Conformable variational iteration method,conformable fractional reduced differential transform method and conformable homotopy analysis method for non-linear fractional partial differential equations

In this paper, we introduce conformable variational iteration method (C-VIM), conformable fractional reduced differential transform method (CFRDTM) and conformable homotopy analysis method (C-HAM). Between these methods, the C-VIM is introduced for the first time for fractional partial differential equations (FPDEs). These methods are new versions of well-known VIM, RDTM and HAM. In addition, above-mentioned techniques are based on new defined conformable fractional derivative to solve linear and non-linear conformable FPDEs. Firstly, we present some basic definitions and general algorithm for proposal methods to solve linear and non-linear FPDEs. Secondly, to understand better, the presented new methods are supported by some examples. Finally, the obtained results are illustrated by the aid of graphics and the tables. The applications show that these new techniques C-VIM, CFRDTM and C-HAM are extremely reliable and highly accurate and it provides a significant improvement in solving linear and non-linear FPDEs.     

Triply-Logarithmic Parallel Upper and Lower Bounds for Minimum and Range Minima over Small Domains

We consider the problem of computing the minimum ofnvalues, and several well-known generalizations [prefix minima, range minima, and all nearest smaller values (ANSV)] for input elements drawn from the integer domain [1···s], wheres ≥ n. In this article we give simple and efficient algorithms for all of the preceding problems. These algorithms all takeO(log log log s) time using an optimal number of processors andO(ns^ε) space (for constant ε < 1) on the COMMON CRCW PRAM. The best known upper bounds for the range minima and ANSV problems were previouslyO(log log n) (using algorithms for unbounded domains). For the prefix minima and for the minimum problems, the improvement is with regard to the model of computation. We also prove a lower bound of Ω(log log n) for domain sizes = 2^{Ω(log n log log n)}. Since, forsat the lower end of this range, log log n = Ω(log log log s), this demonstrates that any algorithm running ino(log log log s) time must restrict the range ofson which it works.     

On the generation of non-continuous semigroups

A generalization of a part of a theorem of W. Arendt about integrated semigroups is given, and a generation theorem for non-continuous semigroups is presented.     

Liquid crystalline mesophases of pluronics (L64, P65, and P123) and transition metal nitrate salts ([M(H2O)6](NO3)2)

The triblock poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) copolymers, Pluronics (L64, P65, and P123), form liquid crystalline (LC) mesophases with transition metal nitrate salts (TMS), [M(H(2)O)(n)](NO(3))(2), in the presence and absence of free water in the media. In this assembly process, M-OH(2) plays an important role as observed in a TMS:C(n)EO(m) (C(n)EO(m) is oligo(ethylene oxide) nonionic surfactants) system. The structure of the LC mesophases and interactions of the metal ion-nitrate ion and metal ion-Pluronic were investigated using microscopy (POM), diffraction (XRD), and spectroscopy (FTIR and micro-Raman) techniques. The TMS:L64 system requires a shear force for mesophase ordering to be observed using X-ray diffraction. However, TMS:P65 and TMS:P123 form well structured LC mesophases. Depending on the salt/Pluronic mole ratio, hexagonal LC mesophases are observed in the TMS:P65 systems and cubic and tetragonal LC mesophases in the TMS:P123 systems. The LC mesophase in the water/salt/Pluronic system is sensitive to the concentration of free (H(2)O) and coordinated water (M-OH(2)) molecules and demonstrates structural changes. As the free water is evaporated from the H(2)O:TMS:Pluronic LC mesophase (ternary mixture), the nitrate ion remains free in the media. However, complete evaporation of the free water molecules enforces the coordination of the nitrate ion to the metal ion in all TMS:Pluronic systems.     

Bimolecular homolytic substitution (S(H)2) reactions with hydrogen atoms. time-resolved electron spin resonance detection in the pulse radiolysis of alpha-(methylthio)acetamide

Pulse radiolysis of aqueous solutions of alpha-(methylthio)acetamide produced unexpectedly large quantities of acetamide radicals that were identified by time-resolved electron spin resonance (TRESR) spectroscopy. The pH dependence of the TRESR-measured radical yields, results from selective scavenging reactions, and density functional theory predictions of the reaction thermochemistry prove that bimolecular homolytic substitution, S(H)2, of the acetamide radical fragment by a H atom is the most likely formation pathway.     

Synthesis of arylated anthraquinones by site-selective Suzuki–Miyaura reactions of the bis(triflates) of 1,3-di(hydroxy)anthraquinones

Arylated anthraquinones were prepared by Suzuki–Miyaura reactions of the bis(triflates) of various 1,3-(dihydroxy)anthraquinones. While the reactions of the bis(triflates) of parent 1,3-(dihydroxy)anthraquinone and of 2-chloro-1,3-di(hydroxy)anthraquinone proceeded with very good site-selectivity, the corresponding reactions of the bis(triflate) of 2-fluoro-1,3-diarylanthraquinones were not site-selective, which was explained based on the π-donating effect of the fluorine atom.