Arithmetic Progressions in Sets of Fractional Dimension

Let E ì \mathbbR{E \subset\mathbb{R}} be a closed set of Hausdorff dimension α. Weprove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains non-trivial 3-term arithmetic progressions.     

Modelling of the Electrical Membrane Potential for Concentration Polarization Conditions

Based on Kedem–Katchalsky formalism, the model equation of the membrane potential (Δψs) generated in a membrane system was derived for the conditions of concentration polarization. In this system, a horizontally oriented electro-neutral biomembrane separates solutions of the same electrolytes at different concentrations. The consequence of concentration polarization is the creation, on both sides of the membrane, of concentration boundary layers. The basic equation of this model includes the unknown ratio of solution concentrations (Ci/Ce) at the membrane/concentration boundary layers. We present the calculation procedure (Ci/Ce) based on novel equations derived in the paper containing the transport parameters of the membrane (Lp, σ, and ω), solutions (ρ, ν), concentration boundary layer thicknesses (δl, δh), concentration Raileigh number (RC), concentration polarization factor (ζs), volume flux (Jv), mechanical pressure difference (ΔP), and ratio of known solution concentrations (Ch/Cl). From the resulting equation, Δψs was calculated for various combinations of the solution concentration ratio (Ch/Cl), the Rayleigh concentration number (RC), the concentration polarization coefficient (ζs), and the hydrostatic pressure difference (ΔP). Calculations were performed for a case where an aqueous NaCl solution with a fixed concentration of 1 mol m⁻³ (Cl) was on one side of the membrane and on the other side an aqueous NaCl solution with a concentration between 1 and 15 mol m⁻³ (Ch). It is shown that (Δψs) depends on the value of one of the factors (i.e., ΔP, Ch/Cl, RC and ζs) at a fixed value of the other three.     

Right Nuclei of Quasigroup Extensions

The aim of this article is the study of right nuclei of quasigroups with right unit element. We investigate an extension process in this category of quasigroups, which is defined by a slight modification of non-associative Schreier-type extensions of groups or loops. The main results of the article give characterizations of quasigroup extensions satisfying particular nuclear conditions. We apply these results for constructions of right nuclear quasigroup extensions with right inverse property.     

Secondary structure elements in polylactic acid models

Geometry optimization results are reported on putative elements of secondary structure in decameric units of polylactic acid (PLA) analogous to those seen in protein structure—helical structures (α, π, 3₁₀) as well as a β-sheet—employing molecular mechanics, semiempirical, ab initio and density functional methods. The four possible structures of the deca-PLA are generally predicted, with all methods to be within ~15 kcal/mol of each other, with the more stable conformation varying depending on the method employed. The highest-level method employed here (M062x/6-311+G**) predicts that the α, π and 3₁₀ structures have very similar energies, with π slightly favored by values within the error limits of the method; this is in contrast with results obtained with less accurate semiempirical and empirical methods, which predict larger differences and other structures as favorites. Relative energies of poly-l and poly-d,l lactic acid structures indicate the former to be energetically-favored over the latter. Three types of weak interactions appear to dictate the relative stabilities of secondary structure elements in polylactic acid structures.     

Moufang’s theorem for non-Moufang loops

We introduce a class of non-Moufang loops satisfying Moufang’s theorem.     

Distance sets of well-distributed planar sets for polygonal norms

LetX be a two-dimensional normed space, and letBX be the unit ball inX. We discuss the question of how large the set of extremal points ofBX may be ifX contains a well-distributed set whose distance set Δ satisfies the estimate |Δ∩[0,N]|≤CN ^3/2-ε. We also give a necessary and sufficient condition for the existence of a well-distributed set with |Δ∩[0,N]|≤CN.     

The surface properties of iron catalyst for ammonia synthesis

The reduction process and the surface properties of industrial prereduced triply and doubly promoted iron catalyst for ammonia synthesis have been studied using thermogravimetry, thermal desorption of N₂ and X-ray diffraction. The properties of both catalysts have been discussed on the basis of the double layer model of iron catalyst. A concept of facet formation on the active surface of the catalysts has been proposed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Liquid chromatography determination of natural dyes in extracts from historical Scottish textiles excavated from peat bogs

Textiles excavated from Scottish sites belonging now to the collections of the National Museums of Scotland, including seventeenth century textiles from peat bogs in the Scottish Highlands and Islands, were selected for analysis by high performance liquid chromatography with photodiode array detection (PDA HPLC) to detect whether any dyes remained and, if so, to identify their biological sources. Dye components were identified in 36 of the 81 samples analysed. Although it was not possible to identify the exact sources of the dyestuffs because of the wide-spread occurrence of these natural dyes components, the study has shown that textiles previously not thought to have been coloured had detectable traces of dye. Before the historical textiles were analyzed, an improved extraction procedure that combined the routine acid hydrolysis method with one using dimethylformamide (DMF) was applied. The DMF method enabled increased recovery of major flavonoid and anthraquinoid compounds, and very high efficiency of recovery of indigotin even in textiles with no colour visible, thereby complementing the acid hydrolysis method already in use. Extracts from historical thread samples were analysed by PDA HPLC using a reversed-phase gradient system comprising of a C18 column (150 mm x 4.6 mm i.d., 25 +/- 1 degrees C) with water, methanol and o-phosphoric acid at an eluent flow rate of 1.2 ml/min. A preliminary investigation to improve the detection limits further for a selection of natural dyes was made by comparing results from the 4.6mm internal diameter (i.d.) column with a narrow bore C18 column (2.1 mm i.d.). An increase in the detector response was observed for narrow-bore column proving its possibility of enhancement of sensitivity.     

Synthesis and alkaline earth metal cation extraction by proton di-ionizable p-tert-butylcalix[4]arene-crown-5 compounds in cone, partial-cone and 1,3-alternate conformations

Synthetic strategies for novel, proton di-ionizable p-tert-butylcalix[4]arene-crown-5 compounds in cone, partial-cone and 1,3-alternate conformations are reported. Selective linkage of the two diametrical phenolic oxygens in p-tert-butylcalix[4]arene with tetraethylene glycol ditosylate gave 1,3-bridged p-tert-butylcalix[4]arene-crown-5. The two remaining phenolic units were alkylated using NaH and KH as the bases to give the cone and partial-cone conformers, respectively. Preparation of the 1,3-alternate conformers utilized a different sequence in which O-alkylation was followed by crown ether ring formation. Structures of these new ligands were elucidated by (1)H and (13)C NMR spectroscopy. These proton-ionizable ligands were tested for their solvent extraction properties toward alkaline earth metal cations. Surprising differences in their extraction behaviors are noted compared to those reported previously for di-ionizable p-tert-butylcalix[4]arenecrown-6 analogues.