Necessary Conditions for Nonsmooth Optimization Problems with Operator Constraints in Metric Spaces

This paper concerns nonsmooth optimization problems involving operator constraints given by mappings on complete metric spaces with values in nonconvcx subsets of Banach spaces. We derive general first-order necessary optimality conditions for such problems expressed via certain constructions of generalized derivatives for mappings on metric spaces and axiomatically defined subdifferentials for the distance function to nonconvex sets in Banach spaces. Our proofs arc based on variational principles and perturbation/approximation techniques of modern variational analysis. The general necessary conditions obtained are specified in the case of optimization problems with operator constraints dDScribcd by mappings taking values in approximately convex subsets of Banach spaces, which admit uniformly Gateaux differentiable renorms (in particular, in any separable spaces)

Narrow-line-width UV bursts in the transition region above Sunspots observed by IRIS

Various small-scale structures abound in the solar atmosphere above active regions, playing an important role in the dynamics and evolution therein. We report on a new class of small-scale transition region structures in active regions, characterized by strong emissions but extremely narrow Si IV line profiles as found in observations taken with the Interface Region Imaging Spectrograph (IRIS). Tentatively named as Narrow-line-width UV bursts (NUBs), these structures are located above sunspots and comprise of one or multiple compact bright cores at sub-arcsecond scales. We found six NUBs in two datasets (a raster and a sit-and-stare dataset). Among these, four events are short-living with a duration of $\sim$10 mins while two last for more than 36 mins. All NUBs have Doppler shifts of 15--18 km/s, while the NUB found in sit-and-stare data possesses an additional component at $\sim$50 km/s found only in the C II and Mg II lines. Given that these events are found to play a role in the local dynamics, it is important to further investigate the physical mechanisms that generate these phenomena and their role in the mass transport in sunspots.Comment: 8 pages, 4 figures and 1 table, accepted for publication in ApJ

Plasma parameters and geometry of cool and warm active region loops

How the solar corona is heated to high temperatures remains an unsolved mystery in solar physics. In the present study we analyse observations of 50 whole active-region loops taken with the Extreme-ultraviolet Imaging Spectrometer (EIS) on board the Hinode satellite. Eleven loops were classified as cool (<1 MK) and 39 as warm (1-2 MK) loops. We study their plasma parameters such as densities, temperatures, filling factors, non-thermal velocities and Doppler velocities. We combine spectroscopic analysis with linear force-free magnetic-field extrapolation to derive the three-dimensional structure and positioning of the loops, their lengths and heights as well as the magnetic field strength along the loops. We use density-sensitive line pairs from Fe XII, Fe XIII, Si X and Mg VII ions to obtain electron densities by taking special care of intensity background-subtraction. The emission-measure loci method is used to obtain the loop temperatures. We find that the loops are nearly isothermal along the line-of-sight. Their filling factors are between 8% and 89%. We also compare the observed parameters with the theoretical RTV scaling law. We find that most of the loops are in an overpressure state relative to the RTV predictions. In a followup study, we will report a heating model of a parallel-cascade-based mechanism and will compare the model parameters with the loop plasma and structural parameters derived here.Comment: ApJ, accepted for publicatio