The Necessary And Sufficient Condition for Generalized Demixing

Demixing is the problem of identifying multiple structured signals from a superimposed observation. This work analyzes a general framework, based on convex optimization, for solving demixing problems. We present a new solution to determine whether or not a specific convex optimization problem built for generalized demixing is successful. This solution will also bring about the possibility to estimate the probability of success by the approximate kinematic formula

Complexity Analysis of Balloon Drawing for Rooted Trees

In a balloon drawing of a tree, all the children under the same parent are placed on the circumference of the circle centered at their parent, and the radius of the circle centered at each node along any path from the root reflects the number of descendants associated with the node. Among various styles of tree drawings reported in the literature, the balloon drawing enjoys a desirable feature of displaying tree structures in a rather balanced fashion. For each internal node in a balloon drawing, the ray from the node to each of its children divides the wedge accommodating the subtree rooted at the child into two sub-wedges. Depending on whether the two sub-wedge angles are required to be identical or not, a balloon drawing can further be divided into two types: even sub-wedge and uneven sub-wedge types. In the most general case, for any internal node in the tree there are two dimensions of freedom that affect the quality of a balloon drawing: (1) altering the order in which the children of the node appear in the drawing, and (2) for the subtree rooted at each child of the node, flipping the two sub-wedges of the subtree. In this paper, we give a comprehensive complexity analysis for optimizing balloon drawings of rooted trees with respect to angular resolution, aspect ratio and standard deviation of angles under various drawing cases depending on whether the tree is of even or uneven sub-wedge type and whether (1) and (2) above are allowed. It turns out that some are NP-complete while others can be solved in polynomial time. We also derive approximation algorithms for those that are intractable in general

Nonlinear evolution of streaming instabilities in accreting protoplanetary disks

The streaming instability (SI) is one of the most promising candidates for triggering planetesimal formation by producing dense dust clumps that undergo gravitational collapse. Understanding how the SI operates in realistic protoplanetary disks (PPDs) is therefore crucial to assess the efficiency of planetesimal formation. Modern models of PPDs show that large-scale magnetic torques or winds can drive laminar gas accretion near the disk midplane. In a previous study, we identified a new linear dust-gas instability, the azimuthal drift SI (AdSI), applicable to such accreting disks and is powered by the relative azimuthal motion between dust and gas that results from the gas being torqued. In this work, we present the first nonlinear simulations of the AdSI. We show that it can destabilize an accreting, dusty disk even in the absence of a global radial pressure gradient, which is unlike the classic SI. We find the AdSI drives turbulence and the formation of vertically-extended dust filaments that undergo merging. In dust-rich disks, merged AdSI filaments reach maximum dust-to-gas ratios exceeding 100. Moreover, we find that even in dust-poor disks the AdSI can increase local dust densities by two orders of magnitude. We discuss the possible role of the AdSI in planetesimal formation, especially in regions of an accreting PPD with vanishing radial pressure gradients.Comment: Accepted by Ap

Emergence of General Relativity from Loop Quantum Gravity

Abstract I show that general relativity emerges from loop quantum gravity, in a relational description of gravitation field in terms of coordinates defined by matter. Local Dirac observables and coherent states are constructed for an explicit evaluation of the dynamics. The dynamics of large scales conforms with general relativity, up to the corrections near singularities