Quantum Correlations

This chapter analyzes the phenomenon of quantum mechanical correlations using the BST notions of transitions, propensities, and funny business. It considers two ways of understanding such correlations: First, as modal correlations (exhibited, e.g., in the GHZ setup) and, second, as probabilistic correlations (exhibited, e.g., in the Bell-Aspect setup). Having introduced the notion of structure extensions, it asks if it is possible to extend an initial BST structure harboring correlations (modal or probabilistic) in such a way that the resulting extended structure harbors no such correlations. An important distinction, rigorously stated in BST, is that between agents-induced indeterminism and Nature-induced indeterminism. A main result concerning structure extensions is that in the mentioned cases, the procedure is not possible unless the extended structure violates the initially given division between cases of agents-induced indeterminism and cases of Nature-induced indeterminism.

Intuitionistic fuzzy incidence graphs

Connectivity parameters have a crucial role in the study of different networks in the physical world. The notion of connectivity plays a key role in both theory and application of different graphs. In this article, a prime idea of connectivity concepts in intuitionistic fuzzy incidence graphs (IFIGs) with various examples is examined. IFIGs are essential in interconnection networks with influenced flows. Therefore, it is of paramount significance to inspect their connectivity characteristics. IFIGs is an extended structure of fuzzy incidence graphs (FIGs). Depending on the strength of a pair, this paper classifies three different types of pairs such as an α - strong, β - strong, and δ-pair. The benefit of this kind of stratification is that it helps to comprehend the fundamental structure of an IFIG thoroughly. The existence of a strong intuitionistic fuzzy incidence path among vertex, edge, and pair of an IFIG is established. Intuitionistic fuzzy incidence cut pairs (IFICPs) and intuitionistic fuzzy incidence trees (IFIT) are characterized using the idea of strong pairs (SPs). Complete IFIG is defined, and various other structural properties of IFIGs are also investigated. The proof that complete IFIG does not contain any δ-pair is also provided. A real-life application of these concepts related to the network of different computers is also provided.

Crystal structures of the gold NHC complex bis(4-bromo-1,3-diethylimidazol-2-ylidene)gold(I) iodide and its 1:1 adduct with trans-bis(4-bromo-1,3-diethyl-imidazol-2-ylidene)diiodidogold(III) iodide

The first title compound, [Au(C7H11BrN2)2]I, crystallizes in the space group P\overline{1} without imposed symmetry. The cations and anions are linked to form chains by Br...I...Br halogen-bond linkages. The second title compound, [Au(C7H11BrN2)2][AuI2(C7H11BrN2)2]I2, is an adduct of the first and its formally I2-oxidized AuIII analogue. It also crystallizes in space group P\overline{1}, whereby both gold atoms occupy inversion centres. The extended structure is a reticular layer involving Br...I...Br and I...I...Au linkages.

N,N′-Dicyclohexyl-N-(phthaloylglycyl)urea

The molecular structure of the title compound {systematic name: 1,3-dicyclohexyl-1-[2-(1,3-dioxo-2,3-dihydro-1H-isoindol-2-yl)acetyl]urea}, C23H29N3O4, derived from N,N′-dicyclohexylurea, shows that the tertiary N atom substituted by a cyclohexyl and phthaloylglycyl groups adopts a perfectly planar geometry (bond-angle sum = 360.0°). In the same way as for N,N′-dicyclohexylurea, the extended structure of the title compound features N—H...O hydrogen bonds, which generate chains of molecules running in the [001] direction.

Methyl 2-(3-chloro-2-methylanilino)pyridine-3-carboxylate

The title compound, C14H13ClN2O2, was obtained during an attempt to grow single crystals of 4-acetylphenyl 2-[(3-chloro-2-methylphenyl)amino]nicotinate in methanol, and was probably generated by alcoholysis. Two intramolecular hydrogen bonds are formed, one between the N—H group and the carbonyl O atom of the ester and the other between the ortho sp 2CH group of the benzene ring and the pyridine N atom. Aromatic π–π stacking [shortest centroid–centroid separation = 3.598 (2) Å] is observed in the extended structure.

Crystal structure and spectroscopic properties of (E)-1,3-dimethyl-2-[3-(4-nitrophenyl)triaz-2-enylidene]-2,3-dihydro-1H-imidazole

The title compound (E)-1,3-dimethyl-2-[3-(4-nitrophenyl)triaz-2-enylidene]-2,3-dihydro-1H-imidazole, C11H12N6O2, has monoclinic (C2/c) symmetry at 100 K. This triazene derivative was synthesized by the coupling reaction of 1,3-dimethylimidazolium iodide with 1-azido-4-nitro benzene in the presence of sodium hydride (60% in mineral oil) and characterized by 1H NMR, 13C NMR, IR, mass spectrometry, and single-crystal X-ray diffraction. The molecule consists of six-membered and five-membered rings, which are connected by a triazene moiety (–N=N—N–). In the solid-state, the molecule is found to be planar due to conjugation throughout the molecule. The extended structure shows two layers of molecules, which present weak intermolecular interactions that facilitate the stacked arrangement of the molecules forming the extended structure. Furthermore, there are several weak pseudo-cyclical interactions between the nitro oxygen atoms and symmetry-adjacent H atoms, which help to arrange the molecules.

The crystal structure of the decaaluminum alkoxide cluster Al10O4(OH)8 L 14 (L = 1,1,1,3,3,3-hexafluoropropan-2-olate)

In the title centrosymmetric cluster compound, hexakis(μ2-1,1,1,3,3,3-hexafluoropropan-2-olato)octakis(1,1,1,3,3,3-hexafluoropropan-2-olato)octa-μ2-hydroxido-di-μ4-oxido-di-μ3-oxido-decaaluminium, [Al10(C3HF6O)14(OH)8O4] (C42H22Al10F84O26), there is a central μ4-OAl4 moiety, which has six edges of which three contain μ(O)-1,1,1,3,3,3-hexafluoropropan-2-olate (L) ligands and two contain μ-OH groups each bridging two Al atoms along an edge. The sixth edge is occupied by a group containing a fifth aluminium atom [bis-μ(OH)-, μ3(O)—AlL]. This last μ3(O) group generates a centrosymmetric Al2O2 dimer, thus the μ3(O) atom is linked to two Al atoms in the asymmetric unit as well as a third Al atom through a center of inversion. Three of the hexafluoropropyl groups of the C3HF6O− ligands are disordered and each was refined with two conformations with occupancies of 0.770 (3)/0.230 (3), 0.772 (3)/0.228 (3) and 0.775 (3)/0.225 (3). The five unique Al centers have coordination numbers varying from four to six with bond angles that show considerable distortions from regular geometry: for the four-coordinate atom, τ4′ = 0.886, while three Al atoms are five-coordinate (τ5 values = 0.098, 1.028, and 0.338) and one is distorted six-coordinate with O—Al—O bond angles ranging from 74.22 (9) to 171.59 (12)°. The geometry about the central O atom in the OAl4 block is significantly distorted tetrahedral (τ4′ = 0.630) with Al—O—Al angles ranging from 95.50 (9) to 147.74 (13)°. The extended structure features numerous O—H...O, O—H...F, C—H...O and C—H...F hydrogen bonds and short F...F contacts.