Related papers: Differential invariants for a class of diffusion e…
Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential…
The problem of classification into symmetry integrable classes is solved for a family of second order nonlinear evolution equations labeled by arbitrary functions. Four nonequivalent symmetry integrable classes are thus obtained and the…
This article presents a general description of dynamical systems using the language of enriched functors and enriched natural transformations. This framework is essential to establish the equivalence of three descriptions of dynamics -- a…
We introduce an equivalence relation on the global class of morphisms of a category that extends several classical notions of equivalence in mathematics. We show that the standard group-action equivalence is a special case of our framework.…
Full set of autonomous completely solvable differential systems of equations in total differentials is built by basis of infinitesimal operators, universal invariant, and structure constants of admited multiparametric Lie group (abelian and…
We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well-known that this…
Using group theoretical methods, we analyze the generalization of a one-dimensional sixth-order thin film equation which arises in considering the motion of a thin film of viscous fluid driven by an overlying elastic plate. The most general…
In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an…
We consider the one-dimensional porous medium equation $u_t=\left (u^nu_x \right )_x+\frac{\mu}{x}u^nu_x$. We derive point transformations of a general class that map this equation into itself or into equations of a similar class. In some…
We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to…
We study the approximation of functions which are invariant with respect to certain permutations of the input indices using flow maps of dynamical systems. Such invariant functions includes the much studied translation-invariant ones…
Many generative tasks in chemistry and science involve distributions invariant to group symmetries (e.g., permutation and rotation). A common strategy enforces invariance and equivariance through architectural constraints such as…
We show that a noncommutative dynamical system of the type that occurs in quantum theory can often be associated with a dynamical principle; that is, an infinitesimal structure that completely determines the dynamics. The nature of these…
We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds…
This paper is concerned with the uniqueness, existence, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in…
The algebra of invariants of d-tuples of n x n skew-symmetric matrices under the action of the orthogonal group by simultaneous conjugation is considered over an infinite field of characteristic different from two. For n=3 and d>0 a minimal…
The introduction of convolutional layers greatly advanced the performance of neural networks on image tasks due to innately capturing a way of encoding and learning translation-invariant operations, matching one of the underlying symmetries…
In recent years, deep learning techniques have shown great success in various tasks related to inverse problems, where a target quantity of interest can only be observed through indirect measurements by a forward operator. Common approaches…
Admissible point transformations of classes of $r$th order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational form, the Laguerre-Forsyth form, the first…
We generalize the notion of semi-normalized classes of systems of differential equations, study properties of such classes and extend the algebraic method of group classification to them. In particular, we prove the important theorems on…