Kolmogorov Width of Classes of Smooth Functions on the Sphere

Let ^d−1{(x₁,…,x_d) ^d:x²₁+···+x²_d=1} be the unit sphere of the d-dimensional Euclidean space ^d. For r>0, we denote by B^r_p (1p∞) the class of functions f on ^d−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on ^d−1, and P^λ_k(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of B^r_p in L^q( ^d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces X_N of L^q( ^d−1).The main result in this paper is that for r2(d−1)²,where A_NB_N means that there exists a positive constant C, independent of N, such that C⁻¹A_NB_NCA_N.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions.     

Predicting the Mechanical Properties of Binary Blends of Immiscible Polymers

We couple a morphological study of an immiscible binary AB mixture with a micromechanical simulation to determine how the spatial distribution of the A and B domains and the interfacial region (interphase) affects the mechanical behavior of the blend. The morphological studies are conducted through a three-dimensional Cahn-Hilliard (CH) simulation. Through the CH calculations, we obtain the size and structure of the domains for different blend compositions. The output of the CH model serves as the input to the Lattice Spring Model (LSM), which consists of a three-dimensional network of springs. In particular, the location of the different phases is mapped onto the LSM lattice and the appropriate force constants are assigned to the LSM sites. A stress is applied to the LSM lattice and we calculate the elastic response of the material. We find that the local stress and strain fields are highly dependent on the morphology of the system. By integrating the morphological and mechanical models, we can isolate how modifications in the composition of the mixture affect the macroscopic behavior. Thus, we can establish how choices made in the components affect the ultimate performance of the material.     

A microcalorimetric study of the association of hexacyanoferrate(II) ions with calcium and magnesium ions in aqueous solution

Using flow microcalorimetry, the ion association reaction M²⁺(aq)+Fe(CN) ₆ ^4– (aq)=MFe(CN) ₆ ^2– (aq) (M=Ca, Mg) has been studied at 25°C over the ionic strength range 0.02 to 0.08 mol-dm^–3. Analyses of the data to obtain H^o, the enthalpy change at infinite dilution, are described. The value obtained for H^o is sensitive to the kind of functions used to correct for non-ideal behavior.     

Energy migration in novel pH-triggered self-assembled beta-sheet ribbons

Energy migration between tryptophan residues has been experimentally demonstrated in self-assembled peptide tapes. Each peptide contains 11 amino acids with a Trp at position 6. The peptide self-assembly is pH-sensitive and forms amphiphilic tapes, which further stack in ribbons (double tapes) and fibrils in water depending on the concentration. Fluorescence spectra, quenching, and anisotropy experiments showed that when the pH is lowered from 9 to 2, the peptide self-assembly buries the tryptophan in a hydrophobic and restricted environment in the interior of stable ribbons as expected on the basis of the peptide design. These fluorescence data support directly and for the first time the presence of such ribbons which are characterized by a highly packed and stable hydrophobic interior. In common with Trp in many proteins, fluorescence lifetimes are nonexponential, but the average lifetime is shorter at low pH, possibly due to quenching with neighboring Phe residues. Unexpectedly, time-resolved fluorescence anisotropy does not change significantly with self-assembly when in water. In highly viscous sucrose-water mixtures, the anisotropy decay at low pH was largely unchanged compared to that in water, whereas at high pH, the anisotropy decay increased significantly. We concluded that depolarization at low pH was not due to rotational diffusion but mainly due to energy migration between adjacent tryptophan residues. This was supported by a master equation kinetic model of Trp-Trp energy migration, which showed that the simulated and experimental results are in good agreement, although on average only three Trp residues were visited before emission.     

The chemical synthesis of beta-(1-->4)-linked D-mannobiose and D-mannotriose

A D-cellobiose derivative was converted to D-mannobiose via simultaneous epimerization at C-2 and C-2'. Subsequent beta-D-glucosylation and epimerization at C-2" gave D-mannotriose.     

