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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 modified Fletcher-Reeves conjugate gradient method for unconstrained optimization with applications in image restoration

Zainab Hassan Ahmed, Mohamed Hbaib, Khalil K. Abbo (2024)

Applications of Mathematics

The Fletcher-Reeves (FR) method is widely recognized for its drawbacks, such as generating unfavorable directions and taking small steps, which can lead to subsequent poor directions and steps. To address this issue, we propose a modification to the FR method, and then we develop it into the three-term conjugate gradient method in this paper. The suggested methods, named ``HZF'' and ``THZF'', preserve the descent property of the FR method while mitigating the drawbacks. The algorithms incorporate...

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 ℝ 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.

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