Polynomials generated by a three term recurrence relation: bounds for complex zeros

This paper deals with the zeros of polynomials generated by a certain three term recurrence relation. The main objective is to find bounds, in terms of the coefficients of the recurrence relation, for the regions where the zeros are located. In most part, the zeros are explored through an Eigenvalue representation associated with a corresponding Hessenberg rnatrix. Applications to Szego polynomials, para-orthogonal polynomials and polynomials with non-zero complex coefficients are considered. (C) 2004 Elsevier B.V. All rights reserved.

SZEGO and PARA-ORTHOGONAL POLYNOMIALS on THE REAL LINE: ZEROS and CANONICAL SPECTRAL TRANSFORMATIONS

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Szego{double acute} and para-orthogonal polynomials on the real line: Zeros and canonical spectral transformations

We study polynomials which satisfy the same recurrence relation as the Szego{double acute} polynomials, however, with the restriction that the (reflection) coefficients in the recurrence are larger than one in modulus. Para-orthogonal polynomials that follow from these Szego{double acute} polynomials are also considered. With positive values for the reflection coefficients, zeros of the Szego{double acute} polynomials, para-orthogonal polynomials and associated quadrature rules are also studied. Finally, again with positive values for the reflection coefficients, interlacing properties of the Szego{double acute} polynomials and polynomials arising from canonical spectral transformations are obtained. © 2012 American Mathematical Society.

Palindromic and perturbed polynomials: zeros location

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials that is associated with the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros are dense in their associated critical intervals, confirming the conjecture and the numerical experiments made by M. Lapidus and M. van Frankenhuysen in several papers.

Hermite Polynomials and the Zero-Distribution of Riemann's Zeta Function

AMS Subject Classification 2010: 11M26, 33C45, 42A38.

Smale's Conjecture on Mean Values of Polynomials and Electrostatics

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.

Genomics of pyrrolnitrin biosynthetic loci: evidence for conservation and whole-operon mobility within Gram-negative bacteria

Pyrrolnitrin (PRN) is a tryptophan-derived secondary metabolite produced by a narrow range of Gram-negative bacteria. The PRN biosynthesis by rhizobacteria presumably has a key role in their life strategies and in the biocontrol of plant diseases. The biosynthetic operon that encodes the pathway that converts tryptophan to PRN is composed of four genes, prnA through D, whose diversity, genomic context and spread over bacterial genomes are poorly understood. Therefore, we launched an endeavour aimed at retrieving, by in vitro and in silico means, diverse bacteria carrying the prnABCD biosynthetic loci in their genomes. Analysis of polymorphisms of the prnD gene sequences revealed a high level of conservation between Burkholderia, Pseudomonas and Serratia spp. derived sequences. Whole-operon- and prnD-based phylogeny resulted in tree topologies that are incongruent with the taxonomic status of the evaluated strains as predicted by 16S rRNA gene phylogeny. The genomic composition of c. 20 kb DNA fragments containg the PRN operon varied in different strains. Highly conserved and distinct transposase-encoding genes surrounding the PRN biosynthetic operons of Burkholderia pseudomallei strains were found. A prnABCD-deprived genomic region in B. pseudomallei strain K96243 contained the same gene composition as, and shared high homology with, the flanking regions of the PRN operon in B. pseudomallei strains 668, 1106a and 1710b. Our results strongly suggest that the PRN biosynthetic operon is mobile. The extent, frequency and promiscuity of this mobility remain to be understood.

Impact of Matrix-Assisted Laser Desorption Ionization Time-of-Flight Mass Spectrometry on the Clinical Management of Patients With Gram-negative Bacteremia: A Prospective Observational Study.

Background. Early identification of pathogens from blood cultures using matrix-assisted laser desorption ionization time-of-flight (MALDI-TOF) mass spectrometry may optimize the choice of empirical antibiotic therapy in the setting of bloodstream infections. We aimed to assess the impact of this new technology on the use of antibiotic treatment in patients with gram-negative bacteremia. Methods. We conducted a prospective observational study from January to December 2010 to evaluate the sequential and separate impacts of Gram stain reporting and MALDI-TOF bacterial identification performed on blood culture pellets in patients with gram-negative bacteremia. The primary outcome was the impact of MALDI-TOF on empirical antibiotic choice. Results. Among 202 episodes of gram-negative bacteremia, Gram stain reporting had an impact in 42 cases (20.8%). MALDI-TOF identification led to a modification of empirical therapy in 71 of all 202 cases (35.1%), and in 16 of 27 cases (59.3%) of monomicrobial bacteremia caused by AmpC-producing Enterobacteriaceae. The most frequently observed impact was an early appropriate broadening of the antibiotic spectrum in 31 of 71 cases (43.7%). In total, 143 of 165 episodes (86.7%) of monomicrobial bacteremia were correctly identified at genus level by MALDI-TOF. Conclusions. In a low prevalence area for extended spectrum betalactamases (ESBL) and multiresistant gram-negative bacteria, MALDI-TOF performed on blood culture pellets had an impact on the clinical management of 35.1% of all gram-negative bacteremia cases, demonstrating a greater impact than Gram stain reporting. Thus, MALDI-TOF could become a vital second step beside Gram stain in guiding the empirical treatment of patients with bloodstream infection.

Regulatory roles of the GacS/GacA two-component system in plant-associated and other gram-negative bacteria.

