Spinor representation of the group GL(4,R) is needed for correct description of the Fermi fields on Riemann space, such as the space-time of general relativity. It is used for two purposes: to define the connectivity and covariant derivative of spinor field and to define the Lie derivative. Recent publications [1–3] have reminded of this problem. The important problem for definition of Fermi fi...

We carried out a Lie group study of the micro-Raman tissue spectra of the Gurken gradients of Drosophila oogenesis. (2×2) matrix representations resulting from the polarized Raman scattering were employed to assess the roles of the Dally-like protein (Dlp) in follicle cell. It was found that the Gurken expansion caused by over-expressing Dlp revealed a X1 Lie point symmetry, while the Gurken di...

In this thesis, we consider smooth optimal control systems that evolve on Lie groups. Pontryagin’s maximum principle allows us to search for local solutions of the optimal control problem by studying an associated Hamiltonian dynamical system. When the associated Hamiltonian function possess symmetries, we can often study the Hamiltonian system in a vector space whose dimension is lower than th...

Solution by M.C. Nucci (Dipartimento di Matematica e Informatica, Università di Perugia, 06123 Perugia, Italy) Lie’s monumental work on transformation groups, [12], [13] and [14], and in particular contact transformations [15], has provided systematic techniques for obtaining exact solutions of differential equations [16]. Many books have been dedicated to this subject and its generalizations (...

In this paper, a useful classification of all Lie subalgebras of a given Lie algebraup to an inner automorphism is presented. This method can be regarded as animportant connection between differential geometry and algebra and has many applications in different fields of mathematics. After main results, we have applied this procedure for classifying the Lie subalgebras of some examples of Lie al...

In this paper, symmetries and symmetry reduction of two higher-dimensional nonlinear evolution equations (NLEEs) are obtained by Lie group method. These NLEEs play an important role in nonlinear sciences. We derive exact solutions to these NLEEs via the [Formula: see text]-expansion method and complex method. Five types of explicit function solutions are constructed, which are rational, exponen...

By using the self-dual Yang-Mills (SDYM) equation as an example, we study a method for relating symmetries and recursion operators of two partial differential equations connected to each other by non-auto-Backlund transformation. We prove Lie-algebra isomorphism between SDYM those potential (PSDYM) equation, describe construction these systems. Using certain known aspects PSDYM symmetry algebra...

A review of the role of symmetries in solving differential equations is presented. After showing some recent results on the application of classical Lie point symmetries to problems in fluid draining, meteorology, and epidemiology of AIDS, the nonclassical symmetries method is presented. Finally, it is shown that iterations of the nonclassical symmetries method yield new nonlinear equations, wh...

Coset conformal field theories with abelian isotropy groups. Abstract Conformal field theories based on g/u(1) d coset constructions where g is a reductive algebra are studied.It is shown that the theories are equivalent to constrained WZNW models for g. Generators of extended symmetry algebras and primary fields are constructed. In a recent paper [1] currents, primary fields and generators of ...

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 ...