On the impulsive implicit Ψ-Hilfer fractional differential equations with delay

In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam-Hyers-Mittag-Leffler stability results for impulsive implicit Ψ-Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam-Hyers and generalized Ulam-Hyers stability are the specific cases of Ulam-Hyers-Mittag-Leffler stability. Extended version of the Gronwall inequality, abstract Gronwall lemma, and Picard operator theory are the primary devices in our investigation. We provide an example to illustrate the obtained results.     

Simple Equations Method (SEsM): Algorithm,Connection with Hirota Method,Inverse Scattering Transform Method,and Several Other Methods

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.     

雾化辅助化学气相沉积法氧化镓薄膜生长研究

采用自主设计搭建的雾化辅助化学气相沉积系统设备,开展了Ga₂O₃薄膜制备及其特性研究工作。通过X射线衍射研究了沉积温度、系统沉积压差对Ga₂O₃薄膜结晶质量的影响。结果表明,Ga₂O₃在425~650 ℃温度区间存在物相转换关系。随着沉积温度从425 ℃升高至650 ℃,薄膜结晶分别由非晶态、纯α-Ga₂O₃结晶状态向α-Ga₂O₃、β-Ga₂O₃两相混合结晶状态改变。通过原子力显微镜表征探究了生长温度对Ga₂O₃薄膜表面形貌的影响,从475 ℃升高至650 ℃时,薄膜表面粗糙度由26.8 nm下降至24.8 nm。同时,高分辨X射线衍射仪测试表明475 ℃、5 Pa压差条件下的α-Ga₂O₃薄膜样品半峰全宽仅为190.8″,为高度结晶态的单晶α-Ga₂O₃薄膜材料。     

Diblock copolymers of polystyrene‐b‐poly(1,3‐cyclohexadiene) exhibiting unique three‐phase microdomain morphologies

The synthesis and molecular characterization of a series of conformationally asymmetric polystyrene‐block‐poly(1,3‐cyclohexadiene) (PS‐b‐PCHD) diblock copolymers (PCHD: ～90% 1,4 and ～10% 1,2), by sequential anionic copolymerization high vacuum techniques, is reported. A wide range of volume fractions (0.27 ≤ ?_PS ≤ 0.91) was studied by transmission electron microscopy and small‐angle X‐ray scattering in order to explore in detail the microphase separation behavior of these flexible/semiflexible diblock copolymers. Unusual morphologies, consisting of PCHD core(PCHD‐1,4)–shell(PCHD‐1,2) cylinders in PS matrix and three‐phase (PS, PCHD‐1,4, PCHD‐1,2) four‐layer lamellae, were observed suggesting that the chain stiffness of the PCHD block and the strong dependence of the interaction parameter χ on the PCHD microstructures are important factors for the formation of this unusual microphase separation behavior in PS‐b‐PCHD diblock copolymers. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 1564–1572     

Probing the early stages of thermal fractionation by successive self‐nucleation and annealing performed with fast scanning chip‐calorimetry

The thermal fractionation kinetics of a linear low‐density polyethylene (LLDPE) during Successive Self‐Nucleation and Annealing (SSA) is investigated by fast scanning chip‐calorimetry (FSC), by systematically varying the holding times (t_s) at each fractionation temperature (T_s). The range of explored fractionation times spans four orders of magnitude, from 0.001 to 10 s. Discernible thermal fractions are already detected in the very early stages of the process, at t_s shorter than one second. As t_s increases, the melting endotherm after SSA indicates a progressive lamellar thickening and narrowing of the thicknesses distribution of the various crystalline fractions. The largest variations are observed for the families of crystals containing the longest crystallizable sequences, which also undergo a change of their relative content as a consequence of self‐nucleated crystallization at T_s. The quality of the thermal fractionation obtained in 10 seconds with FSC is equivalent to that of conventional differential scanning calorimetry SSA (t_s = 300 s). © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 2200–2209     

Rethinking pedagogy for second-order differential equations: a simplified approach to understanding well-posed problems

Knowing an equation has a unique solution is important from both a modelling and theoretical point of view. For over 70 years, the approach to learning and teaching ‘well posedness’ of initial value problems (IVPs) for second- and higher-order ordinary differential equations has involved transforming the problem and its analysis to a first-order system of equations. We show that this excursion is unnecessary and present a direct approach regarding second- and higher-order problems that does not require an understanding of systems.     

Applications of ‐invariant distributions

In this sequel to the article 17 , a criterion for the regularity of fundamental solutions of differential operators with positive symbol is proved in analogy to Hörmander's criteria, see 9 , 12 . It implies that the temperate fundamental solution of the operator is not regular. Furthermore, a formula for the convolution of two ‐invariant distributions is presented, and, finally, L. Schwartz' question on the surjectivity of linear partial differential operators with constant coefficients on the space is completely answered in the case of ‐invariant differential operators.