the symmetry-broken phase. Our results confirm the equivalence of the present transition to that found in Monte Carlo simulations of kinetic Ising systems in oscillating fields, demonstrating that the nonequilibrium phase transition in both cases belongs to the universality class of the equilibrium Ising model in zero field. This conclusion is in agreement with symmetry arguments [G. Grinstein, C. Jayaprakash, and Y. He, Phys. Rev. Lett. 55, 2527 (1985)] and recent numerical results [G. Korniss, C.J. White, P. A. Rikvold, and M. A. Novotny, Phys. Rev. E (submitted)]. Furthermore, a theoretical result for the structure function of the local magnetization with thermal noise, based on the Ornstein-Zernike approximation, agrees well with numerical results in one dimension.Comment: 16 pp. RevTex, 9 embedded ps figure

Mexametric Assessment of Melanin Level in Children's Skin

Background. High predisposition to sunburn in childhood and associated increased risk of malignant skin tumors development, decrease with age. There is a likelihood of a relationship between the described trends and dynamic age related changes in functional state of melanocytes and melanin.Aim: to evaluate the level of melanin in the skin in children of different age. Materials and methods. The study involved 78 children aged from 7 to 17 years, without any disorders of skin pigmentation. Three groups of observations were formed: Group 1 - 28 children from 7 to 9 years, Group 2 - 25 children from 10 to 12 years, Group 3 - 25 children from 16 to 17years. The melanin level was evaluated in the skin of face, body and extremities using mexametry.Results. The highest level of skin melanin was observed in forearms and lower legs in all age groups (up to 28.3 ± 10.8, 23.3 ± 8.6 and 26.7 ± 10.6 c.u. in Group 1, 2 and 3 respectively), the lowest - in cheeks and chest (up to 8.0 ± 4.7, 4.4 ± 3.4 and 9.5 ± 4.1 c.u. in Group 1, 2 and 3 respectively). There were relationships between skin site and level of melanin, but no relationships between level of melanin and gender or age.Findings. The level of melanin in the skin in children aged from 7 to 17 years is individual and depends on the location of the skin area, but does not depend on gender or age. The distribution of melanin in the skin is stable and does not change over time

Berry phases for the nonlocal Gross-Pitaevskii equation with a quadratic potential

A countable set of asymptotic space -- localized solutions is constructed by the complex germ method in the adiabatic approximation for the nonstationary Gross -- Pitaevskii equation with nonlocal nonlinearity and a quadratic potential. The asymptotic parameter is 1/T, where $T\gg1$ is the adiabatic evolution time. A generalization of the Berry phase of the linear Schr\"odinger equation is formulated for the Gross-Pitaevskii equation. For the solutions constructed, the Berry phases are found in explicit form.Comment: 13 pages, no figure

Theory of Dyakonov-Tamm waves at the planar interface of a sculptured nematic thin film and an isotropic dielectric material

In order to ascertain conditions for surface-wave propagation guided by the planar interface of an isotropic dielectric material and a sculptured nematic thin film (SNTF) with periodic nonhomogeneity, we formulated a boundary-value problem, obtained a dispersion equation therefrom, and numerically solved it. The surface waves obtained are Dyakonov-Tamm waves. The angular domain formed by the directions of propagation of the Dyakonov--Tamm waves can be very wide (even as wide as to allow propagation in every direction in the interface plane), because of the periodic nonhomogeneity of the SNTF. A search for Dyakonov-Tamm waves is, at the present time, the most promising route to take for experimental verification of surface-wave propagation guided by the interface of two dielectric materials, at least one of which is anisotropic. That would also assist in realizing the potential of such surface waves for optical sensing of various types of analytes infiltrating one or both of the two dielectric materials.Comment: accepted for publication in J. Opt.