and Applied Analysis 3 Let y1, . . . , yk be an orthonormal system of coordinates in R, 1 ≤ k ≤ n. Let A be a domain in R, and let B be an n − k -dimensional Riemannian manifold. We consider the manifold N A × B. 2. Boundary Sets Below we introduce the notions of parabolic and hyperbolic type of boundary sets on noncompact Riemannian manifolds and study exhaustion functions of such sets. We als...

Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize that they belong together in a single mathematical system. This paper reviews the rationale for embedding differential forms in the more comprehensive system of Geometric Calculus. The most significant app...

Usually in this type of calculations one does not write explicitly the unit vectors ei. This makes the notation more economical but is possible only either if the vectors are decomposed into the Cartesian unit vectors ex, ey, ez, or (for vectors decomposed into unit vectors e1(ξ), e2(ξ), e3(ξ) associated with some curvelinear coordinates ξ, ξ, ξ see below), or if no differentiations are involve...

In this paper, we consider weighted composition operators betweenmeasurable differential forms and then some classic properties of these operators are characterized.

This paper is intended as an introduction to noncommutative geometry for readers with some knowledge of abstract algebra and differential geometry. We show how to extend the theory of differential forms to the “noncommutative spaces” studied in noncommutative geometry. We formulate and prove the Hochschild-Kostant-Rosenberg theorem and an extension of this result involving the Connes differential.

* Correspondence: tzymath@gmail. com Department of Mathematics and System Science, National University of Defense Technology, Changsha, PR China Abstract This article is devoted to extensions of some existing results about the Caratheodory operator from the function sense to the differential form situation. Similarly as the function sense, we obtain the convergence of sequences of differential ...

In linear system theory, we often encounter the situation of investigating some quadratic functionals which represent Lyapunov functions, energy storage, performance measures, e.t.c. Such a quadratic functional is called a quadratic differential form (QDF) in the context of the behavioral approach. In the past works, a QDF is usually defined in terms of a polynomial matrix. The contribution of ...

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Algebraic computations in differential geometry have usually a strong “analytic” side, and symbolic formula crunching is heavily used, even if at the end, the user needs only numbers, or graphic visualization. We show how to implement in a simple way the domain of differential forms with the p-vector algebra, Hodge “star” operator, and the differentiation. There is no explicit symbolic manipula...