138 research outputs found
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
ΠΡΠΏΡΡΠ°Π½ΠΈΡ ΠΈ ΠΊΠ°Π»ΠΈΠ±ΡΠΎΠ²ΠΊΠ° ΠΌΠΈΠΊΡΠΎΠΌΠ΅Ρ Π°Π½ΠΈΡΠ΅ΡΠΊΠΈΡ Π°ΠΊΡΠ΅Π»Π΅ΡΠΎΠΌΠ΅ΡΡΠΎΠ²
Π ΠΎΠ·Π³Π»ΡΠ½ΡΡΠ° ΠΌΠ΅ΡΠΎΠ΄ΠΈΠΊΠ° ΠΌΠ΅Ρ
Π°Π½ΡΡΠ½ΠΈΡ
ΡΡΠ°ΡΠΈΡΠ½ΠΈΡ
Π²ΠΈΠΏΡΠΎΠ±ΡΠ²Π°Π½Ρ ΡΠ° ΠΊΠ°Π»ΡΠ±ΡΡΠ²Π°Π½Π½Ρ ΠΌΡΠΊΡΠΎΠΌΠ΅Ρ
Π°Π½ΡΡΠ½ΠΈΡ
Π°ΠΊΡΠ΅Π»Π΅ΡΠΎΠΌΠ΅ΡΡΡΠ², ΡΠΊΠ° Π΄ΠΎΠ·Π²ΠΎΠ»ΡΡ Π²ΠΈΠ·Π½Π°ΡΠΈΡΠΈ ΠΎΡΠ½ΠΎΠ²Π½Ρ ΡΡ
ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΈ Π· Π²ΡΠ°Ρ
ΡΠ²Π°Π½Π½ΡΠΌ ΠΏΠΎΡ
ΠΈΠ±ΠΎΠΊ ΡΡ
Π²ΡΡΠ°Π½ΠΎΠ²Π»Π΅Π½Π½Ρ. ΠΠΎΠΆΠ½Π° Π²ΠΈΠΊΠΎΡΠΈΡΡΠΎΠ²ΡΠ²Π°ΡΠΈ ΡΠΎΠ·Π³Π»ΡΠ½ΡΡΡ ΠΌΠ΅ΡΠΎΠ΄ΠΈΠΊΡ Π΄Π»Ρ ΠΊΠ»ΡΠΌΠ°ΡΠΈΡΠ½ΠΈΡ
ΡΠ° Π΅Π»Π΅ΠΊΡΡΠΎΠΌΠ°Π³Π½ΡΡΠ½ΠΈΡ
Π²ΠΈΠΏΡΠΎΠ±ΡΠ²Π°Π½Ρ.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.Π Π°ΡΡΠΌΠΎΡΡΠ΅Π½Π° ΠΌΠ΅ΡΠΎΠ΄ΠΈΠΊΠ° ΠΌΠ΅Ρ
Π°Π½ΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΡΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΈΡΠΏΡΡΠ°Π½ΠΈΠΉ ΠΈ ΠΊΠ°Π»ΠΈΠ±ΡΠΎΠ²ΠΊΠΈ ΠΌΠΈΠΊΡΠΎΠΌΠ΅Ρ
Π°Π½ΠΈΡΠ΅ΡΠΊΠΈΡ
Π°ΠΊΡΠ΅Π»Π΅ΡΠΎΠΌΠ΅ΡΡΠΎΠ², ΠΊΠΎΡΠΎΡΠ°Ρ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΡΡ ΠΎΡΠ½ΠΎΠ²Π½ΡΠ΅ ΠΈΡ
ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΡ Ρ ΡΡΠ΅ΡΠΎΠΌ ΠΏΠΎΠ³ΡΠ΅ΡΠ½ΠΎΡΡΠΈ ΠΈΡ
ΡΡΡΠ°Π½ΠΎΠ²ΠΊΠΈ. ΠΠΎΠΆΠ½ΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡ ΡΠ°ΡΡΠΌΠΎΡΡΠ΅Π½Π½ΡΡ ΠΌΠ΅ΡΠΎΠ΄ΠΈΠΊΡ Π΄Π»Ρ ΠΊΠ»ΠΈΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΈ ΡΠ»Π΅ΠΊΡΡΠΎΠΌΠ°Π³Π½ΠΈΡΠ½ΡΡ
ΠΈΡΠΏΡΡΠ°Π½ΠΈΠΉ
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
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