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Over an arbitrary field of characteristic different from $2$ admitting an anisotropic torsion $3$-fold Pfister form, we apply a construction due to Merkurjev to produce an algebra with orthogonal involution of degree $6$ which admits proper…
We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…
It is known that connected translation invariant $n$-dimensional noncommutative differentials $d x^i$ on the algebra $k[x^1,\cdots,x^n]$ of polynomials in $n$-variables over a field $k$ are classified by commutative algebras $V$ on the…
A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by…
We exploit various inclusions of algebraic groups to give a new construction of groups of type E8, determine the Killing forms of the resulting E8's, and define an invariant of central simple algebras of degree 16 with orthogonal involution…
We study generalized solutions of an evolutionary equation related to some densely defined skew-symmetric operator in a real Hilbert space. We establish existence of a contractive semigroup, which provides generalized solutions, and suggest…
We discuss contact invariant structures on the space of solutions of a third-order ordinary differential equation. Associated to any third-order differential equation modulo contact transformations, Chern introduced a degenerate conformal…
On conformal manifolds of even dimension $n\geq 4$ we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new…
Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special…
The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can…
We propose a new algebraic approach to study compatibility of partial differential equations. The approach uses concepts from commutative algebra, algebraic geometry and Gr\"obner bases to clarify crucial notions concerning compatibility…
We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…
Nonconservative evolution problems describe irreversible processes and dissipative effects in a broad variety of phenomena. Such problems are often characterised by a conservative part, which can be modelled as a Hamiltonian term, and a…
Within the framework of inverse Lie problem, we give some non-trivial examples of coupled Lie remarkable equations, \textit{i.e.}, classes of differential equations that are in correspondence with their Lie point symmetries. In particular,…
We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the…
We investigate the generalized involution models of the projective reflection groups $G(r,p,q,n)$. This family of groups parametrizes all quotients of the complex reflection groups $G(r,p,n)$ by scalar subgroups. Our classification is…
We study a class of evolutionary partial differential systems with two components related to second order (in time) non-evolutionary equations of odd order in spatial variable. We develop the formal diagonalisation method in symbolic…
We expand our group classification of quasilinear evolution equations (Acta Appl.Math., v.69, 2001) to the case of general evolution equation in one spatial variable. This enables obtaining several new classes of evolution equations with…
We introduce a new, more general type of nonlinear gauge transformation in nonrelativistic quantum mechanics that involves derivatives of the wave function and belongs to the class of B\"acklund transformations. These transformations…
We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are…