Zeros of Jacobi functions of second kind

The number of zeros in (- 1, 1) of the Jacobi function of second kind Q(n)((alpha, beta)) (x), alpha, beta > - 1, i.e. The second solution of the differential equation(1 - x(2))y (x) + (beta - alpha - (alpha + beta + 2)x)y' (x) + n(n + alpha + beta + 1)y(x) = 0,is determined for every n is an element of N and for all values of the parameters alpha > - 1 and beta > - 1. It turns out that this number depends essentially on alpha and beta as well as on the specific normalization of the function Q(n)((alpha, beta)) (x). Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind. (c) 2005 Elsevier B.V. All rights reserved.

Monotonicity of zeros of Jacobi polynomials

Denote by x(nk)(alpha, beta), k = 1...., n, the zeros of the Jacobi polynornial P-n((alpha,beta)) (x). It is well known that x(nk)(alpha, beta) are increasing functions of beta and decreasing functions of alpha. In this paper we investigate the question of how fast the functions 1 - x(nk)(alpha, beta) decrease as beta increases. We prove that the products t(nk)(alpha, beta) := f(n)(alpha, beta) (1 - x(nk)(alpha, beta), where f(n)(alpha, beta) = 2n(2) + 2n(alpha + beta + 1) + (alpha + 1)(beta + 1) are already increasing functions of beta and that, for any fixed alpha > - 1, f(n)(alpha, beta) is the asymptotically extremal, with respect to n, function of beta that forces the products t(nk)(alpha, beta) to increase. (c) 2007 Elsevier B.V. All rights reserved.

On the behaviour of zeros of Jacobi polynomials

Denote by x(n,k)(alpha, beta) and x(n,k) (lambda) = x(n,k) (lambda - 1/2, lambda - 1/2) the zeros, in decreasing order, of the Jacobi polynomial P-n((alpha, beta))(x) and of the ultraspherical (Gegenbauer) polynomial C-n(lambda)(x), respectively. The monotonicity of x(n,k)(alpha, beta) as functions of a and beta, alpha, beta > - 1, is investigated. Necessary conditions such that the zeros of P-n((a, b)) (x) are smaller (greater) than the zeros of P-n((alpha, beta))(x) are provided. A. Markov proved that x(n,k) (a, b) < x(n,k)(α, β) (x(n,k)(a, b) > x(n,k)(alpha, beta)) for every n is an element of N and each k, 1 less than or equal to k less than or equal to n if a > alpha and b < β (a < alpha and b > beta). We prove the converse statement of Markov's theorem. The question of how large the function could be such that the products f(n)(lambda) x(n,k)(lambda), k = 1,..., [n/2] are increasing functions of lambda, for lambda > - 1/2, is also discussed. Elbert and Siafarikas proved that f(n)(lambda) = (lambda + (2n(2) + 1)/ (4n + 2))(1/2) obeys this property. We establish the sharpness of their result. (C) 2002 Elsevier B.V. (USA).

Monotonicity of zeros of Jacobi-Sobolev type orthogonal polynomials

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Asymptotics for Jacobi-Sobolev orthogonal polynomials associated with non-coherent pairs of measures

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The X-ray crystallographic structure of Escherichia coli branching enzyme

Branching enzyme catalyzes the formation of alpha-1,6 branch points in either glycogen or starch. We report the 2.3-Angstrom crystal structure of glycogen branching enzyme from Escherichia coli. The enzyme consists of three major domains, an NH2-terminal seven-stranded beta-sandwich domain, a COOH-terminal domain, and a central alpha/beta-barrel domain containing the enzyme active site. While the central domain is similar to that of all the other amylase family enzymes, branching enzyme shares the structure of all three domains only with isoamylase. Oligosaccharide binding was modeled or branching enzyme using the enzyme-oligosaccharide complex structures of various alpha-amylases and cyclodextrin glucanotransferase and residues were implicated in oligosaccharide binding. While most of the oligosaccharides modeled well in the branching enzyme structure, an approximate 50degrees rotation between two of the glucose units was required to avoid steric clashes with Trp(298) of branching enzyme. A similar rotation was observed in the mammalian alpha-amylase structure caused by an equivalent tryptophan residue in this structure. It appears that there are two binding modes for oligosaccharides in these structures depending on the identity and location of this aromatic residue.

