Related papers: Desingularization of First Order Linear Difference…
After tersely reviewing the various meanings that can be given to the property of a system of nonlinear ODEs to be solvable, we identify a special case of the system of two first-order ODEs with homogeneous quadratic right-hand sides which…
An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between $\lambda$-symmetries and first integrals. This relation further leads to a new $\lambda$-symmetry linearization…
Our paper "Solving Third Order Linear Difference Equations in Terms of Second Order Equations" gave two algorithms for solving difference equations in terms of lower order equations: an algorithm for absolute factorization, and an algorithm…
Normal forms allow the use of a restricted class of coordinate transformations (typically homogeneous polynomials) to put the bifurcations found in nonlinear dynamical systems into a few standard forms. We investigate here the consequences…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary…
The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our…
A classic problem, the design of the tallest column, is solved again using a different method. By the use of a similarity solution the equations are transformed and the difficult singularity at the endpoint is peeled away. The resulting…
We study a system of partial differential equations defined by commuting family of differential operators with regular singularities. We construct ideally analytic solutions depending on a holomorphic parameter. We give some explicit…
Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these…
First order algebraic differential equations are considered. An necessary condition for a first order algebraic differential equation to have a rational general solution is given: the algebraic genus of the equation should be zero.…
This work deals with defect structures in models described by scalar fields. The investigations focus on generalized models, with the kinetic term modified to allow for a diversity of possibilities. We develop a new framework, in which we…
We present algorithms to classify isolated hypersurface singularities over the real numbers according to the classification by V.I. Arnold (Arnold et al., 1985). This first part covers the splitting lemma and the simple singularities; a…
We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The method we propose is capable of dealing with either variational and non-variational problems, and…
In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems. These systems present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms and unique…
We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be…
We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees…
We call a singularity of a presymplectic form $\omega$ removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the…
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn…
We show that Ore operators can be desingularized by calculating a least common left multiple with a random operator of appropriate order. Our result generalizes a classical result about apparent singularities of linear differential…