Schwarz—Pick Inequalities for Derivatives of Arbitrary Order

2003 ◽  
Vol 19 (2) ◽  
pp. 265-277 ◽  
Author(s):  
Avkhadiev ◽  
-J. Wirths
Keyword(s):  
Arbitrary Order ◽  
Derivatives Of

scholarly journals CONSTRAINT ALGEBRAS IN GAUGE-INVARIANT SYSTEMS

1995 ◽  
Vol 10 (28) ◽  
pp. 4087-4105 ◽  
Author(s):  
KH. S. NIROV
Keyword(s):  
Wide Class ◽  
Arbitrary Function ◽  
Arbitrary Order ◽  
Local Symmetry ◽  
Poisson Brackets ◽  
Hamiltonian Description ◽  
Gauge Invariant ◽  
Symmetry Transformations ◽  
Constraint Algebra ◽  
Derivatives Of

A Hamiltonian description is constructed for a wide class of mechanical systems having local symmetry transformations depending on time derivatives of the gauge parameters of arbitrary order. The Poisson brackets of the Hamiltonian and constraints with each other and with an arbitrary function are explicitly obtained. The constraint algebra is proved to be of the first class.

scholarly journals Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Jianfei Wang
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Arbitrary Order ◽  
Complex Banach Space ◽  
Holomorphic Mappings ◽  
Partial Derivatives ◽  
Carathéodory Metric ◽  
Homogeneous Ball ◽  
Derivatives Of

LetBXbe the unit ball in a complex Banach spaceX. AssumeBXis homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ballBntoBXassociated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

A Friedrichs inequality and an application

1984 ◽  
Vol 97 ◽  
pp. 185-191 ◽  
Author(s):  
Roger T. Lewis
Keyword(s):  
Differential Operators ◽  
Arbitrary Order ◽  
Type Inequality ◽  
Essential Spectra ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Partial Differential Operators ◽  
Even Order ◽  
Friedrichs Inequality ◽  
Derivatives Of

SynopsisAn inequality whose origins date to the work of G. H. Hardy is presented. This Hardy-type inequality applies to derivatives of arbitrary order of functions whose domain is a subset of ℝn. The Friedrichs inequality is a corollary. The result is then used to establish lower bounds on the essential spectra of even-order elliptic partial differential operators on unbounded domains.

scholarly journals Higher Order Automatic Differentiation with Dual Numbers

2020 ◽  
Author(s):  
László Szirmay-Kalos
Keyword(s):  
Gaussian Curvature ◽  
Automatic Differentiation ◽  
Arbitrary Order ◽  
Higher Order ◽  
Dual Numbers ◽  
Computer Implementation ◽  
Particular Solution ◽  
Derivatives Of ◽  
Curve Fairing ◽  
Curvature Computation

In engineering applications, we often need the derivatives of functions defined by a program. The approach chosen for derivative computation must be algebraic to allow computer implementation. A particular solution to obtain first derivatives is the application of dual numbers. This paper proposes simple and compact generalizations of this idea to obtain derivatives of arbitrary order for single or multi-variate functions and the automatic handling of 0/0 ambiguities in the calculations. We also provide the C++ code that takes advantage of operator overloading and recursion. The method is demonstrated by path animation, Gaussian curvature computation, and curve fairing.

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Virginia Kiryakova ◽  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Special Functions ◽  
Kernel Functions ◽  
Fractional Integrals ◽  
Human Sciences ◽  
New Applications ◽  
Derivatives Of

AbstractIn this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ 1,…,δ m) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.

Arbitrary-Order Derivatives of Quantum Chemical Methods via Automatic Differentiation

2021 ◽  
Vol 12 (12) ◽  
pp. 3232-3239
Author(s):  
Adam S. Abbott ◽  
Boyi Z. Abbott ◽  
Justin M. Turney ◽  
Henry F. Schaefer
Keyword(s):  
Quantum Chemical ◽  
Automatic Differentiation ◽  
Arbitrary Order ◽  
Chemical Methods ◽  
Quantum Chemical Methods ◽  
Derivatives Of

scholarly journals Growth and fixed points of solutions and their arbitrary-order derivatives of higher-order linear differential equations in the unit disc

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yu Chen ◽  
Guan-Tie Deng ◽  
Zhan-Mei Chen ◽  
Wei-Wei Wang
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Unit Disc ◽  
Arbitrary Order ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Iterated Order ◽  
Linear Differential ◽  
Derivatives Of

AbstractIn this paper, we investigate the growth and fixed points of solutions of higher-order linear differential equations in the unit disc. We extend the coefficient conditions to a type of one-constant-control coefficient comparison and obtain the same estimates of iterated order of solutions. We also obtain better estimates by providing a precise value of iterated order of solution instead of a range of that in the case of coefficient characteristic function comparison. Moreover, we utilize iteration to investigate and estimate the fixed points of solutions’ arbitrary-order derivatives with higher-order equations $f^{(k)}+A_{k-1}(z)f^{(k-1)}+{\cdots }+A_{1}(z)f'+A_{0}(z)f=0$ f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 and provide a concise method to judge if the items generated by the iteration do not vanish identically and ensure the iteration proceeds. Our results are an improvement over those by B. Belaïdi, T. B. Cao, G. W. Zhang and A. Chen.

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