Fourier multipliers of Lipschitz spaces

Let G denote an infinite, compact, metrizable, 0-dimensional, Abelian group. The following are characterized: (i) the multipliers from one Lipschitz space Lip(α, p; G) to another Lipschitz space Lip(β, q; G) for 0 < α < β < ∞ and 1 ? p, q ? ∞; and (ii) the multipliers from Lip(α, p; G) to Lip(β, q; G) for 0 < β ? α < ∞ and 1 < q ? 2 ? p < ∞. Two special cases of (i), namely the case q = ∞ and the case p = 1, were obtained by the authors in an earlier publication (1981). A. Zygmund (J. Math. Mech.8 (1959), 889–895) and T. Mizuhara (Tôhoku Math. J.24 (1972), 263–268) have characterized the multipliers of certain Lipschitz spaces defined on the circle group.     

Terphenyl ligand systems in lanthanide chemistry: use of the 2,6-di(1-naphthyl)phenyl ligand for the synthesis of kinetically stabilized complexes of trivalent ytterbium, thulium, and yttrium

The molecular structures of terphenyl derivatives of trivalent ytterbium, thulium, and yttrium of general composition DnpLnCl(2)(THF)(2) [Dnp = 2,6-di(1-naphthyl)phenyl] are reported. The complexes (Ln = Yb: 1; Ln = Tm: 2; Ln = Y: 3) are synthesized by reaction of 1 equiv of DnpLi with 1 equiv of LnCl(3) (Ln = Yb, Tm, or Y) in tetrahydrofuran at room temperature in 50% yield. Attempts to prepare a Dnp scandium compound gave heterobimetallic [(THF)(3)Sc(2)OCl(5)Li(THF)](2) (4) in low yield. 1 crystallizes in the monoclinic space group C2/c. Crystal data for 1 at 203 K: a = 14.333(3) A, b = 16.353(3) A, c = 12.427(2) A, beta = 91.021(4) degrees, Z = 4, D(calcd) = 1.637 g cm(-3), R(1) = 4.44%. 2 crystallizes in the monoclinic space group C2/c. Crystal data for 2 at 203 K: a = 14.333(1) A, b = 16.374(2) A, c = 12.404(1) A, beta = 90.934(2) degrees, Z = 4, D(calcd) = 1.628 g cm(-3), R(1) = 3.00%. 3 crystallizes in the monoclinic space group C2/c. Crystal data for 3 at 203 K: a = 14.348(3) A, b = 16.476(3) A, c = 12.356(2) A, beta = 90.987(4) degrees, Z = 4, D(calcd) = 1.441 g cm(-3), R(1) = 5.62%. 4 crystallizes in the monoclinic space group P2(1)/n. Crystal data for 4 at 203 K: a = 11.0975(9) A, b = 11.0976(9) A, c = 21.3305(18) A, beta = 94.718(2) degrees, Z = 2, D(calcd) = 1.051 g cm(-3), R(1) = 3.45%. Complexes 1-3 represent examples of novel chiral (racemic) organometallic complexes of the lanthanide elements ytterbium and thulium and the group 3 element yttrium, respectively. The molecular structures of monomeric 1-3 exhibit distorted trigonal-bipyramidal coordination environments at the metal center, with the two oxygen atoms of the tetrahydrofuran ligands occupying the axial positions of a trigonal-bipyramidal coordination polyeder. The molecular structure of the scandium compound 4 shows a complex polynuclear heterobimetallic arrangement.     

Factorization of Lipschitz functions and absolute convergence of Vilenkin-Fourier series

LetG be a Vilenkin group (i.e., an infinite, compact, metrizable, zero-dimensional Abelian group). Our main result is a factorization theorem for functions in the Lipschitz spaces \(\mathfrak{L}\mathfrak{i}\mathfrak{p}\) (α,p; G). As colloraries of this theorem, we obtain (i) an extension of a factorization theorem ofY. Uno; (ii) a convolution formula which says that \(\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha , r; G) = \mathfrak{L}\mathfrak{i}\mathfrak{p} (\beta , l; G)*\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha - \beta , r; G)\) for 0<β<α<∞ and 1≤r≤∞; and (iii) an analogue, valid for allG, of a classical theorem ofHardy andLittlewood. We also present several results on absolute convergence of Fourier series defined onG, extending a theorem ofC. W. Onneweer and four results ofN. Ja. Vilenkin andA. I. Rubinshtein. The fourVilenkin-Rubinshtein results are analogues of classical theorems due, respectively, toO. Szász, S. B. Bernshtein, A. Zygmund, andG. G. Lorentz.     

Heterocyclic Substitutions Greatly Improve Affinity and Stability of Folic Acid towards FRα. an In Silico Insight

Folate receptor alpha (FRα) is known as a biological marker for many cancers due to its overexpression in cancerous epithelial tissue. The folic acid (FA) binding affinity to the FRα active site provides a basis for designing more specific targets for FRα. Heterocyclic rings have been shown to interact with many receptors and are important to the metabolism and biological processes within the body. Nineteen FA analogs with substitution with various heterocyclic rings were designed to have higher affinity toward FRα. Molecular docking was used to study the binding affinity of designed analogs compared to FA, methotrexate (MTX), and pemetrexed (PTX). Out of 19 FA analogs, analogs with a tetrazole ring (FOL03) and benzothiophene ring (FOL08) showed the most negative binding energy and were able to interact with ASP81 and SER174 through hydrogen bonds and hydrophobic interactions with amino acids of the active site. Hence, 100 ns molecular dynamics (MD) simulations were carried out for FOL03, FOL08 compared to FA, MTX, and PTX. The root mean square deviation (RMSD) and root mean square fluctuation (RMSF) of FOL03 and FOL08 showed an apparent convergence similar to that of FA, and both of them entered the binding pocket (active site) from the pteridine part, while the glutamic part was stuck at the FRα pocket entrance during the MD simulations. Molecular mechanics Poisson-Boltzmann surface accessible (MM-PBSA) and H-bond analysis revealed that FOL03 and FOL08 created more negative free binding and electrostatic energy compared to FA and PTX, and both formed stronger H-bond interactions with ASP81 than FA with excellent H-bond profiles that led them to become bound tightly in the pocket. In addition, pocket volume calculations showed that the volumes of active site for FOL03 and FOL08 inside the FRα pocket were smaller than the FA–FRα system, indicating strong interactions between the protein active site residues with these new FA analogs compared to FA during the MD simulations.     

Synthesis of novel acetylenic cyclophanes with helical chirality: potential new structures for liquid crystals

The synthesis of a series of novel acetylenic cyclophanes is described. X-ray crystallographic analysis of the core structure revealed a twisted conformation with helical chirality. Preliminary results suggest that these cyclophanes, with appropriate functionality, have the potential to act as unique liquid crystalline materials.     

A Paramagnetic Diniobium Complex with a Very Short Nb-Nb Distance: Evidence for a Pseudo Nb-Nb Triple Bond?

In spite of the short Nb-Nb distance (2.268 ?) and the presumable existence of an Nb identical withNb bond, the paddle-wheel-shaped diniobium(II) complex 1 is paramagnetic. Theoretical calculations indicate that the presence of LiCl moieties on the intermetallic axis lowers the Nb-Nb bond order and is responsible for the observed paramagnetism.