A continuous function space with a Faber basis

Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (P_n)_n=0^∞ with deg P_n=n such that every fC(S) has a unique representation f=∑_i=0^∞ α_iP_i and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(S_q)≠U(S_q)=C(S_q), where A(S_q) is the so-called Wiener algebra and U(S_q) is the set of all fC(S_q) which are uniquely represented by its Fourier series.     

Oxidative dehydrogenation of crotonaldehyde overK2HPMo12O40catalyst

Summary The oxidative dehydrogenation of crotonaldehyde to furan and maleic anhydride was carried out over K₂HPMo₁₂O₄₀catalyst. A positive effect of water vapor on furan formation is explained by ability of the catalyst to isomerize 2E- to 2Z-crotonaldehyde.</o:p>     

Stability properties of collocation methods

Lower bounds for are given for which equidistant s-point collocation methods areA()-stable for arbitrarys.     

The class reconstruction number of maximal planar graphs

The reconstruction numberrn(G) of a graphG was introduced by Harary and Plantholt as the smallest number of vertex-deleted subgraphsG _i =G – v _i in the deck ofG which do not all appear in the deck of any other graph. For any graph theoretic propertyP, Harary defined theP-reconstruction number of a graph G P as the smallest number of theG _i in the deck ofG, which do not all appear in the deck of any other graph inP We now study the maximal planar graph reconstruction numberrn(G), proving that its value is either 1 or 2 and characterizing those with value 1.     

Synthesis and structure of trigonal and tetragonal connectors for a "Tinkertoy" construction set

We describe a convergent synthesis of eight 1,3,5- and 1,2,4,5-substituted benzene derivatives with long rigid arms containing 4-pyridyl, 2,2'-bipyridyl, and 2,2'-bipyrimidyl termini, meant to be used as trigonal or tetragonal connectors for the construction of large molecular structures. The synthesis involved copper-free Pd-mediated coupling of terminal acetylenes to aryl halides. First, one of the termini of 1,3-diethynylbicyclo[1.1.1]pentane was coupled with a brominated aza heterocycle, and second, 3 equiv of the resulting extended arm were coupled with 1,3,5-triiodobenzene or 4 with 1,2,4,5-tetraiodobenzene. An improved large-scale synthesis for 1,3-diethynylbicyclo[1.1.1]pentane is described. The structures of two of the arms were determined by single-crystal X-ray analysis. Several long molecular rods with 4-pyridyl termini were obtained as byproducts, and a single-crystal X-ray structure is reported for the shortest of these.     

Isoperichoric focusing field-flow fractionation and thin layer isoperichoric focusing based on coupling of gradient generated by primary field with secondary field action: general principles and performances

The primary field forces can generate spatially oriented gradient of the effective property of a continuum or pseudo-continuum fluid (carrier liquid). When this gradient is coupled with the action of a secondary field of identical or different nature the isoperichoric focused zones of the dispersed species can appear. Consequently, they can be separated according to differences responding to the property gradient of the carrier liquid. This concept can be applied under static (non-flow) conditions in thin layer focusing as well as under dynamic conditions with the elution due to the carrier liquid flow in focusing field-flow fractionation. The gradient established by the action of the primary field and the concentration distribution of the isoperichoric focused zone formed by the coupled effect of the gradient and of the primary or secondary field are described theoretically. The rigorous relationship describing the shape of the focused zone is compared with the approximate solutions. The performances of the proposed principle were evaluated by model calculations. Potential experimental configurations considering the implementation of the static and dynamic conditions are discussed. The generalized isoperichoric focusing theory can be applied to describe the particular processes operating in analytical and preparative focusing separations of the particles of various, but especially of biological origin.     

Zur Konfiguration des Vitamin-D3-Metaboliten 25, 26-Dihydroxycholecalciferol: Synthese von (25S,26)- und (25R,26)-Dihydroxycholecalciferol

Configuration of the Vitamin-D₃-Metabolite 25,26-Dihydroxycholecalciferol: Synthesis of (25S,26)- and (25R,26)-Dihydroxycholecalciferol For selective synthesis of the title compounds, (25S)- 1b and (25R)- 1b (Scheme 1), the protected cholesterol precursors (25S)- 6 and (25R)- 6 were prepared from stigmasterol-derived steroid-units 4a-d and C₅-side chain building blocks 5a–d by Grignard- or Wittig-coupling (Scheme 2), the configuration at C(25) of the target compounds being already present in the C₅-units. Conversion of the cholesterol intermediates to the corresponding vitamin-D₃ derivatives was carried out via the 7,8-didehydrocholesterol compounds (25S)- 2b and (25R)- 2b (Scheme 1), using the established photochemical-thermal transformation of the 5,7-diene system to the seco-triene system of cholecalciferol. The configuration at C(25) of the cholesterol precursors as assigned on basis of the known configuration of the C₅-units used, was found to be in agreement with the result of a single crystal X-ray analysis on compound 11 . The configuration at C(25) remained untouched on conversion of the cholesterol ring system to the seco-triene system of vitamin D₃ as evident from comparison of the lanthanide-induced CD. Cotton effects observed for (25S)- 3b and (25S) 1b . 25,26-Dihydroxycholecalciferol observed as a natural vitamin-D₃ metabolite has (25S)-configuration.