Quasi static growth of brittle cracks in a linearly elastic flexural plate

In this paper we propose a variational model for the irreversible quasi static growth in brittle fractures for a linearly elastic homogeneous isotropic plate, subject to a time dependent vertical displacement on a part of its lateral surface. The model is based on the Griffith’s criterion for crack growth and is inspired to the model proposed in [11] by Francfort and Marigo in the case of 3-D elasticity. We give a precise mathematical formulation of the model and in this framework we prove an existence result. Mathematics Subject Classification (2000) 74K20, 35J35, 74G65, 35R35, 35A35, 74R10     

由环的幂零性所确定的根

结合环中的环的幂零性不是根性质。为此，本文将结合环中的幂零理想概念扩展为次拟幂零理想和拟幂零理想，定义次拟幂零根ＳＮ和拟幂零根ＱＮ，证明它们均为Ａｍｉｔｓｕｒ－Ｋｕｒｏｓｈ根，且二者相等，进一步，我们给出了ＱＮ－半单环的构造命题和ＱＮ－根的模刻划。     

Nonlinear self-adjointness through differential substitutions

It is known (Ibragimov, 2011; Galiakberova and Ibragimov, 2013) [14,18] that the property of nonlinear self-adjointness allows to associate conservation laws of the equations under study, with their symmetries. In this paper we show that, even when the equation is nonlinearly self-adjoint with a non differential substitution, finding the explicit form of the differential substitution can provide new conservation laws associated to its symmetries. By using the general theorem on conservation laws (Ibragimov, 2007) [11] and the property of nonlinear self-adjointness we find some new conservation laws for the modified Harry-Dym equation. By using a differential substitution we construct a conservation law for the Harry-Dym equation, which has not been derived before using Ibragimov method.     

Hydrothermal Synthesis and Characterization of Quasi‐1‐D Tungsten Disulfide Nanocrystal

Quasi‐1‐D (one‐dimensional) tungsten disulfide (WS₂) nanocrystal was synthesized through a two‐step hydrothermal process. Energy dispersive spectroscopy (EDS) identified that the chemical composition of the final product was WS₂. The produced nanocrystal was further characterized with X‐ray diffraction (XRD), transmission electron microscopy (TEM), and scanning electron microscopy (SEM). The results have shown that most of the products are quasi‐1‐D nanocrystals with the width around 140 nanometers and thickness about 30 nanometers. However, the length of the majority of nanocrystals could reach 1 micron. The selected area electron diffraction (SAED) pattern indicates that the nanocrystal is of a single crystal. N₂ adsorption measurement revealed that the BET specific surface area of this sample is 97 m²g^?1, which indicates that the as‐prepared product has better catalyzing and friction performance.