Related papers: Some applications of the parameterized Picard-Vess…
We discuss relations which exist between analytic functions belonging to the recently introduced class of special functions of the isomonodromy type (SFITs). These relations can be obtained by application of some simple transformations to…
The purpose of this paper is to present some multidimensional fixed-point theorems and their applications. For this, we provide a multidimensional fixed point theorem and then using this theorem we prove the existence and uniqueness of a…
Symbolic dynamics is a coarse-grained description of dynamics. By taking into account the ``geometry'' of the dynamics, it can be cast into a powerful tool for practitioners in nonlinear science. Detailed symbolic dynamics can be developed…
In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…
Let F be a differential field of characteristic zero. In this article, we construct Picard-Vessiot extensions of F whose differential Galois group is isomorphic to the full unipotent subgroup of the upper triangular group defined over the…
The question of defining unique, generally applicable constrained second, and higher-order, derivatives is investigated. It is shown that second-order constrained derivatives obtained via two successive constrained differentiations provide…
A diagrammatic method is presented for averaging over the circular ensemble of random-matrix theory. The method is applied to phase-coherent conduction through a chaotic cavity (a ``quantum dot'') and through the interface between a normal…
We prove two "master" convolution theorems for multivariate determinantal polynomials. The methods used include basic properties of what we call a "minor-orthogonal" ensemble as well as properties of the mixed discriminant of matrices. We…
We derive special forms of the Poisson summation formula for even and odd functions, which are applied to obtain representations for Euler-type numbers and to sum various series related to elliptic functions.
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
The goal of these notes is to provide an introduction to rough partial differential equations. For this purpose, we will present the theory of rough paths to the extend as it is required. Applications to stochastic partial differential…
We extend Painlev\'e's determinateness theorem to the case of first order ordinary differential equations in the complex domain with known terms allowed be multivalued in the dependent variable as well; multivaluedness is supposed to be…
We propose a discrete surface theory in $\mathbb R^3$ that unites the most prevalent versions of discrete special parametrizations. This theory encapsulates a large class of discrete surfaces given by a Lax representation and, in…
I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of…
In this work, we give a variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with deviated…
We use the method of synthetic differential geometry to revisit the geometric reasoning employed by Lie, Klein and others in their study of partial differential equations.
It is shown how the dimension of any arbitrary over-determined system of differential equations can be reduced, which makes the system suitable for numerical solution modeling. Specifically, over-determined equations of hydrodynamics are…
Previously, we have introduced a very small number of examples of what we call Ouroboros functions. Using our already established theory of Ouroboros spaces and their functions, we will provide a set of families of Ouroboros functions that…
In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.
Parameter estimation connects mathematical models to real-world data and decision making across many scientific and industrial applications. Standard approaches such as maximum likelihood estimation and Markov chain Monte Carlo estimate…