scholarly journals Ger-type and Hyers-Ulam stabilities for the first-order linear differential operators of entire functions

2004 ◽  
Vol 2004 (22) ◽  
pp. 1151-1158 ◽  
Author(s):  
Takeshi Miura ◽  
Go Hirasawa ◽  
Sin-Ei Takahasi
Keyword(s):  
Entire Function ◽  
Differential Operator ◽  
Stability Problem ◽  
Differential Operators ◽  
Entire Functions ◽  
Ulam Stability ◽  
Nonzero Constant ◽  
First Order ◽  
Linear Differential Operators ◽  
Linear Differential

Lethbe an entire function andTha differential operator defined byThf=f′+hf. We show thatThhas the Hyers-Ulam stability if and only ifhis a nonzero constant. We also consider Ger-type stability problem for|1−f′/hf|≤ϵ.

scholarly journals Involutive Property of Resolutions of Differential Operators

1966 ◽  
Vol 27 (2) ◽  
pp. 419-427
Author(s):  
Masatake Kuranishi
Keyword(s):  
Vector Bundle ◽  
Differential Operator ◽  
Vector Space ◽  
Cross Sections ◽  
Vector Bundles ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
First Order ◽  
Linear Differential

Let E and E′ be C∞ vector bundles over a C∞ manifold M. Denote by Γ(E) (resp. by Γ(E′) the vector space of C∞ cross-sections of E (resp. of E′) over M. Take a linear differential operator of the first order D: Γ(E) → Γ(E′) induced by a vector bundle mapping σ(D): jl(E) ′ E′, where Jk(E) denotes the vector bundle of k-jets of cross-sections of E.

scholarly journals Splines and fractional differential operators

2020 ◽  
pp. 2040005
Author(s):  
Peter Massopust
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Operator Equations ◽  
B Splines ◽  
Linear Differential Operators ◽  
Fractional Differential ◽  
Linear Differential ◽  
Fractional Differential Operators ◽  
Integral Order

Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form [Formula: see text] where [Formula: see text] is a linear differential operator of integral order. In this paper, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.

Explicit invariant solutions for invariant linear differential operators

1984 ◽  
Vol 98 (1-2) ◽  
pp. 149-166 ◽  
Author(s):  
Zofia Szmydt ◽  
Bogdan Ziemian
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Invariant Solution ◽  
Real Analytic Function ◽  
Linear Differential Operators ◽  
Invariant Distributions ◽  
Real Analytic Manifold ◽  
Real Analytic ◽  
Linear Differential

SynopsisLet F be a real analytic function on a real analytic manifold X. Let P be a linear differential operator on X such that , where Q is an ordinary differential operator with analytic coefficients whose singular points are all regular. For each (isolated) critical value z of F, we construct locally an F-invariant solution u of the equation Pu - v, v being an arbitrary F-invariant distribution supported in F−1(z). The solution u is constructed explicitly in the form of a series of F-invariant distributions.

scholarly journals Linear Differential Operators Invertible in the Integro-differential Sense

2019 ◽  
pp. 20-33
Author(s):  
V. N. Chetverikov
Keyword(s):  
Differential Operator ◽  
Integral Operator ◽  
Differential Operators ◽  
Spatial Variable ◽  
Diagonal Form ◽  
Operator Equations ◽  
Evolutionary Systems ◽  
Linear Differential Operators ◽  
The Matrix ◽  
Linear Differential

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.

ON THE BRÜCK CONJECTURE

2015 ◽  
Vol 93 (2) ◽  
pp. 248-259 ◽  
Author(s):  
TINGBIN CAO
Keyword(s):  
Entire Function ◽  
Differential Equations ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Small Function ◽  
Shared Value ◽  
Nonzero Constant ◽  
Hyper Order ◽  
Linear Differential ◽  
Brück Conjecture

The Brück conjecture states that if a nonconstant entire function $f$ with hyper-order ${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$ shares one finite value $a$ (counting multiplicities) with its derivative $f^{\prime }$, then $f^{\prime }-a=c(f-a)$, where $c$ is a nonzero constant. The conjecture has been established for entire functions with order ${\it\sigma}(f)<+\infty$ and hyper-order ${\it\sigma}_{2}(f)<{\textstyle \frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case ${\it\sigma}_{2}(f)=\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations $f^{(k)}+A(z)f=Q(z)$. The shared value $a$ is extended to be a ‘small’ function with respect to the entire function $f$.

Boolean linear differential operators on elementary cellular automata

2014 ◽  
Vol 25 (06) ◽  
pp. 1450009
Author(s):  
Ángel Martín del Rey
Keyword(s):  
Differential Operator ◽  
Cellular Automata ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Linear Differential Operators ◽  
Elementary Cellular Automata ◽  
Linear Differential

In this paper, the notion of boolean linear differential operator (BLDO) on elementary cellular automata (ECA) is introduced and some of their more important properties are studied. Special attention is paid to those differential operators whose coefficients are the ECA with rule numbers 90 and 150.

scholarly journals Fiberwise linear differential operators

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fabrizio Pugliese ◽  
Giovanni Sparano ◽  
Luca Vitagliano
Keyword(s):  
Vector Bundle ◽  
Differential Operator ◽  
Line Bundle ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Lie Algebroid ◽  
Lie Groupoid ◽  
Linear Differential Operators ◽  
Linear Differential ◽  
Definition Of

Abstract We define a new notion of fiberwise linear differential operator on the total space of a vector bundle E. Our main result is that fiberwise linear differential operators on E are equivalent to (polynomial) derivations of an appropriate line bundle over E ∗ {E^{\ast}} . We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.

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