The sensor kinase GacS and the response regulator GacA are members of a two-component system that is present in a wide variety of gram-negative bacteria and has been studied mainly in enteric bacteria and fluorescent pseudomonads. The GacS/GacA system controls the production of secondary metabolites and extracellular enzymes involved in pathogenicity to plants and animals, biocontrol of soilborne plant diseases, ecological fitness, or tolerance to stress. A current model proposes that GacS senses a still-unknown signal and activates, via a phosphorelay mechanism, the GacA transcription regulator, which in turn triggers the expression of target genes. The GacS protein belongs to the unorthodox sensor kinases, characterized by an autophosphorylation, a receiver, and an output domain. The periplasmic loop domain of GacS is poorly conserved in diverse bacteria. Thus, a common signal interacting with this domain would be unexpected. Based on a comparison with the transcriptional regulator NarL, a secondary structure can be predicted for the GacA sensor kinases. Certain genes whose expression is regulated by the GacS/GacA system are regulated in parallel by the small RNA binding protein RsmA (CsrA) at a posttranscriptional level. It is suggested that the GacS/GacA system operates a switch between primary and secondary metabolism, with a major involvement of posttranscriptional control mechanisms.

Proinflammatory activity of cell-wall constituents from gram-positive bacteria.

Innate immunity reacts to conserved bacterial molecules. The outermost lipopolysaccharide (LPS) of Gram-negative organisms is highly inflammatory. It activates responsive cells via specific CD14 and toll-like receptor-4 (TLR4) surface receptor and co-receptors. Gram-positive bacteria do not contain LPS, but carry surface teichoic acids, lipoteichoic acids and peptidoglycan instead. Among these, the thick peptidoglycan is the most conserved. It also triggers cytokine release via CD14, but uses the TLR2 co-receptor instead of TLR4 used by LPS. Moreover, whole peptidoglycan is 1000-fold less active than LPS in a weight-to-weight ratio. This suggests either that it is not important for inflammation, or that only part of it is reactive while the rest acts as ballast. Biochemical dissection of Staphylococcus aureus and Streptococcus pneumoniae cell walls indicates that the second assumption is correct. Long, soluble peptidoglycan chains (approximately 125 kDa) are poorly active. Hydrolysing these chains to their minimal unit (2 sugars and a stem peptide) completely abrogates inflammation. Enzymatic dissection of the pneumococcal wall generated a mixture of highly active fragments, constituted of trimeric stem peptides, and poorly active fragments, constituted of simple monomers and dimers or highly polymerized structures. Hence, the optimal constraint for activation might be 3 cross-linked stem peptides. The importance of structural constraint was demonstrated in additional studies. For example, replacing the first L-alanine in the stem peptide with a D-alanine totally abrogated inflammation in experimental meningitis. Likewise, modifying the D-alanine decorations of lipoteichoic acids with L-alanine, or deacylating them from their diacylglycerol lipid anchor also decreased the inflammatory response. Thus, although considered as a broad-spectrum pattern-recognizing system, innate immunity can detect very subtle differences in Gram-positive walls. This high specificity underlines the importance of using well-characterized microbial material in investigating the system.

Parity of the Number of Irreducible Factors for Composite Polynomials

Various results on parity of the number of irreducible factors of given polynomials over finite fields have been obtained in the recent literature. Those are mainly based on Swan’s theorem in which discriminants of polynomials over a finite field or the integral ring Z play an important role. In this paper we consider discriminants of the composition of some polynomials over finite fields. The relation between the discriminants of composed polynomial and the original ones will be established. We apply this to obtain some results concerning the parity of the number of irreducible factors for several special polynomials over finite fields.

On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials

In this work, we have mainly achieved the following: 1. we provide a review of the main methods used for the computation of the connection and linearization coefficients between orthogonal polynomials of a continuous variable, moreover using a new approach, the duplication problem of these polynomial families is solved; 2. we review the main methods used for the computation of the connection and linearization coefficients of orthogonal polynomials of a discrete variable, we solve the duplication and linearization problem of all orthogonal polynomials of a discrete variable; 3. we propose a method to generate the connection, linearization and duplication coefficients for q-orthogonal polynomials; 4. we propose a unified method to obtain these coefficients in a generic way for orthogonal polynomials on quadratic and q-quadratic lattices. Our algorithmic approach to compute linearization, connection and duplication coefficients is based on the one used by Koepf and Schmersau and on the NaViMa algorithm. Our main technique is to use explicit formulas for structural identities of classical orthogonal polynomial systems. We find our results by an application of computer algebra. The major algorithmic tools for our development are Zeilberger’s algorithm, q-Zeilberger’s algorithm, the Petkovšek-van-Hoeij algorithm, the q-Petkovšek-van-Hoeij algorithm, and Algorithm 2.2, p. 20 of Koepf's book "Hypergeometric Summation" and it q-analogue.

Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM

We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a delta-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on delta. We apply the obtained estimates to show exponential convergence with rate O(exp(−b square root N)), N being the number of degrees of freedom and b>0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(−b cubic root N )), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

L-orthogonal polynomials associated with related measures

A positive measure psi defined on [a, b] such that its moments mu(n) = integral(b)(a)t(n) d psi(t) exist for n = 0, +/-1, +/-2. can be called a strong positive measure on [a, b] When 0 <= a < b <= infinity the sequence of polynomials {Q(n)} defined by integral(b)(a) t(-n+s) Q(n)(t) d psi(t) = 0, s = 0, ., n - 1, exist and they are referred here as L-orthogonal polynomials We look at the connection between two sequences of L-orthogonal polynomials {Q(n)((1))} and {Q(n)((0))} associated with two closely related strong positive measures and th defined on [a, b]. To be precise, the measures are related to each other by (t - kappa) d psi(1)(t) = gamma d psi(0)(t). where (t - kappa)/gamma is positive when t is an element of (n, 6). As applications of our study. numerical generation of new L-orthogonal polynomials and monotonicity properties of the zeros of a certain class of L-orthogonal polynomials are looked at. (C) 2010 IMACS Published by Elsevier B V All rights reserved