• Korean Journal of Mathematics
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• 제17권3호
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• pp.283-291
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• 2009

We investigate the existence of multiple solutions (${\xi},{\eta}$) for perturbations of the parabolic system with Dirichlet boundary condition $$(0.1)\;\begin{array}{lcr}{\xi}_t=-L{\xi}+g_1(3{\xi}+{\eta})-s{\phi}_1-h_1(x,t)\;in\;{\Omega}{\times}(0,\;2{\pi}),\\{\eta}_t=-L{\eta}+g_2(3{\xi}+{\eta})-s{\phi}_1-h_2(x,t)\;in\;{\Omega}{\times}(0,\;2{\pi}).\end{array}$$ We show the existence of multiple solutions (${\xi},{\eta}$) for perturbations of the parabolic system when the nonlinearity $g^{\prime}_1,\;g^{\prime}_2$ are bounded and $3g^{\prime}_1(-{\infty})+g^{\prime}_2(-{\infty})<{\lambda}_1,\;{\lambda}_n<3g^{\prime}_1(+{\infty})+g^{\prime}_2(+{\infty})<{\lambda}_{n+1}$.

• 대한수학회보
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• 제51권2호
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• pp.409-421
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• 2014

This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: $$\{-div\((1+\frac{|{\nabla}u|^{p(x)}}{\sqrt{1+|{\nabla}u|^{2p(x)}}}})|{\nabla}u|^{p(x)-2}{\nabla}u\)={\lambda}f(x,u)\;a.e.\;in\;{\Omega}\\u=0,\;on\;{\partial}{\Omega}$$ where ${\Omega}{\subset}R^N$ is a bounded domain with smooth boundary ${\partial}{\Omega}$, ${\lambda}$ > 0 is a parameter. The purpose of this paper is to obtain the existence results of nontrivial solutions for every parameter ${\lambda}$. Firstly, by using the mountain pass theorem a nontrivial solution is constructed for almost every parameter ${\lambda}$ > 0. Then we consider the continuation of the solutions. Our results are a generalization of that of Manuela Rodrigues.

• Kyungpook Mathematical Journal
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• 제28권2호
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• pp.147-154
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• 1988

For a positive integer p, $A_p$ will denote the class of functions $f(z)=z^p+\sum\limits^{\infty}_{n=p+1}a_nz^n$ which are analytic in the unit disc U = {z: |z| <1}. For $0{\leq}{\alpha}{\leq}1$, 0<${\beta}{\leq}1$, $0{\leq}{\lambda}$ $S_p({\alpha},{\beta},{\lambda})$ denote the class of functions $f(z){\in}A_p$ which satisfy the condition $\left|\frac{{\frac{zf^{\prime}(z)}{f(z)}}-p}{{{\alpha}{\frac{zf^{\prime}(z)}{f(z)}}+p-{\lambda}(1+{\alpha})}}\right|$<${\beta}$ for $z{\in}U$ In this paper we obtain a representation theorem for the class $S_p({\alpha},{\beta},{\lambda})$ and also derive distortion theorem and sharp estimates for the coefficients of this class.

• Journal of applied mathematics & informatics
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• 제24권1_2호
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• pp.421-426
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• 2007

The problem of vertex labeling with a condition at distance two in a graph, is a variation of Hale's channel assignment problem, which was first explored by Griggs and Yeh. For positive integer $p{\geq}q$, the ${\lambda}_{p,q}$-number of graph G, denoted ${\lambda}(G;p,q)$, is the smallest span among all integer labellings of V(G) such that vertices at distance two receive labels which differ by at least q and adjacent vertices receive labels which differ by at least p. Van den Heuvel and McGuinness have proved that ${\lambda}(G;p,q){\leq}(4q-2){\Delta}+10p+38q-24$ for any planar graph G with maximum degree ${\Delta}$. In this paper, we studied the upper bound of ${\lambda}_{p,q}$-number of some planar graphs. It is proved that ${\lambda}(G;p,q){\leq}(2q-1){\Delta}+2(2p-1)$ if G is an outerplanar graph and ${\lambda}(G;p,q){\leq}(2q-1){\Delta}+6p-4q-1$ if G is a Halin graph.

• 한국생물공학회:학술대회논문집
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• 한국생물공학회 2005년도 생물공학의 동향(XVII)
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• pp.335-339
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• 2005

정어리에서 분리한 균주 중 가장 항산화력이 강한 균주를 5개를 선발하여 이를 antiangiogenesis test를 통하여 효과가 가장 뛰어난 균주인 ${\lambda}-28$번 균주를 선발하였다. 균주의 최적 배지를 제조하기 각기 다른 온도, 영양성분등을 투여하여 최적의 조건을 알아보았다. 온도에 있어서는 균주의 특성상 바다온도와 가장 비슷한 $25^{\circ}C$에서 가장 좋은 성장률을 보였고 영양적인 측면에서 보면 탄소원에서는 glucose가 균주 성장률이나 항산화력에서 다른 첨가물에 비해 유효한 효과를 보였고, 무기 탄소원에서 ammonium chloride가 좋은 항산화력을 보였다. 유기 탄소원을 첨가한 실험에서는 유효한 효과가 나타나지 않았다.

• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.867-876
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• 2016

The object of the present paper is to investigate the starlikeness of the class of functions $f(z)=z^p+{\sum\limits_{k=n}^{\infty}}a_p+k^{z^{p^{+k}}} (p,n{\in}{\mathbb{N}}=\{1,2,{\ldots}\})$ which are analytic and p-valent in the unit disc U and satisfy the condition $\|(1-{\lambda}({\frac{f(z)}{z^p}})^{\alpha}+{\lambda}{\frac{zf^{\prime}(z)}{pf(z)}}({\frac{f(z)}{z^p}})^{\alpha}-1\|$ < ${\mu}$ (0 < ${\mu}{\leq}1$, ${\lambda}{\geq}0$, ${\alpha}$ > 0, $z{\in}U$). The starlikeness of certain integral operator are also discussed. The results obtained generalize the related works of some authors and some other new results are also obtained.

• 대한수학회보
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• 제53권6호
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• pp.1795-1804
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• 2016

Let $M^n$, $2{\leq}n{\leq}6$ be a complete noncompact hypersurface immersed in ${\mathbb{H}}^{n+1}$. We show that there exist two certain positive constants 0 < ${\delta}{\leq}1$, and ${\beta}$ depending only on ${\delta}$ and the first eigenvalue ${\lambda}_1(M)$ of Laplacian such that if M satisfies a (${\delta}$-SC) condition and ${\lambda}_1(M)$ has a lower bound then $H^1(L^2(M))=0$. Excepting these two conditions, there is no more additional condition on the curvature.

• Korean Journal of Mathematics
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• 제13권1호
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• pp.1-11
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• 2005

The purpose of the present paper is to introduce a new concept of the E($^*{\kappa}$)-connection ${\Gamma}^{\nu}_{{\lambda}{\mu}}$, which is both Einstein and ($^*{\kappa}$)-connection, and to obtain a necessary and sufficient condition for the existence of the unique E($^*{\kappa}$)-connection in $n-^*g$-UFT. Next, under this condition, we shall obtain a surveyable tensorial representation of the unique E($^*{\kappa}$)-connection in $n-^*g$-UFT.

• 대한수학회보
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• 제35권4호
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• pp.641-652
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• 1998

Chung and et al. ([2].1991) introduced a new concept of a manifold, denoted by $^{\ast}g-SEX_n$, in Einstein's n-dimensional $^{\ast}g$-unified field theory. The manifold $^{\ast}g-SEX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^{\ast}g^{\lambda \nu}$ through the SE-connection which is both Einstein and semi-symmetric. In this paper, they proved a necessary and sufficient condition for the unique existence of SE-connection and sufficient condition for the unique existence of SE-connection and presented a beautiful and surveyable tensorial representation of the SE-connection in terms of the tensor $^{\ast}g^{\lambda \nu}$. Recently, Chung and et al.([3],1998) obtained a concise tensorial representation of SE-curvature tensor defined by the SE-connection of $^{\ast}g-SEX_n$ and proved deveral identities involving it. This paper is a direct continuations of [3]. In this paper we derive surveyable tensorial representations of constracted curvature tensors of $^{\ast}g-SEX_n$ and prove several generalized identities involving them. In particular, the first variation of the generalized Bianchi's identity in $^{\ast}g-SEX_n$, proved in theorem (2.10a), has a great deal of useful physical applications.

• 호남수학학술지
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• 제44권4호
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• pp.521-534
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• 2022

In this paper, we introduce a new class 𝓡^k_λ(λ ≥ 1, k is any positive real number) of univalent complex functions, which consists of functions f of the form f(z) = z + Σ^∞_n=2 a_nzⁿ (|z| < 1) satisfying the subordination condition $$(1-{\lambda}){\frac{f(z)}{z}}+{\lambda}f^{\prime}(z){\prec}{\frac{1+r^2_kz^2}{1-k{\tau}_kz-{\tau}^2_kz^2}},\;{\tau}_k={\frac{k-{\sqrt{k^2+4}}}{2}$$, and investigate the Fekete-Szegö problem for the coefficients of f ∈ 𝓡^k_λ which are connected with k-Fibonacci numbers $F_{k,n}={\frac{(k-{\tau}_k)^n-{\tau}^n_k}{\sqrt{k^2+4}}}$ (n ∈ ℕ ∪ {0}). We obtain sharp upper bound for the Fekete-Szegö functional |a₃-𝜇a²₂| when 𝜇 ∈ ℝ. We also generalize our result for 𝜇 ∈ ℂ.