Dominated Compactness Theorem in Banach Function Spaces and its Applications

A famous dominated compactness theorem due to Krasnosel’skiĭ states that compactness of a regular linear integral operator in L ^p follows from that of a majorant operator. This theorem is extended to the case of the spaces , with variable exponent p(·), where we also admit power type weights . This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces is applied to fractional integral operators over bounded open sets. Submitted: June 6, 2007. Accepted: November 20, 2007.     

Stereospecific and stereoselective alkyl and silylcyclopropanation of alpha,beta-unsaturated amides

A novel chromium-promoted alkyl- and silyl cyclopropanation of (E)- or (Z)-alpha,beta-unsaturated amides in which the C-C double bond is di-, or trisubstituted is described. This process takes place with total stereospecificity, and the new stereogenic center is generated with high or total stereoselectivity. A mechanism is proposed to explain the cyclopropanation reaction.     

General, stereoselective synthesis of (Z)-beta,gamma-unsaturated nitriles promoted by samarium diiodide

A method to obtain (Z)-beta,gamma-unsaturated nitriles in high or good yields and with moderate or high stereoselectivity is described. The products were achieved through the photoinduced metalation of 3-acetoxy-4-chloronitriles with SmI2. The starting compounds were readily prepared, and a mechanism is proposed to explain this stereoselective beta-elimination reaction.     

Efficient synthesis of 2-(trifluoromethyl)nicotinic acid derivatives from simple fluorinated precursors

Novel routes to 2-trifluoromethyl-nicotinic acid derivatives have been developed involving synthesis of the pyridine ring. These pyridyl compounds serve as key intermediates in the manufacture of the recently discovered COMT inhibitor, 3-(5-(3,4-dihydroxy-5-nitrophenyl)-1,2,4-oxadiazol-3-yl)-2-(trifluoromethyl)pyridine 1-oxide.     

On the densities of cliques and independent sets in graphs

Let r; s≥2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph K _n is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r = 2 or s = 2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.     

A-identities for the {2 \times 2} matrix algebra

Let K be an algebraically closed field of characteristic zero. We prove that the minimum degree for A-identities of \({M_{2}(K)}\) is 6. We also construct an explicit A-polynomial of degree 6 and prove it is an A-identity.     

Amido/amine triazacyclononane-based zirconium complexes: syntheses, reactivity, and structures

The reactions of [Zr(NMe2)4]2 with triamido-triazacyclonane ligand precursors, {NH(Ph)SiMe2}3tacn (H3N3[9]N3) and {NH(C6H4F)SiMe2}3tacn (H3N3-F[9]N3), led to the formation of complexes [Zr(NMe2)2{N(Ph)SiMe2}2{NH(Ph) SiMe2}tacn], 1, and [Zr(NMe2)2{N(o-C6H4F)SiMe2}2{NH(o-C6H4F)SiMe2} tacn], 2, where the zirconium is coordinated to two remaining dimethylamido ligands and to a dianionic tacn-based ligand, [{N(Ph')SiMe2}2{NH(Ph')SiMe2}tacn]2-, that formed from deprotonation of two amine pendent arms of the ligands' precursors. The third pendent arm of H3N3[9]N3 and H3N3-F[9]N3 remains neutral and not bonded to the zirconium. Treatment of 1 with NaH led to the synthesis of [Zr(NMe2){N(Ph)SiMe2}2tacn], 3, that results from the cleavage of the N-Si bond of the original neutral pendent arm. Complexes [ZrCl{N(Ph')SiMe2}2tacn] (Ph' = C6H5, 4, and C6H4F, 5) have been obtained by reactions of ZrCl4 with {MN(Ph')SiMe2}3tacn.2THF (M = Li, Na). Reactions of 4 and 5 with LiC triple bond CPh led to the syntheses of [Zr(CCPh){N(Ph')SiMe2}2tacn] (Ph' = C6H5, 6, and C6H4F, 7). The solid-state structure of 3 shows a chiral metal center.     

New insight on the investigation of the role of water in the solid-state structures of potassium croconate, K(2)C(5)O(5).2H(2)O, and its anhydrate

This work presents a comparative study of dihydrated and anhydrous forms of potassium croconate crystals by vibrational spectroscopy, X-ray powder diffraction, and thermogravimetry. These compounds have different colors (dihydrated is orange, and dehydrated is yellow) due to the presence of coordinated water molecules. X-ray diffraction patterns show that the unit cell of the yellow compound is smaller than that of the orange analogue, suggesting that the croconate ion layers are more closely bonded in this salt. The loss of water is reversible due to the potassium cation size which is intermediate between small (Li+ and Na+) and large (Rb+ and Cs+) alkaline metal ions. However, the hydrated compound (orange) is more stable, and with a small quantity of water the yellow compound is quickly converted to the orange compound. A diagnostic feature of the Raman spectrum for the orange (hydrated) and yellow (anhydrous) analogues is the singlet at 1240 cm(-1) in the former, assigned to a nu(CC) + delta(CCC) + nu(CO) + beta(CO) mode of E'2 symmetry, which splits in the yellow form to a doublet at 1256 and 1232 cm(-1).     

The first cyclopropanation reaction of unmasked alpha,beta-unsaturated carboxylic acids: direct and complete stereospecific synthesis of cyclopropanecarboxylic acids promoted by Sm/CHI3

A samarium-promoted cyclopropanation of unmasked alpha,beta-unsaturated acids is described. This reaction can be carried out on (E)- or (Z)-alpha,beta-unsaturated carboxylic acids. In all cases the process is completely stereospecific and stereoselective. A mechanism has been proposed to explain the cyclopropanation reaction.     

Shannon-information entropy sum in the confined hydrogenic atom

The appearance of critical points in the Shannon entropy sum as a function of confinement radius, in ground and excited state confined hydrogenic systems, is discussed. We illustrate that the Coulomb potential in tandem with the hard sphere confinement are responsible for these points. The positions of these points are observed to vary with the intensity of the potential. The effects of the Coulomb potential on the system are further probed, by examining the differences between the densities of the confined atom and those of the particle confined in a spherical box, for the same confinement radius. These differences are quantified by using Kullback-Leibler and cumulative residual Kullback-Leibler distance measures from information theory. These measures detect that the effects of the Coulomb potential are squeezed out of the system as the confinement radius decreases. That is, the confined atom densities resemble the particle in a box ones, for smaller confinement radii. Furthermore, the critical points in the entropy sum lie in the same regions where there are changes in the distance measures, as the atom behaves more particle in a spherical box-like. The analysis is further complemented by examination of the derivative of the entropy sum with respect to confinement radius. This study illustrates the inhomogeneity in the magnitudes of the derivatives of the entropy sum components and their dependence on the Coulomb potential. A link between the derivative and the entropic force is also illustrated and discussed. Similar behaviors are observed when the virial ratio is compared to the entropic power one, as a function of confinement radius.