Fibered neighborhoods of curves in surfaces

The aim of this article is to study fibered neighborhoods of compact holomorphic curves embedded in surfaces. It is shown that when the self-intersection number of the curve is sufficiently negative the fibration is equivalent to the linear one defined in the normal bundle to the curve. The obstructions to equivalence in the general case are described.     

Quantitative relationship of application rate and pesticide residues in greenhouse tomatoes

The association between application rate of a pesticide and its residue in ripe tomatoes was studied. The average residue level (R) of any pesticide in ripe tomatoes remained in quantitative relation to its dose (D), expressed by the following regression equation: R = 0.24 D (mg/kg), where the numerical factor, 0.24, represents the average residue in mg/kg after application of 1 kg active ingredient per hectare with relative standard deviation of 23%. Quantitative association between these 2 factors enables evaluation of greenhouse tomato growers with respect to their observation of Good Agricultural Practice rules and the Plant Protection Act, obligatory in Poland since 1996, and thus may be a reliable basis for the registration of new agrochemicals.     

Liouvillian integration and Bernoulli foliations

Analytic foliations in the 2-dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable. It is shown that such a condition leads to the existence of a Liouville type first integral, and, under ``generic' extra conditions, it is proven that these foliations can be defined by Bernoulli equations.

     

On the quadratic convergence of Newton’s method under center-Lipschitz but not necessarily Lipschitz hypotheses

We provide new local and semilocal convergence results for Newton’s method in a Banach space. The sufficient convergence conditions do not include the Lipschitz constant usually associated with Newton’s method. Numerical examples demonstrating the expansion of Newton’s method are also provided in this study.     

On the solution of systems of equations with constant rank derivatives

The famous for its simplicity and clarity Newton–Kantorovich hypothesis of Newton’s method has been used for a long time as the sufficient convergence condition for solving nonlinear equations. Recently, in the elegant study by Hu et al. (J Comput Appl Math 219:110–122, 2008), a Kantorovich-type convergence analysis for the Gauss–Newton method (GNM) was given improving earlier results by Häubler (Numer Math 48:119–125, 1986), and extending some results by Argyros (Adv Nonlinear Var Inequal 8:93–99, 2005, 2007) to hold for systems of equations with constant rank derivatives. In this study, we use our new idea of recurrent functions to extend the applicability of (GNM) by replacing existing conditions by weaker ones. Finally, we provide numerical examples to solve equations in cases not covered before (Häubler, Numer Math 48:119–125, 1986; Hu et al., J Comput Appl Math 219:110–122, 2008; Kontorovich and Akilov 2004).     

A derivative free iterative method for solving least squares problems

A derivative free iterative method for approximating a solution of nonlinear least squares problems is studied first in Shakhno and Gnatyshyn (Appl Math Comput 161:253–264, 2005). The radius of convergence is determined as well as usable error estimates. We show that this method is faster than its Secant analogue examined in Shakhno and Gnatyshyn (Appl Math Comput 161:253–264, 2005). Numerical example is also provided in this paper.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

On the weakening of the convergence of Newton’s method using recurrent functions

We use Newton’s method to approximate a locally unique solution of an equation in a Banach space setting. We introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton’s method than before [J. Appell, E. De Pascale, J.V. Lysenko, P.P. Zabrejko, New results on Newton–Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997) 1–17; I.K. Argyros, The theory and application of abstract polynomial equations, in: Mathematics Series, St. Lucie/CRC/Lewis Publ., Boca Raton, Florida, USA, 1998; I.K. Argyros, Concerning the “terra incognita” between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005) 179–194; I.K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer-Verlag Publ., New York, 2008; S. Chandrasekhar, Radiative Transfer, Dover Publ., New York, 1960; F. Cianciaruso, E. De Pascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003) 713–723; N.T. Demidovich, P.P. Zabrejko, Ju.V. Lysenko, Some remarks on the Newton–Kantorovich method for nonlinear equations with Hölder continuous linearizations, Izv. Akad. Nauk Belorus 3 (1993) 22–26. (in Russian); E. De Pascale, P.P. Zabrejko, Convergence of the Newton–Kantorovich method under Vertgeim conditions: A new improvement, Z. Anal. Anwendvugen 17 (1998) 271–280; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations, Dokl. Akad. Nauk BSSR 38 (1994) 20–24. (in Russian); B.A. Vertgeim, On conditions for the applicability of Newton’s method, (Russian), Dokl. Akad. Nauk., SSSR 110 (1956) 719–722; B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957) 166–169. (in Russian); English transl.:; Amer. Math. Soc. Transl. 16 (1960) 378–382] provided that the Fréchet-derivative of the operator involved is p$p$-Hölder continuous (p∈(0,1]$p\in (0,1]$).     

Directional Secant-Type Methods for Solving Equations

A semilocal convergence analysis for directional Secant-type methods in multidimensional space is provided. Using weaker hypotheses than the ones exploited by An and Bai, we provide a semilocal convergence analysis with the following advantages: weaker convergence conditions, larger convergence domain, finer error estimates on the distances involved, and more precise information on the location of the solution. A numerical example, where our results apply to solve an equation but not the ones of An and Bai, is also provided. In a second example, we show how to implement the method.