Page 1 Next

Displaying 1 – 20 of 152

Showing per page

A characterization of graphs which can be approximated in area by smooth graphs

Domenico Mucci (2001)

Journal of the European Mathematical Society

For vector valued maps, convergence in W 1 , 1 and of all minors of the Jacobian matrix in L 1 is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension n 3 can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. 2-dimensional plane intersecting the domain.

A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio (2011)

Annales Polonici Mathematici

Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let π V denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an m -measurable subset of ℝⁿ with m ( A ) < . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle ( V , v ) | V G ( n , m ) , v V such that, for all P ∈ A, one has m ( n - m ) ( V G ( n , m ) | ( V , π V ( P ) ) Z ) > 0 . One can replace “for all P ∈ A” by “for m -a.e. P ∈...

A new proof of the rectifiable slices theorem

Robert L. Jerrard (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

This paper gives a new proof of the fact that a k -dimensional normal current T in m is integer multiplicity rectifiable if and only if for every projection P onto a k -dimensional subspace, almost every slice of T by P is 0 -dimensional integer multiplicity rectifiable, in other words, a sum of Dirac masses with integer weights. This is a special case of the Rectifiable Slices Theorem, which was first proved a few years ago by B. White.

Currently displaying 1 – 20 of 152

Page 1 Next