Growth Estimates for Analytic Vector-Valued Functions in the Unit Ball Having Bounded $\mathbf{L}$-index in Joint Variables

Baksa, V. P.; Bandura, Andriy; Скасків, О. Б.

doi:10.33205/cma.650977

Cited by 4 publications

(3 citation statements)

References 19 publications

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“…The analysis of variance for the applied models showed (Table 6) that the developed models meet the above requirements, except for the mathematical model for the CV(I). All regression models ( 12)-( 17) as mathematical functions have multivariate functions with bounded indexes in any direction [95] and bounded indexes in joint variables [96], and their indexes equal the corresponding maximum degree. The regression model of the fill coefficient has the following form:…”

Section: Prediction Of Mathematic Model Of Welding Current and Arc Vo...mentioning

confidence: 99%

Effect of Exothermic Additions in Core Filler on Arc Stability and Microstructure during Self-Shielded, Flux-Cored Arc Welding

Lozynskyi,

Trembach,

Katinas

et al. 2024

Crystals

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In the conditions of an energy crisis, an important issue is the increase in energy efficiency and productivity of welding and hardfacing processes. The article substantiates the perspective of using exothermic additives introduced into core filler for flux-cored wire arc welding processes as a relatively cheap additional heat source, reducing energy consumption when melting filler materials, and increasing the deposition rate. The mixture design (MD) was selected as the design method to optimize the average values of current and voltage, as well as arc stability parameters depending on core filler composition. This article studies the influence of the introduction of exothermic addition (EA), as well as the ratios CuO/C and CuO/Al on arc stability for the FCAW S process. Parameters characterizing arc stability were determined using an oscillograph, and from the obtained oscillograms, an analysis was conducted on arc voltage and welding current signals during flux-cored arc welding. It was determined that various methods can be used to evaluate arc stability, which can be divided into two groups: graphical (current and voltage cyclograms, box plots with frequency histograms, ellipse parameters plotted on current, and voltage cyclograms) and statistical (standard variation and coefficients of variation for welding current and arc voltage). In this paper, a comprehensive evaluation of arc stability depending on the composition of the cored wire filler was carried out. It was determined that the most stable current parameters were observed for the flux-cored wire electrode with an average exothermic addition content at the level of EA = 26.5–28.58 wt.% and a high carbon content (low values of CuO/C = 3.75). Conversely, the lowest values of arc stability (CV(U) and Std(U)) were observed during hardfacing with a flux-cored wire electrode with a high CuO/Al ratio ≥ 4.5 and a content of exothermic addition in the core filler below the average EA < 29 wt.%. Mathematical models of mean values, standard deviation, coefficient of variation for welding current, and arc voltage were developed. The results indicated that the response surface prediction models had good accuracy and prediction ability. The developed mathematical models showed that the ratio of oxidizing agent to reducing agent in the composition of exothermic addition (CuO/Al) had the greatest influence on the welding current and arc voltage characteristics under investigation. The percentage of exothermic mixture in the core filler (EA) only affected the average welding current (Iaw) and the average arc voltage (Uaw). The graphite content expressed through the CuO/C ratio had a significant impact on welding current parameters as well as the coefficient of variation of arc voltage (CV(U)). Two welding parameters were selected for optimization: the mean welding current (Iaw) and the standard deviation of arc voltage (Std(U)). The best arc stability when using exothermic addition CuO-Al in the core filler was observed at CuO/Al = 3.6–3.9, CuO/C = 3.5–4.26, and at an average EA content of 29–38 wt.%. The significant influence of the CuO/Al and CuO/C ratios on arc voltage parameters can also be explained by their impact on the elemental composition of the welding arc (copper, cupric oxide (CuO), and Al2O3). The more complete this reaction, the higher the amount of easily vaporized copper (Cu) in the arc plasma, enhancing arc stability. The influence of core filler composition on the microstructure of deposited metal of the Fe-Cr-Cu-Ti alloy system was investigated.

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Section: Prediction Of Mathematic Model Of Welding Current and Arc Vo...mentioning

confidence: 99%

Effect of Exothermic Additions in Core Filler on Arc Stability and Microstructure during Self-Shielded, Flux-Cored Arc Welding

Lozynskyi,

Trembach,

Katinas

et al. 2024

Crystals

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“…The class of all f functions given by ( 1) is denoted by A and the class of all f functions in A, which are univalent, is denoted by S (for more details, see [1]; see also some of the recent studies [2][3][4]). It is well known that every f function in the class S has an inverse map f −1 given by f −1 (w) = w − a 2 w 2 + (2a 2 2 − a 3 )w 3 − (5a 3 2 − 5a 2 a 3 + a 4 )w 4 + .…”

Section: Introductionmentioning

confidence: 99%

Initial Coefficients Estimates and Fekete–Szegö Inequality Problem for a General Subclass of Bi-Univalent Functions Defined by Subordination

Illafe

Yousef

Mohd

et al. 2023

Axioms

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In the present work, we aim to introduce and investigate a novel comprehensive subclass of normalized analytic bi-univalent functions involving Gegenbauer polynomials and the zero-truncated Poisson distribution. For functions in the aforementioned class, we find upper estimates of the second and third Taylor–Maclaurin coefficients, and then we solve the Fekete–Szegö functional problem. Moreover, by setting the values of the parameters included in our main results, we obtain several links to some of the earlier known findings.

