The maximal solution of a restricted subadditive inequality in numerical analysis

For a fixed non-negative integerp, letU _2p = {U _2p (n)},n ≥ 0, denote the sequence that is defined by the initial conditionsU _2p (0) =U _2p (1) =U _2p (2) = =U _2p (2p) = 1 and the restricted subadditive recursion $$U_{2p} (n + 2p + 1) = \mathop {\min }\limits_{0 \leqslant l \leqslant p} (U_{2p} (n + l) + U_{2p} (n + 2p - l)),n \geqslant 0$$ U _2p is of importance in the theory of sequential search for simple real zeros of real valued continuous 2p-th derivatives In this paper, several closed form expressions forU _2p (n), n > 2p, are determined, thereby providing insight into the structure ofU _2p Two of the properties thus illuminated are (a) the existence of exactlyp + 1 limit points (1 + 1/(p + 1 +i), 0 ≤i ≤p) of the associated sequence {U _2p (n + 1)/U _2p (n)},n ≥ 0, and (b) the relevance toU _2p of the classic number theoretic function ord     

Partitioning Boolean lattices into chains of subsets

We prove that the Boolean lattice of all subsets of an n-set can be partitioned into chains of size four if and only if n9.Research supported in part by N.S.F. grant DMS-8401281.Research supported in part by N.S.F. grant DMS-8406451.     

Über Pterinchemie. 79. Mitteilung. 13C-und 15N-NMR.-spektroskopische Untersuchungen an der (6RS)-5, 10-Methylen-5, 6, 7, 8-tetrahydro-L-folsäure; eindeutiger Beweis ihrer Struktur

¹³C- and ¹⁵N-NMR.-Spectroscopic Studies on (6RS)-5, 10-Methylene-5, 6, 7, 8-tetrahydro-L -folic Acid; Unequivocal Proof of thier Structure The existence of a 5, 10-methylene bridge in the title compound has been established with the help of ¹⁵N-NMR, and ¹³C-NMR. spectroscopy. The simultaneous coupling of ¹³C(11) with ¹⁵N(5) and ¹⁵N(10) proves that C(11) must be bound as a methylene bridge to both N-atoms. Some other aspects of the NMR. spectra of the compound are discussed. Synthesis of the title product with the label ¹⁵N is described.     

Fragmentation of Homoallylic Alkoxides. Preparation of 1-(3′-cyclopentenyl)-2-alkanones from 2-substituted bicyclo[2.2.1]hept-5-en-2-ols

The potassium 2-substituted bicyclo[2.2.1]hept-5-en-2-alkoxides derived from alcohols 2–9 at 30° in hexamethylphosphoric triamide (HMPA) afford 1-(3′-cyclopentenyl)-2-alkanones 10–19 via heterolytic C(1), C(2)-allylic bond cleavage in the substrate alkoxide followed by intramolecular protonation of the resultant transient allylic anion.     

Sizes and interfacial free energies of crystallites formed from fractionated linear polyethylene

Replicas of fracture surfaces of fractions of linear polyethylene, which were crystallized at elevated temperatures for extended time periods, were examined by electron microscopy. Striated. lamella-type crystallites were observed for all molecular weights over the range 3.2 × 10³?5.7 × 10⁵. In agreement with Anderson's previous report, for molecular weights of 12,000 or less, the crystallite thicknesses were comparable to the extended chain length. As the molecular weight increased above this level, however, the crystallite sizes increased only slightly and hence at high molecular weights were very much smaller than the extended chain length. From the measured melting temperatures, crystallite interfacial free energies were calculated from the theory for the melting of finite size crystals comprised of chains of finite length. The crystallite interfacial free energy was found to increase with molecular weight. Based on these results, a crystallization process is outlined which allows for the formation of either extended chain crystallites, or crystallites whose size is much smaller than the extended chain length without any change in nucleation mechanism or arbitrary adjustment in growth mechanism with molecular weight.     

Optimized separation of scopoletin and umbelliferone and cis-trans isomers of ferulic and p-coumaric acids by reverse-phase high-performance liquid chromatography

A systematic optimization strategy for reverse-phase h.p.l.c. is used to produce complete separation of all components of a mixture containing two coumarins, umbelliferone and scopoletin, and the cis-trans isomers of two cinnamic acids, p-coumaric and ferulic. A series of experiments gives retention time data for each of these substances in four citric acid-buffered mobile phases. A mathematical model fitted to this data gives model parameters that provide the basis for a window diagram of relative retentions vs. mobile phase pH. The window diagram gives an estimate of the mobile phase pH required for optimum separation.     

The Raman active internal vibrational modes of potassium nitrate

The Raman active internal vibrational modes of single crystal orthorhombic potassium nitrate have been studied in various polarizations. The full multiplet structure predicted by factor group analysis for the v₂ and v₃ regions has been observed for the first time. The expected site group splitting of the v₄ mode was not observed and can be assumed to be less than 0.5 cm^?1.     

The role of surface oxides in formic acid oxidation on Au

Compared to Pt or Pd electrodes, Au is a poor catalyst for the direct anodic oxidation of HCOOH, but the formation of Au surface oxides in acidic solutions is accompanied by a fast oxidation of HCOOH. This fast reaction is not simply a secondary reaction of Au surface oxides since those oxides are kinetically stable in HCOOH solutions. They do oxidize HCOOH only via a slow and purely electrochemical process which occurs on free Au sites and is “driven” by oxide reduction. The fast HCOOH oxidation is due to a highly reactive intermediate which is able either to form stable Au oxides Au_nO_m or to react with HCOOH. Our results are consistent with the model that by the charge transfer step a reactive non-equilibrium {Au…O> species is formed which converts to stable equilibrium oxides Au_nO_m after migration and rearrangement steps. Pre-equilibrium <Au…O> oxidizes HCOOH and this oxidation is of lower order with respect to <Au…O> compared with the formation of Au_nO_m.