Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of {\em both} time and space, not much was known about achieving such type of algorithm. In this paper we study this issue for a problem which lies at the core of many graph mining applications called {\em densest subgraph problem}. We develop an algorithm that achieves time- and space-efficiency for this problem simultaneously. It is one of the first of its kind for graph problems to the best of our knowledge. In a graph $G = (V, E)$, the "density" of a subgraph induced by a subset of nodes $S \subseteq V$ is defined as $|E(S)|/|S|$, where $E(S)$ is the set of edges in $E$ with both endpoints in $S$. In the densest subgraph problem, the goal is to find a subset of nodes that maximizes the density of the corresponding induced subgraph. For any $\epsilon>0$, we present a dynamic algorithm that, with high probability, maintains a $(4+\epsilon)$-approximation to the densest subgraph problem under a sequence of edge insertions and deletions in a graph with $n$ nodes. It uses $\tilde O(n)$ space, and has an amortized update time of $\tilde O(1)$ and a query time of $\tilde O(1)$. Here, $\tilde O$ hides a O(\poly\log_{1+\epsilon} n) term. The approximation ratio can be improved to $(2+\epsilon)$ at the cost of increasing the query time to $\tilde O(n)$. It can be extended to a $(2+\epsilon)$-approximation sublinear-time algorithm and a distributed-streaming algorithm. Our algorithm is the first streaming algorithm that can maintain the densest subgraph in {\em one pass}. The previously best algorithm in this setting required $O(\log n)$ passes [Bahmani, Kumar and Vassilvitskii, VLDB'12]. The space required by our algorithm is tight up to a polylogarithmic factor.Comment: A preliminary version of this paper appeared in STOC 201

Gapped Surface States in a Strong-Topological-Semimetal

A three-dimensional strong-topological-insulator or -semimetal hosts topological surface states which are often said to be gapless so long as time-reversal symmetry is preserved. This narrative can be mistaken when surface state degeneracies occur away from time-reversal-invariant momenta. The mirror-invariance of the system then becomes essential in protecting the existence of a surface Fermi surface. Here we show that such a case exists in the strong-topological-semimetal Bi$_4$Se$_3$. Angle-resolved photoemission spectroscopy and \textit{ab initio} calculations reveal partial gapping of surface bands on the Bi$_2$Se$_3$-termination of Bi$_4$Se$_3$(111), where an 85 meV gap along $\bar{\Gamma}\bar{K}$ closes to zero toward the mirror-invariant $\bar{\Gamma}\bar{M}$ azimuth. The gap opening is attributed to an interband spin-orbit interaction that mixes states of opposite spin-helicity.Comment: 5 pages, 3 figure

On the Convergence of the Born Series in Optical Tomography with Diffuse Light

We provide a simple sufficient condition for convergence of Born series in the forward problem of optical diffusion tomography. The condition does not depend on the shape or spatial extent of the inhomogeneity but only on its amplitude.Comment: 23 pages, 7 figures, submitted to Inverse Problem

Hyperdiffusion as a Mechanism for Solar Coronal Heating

A theory for the heating of coronal magnetic flux ropes is developed. The dissipated magnetic energy has two distinct contributions: (1) energy injected into the corona as a result of granule-scale, random footpoint motions, and (2) energy from the large-scale, nonpotential magnetic field of the flux rope. The second type of dissipation can be described in term of hyperdiffusion, a type of magnetic diffusion in which the helicity of the mean magnetic field is conserved. The associated heating rate depends on the gradient of the torsion parameter of the mean magnetic field. A simple model of an active region containing a coronal flux rope is constructed. We find that the temperature and density on the axis of the flux rope are lower than in the local surroundings, consistent with observations of coronal cavities. The model requires that the magnetic field in the flux rope is stochastic in nature, with a perpendicular length scale of the magnetic fluctuations of order 1000 km.Comment: 9 pages (emulateapj style), 4 figures, ApJ, in press (v. 679; June 1, 2008

Polar Cremona Transformations and Monodromy of Polynomials

Consider the gradient map associated to any non-constant homogeneous polynomial f\in \C[x_0,...,x_n] of degree $d$, defined by \phi_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x)) where D(f)=\{x\in \CP^n; f(x)\neq 0\} is the principal open set associated to $f$ and $f_i=\frac{\partial f}{\partial x_i}$. This map corresponds to polar Cremona transformations. In Proposition \ref{p1} we give a new lower bound for the degree $d(f)$ of $\phi_f$ under the assumption that the projective hypersurface $V:f=0$ has only isolated singularities. When $d(f)=1$, Theorem \ref{t4} yields very strong conditions on the singularities of $V$.Comment: 8 page

Dissociation of Action and Object Naming: Evidence From Cortical Stimulation Mapping

This cortical stimulation mapping study investigates the neural representation of action and object naming. Data from 13 neurosurgical subjects undergoing awake cortical mapping is presented. Our findings indicate clear evidence of differential disruption of noun and verb naming in the context of this naming task. At the individual level, evidence was found for punctuate regions of perisylvian cortex subserving noun and verb function. Across subjects, however, the location of these sites varied. This finding may help explain discrepancies between lesion and functional imaging studies of noun and verb naming. In addition, an alternative coding of these data served to highlight the grammatical class vulnerability of the target response. The use of this coding scheme implicates a role for the supramarginal gyrus in verb-naming behavior. These data are discussed with respect to a functional-anatomical pathway underlying verb naming

Co-registered combined OCT and THz imaging to extract depth and refractive index of a tissue-equivalent test object

Terahertz (THz) imaging and optical coherence tomography (OCT) provide complementary information with similar length scales. In addition to OCT’s extensive use in ophthalmology, both methods have shown some promise for other medical applications and non-destructive testing. In this paper, we present an iterative algorithm that combines the information from OCT and THz imaging at two different measurement locations within an object to determine both the depth of the reflecting layers at the two locations and the unknown refractive index of the medium for both the OCT wavelengths and THz frequencies. We validate this algorithm using a silicone test object with embedded layers and show that the depths and refractive index values obtained from the algorithm agreed with the measured values to within 3.3%. We further demonstrate for the first time that OCT and THz images can be co-registered and aligned using unsupervised image registration. Hence we show that a combined OCT/THz system can provide unique information beyond the capability of the separate modalities alone, with possible applications in the medical, industrial and pharmaceutical sectors

The effect of pore size on permeability and cell attachment in collagen scaffolds for tissue engineering.

The permeability of scaffolds and other three-dimensional constructs used for tissue engineering applications is important as it controls the diffusion of nutrients in and waste out of the scaffold as well as influencing the pressure fields within the construct. The objective of this study was to characterize the permeability/fluid mobility of collagen-GAG scaffolds as a function of pore size and compressive strain using both experimental and mathematical modeling techniques. Scaffolds containing four distinct mean pore sizes (151, 121, 110, 96 microns) were fabricated using a freeze-drying process. An experimental device was constructed to measure the permeability of the scaffold variants at different levels of compressive strain (0, 14, 29 and 40% while a low-density open-cell foam cellular solids model utilizing a tetrakaidecahedral unit cell was used to accurately model the permeability of each scaffold variant at all level of applied strain. The results of both the experimental and the mathematical analysis revealed that scaffold permeability increases with increasing pore size and decreases with increasing compressive strain. The excellent comparison between experimentally measured and predicted scaffold permeability suggests that cellular solids modelling techniques can be utilized to predict scaffold permeability under a variety of physiological loading conditions as well as to predict the permeability of future scaffolds with a wide variety of pore microstructures