Related papers: The Gateaux Derivative of Map over Division Ring
A procedure to obtain differentiation matrices is extended straightforwardly to yield new differentiation matrices useful to obtain derivatives of complex rational functions. Such matrices can be used to obtain numerical solutions of some…
The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
In this paper we determine the number of the meaningful compositions of higher order of the differential operations and Gateaux directional derivative.
In this paper, we consider characterisations of the class of unitary matrix integrals $\big\langle (\det U)^q {\rm e}^{s^{1/2} \operatorname{Tr}(U + U^\dagger)} \big\rangle_{U(l)}$ in terms of a first-order matrix linear differential…
Quaternion derivatives in the mathematical literature are typically defined only for analytic (regular) functions. However, in engineering problems, functions of interest are often real-valued and thus not analytic, such as the standard…
Looking forward to introducing an analysis in Galois Fields, discrete functions are considered (such as transcendental ones) and MacLaurin series are derived by Lagrange's Interpolation. A new derivative over finite fields is defined which…
We consider some possible approaches to the fractional-order generalization of definition of variation (functional) derivative. Some problems of formulation of a fractional-order variational derivative are discussed. To give a consistent…
We construct a counterexample to a well-known extension theorem for slice regular functions, which motivates us to develop a theory of Riemann slice-domains by introducing a new topology on quaternions. By some paths describing axial…
This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper in J.…
Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications:…
In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…
This short note provides an explicit description of the Fr\'echet derivatives of the principal square root matrix functional at any order. We present an original formulation that allows to compute sequentially the Fr\'echet derivatives of…
We propose a new way to compute the radius of convergence for Quaternionic hyperholomorphic functions and for Octonion analytic functions. We extend the theorem of Cauchy Hadamard and the theorem of Abel on convergence of series to…
This paper is in concern with Cauchy problems involving the fractional derivatives with respect to another function. Results of existence, uniqueness, and Taylor series among others are established in appropriate functional spaces. We prove…
In this paper, the fractional differential matrices based on the Jacobi-Gauss points are derived with respect to the Caputo and Riemann-Liouville fractional derivative operators. The spectral radii of the fractional differential matrices…
Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in…
Cartesian differential categories provide a categorical framework for multivariable differential calculus and also the categorical semantics of the differential $\lambda$-calculus. Taylor series expansion is an important concept for both…
Motivated by the general problem of extending the classical theory of holomorphic functions of a complex variable to the case of quater- nion functions, we give a notion of an H-derivative for functions of one quaternion variable. We show…
Differential categories provide the categorical foundations for the algebraic approaches to differentiation. They have been successful in formalizing various important concepts related to differentiation, such as, in particular,…