Why using symbolic computation for proving the convergence of iterative methods?

Author: Gheorghe Ardelean / Laszlo Balog

Abstract: In [YoonMe Ham et al., Some higher-order modifications of Newton's method for solving nonlinear equations, J. Comput. Appl. Math., 222 (2008) 477--486], some higher-order modifications of Newton's method for solving nonlinear equations are presented. In [Liang Fang et al., Some modifications of Newton's method with higher-order convergence for solving nonlinear equations, J. Comput. Appl. Math., 228 (2009) 296--303], the authors point out some flaws in the results of YoonMe Ham et al. and present some modified variants of the method. In this paper we point out that the paper of Liang Fang et al. itself contains some flaw results and we correct them by using symbolic computation in Mathematica. Moreover, we show that the main result in Theorem 3 of Liang Fang et al. is wrong. The order of convergence of the method is not $3m+2$, but is $2m+4$. We give the general expression of convergence error too.
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No. of downloads: 215

Interpolation operators on a triangle with two and three curved edges

Author: Alina Baboș

Abstract: We construct Lagrange, Hermite and Birkhoff-type operators, which
interpolate a given function and some of its derivatives on the
border of a triangle with two and three curved edges. We also
consider some of their product and boolean sum operators. We study the interpolation properties and the degree of exactness of the constructed operators.
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No. of downloads: 162

The role of the Pompeiu-Hausdorff metric in fixed point theory

Author: Vasile Berinde / Mădălina Păcurar

Abstract: The main aim of this note is to highlight the role of the Pompeiu-Hausdorff metric in fixed point theory and, subsidiarily, to touch some issues related to the history of this fundamental concept in modern mathematics. This will allow us to conclude that what is nowadays almost generally called Hausdorff metric (distance) and very seldom Hausdorff-Pompeiu metric (distance) or Pompeiu-Hausdorff metric (distance), should be fairly and correctly named Pompeiu-Hausdorff metric (distance).
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No. of downloads: 177

QSTR mathematical models for the toxicity of aliphatic carboxylic acids on tetrahymena pyriformis

Author: Zoița Mărioara Berinde

Abstract: The present work represents an attempt to improve QSTR models for aquatic toxicity of $38$ aliphatic carboxylic acids tested in the Tetrahymena pyriformis population growth assay, using the topological index ZEP and the following nine main electrotopological and molecular descriptors: acidic dissociation constant, 1-octanol/water partition coefficient, the energy of the lowest unoccupied molecular orbital, acidic dissociation constant, molar refractivity, refraction index, surface tension, polarizability and, electrotopological states. Several different relations between toxicity [$\log$(IGC50-1)] and the molecular and topological properties were examined, and a group of multiple linear regression models with high fitness scores were generated
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No. of downloads: 122

Schurer operators of King type

Author: Petru I. Braica / Ovidiu T. Pop / Dan Bărbosu

Abstract: A class of linear and positive operators defined by finite sum which generalizes the classical Schurer's operators in the King sense is constructed. For the mentioned class of operators, uniform convergences results, error estimations in terms of modulus of continuity and Voronovskaja type theorems are established.
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No. of downloads: 180

Fixed points for mappings defined on pseudometric spaces

Author: Mitrofan M. Choban

Abstract: In this article, the distinct classes of continuous pseudometrics and metrics
(perfect, quasi-perfect, sequentially complete pseudometrics and metrics)
are defined and studied in depth.
The conditions under which the set of fixed points of a given mapping of a space with concrete
pseudometric is non-empty are determined.
Some examples are proposed. For spaces with pseudometrics there are proved the Bishop-Phelps,
Takahashi, Caristy and other theorems.
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No. of downloads: 156

Cubic differential systems with $2$-invariant straight lines

Author: Dumitru Cozma

Abstract: For cubic differential system with a singular point $O(0, 0)$ of a
center or a focus type having three invariant straight lines of
which one is 2-invariant it is proved that the origin is a center
if and only if the first five Lyapunov quantities vanish.
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No. of downloads: 137

On a functional Fredholm integral equation, via the technique of nonexpansive operators

