• Honam Mathematical Journal
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• v.43 no.2
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• pp.183-197
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• 2021

Recently several authors have extended the Beta function by using its integral representation. However, in many cases no expression of these extended functions in terms of classic special functions is known. In the present paper, we introduce a further extension by defining a family of functions G_r,s : ℝ^*₊ → ℂ, with r, s ∈ ℂ and ℜ(r) > 0. For given r, s, we prove that this function satisfies a second-order linear differential equation with rational coefficients. Solving this ODE, we express G_r,s as a combination of confluent hypergeometric functions. From this we deduce a new integral relation satisfied by Tricomi's function. We then investigate additional specific properties of G_r,1 which take the form of new non trivial integral relations involving exponential and error functions. We discuss the connection between G_r,1 and Stokes' first problem (or Rayleigh problem) in fluid mechanics which consists in determining the flow created by the movement of an infinitely long plate. For $r{\in}{\frac{1}{2}}{\mathbb{N}}^*$, we find additional relations between G_r,1 and Hermite polynomials. In view of these results, we believe the family of extended beta functions G_r,s will find further applications in two directions: (i) for improving our knowledge of confluent hypergeometric functions and Tricomi's function, (ii) and for engineering and physics problems.

• The Korean Journal of Physiology and Pharmacology
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• v.22 no.3
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• pp.343-348
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• 2018

Recent human genetic studies have shown that $G{\beta}5$ is related to various clinical symptoms, such as sinus bradycardia, cognitive disability, and attention deficit hyperactivity disorder. Although the calcium signaling cascade is closely associated with a heterotrimeric G-protein, the function of $G{\beta}5$ in calcium signaling and its relevance to clinical symptoms remain unknown. In this study, we investigated the in vitro changes of store-operated calcium entry (SOCE) with exogenous expression of $G{\beta}5$. The cells expressing $G{\beta}5$ had enhanced SOCE after depletion of calcium ion inside the endoplasmic reticulum. $G{\beta}5$ also augmented Stim1- and Orai1-dependent SOCE. An ORAI1 loss-of-function mutant did not show inhibition of $G{\beta}5$-induced SOCE, and a STIM1-ERM truncation mutant showed no enhancement of SOCE. These results suggested a novel role of GNB5 and Stim1, and provided insight into the regulatory mechanism of SOCE.

• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.311-322
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• 2006

For the Briot-Bouquet differential equations of the form given in [1] $${{\mu}(z)+\frac {z{\mu}'(z)}{z\frac {f'(z)}{f(z)}\[\alpha{\mu}(z)+\beta]}=g(z)$$ we can reduce them to $${{\mu}(z)+F(z)\frac {v'(z)}{v(z)}=h(z)$$ where $$v(z)=\alpha{\mu}(z)+\beta,\;h(z)={\alpha}g(z)+\beta\;and\;F(z)=f(z)/f'(z)$$. In this paper we are going to give conditions in order that if u and v satisfy, respectively, the equations (1) $${{\mu}(z)+F(z)\frac {v'(z)}{v(z)}=h(z)$$, $${{\mu}(z)+G(z)\frac {v'(z)}{v(z)}=g(z)$$ with certain conditions on the functions F and G applying the concept of strong subordination $g\;\prec\;\prec\;h$ given in [2] by the author, implies that $v\;\prec\;{\mu},\;where\;\prec$ indicates subordination.

• Bulletin of the Korean Mathematical Society
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• v.34 no.4
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• pp.627-636
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• 1997

Let $H(U)$ be the space of all analytic functions in the unit disk $U$ and let $K \subset H(U)$. For the operator $A_{\beta,\gamma} : K \longrightarrow H(U)$ defined by $$ A_{\beta,\gamma}(f)(z) = [\frac{z^\gamma}{\beta + \gamma} \int_{0}^{z} f^\beta (t)t^{\gamma-1} dt]^{1/\beta} $$ and $\beta,\gamma \in C$, we determined conditions on g(z), $\beta and \gamma$ such that $$ z[\frac{z}{f(z)]^\beta \prec z[\frac{z}{g(z)]^\beta implies z[\frac{z}{A_{\beta,\gamma}(f)(z)]^\beta \prec z[\frac{z}{A_{\beta,\gamma}(g)(z)]^\beta $$ and we presented some particular cases of our main result.

