Related papers: A short proof of solution formulas for the linear …
The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular…
We study distribution of zeros of a complex polynomial whose coefficients has been modified. We give a new proof of the theorem of Rubinstein, and with similar method we prove a new theorem that is not generalization of the previous…
The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate…
In this paper, we study the polynomial stability of analytical solution and convergence of the semi-implicit Euler method for non-linear stochastic pantograph differential equations. Firstly, the sufficient conditions for solutions to grow…
In this short note we have proved an enhanced version of a theorem of Lorentz [1] and its generalization to the multivariate case which gives a non- uniform estimate of degree of approximation by a polynomial with positive coefficients. The…
In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However,…
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference…
In this paper rational solutions of the fifth Painlev\'e equation are discussed. There are two classes of rational solutions of the fifth Painlev\'e equation, one expressed in terms of the generalised Laguerre polynomials, which are the…
A problem based on the Extended Euclidean Algorithm applied to a class of polynomials with many factors is presented and believed to be hard. If so, it is a one-way function well suited for applications in digital signicatures.
By exploiting a recently developed connection between Heun's differential equation and the generalized associated Lam\'e equation, we not only recover the well known periodic solutions, but also obtain a large class of new, quasi-periodic…
Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the…
We present the solutions of equations of degrees 3 and 4 using Galois theory and some simple Fourier analysis for finite groups, together with historical comments on these and other solution methods.
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…
In this paper we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent…
A second order finite-difference equation has two linearly independent solutions. It is shown here that, like in the continuous case, at most one of the two can be a polynomial solution. The uniqueness in the classical continuous…
Given the matrix equation ${\bf A X} + {\bf X B} + f({\bf X }) {\bf C} ={\bf D}$ in the unknown $n\times m$ matrix ${\bf X }$, we analyze existence and uniqueness conditions, together with computational solution strategies for $f \,:…
In this paper, we propose a procedure for constructing an infinite number of families of solutions of given linear differential equations with partial derivatives with constant coefficients. We use monogenic functions that are defined on…
Recently, Nunge studied Eulerian polynomials on segmented permutations, namely \emph{generalized Eulerian polynomials}, and further asked whether their coefficients form unimodal sequences. In this paper, we prove the stability of the…
In this work, we derive numerous identities for multivariate q-Euler polynomials by using umbral calculus.
In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…