Related papers: Apparent Singularities of D-finite Systems
Interrelation between Thom's catastrophes and differential equations revisited. It is shown that versal deformations of critical points for singularities of A,D,E type are described by the systems of Hamilton-Jacobi type equations. For…
This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are…
A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure…
We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
Algorithms of control of differential equations solutions are under investigation in the article. Idealized and real modifications of the algorithms are distinguished. An equation, which can be the base equation for investigation of the…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic…
We consider equations of the type: \[\partial_t \omega = \omega R(\omega),\] for general linear operators $R$ in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localized…
Singularities appear in numerous important mathematical models used in Physics. And in most of such cases singularities are involved in essentially nonlinear contexts. For more than four decades, general enough nonlinear theories of…
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…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
We give a specific method to solve with quadratic complexity the linear systems arising in known algorithms to deal with the sign determination problem. In particular, this enable us to improve the complexity bound for sign determination in…
Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from…
Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the…
Generalizations of numeration systems in which N is recognizable by a finite automaton are obtained by describing a lexicographically ordered infinite regular language L over a finite alphabet A. For these systems, we obtain a…
Preliminary results of our investigations on solving indefinite qua\-dra\-tic programs by dynamical systems are given. First, dynamical systems corresponding to two fundamental DC programming algorithms to deal with indefinite quadratic…
We introduce middle convolution for systems of linear differential equations with irregular singular points, and we presend a tentative definition of the index of rigidity for them. Under some assumption, we show a list of terminal patterns…
This article generalizes a recently introduced procedure to solve nonlinear systems of equations, radically departing from the conventional Newton-Raphson scheme. The original nonlinear system is first unfolded into three simpler…
The existence and multiplicity of positive periodic solutions for second order non-autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. Our…