Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of orderr is applied to a smooth system of ordinary differential equations.Given any integer m 1, the method may be shown to be of orderr + m as an approximation to a certain modified equation. Ifthe method and the system have a particular qualitative propertythen it is important to determine whether the modified equationsinherit this property. In this article, a technique is introducedfor proving that the modified equations inherit qualitativeproperties from the method and the underlying system. The techniqueuses a straightforward contradiction argument applicable toarbitrary one-step methods and does not rely on the detailedstructure of associated power series expansions. Hence the conclusionsapply, but are not restricted, to the case of Runge-Kutte methods.The new approach unifies and extends results of this type thathave been derived by other means: results are presented forintegral preservation, reversibility, inheritance of fixed points.Hamiltonian problems and volume preservation. The techniquealso applies when the system has an integral that the methodpreserves not exactly, but to order greater than r. Finally,a negative result is obtained by considering a gradient systemand gradient numerical method possessing a global property thatis not shared by the associated modified equations.     

How Fast can a Subharmonic Function Decay Along a Ray?

Much is known about the connection between the growth and decayof subharmonic functions. The results indicate that there isa general principle: asubharmonic function cannot decay ‘toofast’ relative to its growth.Three theorems are provedwhich, together with work previously published elsewhere, givea fairly complete account of how this principle works out fora subharmonic function having extremal decay along a ray. 1991Mathematics Subject Classification: 30D20, 31A05.     

Conformationally rigid chelate rings in metal complexes of pyridyloxy-substituted 2,2'-dioxybiphenyl-cyclotetra- and cyclotriphosphazene platforms

Divalent metal halides react with pyridyloxy-substituted 2,2'-dioxybiphenyl-cyclotri- and cyclotetraphosphazene ligands to form the complexes, [MLX2] [M=Co or Cu; L=(2,2'-dioxybiphenyl)tetrakis(2-pyridyloxy)cyclotriphosphazene (L1) or (2,2'-dioxybiphenyl)tetrakis(4-methyl-2-pyridyloxy)cyclotriphosphazene (L2); X=Cl or Br], [ZnLCl2] [L=bis(2,2'-dioxybiphenyl)bis(2-pyridyloxy)cyclotriphosphazene (L3) or bis(2,2'-dioxybiphenyl)bis(4-methyl-2-pyridyloxy)cyclotriphosphazene (L4)], [MLCl2] [M=Cu or Hg; L=tris(2,2'-dioxybiphenyl)bis(2-pyridyloxy)cyclotetraphosphazene (L5)] and [Cu2LCl4] (L=trans-bis(2,2'-dioxybiphenyl)tetrakis(2-pyridyloxy)cyclotetraphosphazene (L6)]. Single-crystal X-ray structures show the L2 ligand complexes to have a N3Cl2 five-coordinate, trigonal-bipyramidal donor set with the phosphazene ring and pendant pyridyloxy nitrogens binding to the metal ions. The coordinated L2 ligand in the complex, [CoL2Cl2], slowly hydrolyses in acetonitrile with the loss of a pyridine pendant arm to form a dimetallic species, which has been characterized by crystallography as [{CoL2aCl}]2.4MeCN (L2a=[N3P3(biph)(OPy)3(O)]-, biph=2,2'-dioxybiphenyl, OPy=2-oxopyridine). The ligands, L3, L4, L5, and L6, bind to the metal halides via gem-substituted pyridyloxy nitrogens only. The resulting rigid eight-membered chelate rings all have distorted boat conformations, which force distorted-tetrahedral N2Cl2 coordination environments onto the metal ions. The spectroscopic (ESR and electronic) and magnetic properties of the complexes are reported.     

Analogy between unconventional superconductivity and unconventional states of liquid crystals

An analogy is stressed between the order-parameter symmetries of the two-dimensional d-pairing wave superconductors and of liquid-crystal mesophases formed from achiral bent-shaped molecules. It leads to a definition of a class of liquid-crystal states which are the analogs of the unconventional superconducting states, and are characterized by a loss of discrete symmetry operations of the parent state.