Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of G~teaux, Fr6chet, and Hadamard are singled out from the general framework of cr-directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equ...

Steerable filters have been used to analyze local orientation patterns in imagery. Such filters are typically based on directional derivatives, whose symmetry produces orientation responses that are periodic with period pi, independent of image structure. We present a more general set of steerable filters that alleviate this problem.

This paper deals with the number of meaningful compositions of higher order of differential operations and the Gateaux directional derivative.

In this paper, we introduce a new differential-difference operator Tξ (ξ ∈ R ) by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between Tξ and the directional derivative ∂ξ. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this o...

where Δpu = div(|∇u|p−2∇u) is the p-Laplacian of u with 1 < p < ∞, and λ is a real parameter. Moreover, Ω is a smooth bounded domain in RN , N ≥ 1, whose boundary ∂Ω is made of two disjoint nonempty closed sets Γ1 and Γ2 which are smooth manifolds of dimension N − 1, m and n are weights which may be indefinite and unbounded, ∂/∂ν denotes the exterior normal derivative, and u± = max{±u,0}. The m...

where σn−1 denotes the surface measure of the unit sphere S n−1 = {z ∈ R : |z| = 1}, and |z| denotes the standard Euclidean norm of z ∈ R. This formula is given in equation (18) on p125 of [Ste1], and it is proved by using the Fundamental Theorem of Calculus to first write f(x) as the integral along any ray emanating from x of the directional derivative of f in the direction of the ray, and the...

In this paper we study existence, uniqueness and solution estimates to the mixed problem r ru = 0 with Dirichlet to Neumann map boundary conditions and Neumann boundary conditions. We then show how this can be used in the reconstruction of , given the relationship between u and its normal derivative on the boundary portion where we do not apply the Dirichlet to Neumann map. A numerical reconstr...

In this paper, the directional lower and upper derivatives of the maxmin function are investigated by using the directional lower and upper derivative sets of the max-min set valued map. Sufficient conditions ensuring the existence of the directional derivative of the max-min function are obtained.

Directional drilling has evolved over the last few decades to become standard operational practice in the oil and gas exploration industry. One method of steering utilizes a technique known as magnetic Measurement While Drilling (MWD). Vector measurements of geomagnetic fields along a well-bore are used to determine it orientation. This method is advantageous in that it is inexpensive and effic...

In this paper, based on a partial order, we study the characterizations of directional derivatives and the subdifferential of fuzzy function. At the same time, we also discuss the relation between the directional derivative and the subdifferential.