• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.873-882
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• 2022

In this paper, we construct the multiple (h, p, q)-Hurwitz-Euler eta function by generalizing the multiple Hurwitz-Euler eta function. We get some explicit formulas and properties of the higher-order (h, p, q)-Euler numbers and polynomials.

• Kyungpook Mathematical Journal
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• v.64 no.1
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• pp.185-196
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• 2024

In this article, we investigate the growth of meromorphic solutions of $${\alpha}(z)(\frac{{\Delta}_c{\eta}}{{\eta}})^2\,+\,(b_2(z){\eta}^2(z)\;+\;b_1(z){\eta}(z)\;+\;b_0(z))\frac{{\Delta}_c{\eta}}{{\eta}} \atop =d_4(z){\eta}^4(z)\;+\;d_3(z){\eta}^3(z)\;+\;d_2(z){\eta}^2(z)\;+\;d_1(z){\eta}(z)\;+\;d_0(z),$$ where a(z), b_i(z) for i = 0, 1, 2 and d_j (z) for j = 0, ..., 4 are given functions, △_cη = η(z + c) - η(z) with c ∈ ℂ\{0}. In particular, when the a(z), the b_i(z) and the d_j(z) are polynomials, and d₄(z) ≡ 0, we shall show that if η(z) is a transcendental entire solution of finite order, and either deg a(z) ≠ deg d₀(z) + 1, or, deg a(z) = deg d₀(z) + 1 and ρ(η) ≠ ½, then ρ(η) ≥ 1.

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.13-21
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• 2011

In this paper, a pair of Wolfe type nondifferentiable sec-ond order symmetric minimax mixed integer dual problems is formu-lated. Symmetric and self-duality theorems are established under $\eta_1$-bonvexity/$\eta_2$-boncavity assumptions. Several known results are obtained as special cases. Examples of such primal and dual problems are also given.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.229-244
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• 2005

In this paper, we study the third order ordinary differential equation : $$x'(t)=f(t,x(t),x'(t),x'(t)),t{\in}(0,1)$$ subject to the boundary value conditions: $$x'(0)=x'(\xi),x'(1)=^{m-3}{\Sigma}_{i=1}{{\beta}x'({\eta}i),x'(1)=0}$$. Here ${\beta}_{i}{\in}R,\;^{m-3}{\Sigma}_{i=1}\;{\beta}_{i}\;=\;1,\;0<{\eta}_1<{\eta}_2<{\cdots}<{\eta}_{m-3}<1,\;0<\xi<1$. This is the case dimKer L = 2. When the ${\beta}_i$ have different signs, we prove some existence results for the m-point boundary value problem at resonance by use of the coincidence degree theory of Mawhin [12, 13]. Since all the existence results obtained in previous papers are for the case dimKerL = 1, our work is new.

• Bulletin of the Korean Chemical Society
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• v.17 no.2
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• pp.138-142
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• 1996

The transition metal carbonylate, PPN+(${\eta}^5-MeCp)Mn(CO)_2Cl^-$ undergoes a novel ligand substitution reaction with PR3 (R=Me, Et, OEt, $C_6H_5$ in THF at elevated temperatures (40 $^{\circ}C$ up to 60 $^{\circ}C)$ under the pseudo-first-order reaction conditions (usually 20-fold excess of PR3 with respect to metal carbonylate concentrations) where chloride is displaced by PR3. The reaction follows overall first order dependence on [(${\eta}^5-MeCp)Mn(CO)_2Cl^-$]; however, the negative entropy changes of activation (${\Delta}S^{\neq}$=-19.3 e.u. for $P(OEt)_3$; ${\Delta}S^{\neq}$=-16.4 e.u. for $PPh_3$) suggest the existence of the intermediate, ((η3-MeCp)Mn(CO)2(THF)Cl-, which eventually transforms to the product (${\eta}^5-MeCp)Mn(CO)_2(PR_3)$.

• Bulletin of the Korean Chemical Society
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• v.19 no.4
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• pp.462-466
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• 1998

The homobimetallic anion, $({\eta}^5-MeCp)Mn(CO)_2Mn(CO)_5-M^+\; (M^+=Na^+, PPN^+$) was disrupted by $PR_3\;(R=C_6H_5,\;C_2H_5,\;OCH_3)$ in THF at various temperatures (r.t. ∼65℃) under the pseudo first order reaction conditions where excess of $PR_3$ was employed under a nitrogen atmosphere. For the reaction involving $PPN^+$ analog, Mn-Mn heterolytic cleavage occurred, leading to $PPN^+Mn(CO)_5^-\; and \;({\eta}^5-MeCp)Mn(CO)_2PR_3$ as products; however, in case of $Na^+\; analog,\; Na^+$ seems to play a novel counter ion effect on the disruption reaction by transferring one terminal CO from the $Mn(CO)_5$ moiety on to the $({\eta}^5-MeCp)Mn(CO)_2$ of the corresponding homobimetallic complex, eventually resulting in $Na^+Mn(CO)_4PR_3^-\;and\;({\eta}^5-MeCp)Mn(CO)_3$. This reaction is of overall first order with respect to [homobimetallic complex] with the activation parameters (ΔH≠=23.0±0.7 kcal/mol, ΔS≠= - 8.7±0.8 e.u. for $Na^+$ analog; ΔH≠=28.8±0.4 kcal/mol, ΔS≠=15.7±0.6 e.u. for $PPN^+$ analog reaction).

