On the zeros of polynomials: An extension of the Enestrom-Kakeya theorem

This paper presents an extension of the Enestrom-Kakeya theorem concerning the roots of a polynomial that arises from the analysis of the stability of Brown (K, L) methods. The generalization relates to relaxing one of the inequalities on the coefficients of the polynomial. Two results concerning the zeros of polynomials will be proved, one of them providing a partial answer to a conjecture by Meneguette (1994)[6]. (C) 2011 Elsevier B.V. All rights reserved.

The Effect of a Supragingival Plaque-Control Regimen on the Subgingival Microbiota in Smokers and Never-Smokers: Evaluation by Real-Time Polymerase Chain Reaction

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

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).

Sharp bounds for the extreme zeros of classical orthogonal polynomials

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

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

BASIC HYPERGEOMETRIC FUNCTIONS and ORTHOGONAL LAURENT POLYNOMIALS

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

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

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A Characterization of L-orthogonal Polynomials from Three Term Recurrence Relations

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

Blumenthal's theorem for Laurent orthogonal polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B-n(x) = (x - beta(n))beta(n-1)(x) - alpha(n)xB(n-2)(x), with positive recurrence coefficients alpha(n+1),beta(n) (n = 1, 2,...). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where alpha(n) --> alpha and beta(n) --> beta and show that the zeros of beta(n) are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials. (C) 2002 Elsevier B.V. (USA).

Chain sequences and symmetric generalized orthogonal polynomials

in this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K-n((lambda.,M,k)) associated with the probability measure dphi(lambda,M,k;x), which is the Gegenbauer measure of parameter lambda + 1 with two additional mass points at +/-k. When k = 1 we obtain information on the polynomials K-n((lambda.,M)) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K-n((lambda,M,k)) in relation to M and k are also given. (C) 2002 Elsevier B.V. B.V. All rights reserved.

The influence of cervical finish line, internal relief, and cement type on the cervical adaptation of metal crowns

Objective: the aim of this investigation was to evaluate the cervical adaptation of metal crowns under several conditions, namely (1) variations in the cervical finish line of the preparation, (2) application of internal relief inside the crowns, and (3) cementation using different luting materials. Method and Materials: One hundred eighty stainless-steel master dies were prepared simulating full crown preparations: 60 in chamfer (CH), 60 in 135-degree shoulder (OB), and 60 in rounded shoulder (OR). The finish lines were machined at approximate dimensions of a molar tooth preparation (height: 5.5 mm; cervical diameter: 8 mm; occlusal diameter: 6.4 mm; taper degree: 6; and cervical finish line width: 0.8 mm). One hundred eighty corresponding copings with the same finish lines were fabricated. A 30-mu m internal relief was machined 0.5 mm above the cervical finish line in 90 of these copings. The fit of the die and the coping was measured from all specimens (L0) prior to cementation using an optical microscope. After manipulation of the 3 types of cements (zinc phosphate, glass-ionomer, and resin cement), the coping was luted on the corresponding standard master die under 5-kgf loading for 4 minutes. Vertical discrepancy was again measured (L1), and the difference between L1 and L0 indicated the cervical adaptation. Results: Significant influence of the finish line, cement type, and internal relief was observed on the cervical adaptation (P < .001). The CH type of cervical finish line resulted in the best cervical adaptation of the metal crowns regardless of the cement type either with or without internal relief (36.6 +/- 3 to 100.8 +/- 4 mu m) (3-way analysis of variance and Tukey's test, alpha = .05). The use of glass-ionomer cement resulted in the least cervical discrepancy (36.6 +/- 3 to 115 +/- 4 mu m) than those of other cements (45.2 +/- 4 to 130.3 +/- 2 mu m) in all conditions. Conclusion: the best cervical adaptation was achieved with the chamfer type of finish line. The internal relief improved the marginal adaptation significantly, and the glass-ionomer cement led to the best cervical adaptation, followed by zinc phosphate and resin cement.

Effects of light-curing time on the cytotoxicity of a restorative resin composite applied to an immortalized odontoblast-cell line

This in vitro study evaluated the cytotoxic effects of a restorative resin composite applied to an immortalized odontoblast-cell line (MDPC-23). Seventy-two round resin discs (2-mm thick and 4 mm in diameter) were light-cured for 20 or 40 seconds and rinsed, or not, with PBS and culture medium. The resin discs were divided into four experimental groups: Group 1: Z-100/20 seconds; Group 2: Z-100/20 seconds/rinsed; Group 3: Z100/40 seconds; Group 4: Z-100/40 seconds/rinsed. Circular filter paper was used as a control material (Group 5). The round resin discs and filter papers were placed in the bottom of wells of four 24-well dishes (18 wells for each experimental and control group). MDPC-23 cells (30,000 cells/cm(2)) were plated in the wells and allowed to incubate for 72 hours. The zone of inhibition around the resin discs was measured under inverted light microscopy; the MTT assay was carried out for mitochondrial respiration and cell morphology was measured under SEM. The scores obtained from inhibition zone and MTT assay were analyzed with the Kruskal-Wallis followed by Dunnett tests. In Groups 1, 2, 3 and 4, the thickness of the inhibition zone was 1,593 +/- 12.82 mum, 403 +/- 15.49 mum, 1,516 +/- 9.81 mum and 313 +/- 13.56 mum, respectively. There was statistically significant difference among the experimental and control groups at the 0.05 level of significance. The MTT assay demonstrated that the resin discs of the experimental groups 1, 2, 3 and 4 reduced the cell metabolism by 83%, 40.1%, 75.5% and 24.5%. Only between the Groups 2 and 4 was there no statistically significant difference for mitochondrial respiration. Close to the resin discs, the MDPC-23 cells exhibited rounded shapes, with only a few cellular processes keeping the cells attached to the substrate or, even disruption of plasma membrane. Adjacent to the inhibition zone, the cultured cells exhibited multiple fine cellular processes on the cytoplasmic membrane organized in epithelioid nodules, similar to the morphology observed to the control group. Based on the results, the authors may conclude that the Z-100 resin composite light cured for 20 seconds was more cytopathic to MDPC-23 cells than Z-100 light cured for 40 seconds. The cytotoxic effects of the resin discs decreased after rinsing them with PBS and culture medium. This was confirmed by MTT assay and upon evaluation of the inhibition zone, which was narrower following rinsing of the resin discs.

Szego type polynomials and para-orthogonal polynomials

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

On the injectivity of C1 maps of the real plane

Let X : ℝ2 → ℝ2 be a C1 map. Denote by Spec(X) the set of (complex) eigenvalues of DXp when p varies in ℝ2. If there exists ε > 0 such that Spec(X) ∩ (-ε, ε) = ∅, then X is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.