Related papers: Continuous versus discrete structures I -- Discret…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
In this paper we introduce and investigate the notions of diagrams and discrete extensions in the study of finitary $2$-representations of finitary $2$-categories.
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…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…
We consider notions of metrized categories, and then approximate categorical structures defined by a function of three variables generalizing the notion of $2$-metric space. We prove an embedding theorem giving sufficient conditions for an…
Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are…
Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.
We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed…
We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of…
In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds…
Two essential methods, the symmetry analysis and of the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations are discussed. The main similarities and differences of these two different…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…
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…
We consider semiclassical orthogonal polynomials on the unit circle associated with a weight function that satisfy a Pearson-type differential equation involving two polynomials of degree at most three. Structure relations and difference…
Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus we generalize a notion of Braess and Sch\"oberl, originally studied for a posteriori error estimation. We construct…
In this article we discuss the numerical analysis for the finite difference scheme of the one-dimensional nonlinear wave equations with dynamic boundary conditions. From the viewpoint of the discrete variational derivative method we propose…
In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there…
We describe differential invariants of infinite-dimensional algebras being equivalence algebras of some classes of PDE and study structure of these algebras.