Related papers: Feuilletages en droites, \'{e}quations des eikonal…
In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials…
We initiate the study of inflection curves of rational vector fields on the Riemann sphere. For a rational vector field $v_R=-R(z)\frac{\partial}{\partial z}, \qquad R(z)=\frac{Q(z)}{P(z)} $ we define its affine regular inflection locus by…
In this paper we present an algorithm to compute the (real and complex) straight lines contained in a rational surface, defined by a rational parameterization. The algorithm relies on the well-known theorem of Differential Geometry that…
The purpose of this paper is to put into a noncommutative context basic notions related to vector fields from classical differential geometry. The manner of exposition is an attempt to make the material as accessible as possible to…
We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
We describe a construction of explicit affine extractors over large finite fields with exponentially small error and linear output length. Our construction relies on a deep theorem of Deligne giving tight estimates for exponential sums over…
In this study, a recursive solution technique in conjunction with generalized integrating factors is presented and applied to address first and second order linear differential equations. This approach demonstrates practical utility in…
We study restriction problem in vector spaces over finite fields. We obtain finite field analogue of Mockenhaupt-Mitsis-Bak-Seenger restriction theorem, and we show that the range of the exponentials is sharp.
We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss…
Here we examine some Erdos-Falconer-type problems in vector spaces over finite fields involving right angles. Our main goals are to show that a) a subset A of F_q^d of size >> q^[(d+2)/3] contains three points which generate a right angle,…
We obtain exact polynomial solutions for two-dimensional coherent complex scalar fields propagating through arbitrary aberrated shift-invariant linear imaging systems. These are used to model nodal-line dynamics of coherent fields output by…
Pure-radiation solutions are found, exploiting the analogy with the Euler- Darboux equation for aligned colliding plane waves and the Euler-Tricomi equation in hydrodynamics of two-dimensional flow. They do not depend on one of the…
We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define…
We demonstrate theory and computations for finite-energy line defect solutions in an improvement of Ericksen-Leslie liquid crystal theory. Planar director fields are considered in two and three space dimensions, and we demonstrate straight…
In this paper we solve the problem of analytic classification of plane curves singularities with two branches by presenting their normal forms. This is accomplished by means of a new analytic invariant that relates vectors in the tangent…
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…
A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…
Generation of electromagnetic fields by moving charges is a fascinating topic where the tight connection between classical electrodynamics and special relativity becomes particularly apparent. One can gain direct insight into the…
A system of linear equations is normally understood as a linear mapping between two vector spaces. However, most direct solutions (e.g., QR, LU, ...) rely on the inelegant approach of back-substitution: a significant departure from such a…