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.911-927
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• 2019

Let $\mathscr{B}^{{\eta},{\mu}}_{p,n}\;({\alpha});\;({\eta},{\mu}{\in}{\mathbb{R}},\;n,\;p{\in}{\mathbb{N}})$ denote all functions f class in the unit disk ${\mathbb{U}}$ as $f(z)=z^p+\sum_{k=n+p}^{\infty}a_kz^k$ which satisfy: $$\|\[{\frac{f^{\prime}(z)}{pz^{p-1}}}\]^{\eta}\;\[\frac{z^p}{f(z)}\]^{\mu}-1\| <1-{\frac{\alpha}{p}};\;(z{\in}{\mathbb{U}},\;0{\leq}{\alpha}<p)$$. And $\mathscr{M}^{{\eta},{\mu}}_{p,n}\;({\alpha})$ indicates all meromorphic functions h in the punctured unit disk $\mathbb{U}^*$ as $h(z)=z^{-p}+\sum_{k=n-p}^{\infty}b_kz^k$ which satisfy: $$\|\[{\frac{h^{\prime}(z)}{-pz^{-p-1}}}\]^{\eta}\;\[\frac{1}{z^ph(z)}\]^{\mu}-1\|<1-{\frac{\alpha}{p}};\;(z{\in}{\mathbb{U}},\;0{\leq}{\alpha}<p)$$. In this paper several sufficient conditions for some classes of functions are investigated. The authors apply Jack's Lemma, to obtain this conditions. Furthermore, sufficient conditions for strongly starlike and convex p-valent functions of order ${\gamma}$ and type ${\beta}$, are also considered.

• Ocean Systems Engineering
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• v.8 no.1
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• pp.41-55
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• 2018

Accidental oil and gas leak is a critical concern for the offshore industry because it can lead to severe consequences and as a result, it is imperative to evaluate the probabilities of occurrence of the consequences of the leakage in order to assess the risk. Event Tree Analysis (ETA) is a technique to identify the consequences that can result from the occurrence of a hazardous event. The probability of occurrence of the consequences is evaluated by the ETA, based on the failure probabilities of the sequential events. Conventional ETA deals with events with crisp failure probabilities. In offshore applications, it is often difficult to arrive at a single probability measure due to lack of data or imprecision in data. In such a scenario, fuzzy set theory can be applied to handle imprecision and data uncertainty. This paper presents fuzzy ETA (FETA) methodology to compute the probability of the outcomes initiated due to oil/gas leak in an actual offshore-onshore installation. Post FETA, sensitivity analysis by Fuzzy Weighted Index (FWI) method is performed to find the event that has the maximum contribution to the severe sequences. It is found that events of 'ignition', spreading of fire to 'equipment' and 'other areas' are the highest contributors to the severe consequences, followed by failure of 'leak detection' and 'fire detection' and 'fire water not being effective'. It is also found that the frequency of severe consequences that are catastrophic in nature obtained by ETA is one order less than that obtained by FETA, thereby implying that in ETA, the uncertainty does not propagate through the event tree. The ranking of severe sequences based on their probability, however, are identical in both ETA and FETA.

• Journal of the Korean Society for Railway
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• v.12 no.6
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• pp.936-943
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• 2009

In this study, a risk assessment model based on the ETA (Event Tree Analysis) and FTA (Fault Tree Analysis) is developed according to the procedure of hazard analysis and risk assessment in order to estimate the risk quantitatively. The FTA technique is applied to estimate the branch probability (frequency) and the ETA technique is applied to estimate the consequence for each branch path on the ET (Event Tree). A risk assessment model is developed by the combination of those ETA and FTA. In addition, the reliability and the validity of the risk assessment model are verified by comparing the risk estimated through the developed model with the actual equivalent fatality.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.329-338
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• 2014

In this paper, we study the method of upper and lower solutions and develop the generalized quasilinearization technique for the existence and approximation of solutions to some three-point nonlocal boundary value problems associated with higher order fractional differential equations of the type $$^c{\mathcal{D}}^q_{0+}u(t)+f(t,u(t))=0,\;t{\in}(0,1)$$ $$u^{\prime}(0)={\gamma}u^{\prime}({\eta}),\;u^{\prime\prime}(0)=0,\;u^{\prime\prime\prime}(0)=0,{\ldots},u^{(n-1)}(0)=0,\;u(1)={\delta}u({\eta})$$, where, n-1 < q < n, $n({\geq}3){\in}\mathbb{N}$, 0 < ${\eta},{\gamma},{\delta}$ < 1 and $^c\mathcal{D}^q_{0+}$ is the Caputo fractional derivative of order q. The nonlinear function f is assumed to be continuous.