In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in detail in the simplest case of operators of order ...1. Namely, assuming that such an operator T has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of T, we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any T with non-constant coefficients, there exists a unique minimal under inclusion invariant set MCHT and find explicitly what operators T have the property that MCHT=C.
Graded jet geometry Vysoký, Jan
Journal of geometry and physics,
September 2024, 2024-09-00, Letnik:
203
Journal Article
Recenzirano
Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary ...for a geometrical formulation of Lagrangian mechanics and the calculus of variations. It is thus only natural to require their generalization in geometry of Z-graded manifolds and vector bundles.
Our aim is to construct the k-th order jet bundle JEk of an arbitrary Z-graded vector bundle E over an arbitrary Z-graded manifold M. We do so by directly constructing its sheaf of sections, which allows one to quickly prove all its usual properties. It turns out that it is convenient to start with the construction of the graded vector bundle of k-th order (linear) differential operators DEk on E. In the process, we discuss (principal) symbol maps and a subclass of differential operators whose symbols correspond to completely symmetric k-vector fields, thus finding a graded version of Atiyah Lie algebroid. Necessary rudiments of geometry of Z-graded vector bundles over Z-graded manifolds are recalled.
A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of ...minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's $E$-functions take algebraic values. We present algorithms and implementations for these questions, and discuss examples and experiments.
Multi-degree Tchebycheffian splines are splines with pieces drawn from extended (complete) Tchebycheff spaces, which may differ from interval to interval, and possibly of different dimensions. These ...are a natural extension of multi-degree polynomial splines. Under quite mild assumptions, they can be represented in terms of a so-called multi-degree Tchebycheffian B-spline (MDTB-spline) basis; such basis possesses all the characterizing properties of the classical polynomial B-spline basis. We present a practical framework to compute MDTB-splines, and provide an object-oriented implementation in Matlab. The implementation supports the construction, differentiation, and visualization of MDTB-splines whose pieces belong to Tchebycheff spaces that are null-spaces of constant-coefficient linear differential operators. The construction relies on an extraction operator that maps local Tchebycheffian Bernstein functions to the MDTB-spline basis of interest.
The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ...ut=uxx+αβH(u−θ)−u−w, wt=ε(u−γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut=uxx+αβH(u−θ)−u, under different conditions on the model constants.
To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0<2(1+αγ)θ<αβγ; the existence and stability of two standing wave fronts if 2(1+αγ)θ=αβγ and γ2ε>1; the existence and instability of two standing wave fronts if 2(1+αγ)θ=αβγ and 0<γ2ε<1; the existence and instability of an upside down standing pulse solution if 0<(1+αγ)θ<αβγ<2(1+αγ)θ. To establish the bifurcation for the scalar equation, we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0<2θ<β; the existence and stability of two standing wave fronts if 2θ=β; the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0<θ<β<2θ. By the way, we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation ut=uxx+αβH(u−θ)−u−w0, where w0=α(β−2θ)>0 is a positive constant, if 0<2θ<β.
To achieve the main goals, we will make complete use of the special structures of the model equations and we will construct Evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators. It turns out that a complex number λ0 is an eigenvalue of the linear differential operator, if and only if λ0 is a zero of the Evans function. The stability, instability and bifurcations of the nonlinear waves follow from the zeros of the Evans functions.
A very important motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.
Inspired by several numerical methods for finding multiple solutions, a partial Newton-correction method (PNCM) is proposed to find multiple fixed points of semi-linear differential operators. First ...a new augmented singular transform is developed to form a barrier so that an algorithm search outside the subspace generated by previously found fixed points cannot pass the barrier and penetrate into the inside to reach an old fixed point. Thus a fixed point found by an algorithm must be new. Its mathematical validations are established. A flow chart of PNCM is presented. Then a more accurate Legendre–Gauss–Lobatto pseudospectral scheme is constructed and convertes a semi-linear fixed point problem into a linear partial differential equation and an algebraic equation. It greatly simplifies the computation. Finally numerical results are presented to show the effectiveness of these approaches.
The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems ...of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.
In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.
The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.
This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered ...modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.
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