Related papers: Equivalence Groups and Differential Invariants for…
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…
As a first step towards a theory of differential equations involving para-Grassmann variables the linear equations with constant coefficients are discussed and solutions for equations of low order are given explicitly. A connection to…
This work contributes to an understanding of the domain size's effect on the existence and uniqueness of the linear convection--diffusion equation with integral-type boundary conditions, where boundary conditions depend non-locally on…
The explicit construction of direct and inverse Fourier's vector transform with discontinuous coefficients is presented. The technique of applying Fourier's vector transform with discontinuous coefficients for solving problems of…
We find an explicit form of weak solutions to a Riemann problem for a degenerate semilinear parabolic equation with piecewise constant diffusion coefficient. It is demonstrated that the phase transition lines (free boundaries) correspond to…
The dynamics of nonlinear reaction-diffusion systems is dominated by the onset of patterns and Fisher equation is considered to be a prototype of such diffusive equations. Here we investigate the integrability properties of a generalized…
Units equivariance (or units covariance) is the exact symmetry that follows from the requirement that relationships among measured quantities of physics relevance must obey self-consistent dimensional scalings. Here, we express this…
We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is…
symmetry, group classification, differential invariants, Lie-classical method,infinitesimal criterion method, RDC equation, KPP equation, similarity solutions.
We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which…
Functional bases of second-order differential invariants of the Euclid, Poincar\'e, Galilei, conformal, and projective algebras are constructed. The results obtained allow us to describe new classes of nonlinear many-dimensional invariant…
We study weighted inequalities of Hardy and Hardy-Poincar\'e type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are…
We derive a nice representation for point symmetry transformations of the (1+1)-dimensional linear heat equation and properly interpret them. This allows us to prove that the pseudogroup of these transformations has exactly two connected…
All deformations of two dimensional centrally extended Galilei group are classified. The corresponding quantum Lie algebras are found.
In this paper, after a brief review of the general theory concerning regularized derivatives and integrals of a function with respect to another function, we provide a peculiar fractional generalization of the $(1+1)$-dimensional Dodson's…
Fourier transforms of Lorentz invariant functions in Minkowski space, with support on both the timelike and the spacelike domains are performed by means of direct integration. The cases of 1+1 and 1+2 dimensions are worked out in detail,…
Fourdimensional bicovariant differential calculus on quantum E(2) group is constructed.
A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to…
We define and study multivariate exponential functions, symmetric with respect to the alternating group A_n, which is a subgroup of the permutation (symmetric) group S_n. These functions are connected with multivariate exponential…
In this study we present an extension of the replicator equation with diffusion to multiplex graphs. We derive an exact formula for the diffusion term, which shows that, while diffusion is linear for numbers of agents, it is necessary to…