We find the minimal weighted energy ∫ Ω a(x)∣Du∣ of maps u ∶ Ω ⊂ Rm+n → S with prescribed singularities, where a(⋅) is a continuous positive weight. Our result extends previous ones of H. Brezis, J-M. Coron and E.H. Lieb (1986), G. Alberti, S. Baldo and G. Orlandi (2003), and V. Millot (2005).

This publication gives an introductory overview of the role fire in forest systems, benefits using prescribed as a part management, and necessary safety regulatory precautions that landowners land managers need to adhere fire.

The distributional k-dimensional Jacobian of a map u in the Sobolev space W 1,k−1 which takes values in the the sphere S can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M o...

AIM To determine the best method of estimating the optimum magnification needed by visually impaired patients. METHODS The magnification of low vision aids prescribed to 187 presbyopic visually impaired patients for reading newspapers or books was compared with logMAR distance and near acuity (at 25 cm) and magnification predicted by +4 D step near additions. RESULTS Distance letter (r = 0....

The correlation function theory on the basis of prescribed boundary conditions provides a deeper understanding in studying the dynamical parameters of galaxy clusters. The approach approximates that the moderate dense systems discussed by a two point correlation function is helpful for describing the dynamical nature of galaxy clusters. The projected theory of two point correlation function for...