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Exercises and solutions in PDF

On The Critical Points of some Iteration Methods for Solving Algebraic Equations. Global Convergence Properties

In this work we give su±cient conditions for k-th approximations of the polynomial roots of f(x) when the Maehly{Aberth{Ehrlich, Werner-Borsch-Supan, Tanabe, Improved Borsch-Supan iteration methods fail on the next step. For these methods all non-attractive sets are found. This is a subsequent improvement of previously developed techniques and known facts. The users of these methods can use the results presented here for software implementation in Distributed Applications and Simulation Environ- ments. Numerical examples with graphics are shown.

On the Critical Points of Kyurkchiev’s Method for Solving Algebraic Equations

This paper is dedicated to Prof. Nikolay Kyurkchiev on the occasion of his 70th anniversary This paper gives sufficient conditions for kth approximations of the zeros of polynomial f (x) under which Kyurkchiev’s method fails on the next step. The research is linked with an attack on the global convergence hypothesis of this commonly used in practice method (as correlate hypothesis for Weierstrass–Dochev’s method). Graphical examples are presented.

On Arrangements of Real Roots of a Real Polynomial and Its Derivatives

2000 Mathematics Subject Classification: 12D10.

A method for solving polynomial equations by continued fractions.

Bibliography: p. 16.

On the Hyperbolicity Domain of the Polynomial x^n + a1x^(n-1) + 1/4+ an

∗ Partially supported by INTAS grant 97-1644

On the Factorization of the Poincaré Polynomial: A Survey

2000 Mathematics Subject Classification: 13P05, 14M15, 14M17, 14L30.

Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences

ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.

Short-term Physiological Changes In Roots And Leaves Of Sugarcane Varieties Exposed To H2o2 In Root Medium.

The aim of this study was to evaluate the differential sensitivity of sugarcane genotypes to H2O2 in root medium. As a hypothesis, the drought tolerant genotype would be able to minimize the oxidative damage and maintain the water transport from roots to shoots, reducing the negative effects on photosynthesis. The sugarcane genotypes IACSP94-2094 (drought tolerant) and IACSP94-2101 (drought sensitive) were grown in a growth chamber and exposed to three levels of H2O2 in nutrient solution: control; 3mmolL(-1) and 80mmolL(-1). Leaf gas exchange, photochemical activity, root hydraulic conductance (Lr) and antioxidant metabolism in both roots and leaves were evaluated after 15min of treatment with H2O2. Although, root hydraulic conductance, stomatal aperture, apparent electron transport rate and instantaneous carboxylation efficiency have been reduced by H2O2 in both genotypes, IACSP94-2094 presented higher values of those variables as compared to IACSP94-2101. There was a significant genotypic variation in relation to the physiological responses of sugarcane to increasing H2O2 in root tissues, being root changes associated with modifications in plant shoots. IACSP94-2094 presented a root antioxidant system more effective against H2O2 in root medium, regardless H2O2 concentration. Under low H2O2 concentration, water transport and leaf gas exchange of IACSP94-2094 were less affected as compared to IACSP94-2101. Under high H2O2 concentration, the lower sensitivity of IACSP94-2094 was associated with increases in superoxide dismutase activity in roots and leaves and increases in catalase activity in roots. In conclusion, we propose a general model of sugarcane reaction to H2O2, linking root and shoot physiological responses.

Fracture resistance of weakened roots restored with composite resin and glass fiber post

This study evaluated the fracture resistance of weakened roots restored with glass fiber posts, composite resin cores and complete metal crowns. Thirty maxillary canines were randomly divided into 3 groups of 10 teeth each: teeth without weakened roots (control); teeth with partially weakened roots (PWR) and teeth with and largely weakened roots (LWR). The control group was restored with glass fiber posts and a composite resin core. Teeth in the PWR and LWR groups were flared internally to standardized dimensions in order to simulate root weakness. Thereafter, the roots were partially filled with composite resin and restored in the same way as in the control group. The specimens were exposed to 250,000 cycles in a controlled chewing simulator. All intact specimens were subjected to a static load (N) in a universal testing machine at 45 degrees to the long axis of the tooth until failure. Data were analyzed by one-way ANOVA and Dunnett's test for multiple comparisons (p=0.05). There were statistically significant difference differences (p<0.01) among the groups (control group = 566.73 N; PWR = 409.64 N; and LWR = 410.91 N), with significantly higher fracture strength for the control group. There was no statistically significant difference (p>0.05) between the weakened groups. The results of this study showed that thicker root dentin walls significantly increase the fracture resistance of endodontically treated teeth.

Evaluation of the maxillary premolar roots dissociation using radiographic holders with conventional and digital radiography

This in vivo study evaluated the dissociation quality of maxillary premolar roots combining variations of vertical and horizontal angulations by using X-ray holders (Rinn -XCP), and made a comparison between two types of intraoral radiography systems - conventional film (Kodak Insight, Rochester, USA) and digital radiography (Kodak RVG 6100, Kodak, Rochester, USA). The study sample was comprised of 20 patients with a total of 20 maxillary premolars that were radiographed, using the paralleling angle technique (GP), with a 20º variation of the horizontal angle (GM) and 25º variation of the horizontal angle combined with 15º vertical angle (GMV). Each image was independently analyzed by two experienced examiners. These examiners assigned a score to the diagnostic capability of root dissociation and the measurement of the distance between the apexes. Statistical data was derived using the Wilcoxon Signed Rank test, Friedman and T test. The means of the measured distances between buccal and lingual root apexes were greater for the GMV, which ranged from 2.3 mm to 3.3 mm. A statistically significant difference was found between GM and GMV when compared to GP with p < 0.01. An established best diagnostic dissociation roots image was found in the GMV. These results support the use of the anterior X-ray holders which offer a better combined deviation (GMV) to dissociate maxillary premolar roots in both radiography systems.

