Related papers: On equivariant binary differential equations
For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and…
We study equivariant birationality from the perspective of derived categories. We produce examples of nonlinearizable but stably linearizable actions of finite groups on smooth cubic fourfolds.
In this paper, we introduce the notion of Leibniz-dendriform bialgebras and establish their equivalence with phase spaces and matched pairs of Leibniz algebras. The study of the coboundary case leads naturally to the Leibniz-dendriform…
Nonlinear ODEs invariant under the group SL(2,R) are solved numerically. We show that solution methods incorporating the Lie point symmetries provide better results than standard methods.
We describe a way of solving a partial differential equation using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential…
Invariants of general linear system of two hyperbolic partial differential equations (PDEs) are derived under transformations of the dependent and independent variables by real infinitesimal method earlier. Here a subclass of the general…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
Some connections between classical and nonclassical symmetries of a partial differential equation (PDE) are given in terms of determining equations of the two symmetries. These connections provide additional information for determining…
We consider a holomorphic 1-form $\omega$ with an isolated zero on an isolated complete intersection singularity $(V,0)$. We construct quadratic forms on an algebra of functions and on a module of differential forms associated to the pair…
In this article, Lie symmetry analysis is used to investigate invariance properties of some nonlinear fractional partial differential equations with conformable fractional time and space derivatives. The analysis is applied to Korteweg-de…
We study reflected solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs in short). The "reflected" keeps the solution above a given stochastic process. We get the uniqueness and existence by penalization.…
A comprehensive symmetry analysis of the N=1 supersymmetric sine-Gordon equation is performed. Two different forms of the supersymmetric system are considered. We begin by studying a system of partial differential equations corresponding to…
In this paper, the relationships between Lie symmetry groups and fundamental solutions for a class of conformable time fractional partial differential equations (PDEs) with variable coefficients are investigated. Specifically, the…
The main physical result of this paper are exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation.…
Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than…
Studied here is the effect of the presence of symmetry groups in a system of algebraic equations on the numerical resolution with fixed-point algorithms. It is proved that the symmetries imply two important properties of the system: the…
Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in…
We prove that a general class of nonlinear, non-autonomous ODEs in Fr\'echet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that…
Lie symmetry transformations that leave a differential equation invariant play a fundamental role in science and mathematics. Such Lie symmetry groups uniquely determine their Lie symmetry algebras. Exact differential elimination algorithms…
Symmetry is a powerful tool for finding analytical solutions to differential equations, both partial and ordinary, via the similarity variables or via the invariance of the equation under group transformations. It is the largest group of…