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The current-induced motion of magnetic domain walls (DWs) confined to nanostructures is of great interest for fundamental studies as well as for technological applications in spintronic devices. Here, we present magnetic images showing the depinning properties of pulse-current-driven domain walls in well-shaped Permalloy nanowires obtained using photoemission electron microscopy combined with x-ray magnetic circular dichroism. In the vicinity of the threshold current density (Jth=4.2 × 1011 A.m-2) for the DW motion, discontinuous DW depinning and motion have been observed as a sequence of "Barkhausen jumps". A one-dimensional analytical model with a piecewise parabolic pinning potential has been introduced to reproduce theDWhopping between two nearest neighbour sites, which reveals the dynamical nature of the current-driven DW motion in the depinning regime. [ABSTRACT FROM AUTHOR]

This paper deals with a constraint multiobjective programming problem and its dual problem in the lexicographic order. We establish some duality theorems and present several existence results of a Lagrange multiplier and a lexicographic saddle point theorem. Meanwhile, we study the relations between the lexicographic saddle point and the lexicographic solution to a multiobjective programming problem. [ABSTRACT FROM AUTHOR]