Related papers: Derivation of some definite integrals
A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…
We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…
Automatic differentiation is everywhere, but there exists only minimal documentation of how it works in complex arithmetic beyond stating "derivatives in $\mathbb{C}^d$" $\cong$ "derivatives in $\mathbb{R}^{2d}$" and, at best, shallow…
Functional integrals are central to modern theories ranging from quantum mechanics and statistical thermodynamics to biology, chemistry, and finance. In this work we present a new method for calculating functional integrals based on a…
We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…
This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant…
In approximation theory classical discrete operators, like generalized sampling, Sz\'{a}sz-Mirak'jan, Baskakov and Bernstein operators, have been extensively studied for scalar functions. In this paper, we look at the approximation of…
We present a direct derivation of the typical time derivatives used in a continuum description of complex fluid flows, harnessing the principles of the kinematics of line elements. The evolution of the microstructural conformation tensor in…
Stewart has presented a position-space derivation of the identity for the volume integral of a product of two vector fields noted by Gubarev, Stodolsky, and Zakharov, and applied the results to classical electromagnetic theory. I present a…
The details of second-order partial derivatives of rigid-body Inverse/Forward dynamics are provided. Several properties and identities using Spatial Vector Algebra are listed, along with their detailed derivations. The expressions build…
Recently developed applications in the field of machine learning and computational physics rely on automatic differentiation techniques, that require stable and efficient linear algebra gradient computations. This technical note provides a…
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
The subject of this paper is the problem of arrangement of real algebraic curves on real algebraic surfaces. In this paper we extend Rokhlin, Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences for curves on surfaces and give some…
This paper is devoted to studying the first-order variational analysis of non-convex and non-differentiable functions that may not be subdifferentially regular. To achieve this goal, we entirely rely on two concepts of directional…
Fractional order derivatives and integrals (differintegrals) are viewed from a frequency-domain perspective using the formalism of Riesz, providing a computational tool as well as a way to interpret the operations in the frequency domain.…
Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. Both old and new results on these invariants are collected.
If a higher derivative theory arises from a transformation of variables that involves time derivatives, a tailor-made Hamiltonian formulation is shown to exist. The details and advantages of this elegant Hamiltonian formulation, which…
We present a method derived from Laplace transform theory that enables the evaluation of fractional integrals. This method is adapted and extended in a variety of ways to demonstrate its utility in deriving alternative representations for…
We derive analytical shape derivative formulas of the system matrix representing electric field integral equation discretized with Raviart-Thomas basis functions. The arising integrals are easy to compute with similar methods as the entries…
We develop an intrinsic geometrical setting for higher order constrained field theories. As a main tool we use an appropriate generalization of the classical Skinner-Rusk formalism. Some examples of application are studied, in particular,…