104 research outputs found
Archaeological Sites as a Component of the Modern Religious Worldview of the Southern Ural Population (Based on the Example of the Akhunovo Megalithic Complex)
The present research examines the problem of the formation of new sacred sites around one of the most well-known archaeological complexes of the Bronze Age in the Southern Urals. To do so, the author applied the results of field research done of his own at the Akhunovo cromlech and analysed scientific literature and internet resources. Proceeding from the received data, the author traced the process of formation of sacred insight of the monument, determined the main reasons and participants of the sacralisation process and established probable negative outcomes of this phenomenon
Group classification of heat conductivity equations with a nonlinear source
We suggest a systematic procedure for classifying partial differential
equations invariant with respect to low dimensional Lie algebras. This
procedure is a proper synthesis of the infinitesimal Lie's method, technique of
equivalence transformations and theory of classification of abstract low
dimensional Lie algebras. As an application, we consider the problem of
classifying heat conductivity equations in one variable with nonlinear
convection and source terms. We have derived a complete classification of
nonlinear equations of this type admitting nontrivial symmetry. It is shown
that there are three, seven, twenty eight and twelve inequivalent classes of
partial differential equations of the considered type that are invariant under
the one-, two-, three- and four-dimensional Lie algebras, correspondingly.
Furthermore, we prove that any partial differential equation belonging to the
class under study and admitting symmetry group of the dimension higher than
four is locally equivalent to a linear equation. This classification is
compared to existing group classifications of nonlinear heat conductivity
equations and one of the conclusions is that all of them can be obtained within
the framework of our approach. Furthermore, a number of new invariant equations
are constructed which have rich symmetry properties and, therefore, may be used
for mathematical modeling of, say, nonlinear heat transfer processes.Comment: LaTeX, 51 page
Group classification of (1+1)-Dimensional Schr\"odinger Equations with Potentials and Power Nonlinearities
We perform the complete group classification in the class of nonlinear
Schr\"odinger equations of the form
where is an arbitrary
complex-valued potential depending on and is a real non-zero
constant. We construct all the possible inequivalent potentials for which these
equations have non-trivial Lie symmetries using a combination of algebraic and
compatibility methods. The proposed approach can be applied to solving group
classification problems for a number of important classes of differential
equations arising in mathematical physics.Comment: 10 page
New results on group classification of nonlinear diffusion-convection equations
Using a new method and additional (conditional and partial) equivalence
transformations, we performed group classification in a class of variable
coefficient -dimensional nonlinear diffusion-convection equations of the
general form We obtain new interesting cases of
such equations with the density localized in space, which have large
invariance algebra. Exact solutions of these equations are constructed. We also
consider the problem of investigation of the possible local trasformations for
an arbitrary pair of equations from the class under consideration, i.e. of
describing all the possible partial equivalence transformations in this class.Comment: LaTeX2e, 19 page
Nonlocal symmetries of integrable two-field divergent evolutionary systems
Nonlocal symmetries for exactly integrable two-field evolutionary systems of
the third order have been computed. Differentiation of the nonlocal symmetries
with respect to spatial variable gives a few nonevolutionary systems for each
evolutionary system. Zero curvature representations for some new nonevolution
systems are presented
Symmetries of Differential Equations via Cartan's Method of Equivalence
We formulate a method of computing invariant 1-forms and structure equations
of symmetry pseudo-groups of differential equations based on Cartan's method of
equivalence and the moving coframe method introduced by Fels and Olver. Our
apparoach does not require a preliminary computation of infinitesimal defining
systems, their analysis and integration, and uses differentiation and linear
algebra operations only. Examples of its applications are given.Comment: 15 pages, LaTeX 2.0
Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations
A complete group classification of a class of variable coefficient
(1+1)-dimensional telegraph equations , is
given, by using a compatibility method and additional equivalence
transformations. A number of new interesting nonlinear invariant models which
have non-trivial invariance algebras are obtained. Furthermore, the possible
additional equivalence transformations between equations from the class under
consideration are investigated. Exact solutions of special forms of these
equations are also constructed via classical Lie method and generalized
conditional transformations. Local conservation laws with characteristics of
order 0 of the class under consideration are classified with respect to the
group of equivalence transformations.Comment: 23 page
- …