On classification and invariants of second order non-parabolic linear partial differential equations in two variables

The paper deals with second order abstract linear partial differential equations (LPDE) over a partial differential field with two commuting differential operators. In terms of usual differential equations the main content can be presented as follows. The classification and invariants problems for second order LPDEs with respect to transformations $$x=x(\xi,\eta),\ y=y(\xi,\eta),\ u=h(x,y)v(\xi,\eta),$$ where $x,y$ are independent and $u$ is the dependent variable of the LPDE, are considered. Solutions to these problems are given for different subclasses of non-parabolic LPDE which appear naturally in the equivalence problem of LPDE. A criterion for reducibility of such LPDE to LPDE with constant coefficients is offered as well

A note on a separating system of rational invariants for finite dimensional generic algebras

The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by attaching them some quadratic forms

The automorphism groups and derivation algebras of two-dimensional algebras

The automorphisms groups and derivation algebras of all two-dimensional algebras over algebraically closed fields are described.Comment: A type-mistake in $A_{3,2}(\mathbf{c})$ case, page 4, and its consequences are are correcte

Complete Classification of Two-Dimensional Algebras

A complete classification of two-dimensional algebras over algebraically closed fields is providedComment: A type-mistake in $A_{3,2}(\mathbf{c})$ case, page 7, is correcte