Hyperplane arrangements dissect $\mathbb {R}^{n}$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as sum nonnegative integers Whitney numbers the first kind. His generalizes to count within any cone defined intersection collection halfspaces from arrangement, leading notion for each cone. This paper focuses on cones braid consisting reflecting hy...

In this paper we study the global dynamics of inverted spherical pendulum with a vertically vibrating suspension point in presence an external horizontal periodic force field. We do not assume that field is weak or rapidly oscillating. prove there always exists non-falling solution, i.e., initial condition such rod never becomes along corresponding solution. also show numerically asymptotically...

It is shown that δ-connectedness is not a Whitney reversible property. This answers in the negative a question posed by Sam B. Nadler, Jr. in 1978. A topological property P is said to be: (a) a Whitney property provided that if a continuum X has property P, so does μ−1(t) for each Whitney map μ for C(X) and each t ∈ [0, μ(X)) ([6, p. 165]); (b) a Whitney reversible property provided that whenev...

Motivated by boundary value problems for partial differential equations, classical trace and extension theorems characterize traces of spaces of generalized smoothness (e.g., Sobolev, Besov, etc.) to smooth submanifolds of a Euclidean space. But in many cases one needs similar results for subsets of a more complicated geometric structure (for instance, after the change of variables initial data...

The aim of this work is to generalize the idea of the discretizations on sparse grids to discrete differential forms. The extension to general l-forms in d dimensions includes the well known Whitney elements, as well as H(div; Ω)and H(curl; Ω)-conforming mixed finite elements. The formulation of Maxwell’s equations in terms of differential forms gives a crucial hint how they should be discretiz...