Author: Monica Lauran

Abstract: In this paper we present one result relative to existence of solutions of the functional Fredholm integral equation with deviating argument by nonexpansive operators technique
\[
u(x,y)= f(x,y,g(u)(x,y)) + \int\limits_{a}^{b}\int\limits_{a}^{b}K(x,y,s,t,u(s,t))dsdt, \,\,\ x,y\in [a,b]
\]
in a subset of a Banach space.
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No. of downloads: 142

Some remarks on the Krasnoselskij iterative method

Author: Dan Ștefan Marinescu / Mihai Monea

Abstract: In this paper we deal with some convergence results regarding the Krasnoselskij iterative scheme. New proofs for known results like Proposition 3.2 or Theorem 1.1 as well as new results like Proposition 3.4 or Proposition 4.5, are presented.
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No. of downloads: 142

Application of pollution indexes, cluster analysis and isocontent chart to the study of soil pollution

Author: Cristina Mihali / Gabriela Oprea / Claudia Butean / Angela Michnea / Nicolae Pop

Abstract: Statistical methods were applied to the assessment of the multi-elements pollution degree of the soil with heavy metals and arsenic (microelements) in Baia Mare area, NW Romania. The content of several microelements were analyzed in soil samples from four zones in the studied area. Pollution indices were used to define the pollution degree compared to the Romanian regulation. Cluster analysis was used to classify the analyzed microelements (Cu, Zn, Pb, Cd, As, Co, Ni, Cr, Sn, Sb). Based on iso-contents maps for Pb and As, their contents in soil in whole Baia Mare area are assessed. The sampling point coordinates (latitude and longitude) were registered with a GPS (Global Positioning System) instrument. Considering the coordinates of each sampling point and the corresponding concentration of the considered microelement in the sampling point, a matrix XYZ was obtained where X was the longitude; Y was the latitude and Z the concentration of the microelements. A continuous surface was generated beginning with each sampling point. The spatial distribution charts drawn for the most hazardous microelements Pb and As indicated some area high polluted around the pollution source (a lead smelter).
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No. of downloads: 138

On the boundary of algebraic radicals in topological $(n,m)$-semirings

Author: Adina Pop

Abstract: The aim of this paper is to investigate some properties for an ideal radical and an ideal radical boundary in a commutative Hausdorff topological $(n,m)$-semiring. We are going to give some generalizations of results due to Shum [Shum, K. P, On the boundary of algebraic radicals in topological semigroups, Acta Mathematica Acad. Scient. Hung., 25(1974), No. (1-2), 15--19], Chow [Chow, H. L., Remarks on boundaries in semigroups, Period. Math. Hungar., 7 (1976), No. 2, 137--139] relative to semigroups and due Maria S. Pop [Pop, M. S., On boundary in topological $n$-semigroups, Mathematica, 22 (45) (1980), No. 1, 127--130] relative to $n$-semigroups.
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No. of downloads: 136

Testing artificial neural network for hand gesture recognition

Author: JÃ²zsef SÃ¼tÅ‘ / Ștefan Oniga

Abstract: Neural networks are well applicable for gesture recognition. In this article we want to present the result of an artificial feed forward network for a simplified hand gesture recognition problem. In neural networks, the learning algorithm is very important because the performance of neural network depends on it. One of the most known learning algorithm is the backpropagation. There are some mathematical software which provides acceptable result for a given problem by a backpropagation based network, but in some cases a high-level programming language implemented program can provide better solution. The main topics of the article cover the structure of the test environment, the mathematical background of the implemented methods, some programming remarks and the test results.
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No. of downloads: 155

Graphical representations for errors arising by approximation with bivariate operator of Stancu type

Author: Ioana Tașcu / Andrei Horvat-Marc

Abstract: In this paper, we will obtain some graphic representations for the polynomial approximation of particular bivariate functions. We present three examples of approximations by Stancu operator. The selected functions include an example of production function. These could be further used for teaching purposes.
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No. of downloads: 150

Approximating solutions of generalized pseudocontractive variational inequalities by admissible perturbation type iterative methods

Author: Cristina Țicală

Abstract: In this paper we give the solvability class of generalized strongly nonlinear variational inequalities modified by the use of the new concept of admissible perturbation operator on nonempty closed convex sets in Hilbert spaces.
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