Syntheses and Properties of New Layered Alkali-Metal/Ammonium Vanadium(V) Methylphosphonates: M(VO(2))(3)(PO(3)CH(3))(2) (M = NH(4), K, Rb, Tl). Single-Crystal Structures of K(VO(2))(3)(PO(3)CH(3))(2) and NH(4)(VO(2))(3)(PO(3)CH(3))(2)

The hydrothermal syntheses of a family of new alkali-metal/ammonium vanadium(V) methylphosphonates, M(VO(2))(3)(PO(3)CH(3))(2) (M = K, NH(4), Rb, Tl), are described. The crystal structures of K(VO(2))(3)(PO(3)CH(3))(2) and NH(4)(VO(2))(3)(PO(3)CH(3))(2) have been determined from single-crystal X-ray data. Crystal data: K(VO(2))(3)(PO(3)CH(3))(2), M(r) = 475.93, trigonal, R32 (No. 155), a = 7.139(3) ?, c = 19.109(5) ?, Z = 3; NH(4)(VO(2))(3)(PO(3)CH(3))(2), M(r) = 454.87, trigonal, R32 (No. 155), a = 7.150(3) ?, c = 19.459(5) ?, Z = 3. These isostructural, noncentrosymmetric phases are built up from hexagonal tungsten oxide (HTO) like sheets of vertex-sharing VO(6) octahedra, capped on both sides of the V/O sheets by PCH(3) entities (as [PO(3)CH(3)](2-) methylphosphonate groups). In both phases, the vanadium octahedra display a distinctive two short + two intermediate + two long V-O bond distance distribution within the VO(6) unit. Interlayer potassium or ammonium cations provide charge balance for the anionic (VO(2))(3)(PO(3)CH(3))(2) sheets. Powder X-ray, TGA, IR, and Raman data for these phases are reported and discussed. The structures of K(VO(2))(3)(PO(3)CH(3))(2) and NH(4)(VO(2))(3)(PO(3)CH(3))(2) are compared and contrasted with related layered phases based on the HTO motif.     

A Comparison of Two Variational Methods

A discussion is given of the relation between the maximum functionaltheorem and Schwinger's variational principle, and it is shownthat the bound given by Schwinger's principle is better thanor the same as that given by the maximum functional theorem.     

Further Gurtin-type Variational Principles for Linear Initial Value Problems

Following the variational principles for linear initial valueproblems associated with the wave and heat conduction equationsdiscussed by Gurtin and Leitmann and using time convolutionsit is shown that general variational principles exist for theseproblems with sources on the boundaries and within the regionunder consideration.     

Modified Locally One Dimensional Methods for Parabolic Partial Differential Equations in Two Space Variables

In this note we consider the locally one dimensional methodas discussed in Gourlay & Mitchell (1969, 1972) and indicatehow this scheme can be used to solve accurately a parabolicpartial differential equation in two space variables.