We prove Hardy's inequality for the fractional powers of the generalized
sublaplacian and the fractional powers of the Grushin operator. We also find an
integral representation and a ground state representation for the fractional
powers of generalized sublaplacian

Boris-Marko Kukovec (2131279)

Clive L. Oliver (1974418)

Gerhard
A. Venter (2131276)

English

arXiv

The coordination polymer [Cd2(dipic)2(H2O)3]n was prepared
by the reaction of cadmium­(II) chloride or bromide and dipicolinic
acid (dipicH2) at 60 °C under autogenous pressure.
The C2/c polymorph (1) was almost exclusively obtained. However, a few crystals of the P2/c polymorph (2) were occasionally
found in the mixture with the C2/c polymorph, thus making it a disappearing and concomitant polymorph.
The polymeric chains in 1 are connected into dimers by
π–π interaction and O–H···O
hydrogen bonds. These dimers are in turn connected by intermolecular
O–H···O hydrogen bonds into a 2D network. The
polymeric chains in 2 are connected by intermolecular
O–H···O hydrogen bonds into a zigzag chain along
the [001] direction. According to DFT calculations, the hydrogen bonding
is of similar order in both polymorphs (∼7.5 kcal mol–1 per hydrogen bond). However, there is additional stability imparted in 1, as shown by dispersion-corrected DFT, through π–π stacking between polymeric chains, making 1 the thermodynamically favored polymorph. Polymorph 1 was characterized by IR spectroscopy, PXRD analysis, and
TGA and DSC methods. The DSC analysis did not show any sign of phase
transition between 1 and 2. This was also
confirmed by variable temperature PXRD, since the pattern of 1 remained unchanged until the decomposition of 1

Hardy's Inequality for the fractional powers of Grushin operator

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