Structural basis for metal ion coordination and the catalytic mechanism of sphingomyelinases D

Sphingomyelinases D (SMases D) from Loxosceles spider venom are the principal toxins responsible for the manifestation of dermonecrosis, intravascular hemolysis, and acute renal failure, which can result in death. These enzymes catalyze the hydrolysis of sphingomyelin, resulting in the formation of ceramide 1-phosphate and choline or the hydrolysis of lysophosphatidyl choline, generating the lipid mediator lysophosphatidic acid. This report represents the first crystal structure of a member of the sphingomyelinase D family from Loxosceles laeta (SMase I), which has been determined at 1.75-angstrom resolution using the quick cryo-soaking technique and phases obtained from a single iodine derivative and data collected from a conventional rotating anode x-ray source. SMase I folds as an (alpha/beta)(8) barrel, the interfacial and catalytic sites encompass hydrophobic loops and a negatively charged surface. Substrate binding and/or the transition state are stabilized by a Mg2+ ion, which is coordinated by Glu(32), Asp(34), Asp(91), and solvent molecules. In the proposed acid base catalytic mechanism, His(12) and His(47) play key roles and are supported by a network of hydrogen bonds between Asp(34), Asp(52), Trp(230), Asp(233), and Asn(252).

Production of cyclodextrins by CGTase from Bacillus clausii using different starches as substrates

Cyclodextrins ( CDs) are cyclic oligasaccharides composed by D- glucose monomers joined by alpha- 1,4-D glicosidic linkages. The main types of CDs are alpha-,beta-and gamma-CDs consisting of cycles of six, seven, and eight glucose monomers, respectively. Their ability to form inclusion complexes is the most important characteristic, allowing their wide industrial application. The physical property of the CD-complexed compound can be altered to improve stability, volatility, solubility, or bio-availability. The cyclomaltodextrin glucanotransferase ( CGTase, EC 2.4.1.19) is an enzyme capable of converting starch into CD molecules. In this work, the CGTase produced by Bacillus clausii strain E16 was used to produce CD from maltodextrin and different starches ( commercial soluble starch, corn, cassava, sweet potato, and waxy corn starches) as substrates. It was observed that the substrate sources influence the kind of CD obtained and that this CGTase displays a beta- CGTase action, presenting a better conversion of soluble starch at 1.0%, of which 80% was converted in CDs. The ratio of total CD produced was 0: 0.89: 0.11 for alpha/beta/gamma. It was also observed that root and tuber starches were more accessible to CGTase action than seed starch under the studied conditions.

Esterase profile in the postembryonic development of Rhipicephalusmicroplus

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Trajectory of virtual, bound and resonant Efimov states

The pole trajectory of Efimov states for a three-body alpha alpha beta system with alpha alpha unbound and alpha beta bound is calculated using a zero-range Dirac-delta potential. It is shown that a three-body bound state turns into a virtual one by increasing the alpha beta binding energy. This result is consistent with previous results for three equal mass particles. The present approach considers the n-n-(18)C halo nucleus. However, the results have good perspective to be tested and applied in ultracold atomic systems, where one can realize such three-body configuration with tunable two-body interaction.

Effect of 4 mass% Ag addition on the thermal behavior of the Cu-9 mass% Al alloy

The effect of 4 mass% Ag addition on the thermal behavior of the Cu-9 mass% Al alloy was studied using differential scanning calorimetry (DSC), optical microscopy (OM), scanning electron microscopy (SEM), energy dispersive X-ray analysis (EDX) and X-ray diffractometry (XRD). The results showed that the presence of silver causes (Cu)-alpha+(alpha+gamma1)-->(Cu)-alpha+beta transformation to occur in two stages. In the first one, part of the produced beta phase combines with the precipitated Ag to give a silver-rich phase and in the second one the transformation is completed. The formation of this silver-rich phase seems to be enhanced at very low cooling rates.

