The appearance, motion, and disappearance of three-dimensional magnetic null points

N.A.M. acknowledges support from NASA grants NNX11AB61G, NNX12AB25G, and NNX15AF43G; NASA contract NNM07AB07C; and NSF SHINE grants AGS-1156076 and AGS-1358342 to SAO. C.E.P. acknowledges support from the St Andrews 2013 STFC Consolidated grant.While theoretical models and simulations of magnetic reconnection often assume symmetry such that the magnetic null point when present is co-located with a flow stagnation point, the introduction of asymmetry typically leads to non-ideal flows across the null point. To understand this behavior, we present exact expressions for the motion of three-dimensional linear null points. The most general expression shows that linear null points move in the direction along which the magnetic field and its time derivative are antiparallel. Null point motion in resistive magnetohydrodynamics results from advection by the bulk plasma flow and resistive diffusion of the magnetic field, which allows non-ideal flows across topological boundaries. Null point motion is described intrinsically by parameters evaluated locally; however, global dynamics help set the local conditions at the null point. During a bifurcation of a degenerate null point into a null-null pair or the reverse, the instantaneous velocity of separation or convergence of the null-null pair will typically be infinite along the null space of the Jacobian matrix of the magnetic field, but with finite components in the directions orthogonal to the null space. Not all bifurcating null-null pairs are connected by a separator. Furthermore, except under special circumstances, there will not exist a straight line separator connecting a bifurcating null-null pair. The motion of separators cannot be described using solely local parameters because the identification of a particular field line as a separator may change as a result of non-ideal behavior elsewhere along the field line.Publisher PDFPeer reviewe

Hierarchical interpolative factorization for elliptic operators: differential equations

This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF-DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL decomposition that facilitates the efficient inversion of the discretized operator. HIF-DE is based on the multifrontal method but uses skeletonization on the separator fronts to sparsify the dense frontal matrices and thus reduce the cost. We conjecture that this strategy yields linear complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity in 3D can be achieved by skeletonizing the compressed fronts themselves, which amounts geometrically to a recursive dimensional reduction scheme. Numerical experiments support our claims and further demonstrate the performance of our algorithm as a fast direct solver and preconditioner. MATLAB codes are freely available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math. arXiv admin note: substantial text overlap with arXiv:1307.266

Розробка технології збагачення розсипних титанових руд в умовах ТОВ ВКФ «Велта»

Об’єктом дослідження є технологія збагачення розсипних ільменітових руд. Предметом досліджень є ільменітові руди розсипного родовища. Метою дипломної роботи є зменшення втрат ільменіту з відходами магнітного збагачення розсипних руд Берзулівського родовища в умовах ТОВ ВКФ «Велта». В дипломній роботі запропоновано оригінальне рішення, яке полягає у застосуванні високоградієнтної магнітної сепарації з попереднім розділенням їх живлення на класи крупності -0,8+0,25 та -0,25 мм. Перший клас збагачується на роликовому сепараторі зі стрічкою, а другий – на сепараторі відхиляючого типу. Таке рішення дозволило підвищити ефективність схеми збагачення в цілому