Hardy's Inequality for the fractional powers of Grushin operator

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

Structural and DFT Studies on the Polymorphism of a Cadmium(II) Dipicolinate Coordination Polymer

The coordination polymer [Cd₂(dipic)₂(H₂O)₃]_<i>n</i> was prepared by the reaction of cadmium­(II) chloride or bromide and dipicolinic acid (dipicH₂) at 60 °C under autogenous pressure. The <i>C</i>2/<i>c</i> polymorph (<b>1</b>) was almost exclusively obtained. However, a few crystals of the <i>P</i>2/<i>c</i> polymorph (<b>2</b>) were occasionally found in the mixture with the <i>C</i>2/<i>c</i> polymorph, thus making it a disappearing and concomitant polymorph. The polymeric chains in <b>1</b> 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 <b>2</b> 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 <b>1</b>, as shown by dispersion-corrected DFT, through π–π stacking between polymeric chains, making <b>1</b> the thermodynamically favored polymorph. Polymorph <b>1</b> was characterized by IR spectroscopy, PXRD analysis, and TGA and DSC methods. The DSC analysis did not show any sign of phase transition between <b>1</b> and <b>2</b>. This was also confirmed by variable temperature PXRD, since the pattern of <b>1</b> remained unchanged until the decomposition of <b>1</b>

Structural and DFT Studies on the Polymorphism of a Cadmium(II) Dipicolinate Coordination Polymer

The coordination polymer [Cd₂(dipic)₂(H₂O)₃]_<i>n</i> was prepared by the reaction of cadmium­(II) chloride or bromide and dipicolinic acid (dipicH₂) at 60 °C under autogenous pressure. The <i>C</i>2/<i>c</i> polymorph (<b>1</b>) was almost exclusively obtained. However, a few crystals of the <i>P</i>2/<i>c</i> polymorph (<b>2</b>) were occasionally found in the mixture with the <i>C</i>2/<i>c</i> polymorph, thus making it a disappearing and concomitant polymorph. The polymeric chains in <b>1</b> 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 <b>2</b> 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 <b>1</b>, as shown by dispersion-corrected DFT, through π–π stacking between polymeric chains, making <b>1</b> the thermodynamically favored polymorph. Polymorph <b>1</b> was characterized by IR spectroscopy, PXRD analysis, and TGA and DSC methods. The DSC analysis did not show any sign of phase transition between <b>1</b> and <b>2</b>. This was also confirmed by variable temperature PXRD, since the pattern of <b>1</b> remained unchanged until the decomposition of <b>1</b>

Single-Crystal-to-Single-Crystal Transformation of a Novel 2‑Fold Interpenetrated Cadmium-Organic Framework with Trimesate and 1,2-Bis(4-pyridyl)ethane into the Thermally Desolvated Form Which Exhibits Liquid and Gas Sorption Properties

A novel 2-fold interpenetrated, pillared, cadmium metal–organic framework, namely, [Cd­(HBTC)­BPE]_<i>n</i>·<i>n</i>DMF, has been synthesized using 1,3,5-benzenetricarboxylic acid and 1,2-bis­(4-pyridyl)­ethane (BPE). This compound has been desolvated and subjected to various liquids and gases for sorption studies. Structures of the as-synthesized (<b>1</b>), desolvated (<b>2</b>), and resolvated in benzene (<b>3</b>) have been determined by single-crystal X-ray diffraction analysis and further characterized by elemental analysis, IR spectra, and thermogravimetric/differential scanning calorimetry analysis. Single crystal X-ray analysis revealed a 2-fold interpenetrated, three-dimensional (3D) framework which exhibits a 3,5-connected network with the Schläfli symbol of [(6³)­(6⁹.8) and <i>hms</i> topology. Compound <b>1</b> exhibits a temperature-induced single-crystal-to-single-crystal (SC–SC) transformation upon the release of <i>N</i>,<i>N</i>′-dimethylformamide molecules forming compound <b>2</b> (stable up to 300 °C). SC–SC transformation is also observed when it is immersed in benzene, chloroform, 1,4-dioxane, and tetrahydrofuran. The uptake of different solvent molecules was analyzed, and desolvated samples selectively adsorb benzene, chloroform, 1,4-dioxane, and THF molecules over other selected polar solvents. Gas (N₂, CO₂, and N₂O) sorption experiments were also performed and the structure showed 2.5% N₂, 4.5% CO₂, and 3.4% N₂O absorption by mass at room temperature and moderate gas pressures (∼10 bar)

Single-Crystal-to-Single-Crystal Transformation of a Novel 2‑Fold Interpenetrated Cadmium-Organic Framework with Trimesate and 1,2-Bis(4-pyridyl)ethane into the Thermally Desolvated Form Which Exhibits Liquid and Gas Sorption Properties

A novel 2-fold interpenetrated, pillared, cadmium metal–organic framework, namely, [Cd­(HBTC)­BPE]_<i>n</i>·<i>n</i>DMF, has been synthesized using 1,3,5-benzenetricarboxylic acid and 1,2-bis­(4-pyridyl)­ethane (BPE). This compound has been desolvated and subjected to various liquids and gases for sorption studies. Structures of the as-synthesized (<b>1</b>), desolvated (<b>2</b>), and resolvated in benzene (<b>3</b>) have been determined by single-crystal X-ray diffraction analysis and further characterized by elemental analysis, IR spectra, and thermogravimetric/differential scanning calorimetry analysis. Single crystal X-ray analysis revealed a 2-fold interpenetrated, three-dimensional (3D) framework which exhibits a 3,5-connected network with the Schläfli symbol of [(6³)­(6⁹.8) and <i>hms</i> topology. Compound <b>1</b> exhibits a temperature-induced single-crystal-to-single-crystal (SC–SC) transformation upon the release of <i>N</i>,<i>N</i>′-dimethylformamide molecules forming compound <b>2</b> (stable up to 300 °C). SC–SC transformation is also observed when it is immersed in benzene, chloroform, 1,4-dioxane, and tetrahydrofuran. The uptake of different solvent molecules was analyzed, and desolvated samples selectively adsorb benzene, chloroform, 1,4-dioxane, and THF molecules over other selected polar solvents. Gas (N₂, CO₂, and N₂O) sorption experiments were also performed and the structure showed 2.5% N₂, 4.5% CO₂, and 3.4% N₂O absorption by mass at room temperature and moderate gas pressures (∼10 bar)