Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives

Bandura, Andriy; Скасків, О. Б.

doi:10.1155/2017/3253095

Cited by 14 publications

(11 citation statements)

References 19 publications

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“…We also need the following assertions. They are generalizations of corresponding propositions for entire functions of bounded L-index in direction [3,16] and of bounded L-index in joint variables ( [10,17]) and of bounded index ( [23]).…”

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confidence: 99%

“…The least such integer n 0 is called the L-index in joint variables of the function F and is denoted by N (F, L, B n ). There are many papers about entire functions of several variables of bounded index ( [21,22,24,[27][28][29]) and of bounded L-index in joint variables ( [5,[9][10][11]17]). By Q(B n ) we denote the class of functions L, satisfying (1) and the following condition…”

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confidence: 99%

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Analytic functions in a unit ball of bounded $\mathbf{L}$-index: asymptotic and local properties

Bandura¹,

Скасків²

2016

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We have generalized some criteria of boundedness of L-index in joint variables for analytic functions in the unit ball, where L : B n → R n + is a continuous vector-function, B n is the unit ball in C n. One of propositions gives an estimate of the coefficients of power series expansions by a dominating homogeneous polynomial for analytic functions in the unit ball. Also we provide growth estimates of these functions. They describe the behavior of maximum modulus of analytic function on a skeleton in a polydisc by behavior of the function L. Most of our results are based on polydisc exhaustion of the unit ball. Nevertheless, we have generalized criteria of boundedness of L-index in joint variables which describe local behavior of partial derivatives on sphere in C n. The proposition uses a ball exhaustion. An analog of Hayman's theorem is applied to investigation of boundedness of L-index in joint variables for analytic solutions in the unit ball of some linear higher-order systems of PDE's. There were found sufficient conditions providing the boundedness. Growth estimates of analytic solutions in the unit ball are also obtained.

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Analytic functions in a unit ball of bounded $\mathbf{L}$-index: asymptotic and local properties

Bandura¹,

Скасків²

2016

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“…But it is possible to replace the gradient with a directional derivative starting from (1) and to study a similar problem along a direction. There are many papers analyzing the influence of directional derivatives on the properties of such functions [ 94 , 95 , 96 ].…”

Section: Resultsmentioning

confidence: 99%

Multiclass Level-Set Segmentation of Rust and Coating Damages in Images of Metal Structures

Bembenek

Mandziy

Ivasenko

et al. 2022

Sensors

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This paper describes the combined detection of coating and rust damages on painted metal structures through the multiclass image segmentation technique. Our prior works were focused solely on the localization of rust damages and rust segmentation under different ambient conditions (different lighting conditions, presence of shadows, low background/object color contrast). This paper method proposes three types of damages: coating crack, coating flaking, and rust damage. Background, paint flaking, and rust damage are objects that can be separated in RGB color-space alone. For their preliminary classification SVM is used. As for paint cracks, color features are insufficient for separating it from other defect types as they overlap with the other three classes in RGB color space. For preliminary paint crack segmentation we use the valley detection approach, which analyses the shape of defects. A multiclass level-set approach with a developed penalty term is used as a framework for the advanced final damage segmentation stage. Model training and accuracy assessment are fulfilled on the created dataset, which contains input images of corresponding defects with respective ground truth data provided by the expert. A quantitative analysis of the accuracy of the proposed approach is provided. The efficiency of the approach is demonstrated on authentic images of coated surfaces.

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“…An entire function F(z) is called a function of bounded L-index in joint variables [1,2], if there exists a number m ∈ Z+ such that for each J = (j 1 , j 2 , . .…”

Section: Introductionmentioning

confidence: 99%

Analytic functions in the unit ball of bounded L-index in joint variables and of bounded 𝐿-index in direction: a connection between these classes

Bandura

Скасків

2019

Demonstratio Mathematica

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We give negative answer to the question of Bordulyak and Sheremeta for more general classes of entire functions than in the original formulation: Does index boundedness in joint variables for an entire function F imply index boundedness in the variable zj for the function F? This question is addressed for entire functions of bounded L-index in joint variables and entire functions of bounded L-index in direction. We also present a class of analytic functions in the unit ball which has bounded L-index in joint variables and has unbounded l-index in the variables z1 and z2 for any positive continuous function l : B2 → C.

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Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives

Cited by 14 publications

References 19 publications

Analytic functions in a unit ball of bounded $\mathbf{L}$-index: asymptotic and local properties

Analytic functions in a unit ball of bounded $\mathbf{L}$-index: asymptotic and local properties

Multiclass Level-Set Segmentation of Rust and Coating Damages in Images of Metal Structures

Analytic functions in the unit ball of bounded L-index in joint variables and of bounded 𝐿-index in direction: a connection between these classes

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