Related papers: Quaternion Derivatives: The GHR Calculus
In this article we will present a new perspective on the variable order fractional calculus, which allows for differentiation and integration to a variable order, i.e. one differentiates (or integrates) a function along the path of a…
Relational Hoare logics (RHL) provide rules for reasoning about relations between programs. Several RHLs include a rule we call sequential product that infers a relational correctness judgment from judgments of ordinary Hoare logic (HL).…
The "Hotelling rule" (HR) called to be "the fundamental principle of the economics of exhaustible resources" has a logical deficiency which was never paid a sufficient attention to. This deficiency should be taken into account before…
A general formulation of noncommutative or quantum derivatives for operators in a Banach space is given on the basis of the Leibniz rule, irrespective of their explicit representations such as the G\^ateaux derivative or commutators. This…
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…
We study the problem of estimating the derivatives of a regression function, which has a wide range of applications as a key nonparametric functional of unknown functions. Standard analysis may be tailored to specific derivative orders, and…
This short article contains the construction of a construction that generalizes the concept of the derivative of a function of one variable, using the theory of filters. The paper presents a new concept, demonstrates that it really…
In this paper, a generalization of a quadratic manifold approach for the reduction of geometrically nonlinear structural dynamics problems is presented. This generalization is constructed by a linearization of the static force with respect…
The satisfactory development of Quaternionic Analysis has indicated new solutions for physical and mathematical problems. It is worth mentioning the fact that quaternions possess four dimensions, and in this way they may be considered as…
The lambda calculus is a widely accepted computational model of higher-order functional pro- grams, yet there is not any direct and universally accepted cost model for it. As a consequence, the computational difficulty of reducing lambda…
We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we…
Second order circularity, also called properness, for complex random variables is a well known and studied concept. In the case of quaternion random variables, some extensions have been proposed, leading to applications in quaternion signal…
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…
A map $f$ from the quaternion skew field $H$ to itself, can also be thought as a transformation $f:R^4 \to R^4$. In this manuscript, the Jacobian $J(f)$ of $f$ is computed, in the case where $f$ is a quaternion polynomial. As a consequence,…
This paper presents solutions to the following two common quaternion attitude estimation problems: (i) estimation of attitude using measurement of two reference vectors, and (ii) estimation of attitude using rate measurement and measurement…
We show that a substantial portion of stochastic calculus can be developed along similar lines to ordinary calculus, with derivative-based concepts driving the development. We define a notion of stopping derivative, which is a form of right…
The graph Hilbert transform (GHT) is a key tool in constructing analytic signals and extracting envelope and phase information in graph signal processing. However, its utility is limited by confinement to the graph Fourier domain, a fixed…
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on…
One of the most common types of functions in mathematics, physics, and engineering is a sum of products, sometimes called a partition function. After "normalization," a sum of products has a natural graphical representation, called a normal…