• Journal of applied mathematics & informatics
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• 제40권5_6호
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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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• 제64권1호
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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.

• 대한수학회지
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• 제48권1호
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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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• 제17권1_2_3호
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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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• 제17권2호
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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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• 제19권4호
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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).

• 대한수학회보
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• 제56권4호
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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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• 제8권1호
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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.

• 한국철도학회논문집
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• 제12권6호
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• pp.936-943
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• 2009

본 연구에서는 철도사고 위험분석 및 위험도 평가절차에 따라 철도건널목사고에 대한 정량적인 위험도평가를 위한 모델을 사건수목 및 고장수목 분석기법을 이용하여 개발하였다. 위험사건이 발생하여 인명피해로 결과하는 과정에서의 영향인자들을 분석하여 사고진전 시나리오를 구성하였으며, 고장수목분석(FTA, Fault Tree Analysis)을 이용하여 시나리오 경로별 발생확률을 산정하고, 사건수목분석(ETA, Event Tree Analysis)을 이용하여 심각도 값을 산정함으로써 이들의 조합으로 위험도를 산정하는 위험도 평가 모델을 제시하였다. 또한 실제발생한 위험도값과 개발모델을 이용하여 산정된 위험도값의 비교를 통하여 개발모델의 신뢰성 및 타당성을 검증하였다.

• 대한수학회보
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• 제51권2호
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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.