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Left-invariant optimal control problems on Lie groups form an important class of problems with big symmetry group. They are interesting from the theoretical point of view since they often can be completely studied, and general features can…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…
The lattice of fully invariant subgroups of an abelian $p$--group and the lattice of ideals of its endomorphism ring are classified by systems of cardinal invariants.
The idempotent problem of a finitely generated inverse semigroup is the formal language of all words over the generators representing idempotent elements. This note proves that a finitely generated inverse semigroup with regular idempotent…
The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie…
This article focuses on the study of zero-sum invariants of finite non-abelian groups. We address two main problems: the first centers on the ordered Davenport constant and the second on Gao's constant. We establish a connection between the…
The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities, obtained by analytic…
Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver…
We study groups having the property that every non-abelian subgroup is equal to its normalizer. This class of groups is closely related to an open problem posed by Berkovich. We give a full classification of finite groups having the above…
Consider complex semisimple Lie algebras of a given dimension specified by their structure constants. We describe a finite collection of rational functions in the structure constants that form a complete set of invariants: two sets of…
We study integrated semigroups for infinite-dimensional differential-algebraic equations (DAEs) admitting a resolvent index. Building on the notion of integrated semigroups for the abstract Cauchy problem $\frac{d}{d t}x=Ax$, we extend this…
The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector…
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…
We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…
A consistent set of six integrable discrete and continuous dynamical systems are suggested corresponding to arbitrary affine Lie algebra. The set contains a system of partial differential equations which can be treated as a version of…
We have solved completely the problem of the description of quasi-linear hyperbolic differential equations in two independent variables that are invariant under three-parameter Lie groups.
A simple procedure to obtain complete, closed expressions for Lie algebra invariants is presented. The invariants are ultimately polynomials in the group parameters. The construction of finite group elements require the use of projectors,…
We initiate studying inverse spectral problems for Dirac-type functional-differential operators with constant delay. For simplicity, we restrict ourselves to the case when the delay parameter is not less than one half of the interval. For…
This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the…