Related papers: Solution to Rubel's question about differentially …

We study the average condition number for polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with Gaussian entries…

We propose the symmetry reduction method of partial differential equations to the system of differential equations with fewer number of independent variables. We also obtain generalized sufficient conditions for the solution found by…

A geometric approach is used to study the Abel first order differential equation of the first kind. The approach is based on the recently developed theory of quasi-Lie systems which allows us to characterise some particular examples of…

Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its…

We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero…

We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m>1 and the equation is also linear in its derivatives of order m-1…

We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and…

The explicit solution of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODE s with second-degree polynomial right-hand sides, hence featuring 12 a prior arbitrary (time-independent)…

The necessary and sufficient conditions for a function to be totally or partially separable are derived. It is shown that a function is totally separable if and only if each component of the gradient vector of depends only on the…

The paper studies algebraic independence of certain reciprocal sums of Fibonacci and Lucas sequences. Also more general binary recurrences are considered. The main tool is Mahler's method reducing the investigation of the algebraic…

Let rho be a Drinfeld A-module with generic characteristic defined over an algebraic function field. We prove that all of the algebraic relations among periods, quasi-periods, and logarithms of algebraic points on rho are those coming from…

This paper deals with criteria of algebraic independence for the derivatives of solutions of rank one difference equations. The key idea consists in deriving from the commutativity of the differentiation and difference operators a sequence…

We show that irreducibility is not a first-order definable property of real algebraic varieties. The proof is based on the recent o-minimality result for the exponential function. We conjecture that irreducibility is not a definable…

We consider random trigonometric polynomials with general dependent coefficients. We show that under mild hypotheses on the structure of dependence, the asymptotics as the degree goes to infinity of the expected number of real zeros…

This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized…

This paper introduces, up to the author's knowledge, for the first time the generalized initial value problem. In this problem, given an ordinary differential equation defined in some set, the initial conditions are mapped to a subset of…

We consider systems of ordinary differential equations with quadratic homogeneous right hand side. We give a new simple proof of a result already obtained in [8,10] which gives the necessary conditions for the existence of polynomial first…

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…

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to…

The existence of the unique strong solution for a class of stochastic differential equations with non-Lipschitz coefficients was established recently. In this paper, we shall investigate the dependence with respect to the initial values. We…