Related papers: A Note on Algebraic Linear Partial Differential Eq…

The underlying algebra for a noncommutative geometry is taken to be a matrix algebra, and the set of derivatives the adjoint of a subset of traceless matrices. This is sufficient to calculate the dual 1-forms, and show that the space of…

We provide pointwise upper bounds for the transition kernels of semigroups associated with a class of systems of nondegenerate elliptic partial differential equations with unbounded coefficients with possibly unbounded diffusion…

There is a conjecture, that the torsionfreeness of the module of differentials in a point of an algebraic or algebroid curve should imply that the curve is non singular at that point. A report on the main results is given.

Equations are ubiquitous in most mathematical activities. Nevertheless, in this paper it is shown how to do standard mathematics without any equation at all. More than that, it is proven there is a foundational framework for standard…

Global existence of mild solutions to the discrete collisional breakage equations is established for a broad class of collision kernels, without imposing any growth assumptions. In addition, classical solutions are constructed, and…

Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by…

Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically closed field $k$. A base point free complete very special linear system $g^r_n$ on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor $E$…

In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results…

In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of…

We study the effects of imposing linear relations within modules of matrices on average sizes of kernels. The relations that we consider can be described combinatorially in terms of partial colourings of grids. The cells of these grids…

Here we give a complete group classification of the general case of linear systems of three second-order ordinary differential equations excluding the case of systems which are studied in the literature. This is given as the initial step in…

We start with elementary algebraic theory of factorization of linear ordinary differential equations developed in the period 1880-1930. After exposing these classical results we sketch more sophisticated algorithmic approaches developed in…

Answering a long standing question, we give an example of a Hilbert module and a nonzero bounded right linear map having a kernel with trivial orthogonal complement. In particular, this kernel is different from its own double orthogonal…

A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…

We extend Kolchin's results on linear dependence over projective varieties in the constants, to linear dependence over arbitrary complete differential varieties. We show that in this more general setting, the notion of linear dependence…

This paper investigates two classes of quasilinear and essentially nonlinear integral equations with a sum-difference kernel on the half-line. Such equations arise in various areas of physics, including the theory of radiative transfer in…

At present, only some special differential equations have explicit analytical solutions. In general, no one thinks that it is possible to analytically find the exact solution of nonlinear equations. In this article based on the idea that…

We consider a system of linear equations, whose coefficients depend linearly on interval parameters. Its solution set is defined as the set of all solutions of all admissible realizations of the parameters. We study unbounded directions of…

In this work, we construct the algebra of differential forms with the cube of exterior differential equal to zero on one-dimensional space. We prove that this algebra is a graded q-differential algebra where q is a cubic root of unity.…

We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra…