Algebraic description of Jacobians isogeneous to certain Prym varieties with polarization (1,2)

For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally non-equivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the Prym varieties associated to such coverings. Our description is based on previous studies of Prym varieties with polarization (1,2) in connection with separation of variables in a series of classical and new algebraic integrable systems linearized on such varieties. We also consider some special cases of the covering C -> E, in particular, when the corresponding Prym varieties contain pairs of elliptic curves and the Jacobian of C is isogeneous (but not isomorphic) to the product of 3 different elliptic curves. Our description is accompanied with explicit numerical examples.Comment: 51 pages, 3 figures, 3 diagram

Dynamics on strata of trigonal Jacobians and some integrable problems of rigid body motion

We present an algebraic geometrical and analytical description of the Goryachev case of rigid body motion. It belongs to a family of systems sharing the same properties: although completely integrable, they are not algebraically integrable, their solution is not meromorphic in the complex time and involves dynamics on the strata of the Jacobian varieties of trigonal curves. Although the strata of hyperelliptic Jacobians have already appeared in the literature in the context of some dynamical systems, the Goryachev case is the first example of an integrable system whose solution involves a more general curve. Several new features (and formulae) are encountered in the solution given in terms of sigma-functions of such a curve.Comment: 22 pages, 1 figur

Organization of Practical Training under the Conditions of Credit-Modular System for the Fourth Year Students of Faculty of Dentistry

The article deals with the organizational problems of practical training of the IV year students of Faculty of Dentistry in terms of credit-modular system training.This form of training is important in dental specialists training. The experience of medical practical training at the Department of Therapeutic Dentistry of Ivano-Frankivsk National Medical University was described. According to modern requirements the state standards of higher dental education in Ukraine were developed. They were focused not on the content, timing and organizational parameters of the educational process but on the competence first of all, that is the willingness and ability of graduates of dental faculties of higher medical educational establishments to carry out professional activities after obtaining a diploma. The medical practical training of IV year students of the Faculty of Dentistry is based on such methodological principles and professional-active approach to training process goal. The system of medical practical training organization, which was until quite recently, should be adapted according to credit-modular system conditions, namely to combine independent theoretical training with practical skills under the supervision of a teacher and doctor-supervisor. The question of medical practical training possibility during an academic year on a cumulative basis is still undecided, because the increase in the staff number and clinical site of the field-oriented department is necessary in such case. Distance learning, especially during foreign students’ training should be developed and implemented to optimize the methodological support of medical practical training

Geometrical Optics of Beams with Vortices: Berry Phase and Orbital Angular Momentum Hall Effect

We consider propagation of a paraxial beam carrying the spin angular momentum (polarization) and intrinsic orbital angular momentum (IOAM) in a smoothly inhomogeneous isotropic medium. It is shown that the presence of IOAM can dramatically enhance and rearrange the topological phenomena that previously were considered solely in connection to the polarization of transverse waves. In particular, the appearance of a new-type Berry phase that describes the parallel transport of the beam structure along a curved ray is predicted. We derive the ray equations demonstrating the splitting of beams with different values of IOAM. This is the orbital angular momentum Hall effect, which resembles Magnus effect for optical vortices. Unlike the recently discovered spin Hall effect of photons, it can be much larger in magnitude and is inherent to waves of any nature. Experimental means to detect the phenomena is discussed.Comment: 5 pages, 2 figure

The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation

We consider the motion of a planar rigid body in a potential flow with circulation and subject to a certain nonholonomic constraint. This model is related to the design of underwater vehicles. The equations of motion admit a reduction to a 2-dimensional nonlinear system, which is integrated explicitly. We show that the reduced system comprises both asymptotic and periodic dynamics separated by a critical value of the energy, and give a complete classification of types of the motion. Then we describe the whole variety of the trajectories of the body on the plane.Comment: 25 pages, 7 figures. This article uses some introductory material from arXiv:1109.321

Polarization-dependent transformation of a paraxial beam upon reflection and refraction: a real-space approach

We analyze the paraxial beam transformation upon reflection and refraction at a plane boundary. In contrast to the usual approach dealing with the beam angular spectrum, we apply the continuity conditions to explicit spatial representations of the electric and magnetic fields on both sides of the boundary. It is shown that the polarization-dependent distortions of the beam trajectory (in particular, the "longitudinal" Goos-H\"anchen shift and the "lateral" Imbert-Fedorov shift of the beam center of gravity) are directly connected to the incident beam longitudinal component and appear due to its transformation at the boundary.Comment: 10 pages, 1 figure. Formulae (32), (33), footnote 2 and Ref. 27 are added, some sentences are correcte