A Generalization of the Hausdorff-Young Theorem

Considering mixed-norm sequence spaces lp,q, p, q 1, C. N. Kellogg proved the following theorem: if 1 < p 2 then lp,2 and lp,2 , where 1/p + 1/p = 1. This result extends the Hausdorff-Young Theorem.We introduce here multiple mixed-norm sequence spaces , examine their properties and characterize the multipliers of spaces of the form lp,[s;n],q, with the index s repeated n times. By an interpolation-type argument we prove that (l,[2;n],2, lp,[1;n],1) for 1 < p 2. Using these results we obtain a further generalization of the Hausdorff-Young Theorem: if 1 < p 2 then lp,[2;n] and lp,[2;n] for each n = 0, 1, 2, ¨. The spaces lp,[2;n] decrease and lp,[2;n] increase properly with n for 1 < p < 2 and 1/p + 1/p = 1. We also extend a theorem of J. H. Hedlund on multiplers of Hardy spaces and deduce other results.     

Classification of auxin plant hormones by interaction property similarity indices

Although auxins were the first type of plant hormone to be identified, little is known about the molecular mechanism of this important class of plant hormones. We present a classification of a set of about 50 compounds with measured auxin activities, according to their interaction properties. Four classes of compounds were defined: strongly active, weakly active with weak antiauxin behaviour, inactive and inhibitory. All compounds were modeled in two low-energy conformations, P and T, so as to obtain the best match to the planar and tilted conformations, respectively, of indole 3-acetic acid. Each set of conformers was superimposed separately using several different alignment schemes. Molecular interaction energy fields were computed for each molecule with five different chemical probes and then compared by computing similarity indices. Similarity analysis showed that the classes are on average distinguishable, with better differentiation achieved for the T conformers than the P conformers. This indicates that the T conformation might be the active one. Further, a screening was developed which could distinguish compounds with auxin activity from inactive compounds and most antiauxins using the T conformers. The classifications rationalize ambiguities in activity data found in the literature and should be of value in predicting the activities of new plant growth substances and herbicides.     

BMO-boundedness of Lusin's area function

We prove boundedness of the Lusin's area operatorS _T on the space BMO(R ⁿ ).     

Inequalities for Volumes of Simplices in Terms of Their Faces

By generalizing some well-known results, we first obtain an inequality involving the volume and product of s-contents of s-faces of an n-simplex. Using this we generalize two inequalities maximizing the volume of one or two simplices in terms of their edge lengths.     

Intramolecular N−H...O hydrogen bonding,quinoid effect,and partial π-electron delocalization in N-aryl Schiff bases of 2-hydroxy-1-naphthaldehyde: the crystal structures of planar N-(α-naphthyl)- and N-(β-naphthyl)-2-oxy-1-naphthaldimine

The structures of N-(-naphthyl)-2-oxy-1-naphthaldimine1 and N-(-naphthyl)-2-oxy-1-naphthaldimine2 have been investigated by X-ray analysis and by spectroscopic methods. Crystals of1 are monoclinic, space groupPn, with cell dimensionsa=10.823(3),b=5.826(2),c=11.899 (3) Å, and =99.66(3)°. Compound2 crystallizes in the orthorhombic space groupPca2₁ witha=17.564(3),b=6.314(2), andc=13.663(4) Å. The IR spectra exhibited neither N–H nor O–H stretching frequencies. The existence of theintramolecular hydrogen bonding of N–H...O type was predicted by spectroscopic experiment but unequivocally established by diffraction experiment in both cases1 and2. The molecules1 and2 are significantly planar with considerable quinoid effect at the 2-oxy-naphthaldimine moiety. Although essentially planar, both molecules1 and2 show the delocalization of -electrons only in the central part of the molecules including C=N imino group with pendent ring carbon atoms.Intermolecular attractions in the crystals belong to weak van der Waals interactions-between discrete planar molecules spatially arranged into the expectedherringbone motif in the solid state.     

Copper + nickel-in-charcoal (Cu-Ni/C): a bimetallic, heterogeneous catalyst for cross-couplings

A new heterogeneous catalyst composed of copper and nickel oxide particles supported within charcoal has been developed. It catalyzes cross-couplings that traditionally use palladium, nickel, or copper, including Suzuki-Miyaura reactions, Buchwald-Hartwig aminations, vinylalane alkylations, etherifications of aryl halides, aryl halide reductions, asymmetric conjugate reductions of activated olefins, and azide-alkyne "click" reactions.     

A novel access to tetrahydro-beta-carbolines via one-pot hydroformylation/fischer indole synthesis: rearrangement of 3,3-spiroindoleninium cations

The two component one-pot hydroformylation/Fischer indole synthesis sequence of 2,5 dihydropyrroles and phenyl hydrazines allows a facile and convenient access to tetrahydro-beta-carbolines in moderate to good yields.     

Last step of the para route of the Kolbe-Schmitt reaction

This work is an extension of our investigations of the Kolbe--Schmitt reaction mechanism. The last step in the para route of the carboxylation reaction of alkali metal phenoxides is investigated at the B3LYP/LANL2DZ level of theory. Among several examined pathways, two mechanisms are proposed: the one involving a successive rearrangement of hydrogen and the other one based on the formation of free radicals. The former pathway is energetically comparable to the last step of the carboxylation reaction in the ortho position, whereas the latter pathway requires higher activation energy. Bearing in mind that the Kolbe--Schmitt reaction is performed at high temperatures, we assume that both reaction paths are plausible.