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>     

On the dimension to represent a graph by a unit distance graph

It is proved that if a graphG has maximum degreed, then its vertices can be represented by distinct unit vectors inR ^2d so that two vectors are orthogonal if and only if the corresponding vertices are adjacent. As a corollary it follows that if a graph has maximum degreed, then it is isomorphic to a unit distance graph inR ^2d.     

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.     

Indications for the Occurrence of Nitric Oxide Synthases in Fungi and Plants and the Involvement in Photoconidiation of Neurospora crassa*

Indications for the occurrence of nitric oxide synthases in Dictyostelium, Neurospora, Phycomyces and the leguminous plant Mucuna hassjoo as well as a physiological role of nitric oxide in Neurospora crassa are demonstrated. An exogenous nitric oxide donor, sodium nitroprus-side, inhibited light-stimulated conidiation in N. crassa. Specific inhibitors of nitric oxide synthase, like the arginine derivatives N^G-nitro-L-arginine (L-NA) and N^G-nitro-L-arginine-methyl ester (L-NAME), enhanced conidiation in darkness and in the light, whereas the stereo-isomer D-NAME was inactive. This communication reports to our knowledge the first time the presence of enzymatic activity of nitric oxide synthase in fungi and a higher plant and an effect of nitric oxide in fungal photophysiology.     

Electrospray ionization ion-trap multistage mass spectrometric study of sodium cationized aldobiuronic and pseudoaldobiuronic acid derivatives

Fragmentation mechanisms of electrospray ionization (ESI) mass spectrometry of aldobiuronic and pseudoaldobiuronic acid derivatives were elucidated by multistage mass spectrometric (MS(n), n = 2-5) measurements of selected ions. Characteristic under the conditions of ESI-MS analysis is the production of alkali metal (Na and K) cationized adducts. The probability the of locations of Na cations in per-O-methylated compounds was proved by quantum chemical calculations, using the Jaguar program. The most probably position of alkali metal attachment is the carboxy group of the methoxycarbonyl C-5 group of the uronic acid unit. Characteristic cleavages vary according the kind of O-derivatization. In most cases they take place on the acidic part of the dimer and at the interglycosidic oxygen atom. As a result, the criteria for the differentiation of aldobiouronic and pseudoaldobiouronic acids derivatives were elucidated.