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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°.     

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