On comminution and yield in brittle granular mixtures

The mechanics of granular mixtures are pivotal in many industrial applications. Unravelling the relation between yielding and comminution, the action of mechanically induced grain size reduction, in confined mixture systems is a common and open challenge. This paper attacks this problem by adopting the breakage mechanics theory, which was originally proposed for single mineral materials. We present an extension to the theory that allows predicting: (1) the yielding pressure in granular mixtures, (2) the yield pressure increase/hardening with increasing breakage, and (3) the evolution of the grain size distributions of the separate species—all of these novel capabilities are tested and validated with experiments. Of particular appeal is the finding that the average yielding pressure is a simple generalized mean with an exponent −3/2 of the yielding pressures of the homogeneous components.     

Every minor-closed property of sparse graphs is testable

Suppose G is a graph of bounded degree d, and one needs to remove ?n of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G^′. In fact, a similar result is proved for any minor-closed property of bounded degree graphs.The main motivation of the above result comes from theoretical computer-science. Using our main result we infer that for any minor-closed property P, there is a constant time algorithm for detecting if a graph is “far” from satisfying P. This, in particular, answers an open problem of Goldreich and Ron [STOC 1997] [20], who asked if such an algorithm exists when P is the graph property of being planar. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments.     

Tree-indexed random walks on groups and first passage percolation

Summary Suppose that i.i.d. random variables are attached to the edges of an infinite tree. When the tree is large enough, the partial sumsS along some of its infinite paths will exhibit behavior atypical for an ordinary random walk. This principle has appeared in works on branching random walks, first-passage percolation, and RWRE on trees. We establish further quantitative versions of this principle, which are applicable in these settings. In particular, different notions of speed for such a tree-indexed walk correspond to different dimension notions for trees. Finally, if the labeling variables take values in a group, then properties of the group (e.g., polynomial growth or a nontrivial Poisson boundary) are reflected in the sample-path behavior of the resulting tree-indexed walk.Partially supported by a grant from the Landau Center for Mathematical AnalysisPartially supported by NSF grant DMS-921 3595     

Scaling at the selective withdrawal transition through a tube suspended above the fluid surface

In the selective withdrawal experiment fluid is withdrawn through a tube with its tip suspended a distance S above an unperturbed two-fluid interface. At low withdrawal rates, Q, the interface forms a steady state hump and only the upper fluid is withdrawn. When Q is increased (or S decreased), the interface undergoes a transition so that the lower fluid is entrained with the upper one, forming a thin steady-state spout. Near this discontinuous transition the hump curvature becomes very large and displays power-law scaling behavior. This scaling is used to show that steady-state profiles for humps at different flow rates and tube heights can all be scaled onto a single similarity profile.     

Dual double arterial phase dynamic MR imaging with sensitivity encoding (SENSE): which is better for diagnosing hypervascular hepatocellular carcinomas, in-phase or opposed-phase imaging?

The purpose of this study was to investigate the efficacy of sensitivity encoding (SENSE) dynamic magnetic resonance imaging (MRI) with the dual (in-phase and opposed-phase) double arterial phase to detect hypervascular hepatocellular carcinomas (HCCs). MR images of the liver from 44 consecutive patients were obtained. Dynamic MRI with SENSE was performed six times (precontrast, early arterial, late arterial, 1 min, 3 min and 5 min after contrast injection) at 11 s per scan using the gradient recalled echo sequence (TR/TE/flip angle = 168/2.3 and 4.6/70). In-phase and opposed-phase images were obtained simultaneously each scan. For the quantitative analysis, the signal-to-noise ratio (S/N) of HCC and tumor-to-liver contrast-to-noise ratio (C/N) were analyzed for 55 HCCs. The mean S/N of HCCs on in-phase images showed significantly higher values than that on opposed-phase images regarding all phases (P < 0.001). In arterial phases, the mean tumor-to-liver C/N for in-phase images was significantly higher than that for opposed-phase images (P < 0.05). In portal and delayed-phase images, the mean tumor-to-liver C/N in opposed-phase images showed a negative value. In six HCCs with fatty metamorphosis, the mean tumor-to-liver C/N on arterial phase images approached zero in opposed-phase, while it showed a positive value in-phase. In dual double arterial phase dynamic MRI of the liver, in-phase images were superior to opposed-phase images for detecting early enhancement of hypervascular HCCs, while the latter were superior for detecting washout of contrast media from HCCs in the portal and delayed phase. The combination of both images overcomes the difficulty of diagnosing hypervascular HCCs with fatty metamorphosis.     

Mixing times of the biased card shuffling and the asymmetric exclusion process

Consider the following method of card shuffling. Start with a deck of cards numbered 1 through . Fix a parameter between 0 and 1. In this model a ``shuffle' consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability . If the coin comes up heads, then we arrange the two cards so that the lower-numbered card comes before the higher-numbered card. If the coin comes up tails, then we arrange the cards with the higher-numbered card first. In this paper we prove that for all , the mixing time of this card shuffling is , as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.

     

Harmonic and analytic functions on graphs

Harmonic and analytic functions have natural discrete analogues. Harmonic functions can be defined on every graph, while analytic functions (or, more precisely, holomorphic forms) can be defined on graphs embedded in orientable surfaces. Many important properties of the "true" harmonic and analytic functions can be carried over to the discrete setting. We prove that a nonzero analytic function can vanish only on a very small connected piece. As an application, we describe a simple local random process on embedded graphs, which have the property that observing them in a small neighborhood of a node through a polynomial time, we can infer the genus of the surface.