Flexible Clustering with a Sparse Mixture of Generalized Hyperbolic Distributions

Robust clustering of high-dimensional data is an important topic because, in many practical situations, real data sets are heavy-tailed and/or asymmetric. Moreover, traditional model-based clustering often fails for high dimensional data due to the number of free covariance parameters. A parametrization of the component scale matrices for the mixture of generalized hyperbolic distributions is proposed by including a penalty term in the likelihood constraining the parameters resulting in a flexible model for high dimensional data and a meaningful interpretation. An analytically feasible EM algorithm is developed by placing a gamma-Lasso penalty constraining the concentration matrix. The proposed methodology is investigated through simulation studies and two real data sets

Systoles of Arithmetic Hyperbolic Surfaces and 3-manifolds

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with invariant trace field $k$. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic field having small discriminant. When the trace field is $\mathbf{Q}$, using work of Granville and Soundararajan, we establish a stronger result that allows our constant lower bound $x_0$ to grow with the area. As an application, we establish a systolic bound for arithmetic hyperbolic surfaces that is related to prior work of Buser-Sarnak and Katz-Schaps-Vishne. Finally, we establish an analogous density result for commensurability classes of arithmetic hyperbolic 3-orbifolds with small area totally geodesic $2$-orbifolds.Comment: v4: 17 pages. Revised according to referee report. Final version. To appear in Math. Res. Let

Generation of pulsed bipartite entanglement using four-wave mixing

Using four-wave mixing in a hot atomic vapor, we generate a pair of entangled twin beams in the microsecond pulsed regime near the D1 line of $^{85}$Rb, making it compatible with commonly used quantum memory techniques. The beams are generated in the bright and vacuum-squeezed regimes, requiring two separate methods of analysis, without and with local oscillators, respectively. We report a noise reduction of up to $3.8\pm 0.2$ dB below the standard quantum limit in the pulsed regime and a level of entanglement that violates an Einstein--Podolsky--Rosen inequality.Comment: 10 pages, 5 figures, accepted for publication in New Journal Of Physici

Notched impact behavior of polymer blends: Part 1: New model for particle size dependence

A model is proposed to explain the observed relationships between particle size and fracture resistance in high-performance blends, which typically reach maximum toughness at particle diameters of 0.2–0.4 μm. To date there has been no satisfactory explanation for the ductile–brittle (DB) transition at large particle sizes. The model is based on a recently developed criterion for craze initiation, which treats large cavitated rubber particles as craze-initiating Griffith flaws. Using this criterion in conjunction with Westergaard's equations, it is possible to map the spread from the notch tip of three deformation mechanisms: rubber particle cavitation, multiple crazing and shear yielding. Comparison of zone sizes leads to the conclusion that maximum toughness is achieved when the particles are large enough to cavitate a long way ahead of a notch or crack tip, but not so large that they initiate unstable crazes and thus reduce fracture resistance