How Neighborly Can a Centrally Symmetric Polytope Be?

We show that there exist k-neighborly centrally symmetric d-dimensional polytopes with 2(n + d) vertices, where

Regularity for a fourth-order critical equation with gradient nonlinearity

Given Ω a smooth bounded domain of Rn, n?3, we consider functions that are weak solutions to the equation

     

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix \(T_n(f)\), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where \(\mathbf {f}\) is an \(s\times s\) matrix-valued trigonometric polynomial with \(s\ge 1\), and \(T_n(\mathbf {f})\) is the block Toeplitz matrix generated by \(\mathbf {f}\), whose size is \(N(n,s)=sn\). The case \(s=1\) corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the \(\mathbb {Q}_p\) Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.     

Synthesis and reactivity of (C6F5)3B-N-heterocycle complexes. 1. Generation of highly acidic sp3 carbons in pyrroles and indoles

The reaction of pyrroles and indoles with B(C(6)F(5))(3) and BCl(3) produces 1:1 B-N complexes containing highly acidic sp(3) carbons, for example, N-[tris(pentafluorophenyl)borane]-5H-pyrrole (1) and N-[tris(pentafluorophenyl)borane]-3H-indole (2), that are formed by a new formal N-to-C hydrogen shift, the mechanism of which is discussed. With some derivatives, restricted rotation around the B-N bond and/or the B-C bonds was observed by NMR techniques, and some rotational barriers were calculated from experimental data. The acidity of the sp(3) carbons in these complexes is shown by their ability to protonate NEt(3), with formation of pyrrolyl- and indolyl-borate ammonium salts. The driving force for this reaction is given by the restoration of the aromaticity of the heterocycle.     

Synthesis and characterization of high-molecular-weight syndiotactic amorphous polypropylene

Heterocycle-fused titanium indenyl silylamido dimethyl complexes produce very high molecular weight polypropylene having a prevailingly syndiotactic microstructure with syndiotactic pentad contents rrrr up to 40-55% (sam-PP). The samples are basically amorphous and may slowly develop a low level of crystallinity (16-20%) at room temperature. A structural characterization has shown that sam-PP samples crystallize in disordered modifications of the helical form I of syndiotactic polypropylene (s-PP). The stretching of compression-molded films of sam-PP samples produce oriented crystalline fibers in the trans-planar mesomorphic form of s-PP. The low stereoregularity prevents the formation of the ordered trans-planar form III of s-PP, which instead is obtained in stretched fibers of the highly stereoregular and crystalline s-PP. The trans-planar mesomorphic form, obtained in stretched fibers, in turn transforms into the helical form I upon releasing the tension. The analysis of the mechanical properties has shown that sam-PP samples show good elastic behavior in a large range of deformation with remarkable strength, due to the presence of crystallinity. A comparison with the mechanical properties of less syndiotactic and fully amorphous samples is reported. These fully amorphous samples present lower strength and experience rapid viscous flow of the chains at high deformations and/or by application of stresses for long times. The higher strength in the semicrystalline sam-PP samples makes these materials interesting thermoplastic elastomers showing high toughness and ductility.     

Characterization of nutraceuticals and functional foods by innovative HPLC methods

In recent years there is a growing interest in food and food ingredient which may provide health benefits. Food as well as food ingredients containing health-preserving components, are not considered conventional food, but can be defined as functional food. To characterise such foods, as well as nutraceuticals specific, high sensitive and reproducible analytical methodologies are needed. In light of this importance we set out to develop innovative HPLC methods employing reversed phase narrow bore column and high-performance anion-exchange chromatographic methods coupled with pulsed amperometric detection (HPAEC-PAD), which are specific for carbohydrate analysis. The developed methods were applied for the separation and quantification of citrus flavonoids and to characterize fructooligosaccharide (FOS) and fructans added to functional foods and nutraceuticals.     

Ethanol production in immobilized-cell bioreactors from mixed sugar syrups and enzymatic hydrolysates of steam-exploded biomass

We investigated ethanol production from mixed sugar syrups. Hydrolysates were prepared from enzymatic saccharification of steam-pretreated aspen chips. Syrups containing 45 g/L of glucose and 12 g/L of xylose were detoxified through two ion-exchange resins and then fermented with Pichia stipitis and Saccharomyces cerevisiae immobilized in Ca-alginate gel beads. Combinations of different gel fractions in the fermentation volume, amount of yeast cells, and ratios of P. stipitis vs S. cerevisiae within each bead were compared. In the best conditions, by using a total beads volume corresponding to 25% of the working volume, we obtained a yield of 0.39 g_ethanol/g_{initial sugars}. This amount of gel entrapped an initial cell concentration of 6×10¹²cells/L with ratio of S. cerevisiae/P. stipitis of 0.25 g/g. Modified stirredtank reactors were obtained either by adding marbles or by inserting a perforated metal cylinder, which reduced considerably the rupture of beads while visibly improving oxygenation of the medium.     

Studies on the synthesis of heterocyclic compounds. Part VIII. A facile synthesis of new pyrazolo[3,4-b]benzazepine-4,9-diones

2-Carbomethoxybenzoyl chloride reacted with some 5-methylamino-l-phenyl-3-R-pyrazoles to yield N-methyl-l-phenyl-3-R-pyrazol-5-yl)-2-carbomethoxybenzamides. These products were readily transformed into the corresponding acid, which in turn, refluxed in phosphorus oxychloride afforded the tricyclic system, l-phenyl-3-R-pyrazolo[3,4-b]benzazepine-4,9-dione.