Mono-cyclopentadienyl complexes of lanthanum: synthesis and characterization of anilido derivatives

Use of the bulky cyclopentadienyl ligand [η⁵-C₅H₂(SiMe₃)₃-1,2,4]⁻ (Cp?) allows for the isolation of monomeric, mono-ring lanthanide species. As previously reported, (Cp?)K reacts with LaI₃(THF)₄ (THF=tetrahydrofuran) in THF/pyridine to form the mono-ring complex (Cp?)LaI₂(py)₃ (1) (py=pyridine); a minor product of this reaction is the bis-ring species (Cp?)₂LaI(py) (2). The solid state structure of 2 reveals a monomeric compound containing a pseudo-tetrahedral metal center exhibiting no unusual intramolecular contacts. Addition of one equiv of KNHAr (Ar=2,6-ⁱPr₂C₆H₃) to complex 1 in THF generates the mono-anilido compound (Cp?)LaI(NHAr)(THF)₂ (3), which may be converted to the more stable pyridine adduct (Cp?)LaI(NHAr)(py)₂ (4) by the addition of pyridine to 3. An X-ray crystal structure of 3 indicated a trigonal bipyramidal metal center with the anilido group oriented trans to the iodide atom (N1-La1-I1=123.1(3)°). A structural study on the bis-pyridine adduct 4 revealed a similar C_s-symmetric structure with a slightly increased N_anilido-La-I angle of 132.1(2)°. Addition of KNHAr to the di-iodo bipyridine adduct (Cp?)LaI₂(bipy)(py) (5), in which the two iodide atoms are cis-disposed, yields the mono-anilido complex (Cp?)LaI(NHAr)(bipy)(py) (6) (bipy=2,2^′-bipyridine); this compound may also be prepared by the addition of bipy to (Cp?)LaI(NHAr)(py)₂ (4). An X-ray diffraction study shows that the lanthanum center in 6 is octahedrally coordinated by a Cp? ring, an iodide, an anilido group, a pyridine molecule and two nitrogens of a bipy molecule. In this case, the anilido moiety and the iodide ligand are arranged in a cis fashion (N_anilido-La-I=111.2(2)°), resulting in a complex with C₁ symmetry. Both (Cp?)LaI(NHAr)(py)₂ (4) and (Cp?)LaI(NHAr)(bipy)(py) (6) are inactive as catalysts for the hydroamination/cyclization of 2-amino-hex-5-ene.     

Synthesis of a deca-lithium cage containing an [(RN)2As(mu-NR)As(NR)2]4- tetraanion; a homologue of group 15 trianions of the type [E(NR)3]3-

The novel, deca-lithium cage [(mtaNHLi)(As2(Nmta)5)-Li(4).2thf]2 (1) (mtaN = 5-methylthiazolyl, C4H4N2S) contains an imido-bridged tetraanion [(mtaN)2As(mu-Nmta)-As(Nmta)2]4-, which represents a new type of multi-functional imido group 15 ligand framework (homologous with group 15 anions of the type [As(NR)3]3-).     

Templating and selection in the formation of macrocycles containing [[P(micro-NtBu)(2)](micro-NH)](n) frameworks: observation of halide ion coordination

Amination of [ClP(micro-NtBu)](2) (1) using NH(3) in THF gives the cyclophospha(III)zane dimer [H(2)NP(micro-NtBu)](2) (2), in good yield. (31)P NMR spectroscopic studies of the reaction of 1 with 2 in THF/Et(3)N show that almost quantitative formation of the cyclic tetramer [[P(micro-NtBu)](2)(micro-NH)](4) (3) occurs. The remarkable selectivity of this reaction can (in part) be attributed to pre-organisation of 1 and 2, which prefer cis arrangements in the solid state and solution. The macrocycle 3 can be isolated in yields of 58-67 % using various reaction scales. The isolation of the major by-product of the reaction (ca. 0.5-1 % of samples of 3), the pentameric, host-guest complex [[P(micro-NtBu)(2)](2)(micro-NH)](5)(HCl).2 THF] (4.2 THF), gives a strong indication of the mechanism involved. In situ (31)P NMR spectroscopic studies support a stepwise condensation mechanism in which Cl(-) ions play an important role in templating and selection of 3 and 4. Amplification of the pentameric arrangement occurs in the presence of excess LiX (X=Cl, Br, I). In addition, the cyclisation reaction is solvent- and anion-dependent. The X-ray structures of 2 and 4.2 THF are reported.