No abstract

The synthesis and spectroscopic data of (E)-2-{4-[3-(thiophen-3-yl)acryloyl]phenoxy}acetic acid are described. Crystallization from an ethanol–water mixture resulted in the title compound, C30H23KO8S2 or [K(C15H11O4S)(C15H12O4S)] n , containing one molecule of the acid and one molecule of the potassium salt in the asymmetric unit. Both molecules share the H atom between their carboxyl groups and a potassium ion. The C=C bonds display an E configuration. The thiophene and phenyl rings in the two molecules are inclined by 43.3 (2) and 22.7 (2)°. The potassium ion is octahedrally coordinated by six O atoms. This distorted octahedron shares on opposite sides two oxygen atoms with inversion-related octahedra, resulting in chains of octahedra running in the [010] direction, which form ladder-like chains by C—H...π interactions. A Hirshfeld surface analysis indicates that the highest contributions to the surface contacts arise from interactions in which H atoms are involved, with the most important contribution being from H...H (31.6 and 31.9% for the two molecules) interactions.

This paper is devoted to studying non-commensurate fractional order planar systems. Our contributions are to derive sufficient conditions for the global attractivity of non-trivial solutions to fractional-order inhomogeneous linear planar systems and for the Mittag-Leffler stability of an equilibrium point to fractional order nonlinear planar systems. To achieve these goals, our approach is as follows. Firstly, based on Cauchy’s argument principle in complex analysis, we obtain various explicit sufficient conditions for the asymptotic stability of linear systems whose coefficient matrices are constant. Secondly, by using Hankel type contours, we derive some important estimates of special functions arising from a variation of constants formula of solutions to inhomogeneous linear systems. Then, by proposing carefully chosen weighted norms combined with the Banach fixed point theorem for appropriate Banach spaces, we get the desired conclusions. Finally, numerical examples are provided to illustrate the effect of the main theoretical results.