Liapunov functions and boundedness and global existence of solutions

In this paper the properties of a Liapunov function and its derivatives are combined in obtaining sufficient conditions for the boundedness and global existence of solutions of the system of differential equations

x=f(t,x) (1)

where f: [0,∞)XRⁿ→ Rⁿ is continuous. The following terms are introduced: (i) a scalar function which is mildly unbounded relative to a subset of Rⁿ,(ii) a scalar function which is radially unbounded relative to a set, (iii) a Liapunov function with mildly negative definite derivative in a set, (iv) a Liapunov function with strongly negative definite derivative in a set, and (v) a set which is unbounded-departing with respect to system (1). As a special case the autonomous system of differential equations which corresponds to (1) is considered and the semi-invariance property of the positive limiting set of a solution is used in the results     

A square‐planar hydrated cationic tetrakis(methimazole)gold(III) complex

The cationic pseudo‐square‐planar complex tetrakis(1‐methyl‐2,3‐dihydro‐1H‐imidazole‐2‐thione‐κS)gold(III) trichloride sesquihydrate, [Au(C₄H₆N₂S)₄]Cl₃·1.5H₂O, was isolated as dark‐red crystals from the reaction of chloroauric acid trihydrate (HAuCl₄·3H₂O) with four equivalents of methimazole in methanol. The Au^III atoms reside at the corners of the unit cell on an inversion center and are bound by the S atoms of four methimazole ligands in a planar arrangement, with S—Au—S bond angles of approximately 90°.     

Morphology of a phase-separated and a molecular composite PBT/ABPBI polymer blend

The structure and morphology of homopolymers and blends of rigid-rod poly(p-phenylene benzobisthiazole) (PBT) and semiflexible coil poly[2,5(6)benzimidazole] (ABPBI) were examined by wide-angle x-ray diffraction and scanning and transmission electron microscopy. When samples were processed from a solution where the total polymer concentration of 30% PBT/70% ABPBI blend was greater than a critical concentration, large-scale phase separation occurred and 0.1–4 μm ellipsoidal particles were present in a ductile matrix. The ellipsoids were chiefly composed of aggregates of well-oriented 10-nm PBT crystallites, while the matrix material was chiefly ABPBI. When the concentration was less than a critical concentration, the solution was optically homogeneous. In processing of fiber and film samples from the homogeneous solution, large-scale phase separation was inhibited by rapid coagulation in a water bath. After heat treatment, these samples were found to contain crystallites of both PBT and ABPBI with lateral dimensions of ordered regions no larger than 3 nm. The PBT homopolymer was dispersed in the matrix at the molecular level in ordered regions at a scale no larger than 3 nm, resulting in a rigid-rod molecular composite. In the rigid-rod molecular composite fiber both the molecular-level dispersion and high orientation contributed to higher values of strength and modulus compared to the properties of a phase-separated fiber. The strength and modulus of highly oriented fiber were only 25% higher than those of planar isotropically oriented film, suggesting that the level of dispersion of rod molecules is more important than orientation of the reinforcing phase in rigid-rod molecular composites.