Whitney forms are widely known as finite elements for differential forms. Whitney’s original definition yields first order functions on simplicial complexes, and a lot of research has been devoted to extending the nonsimplicial cells higher functions. As result, term become somewhat ambiguous in literature. Our aim here is clarify concept explicitly explain their key properties. We discuss init...

In this paper we will determine the multiple point manifolds of certain self-transverse immersions in Euclidean spaces. Following the triple points, these immersions have a double point self-intersection set which is the image of an immersion of a smooth 5-dimensional manifold, cobordant to Dold manifold $V^5$ or a boundary. We will show there is an immersion of $S^7times P^2$ in $mathbb{R}^{1...

This paper studies the variation of the volume of a subanalytic family of sets. More precisely we are interested in the variation of the density. We prove that the density is continuous along a stratum of a Whitney subanalytic stratification and locally lipschitzian when the stratification satisfies the Kuo-Verdier condition. This problem had been studied by Comte in [C].

Low order Whitney elements are widely used for electromagnetic field problems. Higher order approximations are receiving increasing interest but their definition remains unduly complex. In this paper we propose a new simple construction for Whitney p-elements of polynomial degree higher than one that use only degrees of freedom associated to p-chains. We provide a basis for these elements on si...

I’ll give an outline of the theory of Whitney algebras of a matroid, with the notions of geometric product and the excahnge relations, I’ll give the letterplace superalgebra coding of these algebras, and I’ll show how this coding allows to get the exchange relations directly from the superstraightening laws. I’ll not speak about the Lax Hopf algebra structure of Whitney algebras. All this is pa...

We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery, and Tao. The proof uses the polynomial method introduced by Dvir. In [1], Bennett, Carbery, and Tao formulated a multilinear Kakeya conjecture, and they proved the conjecture except for the endpoint case. In this paper, we slightly sharpen their result by proving the endpoint case of the conjecture. Our method ...

The strong Whitney convergence on bornology introduced by Caserta in [9] is a generalization of the uniform Beer-Levi [5]. This paper aims to study some important topological properties space all real valued continuous functions metric endowed with topologies and bornology. More precisely, we investigate metrizability, various countability properties, countable tightness, Fr?chet property these...

Abstract The notion of conically smooth structure on a stratified space was introduced by Ayala, Francis and Tanaka. This is very well behaved analogue differential in the context topological spaces, satisfying good properties such as existence resolutions singularities handlebody decompositions. In this paper we prove Tanaka’s conjecture that any Whitney admits canonical structure. We thus est...

Let X be a finite graph. Let C(X) be the hyperspace of all nonempty subcontinua of X and let μ : C(X) → R be a Whitney map. We prove that there exist numbers 0 < T0 < T1 < T2 < · · · < TM = μ(X) such that if T ∈ (Ti−1, Ti), then the Whitney block μ(Ti−1, Ti) is homeomorphic to the product μ(T )× (Ti−1, Ti). We also show that there exists only a finite number of topologically different Whitney l...