One can have noninteger topological charge

It is argued that the recently found fractionally charged self-dual solution is relevant.     

The Critical Point of k-Clique Percolation in the Erdős–Rényi Graph

Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erdős–Rényi graph. When the probability p of two nodes being connected is above a certain threshold p _c (k), the complete subgraphs of size k (the k-cliques) are organized into a giant cluster. By making some assumptions that are expected to be valid below the threshold, we determine the average size of the k-clique percolation clusters, using a generating function formalism. From the divergence of this average size we then derive an analytic expression for the critical linking probability p _c (k).     

Characterization, conductivity and photocatalytic studies of AHfM(PO4)3 (A = Na and Ag; M = Ti and Zr) powders synthesized by sol–gel method

Materials belonging to NASICON family of compositions NaHfM(PO₄)₃ and AgHfM(PO₄)₃ (M = Ti and Zr) are prepared by sol–gel and ion exchange methods, respectively. Ethylene glycol is used as a gelating agent. All the compositions are characterized by powder X-ray diffraction, Fourier transform infrared spectroscopy, ³¹P MAS NMR, UV–Vis DRS, XPS and energy dispersive spectral methods. All these phosphates are crystallized in rhombohedral lattice with space group $R\overline{3} c$ R 3 ¯ c . These compounds exhibit characteristic PO₄ vibrational modes in their FT-IR spectra. The ³¹P MAS NMR gave broad signals indicating distribution of chemical environments around P ion. The dc and ac conductivity of AgHfM(PO₄)₃ (M = Ti and Zr) are higher compared to their sodium containing compounds. The Cole–Cole plots of impedance show semicircles between 373 and 623 K. The variation of dc conductivity with temperature follows the Arrhenius equation. The photocatalytic activity of all the samples was studied against methylene blue decomposition using sun light. AgHfM(PO₄)₃ (M = Ti and Zr) have shown higher photoactivity than the sodium containing Nasicons.     

CORRELATION OF DISPERSION STABILITY WITH SURFACTANT CONCENTRATION AND ABRASIVE PARTICLE SIZE FOR CHEMICAL MECHANICAL POLISHING (CMP) SLURRIES

The preparation of stable colloidal slurries is often difficult in industries where many chemical components are added into the slurries. A critically acclaimed example of such an industry is the chemical mechanical polishing (CMP) industry which involves polishing slurries with several chemical additives. In the present work, the stabilization of a slurry used for CMP of metals is investigated in detail. This high ionic strength slurry has been stabilized using an optimaJ combination of sodium dodecyl sulfate (anionic surfactant) and Tween 80 (nonionic surfactant). The amount of surfactant needed to impart stability has been investigated in this study for two different sizes of abrasive particles. It has been found that the amount of surfactant needed to stabilize the slurry increases as the total surface area per gram of panicles increases. Slurry stabilization has been correlated with particle size measurements. It has been found that the average panicle size of the slurry decreases as the stability of the slurry increases. Stable slurries have been found to have particle sizes close to those of the particles before agglomeration. It is proposed that the stabilized CMP slurries can lead to reduced defects in wafers by preventing agglomeration of panicles.