Separatrices splitting for Birkhoff's billiard in symmetric convex domain,closed to an ellipse

An example of a convex domain on the plane with the phenomenon of the transversal intersection of separatrices of the corresponding billiard mapping is presented. This example is constructed as an analytic global symmetric perturbation of an ellipse and we investigate the global symmetric analytic perturbation of the integrable billiard mapping in the ellipse. We establish a theorem on the separatrices splitting of the perturbed billiard mapping and derive the asymptotic formulas for a homoclinic invariant as well as for a "principal" splitting angle of separatrices, arising from the hyperbolic fixed point of the mapping. (c) 1994 American Institute of Physics.     

Veiled assonance between geodesics on ellipsoid and billiard in its focal ellipse

We introduce new special ellipsoidal confocal coordinates in ⁿ (n ≥ 3) and apply them to the geodesic problem on a triaxial ellipsoid in ³ as well as the billiard problem in its focal ellipse.

Using such appropriate coordinates we show that these different dynamical systems have the same common analytic first integral. This fact is not evident because there exists a geometrical spatial gap between the geodesic and billiard flows under consideration, and this separating gap just “veils” the resemblance of the two systems.

In short, a geodesic on the ellipsoid and a billiard trajectory inside its focal ellipse are in a “veiled assonance”—under the same initial data they will be tangent to the same confocal hyperboloid. But this assonance is rather incomplete: the dynamical systems in question differ by their intrinsic action angle-variables, thereby the different dynamics arise on the same phase space (i.e. the same phase curves in the same phase space bear quite different rotation numbers).

Some results of this work have been published before in Russian (Tabanov, 1993) and presented to the International Geometrical Colloquium (Moscow, May 10–14, 1993) and the International Symposium on Classical and Quantum Billiards (Ascona, Switzerland, July 25–30, 1994).     

CONTROLLING THE 3D NANOSCALE ORGANIZATION OF BULK HETEROJUNCTION POLYMER SOLAR CELLS

In this study,the three dimensional nanoscale organization in the photoactive layers of poly(3-hexylthiophene) (P3HT) and a methanofullerene derivative (PCBM) is revealed by transmission electron tomography.After annealing treatment,either at elevated temperature or during slow solvent evaporation,nanoscale interpenetrating networks are formed with high crystalline order and favorable concentration gradients of both components through the thickness of the photoactive layer.Such a tailored morphology acco...     

Radiation from an open resonator formed by axisymmetric,weakly curved mirrors

One considers the eigenoscillations of a resonator formed by axisymmetric, weakly curved mirrors. The constructed eigenfunctions satisfy the Dirichlet condition on the mirrors and the Sommerfeld radiation condition at infinity.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 158–165, 1983.In conclusion, the author wishes to express his sincere gratitude to his scientific advisor V. F. Lazutkin for the formulation of the problem and for his constant interest in the paper.     

Exponentially small splittings in Hamiltonian systems

Families of area preserving analytical maps, depending on a small parameter epsilon, are considered, with the case epsilon=0 corresponding to an integrable map. The asymptotic formulas for the splittings of separatrices are derived by the method of analytical continuation of the separatrices to the complex domain. The main terms of the asymptotics are exponentially small with respect to the size of the perturbation. As epsilon tends to zero, the intersection angle of the separatrices can oscillate. The exponent and the oscillatory multiplier of the asymptotic formulas are determined by the position of poles of the homoclinic (heteroclinic) orbit of the limiting flow. Pre-exponential coefficients in the asymptotic formulas contain a multiplier obtained by the numerical study of separatrices of "model" maps in the complex domain.     

Models for capacitated lot-sizing problem with backlogging,setup carryover and crossover

Setup operations are significant in some production environments. It is mandatory that their production plans consider some features, as setup state conservation across periods through setup carryover and crossover. The modelling of setup crossover allows more flexible decisions and is essential for problems with long setup times. This paper proposes two models for the capacitated lot-sizing problem with backlogging and setup carryover and crossover. The first is in line with other models from the literature, whereas the second considers a disaggregated setup variable, which tracks the starting and completion times of the setup operation. This innovative approach permits a more compact formulation. Computational results show that the proposed models have outperformed other state-of-the-art formulation.     

Preparation, structural analysis and dielectric properties of Bi x La1−x FeO3 perovskite

Rare earth element substituted bismuth ferrites (BiFeO₃) are of enormous importance as magnetoelectric materials. The polycrystalline samples of Bi _x La_1−x FeO₃ (x=0, 0.2, 0.4, 0.6, 0.8) were prepared by solid-state reaction using standard ceramic method. The single-phase formation of these compounds was confirmed by X-ray diffraction (XRD) studies. The samples with x=0, 0.2, 0.4, 0.6 are found to be orthorhombic while the sample with x=0.8 is triclinic. The dielectric constant (ε′) and dissipation factor (tan δ) were measured in the frequency range 100 Hz to 1 MHz at room temperature and as a function of temperature at certain fixed frequencies (1 kHz, 10 kHz, 100 kHz, 1 MHz). All the samples showed dielectric dispersion. The dielectric constant with temperature shows a broad peak; the peak temperature shifts with frequency which reflects the relaxor-type behavior. The peak above 600 K in the measured temperature range corresponds to antiferromagnetic ordering temperature (Néel temperature). The broadness of the peak changes with composition. The ac conductivity as well as ε′ are found to be maximum for the sample x=0.2 at room temperature.