Strichartz type estimates for fractional heat equations

We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing by BMOx(Rn) and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the Strichartz estimates for the fractional heat equations with a time dependent potential of an appropriate integrability. As an application, we prove the global existence and uniqueness of regular solutions in spatial variables for the generalized Navier-Stokes system with Lr(Rn) data.     

Context dependent graph-based method for laser cladding image thresholding

For segmentation method to be useful it must be fast, easy to use, and produce high quality segmentations, but few algorithms can offer this in various conditions and applications. In this paper, we propose a context dependent graph-based method for transition region extraction and thresholding. The graph-based approach is introduced into image thresholding, and context dependent graph is constructed from a given image, which can adaptively extract the pixel context and shape information because of the scalable neighborhood. Then an edge weight function is defined as the measure of possible transition pixels, and a robust fully automatic scheme for the optimal threshold is also presented. The proposed approach is validated both quantitatively and qualitatively. Compared with the traditional state-of-art algorithms on synthetic and real images, as well as laser cladding images, the experimental results suggest that the new proposal is efficient and effective.     

A microfluidic spiral for size-dependent fractionation of magnetic microspheres

Magnetic microspheres (MMS) are useful tools for a variety of medical and pharmaceutical applications. Typically, commercially manufactured MMS exhibit broad size distributions. This polydispersity is problematic for many applications. Since the direct synthesis of monodisperse MMS is often fraught with technical challenges, there is considerable interest in and need associated with the development of techniques for size-dependent fractionation of MMS. In this study we demonstrated continuous size-dependent fractionation of sub-micron scale particles driven by secondary (Dean effect) flows in curved microfluidic channels. Our goal was to demonstrate that such techniques can be applied to MMS containing superparamagnetic nanoparticles. To achieve this goal, we developed and tested a microfluidic chip for continuous MMS fractionation. Our data address two key areas. First, the densities of MMS are typically in the range 1.5–2.5 g/cm³, and thus they tend be non-neutrally buoyant. Our data demonstrate that efficient size-dependent fractionation of MMS entrained in water (density 1 g/cm³) is possible and is not significantly influenced by the density mismatch. In this context we show that a mixture comprising two different monodisperse MMS components can be separated into its constituent parts with 100% and 88% success for the larger and smaller particles, respectively. Similarly, we show that a suspension of polydisperse MMS can be separated into streams containing particles with different mean diameters. Second, our data demonstrate that efficient size-dependent fractionation of MMS is not impeded by magnetic interactions between particles, even under application of homogeneous magnetic fields as large as 35 kA/m. The chip is thus suitable for the separation of different particle fractions in a continuous process and the size fractions can be chosen simply by adjusting the flow velocity of the carrier fluid. These facts open the door to size dependent fractionation of MMS.     

Mixed Valence Tungsten (IV,V) Compounds with Layered Structures (Part IV): Syntheses,Crystal Structures,and Conductivity of BiX2[W2O2X6] (X = Cl,Br)

The reaction of metallic bismuth with either tungsten tetrachlorideoxide WOCl₄ at 650 K or tungsten tetrabromideoxide WOBr₄ at 670 K, respectively, leads to BiX₂[W₂O₂X₆] (X = Cl, Br) as black, lustrous crystal needles. The crystal structure determinations (triclinic, P$\bar{1}$ ) show the two isotypic structures to be closely related to Hg_0.55[W₂O₂Cl₆] with the presence of 1D‐polymeric W₂O₂X₆ double strands. Dinuclear [Bi₂X₄]²⁺ cations are embedded in the host structure via secondary W–X ··· Bi bonds. Unlike the other members of theM_y[W₂O₂X₆] structure family, which crystallize monoclinic and show crystallographic equivalent tungsten atoms, BiX₂[W₂O₂X₆] has independent tungsten sites. Nevertheless, an assignment of an individual oxidation state to the tungsten atoms within the W₂ group (W–W 2.8321(4) Å for X = Cl, 2.8985(4) Å for X = Br) is not possible and a dynamic intervalent state W(IV, V) is assumed. Electrical conductivity measurements for BiCl₂[W₂O₂Cl₆] show semi‐conductive behavior with a very small band gap of 70 meV and a high conductivity of around 0.5 Ω^–1cm^–1 at temperatures above 220 K. A temperature dependence of the activation energy of charge transport is present and interpreted by the Varshni model.