Direct Computation of Operational Matrices for Polynomial Bases

Several numerical methods for boundary value problems use integral and differential operational matrices, expressed in polynomial bases in a Hilbert space of functions. This work presents a sequence of matrix operations allowing a direct computation of operational matrices for polynomial bases, orthogonal or not, starting with any previously known reference matrix. Furthermore, it shows how to obtain the reference matrix for a chosen polynomial base. The results presented here can be applied not only for integration and differentiation, but also for any linear operation.

Resprouting from roots in four Brazilian tree species

Previous studies pointed out that species richness and high density values within the Leguminosae in Brazilian forest fragments affected by fire could be due, at least partially, to the high incidence of root sprouting in this family. However, there are few Studies of the factors that induce root sprouting in woody plants after disturbance. We investigated the bud formation on root cuttings, and considered a man-made disturbance that isolates the root from the shoot apical dominance of three Leguminosae (Bauhinia forficata Link., Centrolobium tomentosum Guill. ex Benth, and Inga laurina (Sw.) Willd) and one Rutaceae (Esenbeckia febrifuga (St. Hit.) Juss. ex Mart.). All these species resprout frequently after fire. We also attempted to induce bud formation on root systems by removing the main trunk, girdling or sectioning the shallow lateral roots from forest tree species Esenbeckia febrifuga and Hymenaea courbaril L. We identified the origin of shoot primordia and their early development by fixing the samples in Karnovsky solution, dehydrating in ethyl alcohol series and embedding in plastic resin. Serial sections were cut on a rotary microtome and stained with toluidine blue O. Permanent slides were mounted in synthetic resin. We observed different modes of bud origin on root cuttings: close to the vascular cambium (C. tomentosum), from the callus (B. forficata and E febrifuga) and from the phloematic parenchyma proliferation (L laurina). Fragments of B. forficala root bark were also capable of forming reparative buds from healing phellogen formed in callus in the bark's inner side. In the attempt of bud induction on root systems, Hymenaea courbaril did not respond to any of the induction tests, probably because of plant age. However, Esenbeckia febrifuga roots formed suckers when the main trunk was removed or their roots were sectioned and isolated from the original plant. We experimentally demonstrated the ability of four tree species to resprout from roots after disturbance. Our results suggest that the release of apical dominance enables root resprouting in the studied species. Rev. Biol. Trop. 57 (3): 789-800. Epub 2009 September 30.

Minimal Nielsen Root Classes and Roots of Liftings

Given a continuous map f : K -> M from a 2-dimensional CW complex into a closed surface, the Nielsen root number N(f) and the minimal number of roots mu(f) of f satisfy N(f) <= mu(f). But, there is a number mu(C)(f) associated to each Nielsen root class of f, and an important problem is to know when mu(f) = mu(C)(f)N(f). In addition to investigate this problem, we determine a relationship between mu(f) and mu((f) over tilde), when (f) over tilde f is a lifting of f through a covering space, and we find a connection between this problems, with which we answer several questions related to them when the range of the maps is the projective plane.

Functional specialization of Eucalyptus fine roots: contrasting potential uptake rates for nitrogen, potassium and calcium tracers at varying soil depths

1. Little is known about the role of deep roots in the nutrition of forest trees and their ability to provide a safety-net service taking up nutrients leached from the topsoil. 2. To address this issue, we studied the potential uptake of N, K and Ca by Eucalyptus grandis trees (6 years of age - 25 m mean height), in Brazil, as a function of soil depth, texture and water content. We injected NO(3)(-)- (15)N, Rb(+) (analogue of K(+)) and Sr(2+) (analogue of Ca(2+)) tracers simultaneously in a solution through plastic tubes at 10, 50, 150 and 300 cm in depth in a sandy and a clayey Ferralsol soil. A complete randomized design was set up with three replicates of paired trees per injection depth and soil type. Recently expanded leaves were sampled at various times after tracer injection in the summer, and the experiment was repeated in the winter. Soil water contents were continuously monitored at the different depths in the two soils. 3. Determination of foliar Rb and Sr concentrations and (15)N atom % made it possible to estimate the relative uptake potential (RUP) of tracer injections from the four soil depths and the specific RUP (SRUP), defined as RUP, per unit of fine root length density in the corresponding soil layer. 4. The highest tracer uptake rates were found in the topsoil, but contrasting RUP distributions were observed for the three tracers. Whilst the RUP was higher for NO(3)(-)- (15)N than for Rb(+) and Sr(2+) in the upper 50 cm of soil, the highest SRUP values for Sr(2+) and Rb(+) were found at a depth of 300 cm in the sandy soil, as well as in the clayey soil when gravitational solutions reached that depth. 5. Our results suggest that the fine roots of E. grandis trees exhibit contrasting potential uptake rates with depth depending on the nutrient. This functional specialization of roots might contribute to the high growth rates of E. grandis trees, efficiently providing the large amounts of nutrients required throughout the development of these fast-growing plantations.