Jacobi multipliers, non-local symmetries and nonlinear oscillators

Constants of motion, Lagrangians and Hamiltonians admitted by a family of relevant nonlinear oscillators are derived using a geometric formalism. The theory of the Jacobi last multiplier allows us to find Lagrangian descriptions and constants of the motion. An application of the jet bundle formulation of symmetries of differential equations is presented in the second part of the paper. After a short review of the general formalism, the particular case of non-local symmetries is studied in detail by making use of an extended formalism. The theory is related to some results previously obtained by Krasil'shchi, Vinogradov and coworkers. Finally the existence of non-local symmetries for such two nonlinear oscillators is proved.Comment: 20 page

Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem

The quasi-geostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Where possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.Comment: Extended version, 24 pages, 1 figur

Grape selection for resistance to biotic and abiotic environmental factors

Most of the viticultural regions of the USSR are located under conditions of limiting biotic and abiotic factors, with frosts, drought, fungal diseases, phylloxera, mites, grape berry moths and some others being of primary importance. The main breeding organizations have been creating for more than 40 years new table and wine cultivars with complex resistance according to long-term programs. These cultivars are own-rooted and capable of wintering in outdoor culture with a limited amount of spray treatments, if any. In crossing, Amur grape and its hybrids, cultivars Seibel and Seyve Villard and some others are used as donors of resistance. Using biophysical and cytoembryological methods, gametes are treated with physical and chemical mutagenic factors in order to increase the variability range of F(1) seedlings, aiming at higher efficiency of selection. The process of selection is accelerated if seedlings are grown hydroponically. Analysis of the F(1) hybrid population determines the nature of the inheritance of valuable agricultural characters and the selection of pairs. The in vitro method is used when seedlings are grown from non-vital seeds, callus embryoids and in accelerated propagation of valuable genotypes providing virus and bacteria elimination. More than 50 cultivars with complex resistance have been bred during 35 years. More than 10 of them have been recommended for culture (Moldova, Lyana, Vostorg, Sukholimanski biely, Pervenets Magaracha, and others), while the remainder are being tested in different viticultural regions of the Soviet Union

Reduction and reconstruction of stochastic differential equations via symmetries

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance

New classes of exact solutions of three-dimensional Navier-Stokes equations

New classes of exact solutions of the three-dimensional unsteady Navier-Stokes equations containing arbitrary functions and parameters are described. Various periodic and other solutions, which are expressed through elementary functions are obtained. The general physical interpretation and classification of solutions is given.Comment: 11 page

On the hierarchy of partially invariant submodels of differential equations

It is noticed, that partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PIS of the higher rank. This introduce a hierarchic structure in the set of all PISs of a given system of differential equations. By using this structure one can significantly decrease an amount of calculations required in enumeration of all PISs for a given system of partially differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. In this framework the complete classification of regular partially invariant solutions of ideal MHD equations is given

Испытания и калибровка микромеханических акселерометров

Розглянута методика механічних статичних випробувань та калібрування мікромеханічних акселерометрів, яка дозволяє визначити основні їх параметри з врахуванням похибок їх встановлення. Можна використовувати розглянуту методику для кліматичних та електромагнітних випробувань.The mechanical and static testing and calibration method of micromechanical accelerometers are viewed. Using this method it is possible to determine almost all parameters of the device. It is possible to use the viewed method for thermal and electromagnetic sensitivity tests.Рассмотрена методика механических статических испытаний и калибровки микромеханических акселерометров, которая позволяет определять основные их параметры с учетом погрешности их установки. Можно использовать рассмотренную методику для климатических и электромагнитных испытаний

Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion--convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.Comment: 25 page

Stokes flow in a rectangular cavity by rotlet forcing

The Stokes flow inside a two-dimensional rectangular cavity |x|a, |y|b is analyzed for a highly viscous, incompressible fluid flow, driven by a single rotlet placed at position (0,c). Specifically, a rigorous solution of the governing two-dimensional biharmonic equation for the stream function is constructed analytically by means of the superposition principle. With this solution, multicellular flow patterns can be described for narrow cavities, in which the number of flow cells is directly related to the value of the aspect ratio A=b/a. The solution also shows that for a certain rotlet position (0,c0), which depends on a and b, the flow has a stagnation point (0,-c0) symmetrically placed inside the rectangle. As the flow would not be affected by placing a second (inactive) rotlet in this stagnation point, this allows us to construct a blinking rotlet model for the rectangular cavity, with the inactive rotlet in the stagnation point of the flow induced by the active rotlet. For rectangular cavities, it holds that more than one of these special rotlet positions can be found for cavities that are elongated to sufficiently large aspect ratios. The blinking rotlet model is applied to illustrate several aspects of stirring in a Stokes flow in a rectangular domain