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“…In recent papers [1,2,3] V. Baksa, A. Bandura, O. Skaskiv considered vector-valued functions having bounded L-index in joint variables which are analytic in the unit ball. They also extended previous investigations of analytic functions in the unit ball ( [6]).…”

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confidence: 99%

Entire multivariate vector-valued functions of bounded $\mathbf{L}$-index: analog of Fricke’s theorem

Bandura

Baksa

2020

Mat. Stud.

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We consider a class of vector-valued entire functions $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$. For this class of functions there is introduced a concept of boundedness of $\mathbf{L}$-index in joint variables. Let $|\cdot|_p$ be a norm in $\mathbb{C}^p$. Let $\mathbf{L}(z)=(l_{1}(z),\ldots,l_{n}(z))$, where $l_{j}(z)\colon \mathbb{C}^{n}\to \mathbb{R}_+$ is a positive continuous function.An entire vector-valued function $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$ is said to be ofbounded $\mathbf{L}$-index (in joint variables), if there exists $n_{0}\in \mathbb{Z}_{+}$ such that $\displaystyle \forall z\in G \ \ \forall J \in \mathbb{Z}^n_{+}\colon \quad\frac{|F^{(J)}(z)|_p}{J!\mathbf{L}^J(z)}\leq \max \left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^K(z)} \colon K\in \mathbb{Z}^n_{+}, \|K\|\leq n_{0} \right \}.$ We assume the function $\mathbf{L}\colon \mathbb{C}^n\to\mathbb{R}^p_+$ such that $0< \lambda _{1,j}(R)\leq\lambda _{2,j}(R)<\infty$ for any $j\in \{1,2,\ldots, p\}$ and $\forall R\in \mathbb{R}_{+}^{p},$where $\lambda _{1,j}(R)=\inf\limits_{z_{0}\in \mathbb{C}^{p}} \inf \left \{{l_{j}(z)}/{l_{j}(z_{0})}\colon z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \},$ $\lambda _{2,j}(R)$ is defined analogously with replacement $\inf$ by $\sup$.It is proved the following theorem:Let $|A|_p=\max\{|a_j|\colon 1\leq j\leq p\}$ for $A=(a_1,\ldots,a_p)\in\mathbb{C}^p$. An entire vector-valued function $F$ has bounded $\mathbf{L}$-index in joint variables if and only if for every $R\in \mathbb{R}^{n}_+$ there exist $n_{0}\in \mathbb{Z}_{+}$, $p_0>0$ such that for all $z_{0}\in \mathbb{C}^{n}$ there exists $K_{0}\in \mathbb{Z}_{+}^{n}$, $\|K_0\|\leq n_{0}$, satisfying inequality $\displaystyle\!\max\!\left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^{K}(z)} \colon \|K\|\leq n_{0},z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \}%\leq \nonumber\\\label{eq:5}\leq p_{0}\frac{|F^{(K_0)}(z_0)|_p}{K_0!\mathbf{L}^{K_0}(z_0)},$ where $\mathbb{D}^{n}[z_{0},R]=\{z=(z_1,\ldots,z_n)\in \mathbb{C}^{n}\colon |z_1-z_{0,1}|<r_{1},\ldots, |z_n-z_{0,n}|<r_{n}\}$ is the polydisc with $z_0=(z_{0,1},\ldots,z_{0,n}),$\ $R=(r_{1},\ldots,r_{n})$. This theorem is an analog of Fricke's Theorem obtained for entire functions of bounded index of one complex variable.

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Growth Estimates for Analytic Vector-Valued Functions in the Unit Ball Having Bounded $\mathbf{L}$-index in Joint Variables

Cited by 4 publications

References 19 publications

Effect of Exothermic Additions in Core Filler on Arc Stability and Microstructure during Self-Shielded, Flux-Cored Arc Welding

Effect of Exothermic Additions in Core Filler on Arc Stability and Microstructure during Self-Shielded, Flux-Cored Arc Welding

Initial Coefficients Estimates and Fekete–Szegö Inequality Problem for a General Subclass of Bi-Univalent Functions Defined by Subordination

Entire multivariate vector-valued functions of bounded $\mathbf{L}$-index: analog of Fricke’s theorem

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