Related papers: The Exterior Derivative - A direct approach
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard-Marchaud fractional derivative, is also considered. The objective is to represent these…
An explicit expression of the k-th derivative of the Bessel function $J_\nu(z)$, with respect to its order $\nu$, is given. Particularizations for the cases of positive or negative $\nu$ are considered.
A simple finite element formulation of the outlet gradient boundary condition is presented in the general context of convective-diffusive transport processes. Basically, the method is based on an upstream evaluation of the dependent…
In this paper, we present a new derivative via the Laplace transform. The Laplace transform leads to a natural form of the fractional derivative which is equivalent to a Riemann-Liouville derivative with fixed terminal point. We first…
We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed…
This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this…
A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product…
In this short note, we provide an elementary complex analytic method for converting known real integrals into numerous strange and interesting looking real integrals.
We develop some basic facts on deformations of exterior differential ideals on a smooth complex algebraic variety. With these tools we study deformations of several types of differential ideals, leading to several irreducible components of…
The goal of this paper is to construct a multivariate generalisation of the Grunwald-Letnikov derivative, a classical fractional derivative operator. To do so, we first produce a formalism of fractional derivatives in terms of infinitesimal…
A new exponentially convergent algorithm is proposed for an abstract the first order differential equation with unbounded operator coefficient possessing a variable domain. The algorithm is based on a generalization of the Duhamel integral…
Derivatives play a critical role in computational statistics, examples being Bayesian inference using Hamiltonian Monte Carlo sampling and the training of neural networks. Automatic differentiation is a powerful tool to automate the…
I explain a direct approach to differentiation and integration. Instead of relying on the general notions of real numbers, limits and continuity, we treat functions as the primary objects of our theory, and view differentiation as division…
This paper presents a theorem which solves the problem of reduction of the determinant order by means of a transformation of it, into other determinant whose each element are a determinant of second order. This implies that, if the process…
In this paper we determine the number of the meaningful compositions of higher order of the differential operations and Gateaux directional derivative.
The algorithm for inverting covariant exterior derivative is provided. It works for a sufficiently small star-shaped region of a fibered set - a local subset of a vector bundle and associated vector bundle. The algorithm contains some…
A \emph{double extrema form} of the calculus of variations is put forward in which only the smallest one of the finite differences is physically meaningful to represent the variational derivatives defined on the discrete points. The most…
Motivated by a recent surge of interest for Dynkin operators in mathematical physics and by problems in the combinatorial theory of dynamical systems, we propose here a systematic study of logarithmic derivatives in various contexts. In…
In the present talk we briefly demonstrate an elegant and effective technique for calculation of the trace expansion in the derivatives of background fields. One of main advantages of the technique is manifestly (super)symmetrical and gauge…