Purification, sequencing and structural characterization of the phospholipase A(1) from the venom of the social wasp Polybia paulista (Hymenoptera, Vespidae)

The biochemical and functional characterization of wasp venom toxins is an important prerequisite for the development of new tools both for the therapy of the toxic reactions due to envenomation caused by multiple stinging accidents and also for the diagnosis and therapy of allergic reactions caused by this type of venom. PLA(1) was purified from the venom of the neotropical social wasp Polybia paulista by using molecular exclusion and cation exchange chromatographies; its amino acid sequence was determined by using automated Edman degradation and compared to the sequences of other vespid venom PLA(1)'s. The enzyme exists as a 33,961.40 da protein, which was identified as a lipase of the GX class, liprotein lipase superfamily, pancreatic lipases (ab20.3) homologous family and RP2 sub-group of phospholipase. P. paulista PLA(1) is 53-82% identical to the phospholipases from wasp species from Northern Hemisphere. The use restrained-based modeling permitted to describe the 3-D structure of the enzyme, revealing that its molecule presents 23% alpha-helix, 28% beta-sheet and 49% coil. The protein structure has the alpha/beta fold common to many lipases; the core consists of a tightly packed beta-sheet constituted of six-stranded parallel and one anti-parallel beta-strand, surrounded by four alpha-helices. P. paulista PLA(1) exhibits direct hemolytic action against washed red blood cells with activity similar to the Cobra cardiotoxin from Naja naja atra. In addition to this, PLA(1) was immunoreactive to specific IgE from the sera of P. paulista-sensitive patients. (c) 2007 Elsevier Ltd. All rights reserved.

Esterase patterns and phylogenetic relationships of Drosophila species in the saltans subgroup (saltans group)

The esterase patterns of sixteen strains from four species in the saltans subgroup were analyzed using polyacrylamide gel electrophoresis. Thirty-four esterase bands were detected. By using alpha and beta naphthyl acetates as substrates, they were classified in 18 alpha-esterases (they hydrolyse the alpha-naphtyl substrate), 15 beta-esterases (they hydrolyse the beta-naphtyl substrate) and 1 alpha/beta-esterase (it hydrolyses the alpha and beta-naphtyl substrates). Among the alpha-esterases, three were detected exclusively in males. Malathion, Eserine and pCMB were used as inhibitors in order to characterize biochemically the esterases. The results indicated the presence of cholinesterases, carboxylesterases and acetylesterases. The degree of mobility of the bands in the gels, their specificity to alpha and beta naphthyl acetates and the results of the inhibition tests allowed us to recognize tentatively nine genetic loci. Phylogenetic relationships among species inferred on the basis of the esterase patterns by PAUP 4.0 b8, with neighbor-joining search and a bootstrap analysis showed that, although the four species are closely related, D. septentriosaltans, D. saltans and D. austrosaltans are closer to each other than to D. prosaltans. These results showed to be consistent with phylogenetic relationships previously inferred from inversion polymorphism.

Quantitative determination of anti-fungal and insecticide amides in adult plants, plantlets and callus from Piper tuberculatum by reverse-phase high-performance liquid chromatography

A rapid, sensitive and reliable reverse-phase HPLC method was used for the quantitative determination of the anti-fungal and insecticide amides, dihydropiplartine (1), piplartine (2), Delta(alpha,beta)-dihydropiperine (3) and pellitorine (4) in plants in natura, in plantlets in vitro and ex vitro, and in callus of Piper tuberculatum. Well-resolved peaks were obtained with good detection response and linearity in the range of 15.0-3000 mug/mL. The plants in natura contained compounds 1-4, the plantlets ex vitro and in vitro accumulated compounds 1-2 and 1-4, respectively, while only amide 4 was found in callus. Copyright (C) 2003 John Wiley Sons, Ltd.

Phase transition in poly(vinylidene fluoride) investigated with micro-Raman spectroscopy

The phase transition from the non-polar a-phase to the polar beta-phase of poly(vinylidene fluoride) (PVDF) has been investigated using micro-Raman spectroscopy, which is advantageous because it is a nondestructive technique. Films of alpha-PVDF were subjected to stretching under controlled rates at 80 degrees C, while the transition to P-PVDF was monitored by the decrease in the Raman band at 794 cm(-1) characteristic of the a-phase, along with the concomitant increase in the 839 cm-1 band characteristic of the P-phase. The alpha ->beta transition in our PVDF samples could be achieved even for the sample stretched to twice (2 X -stretched) the initial length and it did not depend on the stretching rate in the range between 2.0 and 7.0 mm/min. These conclusions were corroborated by differential scanning calorimetry (DSC) and X-ray diffraction experiments for PVDF samples processed under the same conditions as in the Raman scattering measurements. Poling with negative corona discharge was found to affect the a-PVDF morphology, improving the Raman bands related to this crystalline phase. This effect is minimized for films stretched to higher ratios. Significantly, corona-induced effects could not be observed with the other experimental techniques, i.e., X-ray diffraction and infrared spectroscopy.