• The Pure and Applied Mathematics
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• v.5 no.2
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• pp.123-132
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• 1998

Let F and G denote the distribution functions of the failure times and the censoring variables in a random censorship model. Susarla and Van Ryzin(1978) verified consistency of $F_{\alpha}$, he NPBE of F with respect to the Dirichlet process prior D($\alpha$), in which they assumed F and G are continuous. Assuming that A, the cumulative hazard function, is distributed according to a beta process with parameters c, $\alpha$, Hjort(1990) obtained the Bayes estimator $A_{c,\alpha}$ of A under a squared error loss function. By the theory of product-integral developed by Gill and Johansen(1990), the Bayes estimator $F_{c,\alpha}$ is recovered from $A_{c,\alpha}$. Continuity assumption on F and G is removed in our proof of the consistency of $A_{c,\alpha}$ and $F_{c,\alpha}$. Our result extends Susarla and Van Ryzin(1978) since a particular transform of a beta process is a Dirichlet process and the class of beta processes forms a much larger class than the class of Dirichlet processes.

• Communications of the Korean Mathematical Society
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• v.20 no.2
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• pp.321-338
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• 2005

In this paper we investigate a generalization of the Hyers-Ulam-Rassias stability for a functional equation of the form $f(\varphi(X))\;=\;\phi(X)f(X)$, where X lie in n-variables. As a consequence, we obtain a stability result in the sense of Hyers, Ulam, Rassias, and Gavruta for many other equations such as the gamma, beta, Schroder, iterative, and G-function type's equations.

• The Korean Journal of Pharmacology
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• v.18 no.1
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• pp.1-10
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• 1982

In an attempt to delineate the role of beta-adrenoceptors found to be existing in the brain tissue in the central regulation of renal function, isoproterenol, a ${\beta}-adrenergic$ agonist, was administered directly into a lateral ventricle of the rabbit brain and the changes of renal function were observed. Also, the effects of propranolol, a specific ${\beta}-adrenergic$ blocking agent, and its influence upon the isoproterenol action were studied. Isoproterenol, in doses ranging from 5 to $50\;{\mu}g/kg\;i.c.v.$, elicited antidiuresis which seemed to be related to the decreased renal hemodynamics brought about by the systemic hypotension. With moderate doaes of $15\;{\mu}g/kg$ the antidiuresis was less prominent and there was a tendency toward natriuresis, but with higher doses the natriuretic effect became less evident, overrun by the systemic hypotension. Propranolol, $500\;{\mu}g/kg\;i.c.v.$, produced little effect on the renal function, but it eliminated the antidiuretic action of $50\;{\mu}g/kg$ isoproterenol i.c.v. and reversed it to a diuretic and natriuretic one, along with increases in renal plasma flow and glomerular filtration rate. The systemic hypotension also was markedly attenuated by propranolol pretreatment. Thus, it was evident that the renal action of i.c.v. isoproterenol was not blocked by propranolol and became explicit only when the hypotensive action of isoproterenol which seems to he propranolol-sensitive is removed. Various possibilities to account for this disparity in sensitivity were discussed. It is suggested from these observations that the central ${\beta}-adrenoceptors$ might also be involved in the regulation of renal function along with ${\alpha}-adrenoceptors$, though less significant than the latter.

• Toxicological Research
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• v.20 no.3
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• pp.205-211
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• 2004

Concern that some chemicals in our environment may affect human health by disrupt-ing normal endocrine function has prompted a research on interactions of environmental contaminants with steroid hormone receptor. Toxaphene and chlordane are among the 12 persistent organic pollutants identified by the United Nations Environment Programme as requiring urgent attention. We compared the estrogenic activity of two organochlorine pesticides, toxaphene and chlordane, at estrogen receptor a (ER$\alpha$) and estrogen receptor $\beta$ (ER$\beta$). Human hepatoma cells (HepG2) were transiently transfected with rat ER$\alpha$ or ER$\beta$ plus an estrogen-responsive complement C3-luciferase (C3-Luc) reporter gene. After transfection, cells were treated with various concentrations of toxaphene and chlordane to investigate agonism or antagonism of these chemicals. Both toxaphene and chlordane were potent agonists in HepG2 cells for ER$\alpha$. In contrast, these chemicals had a minimal agonist activity with ER$\beta$ and almost abolished 17$\beta$-estradiol-induced ER$\beta$-mediated activity. Therefore, toxaphene and chlordane behaved as an ER$\alpha$ agonist and an ER$\beta$ antagonist with estrogen-responsive reporter plasmid C3-Luc, and exposure to these organochlorine pesticides could have a crictical effect on normal endocrine function.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.429-445
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• 2004

In this paper we investigate the generalized Hyers-Ulam-Rassias stability for a functional equation of the form $f(\varphi(x,y)){\;}={\;}\phi(x,y)f(x,y)$, where x, y lie in the set S. As a consequence we obtain stability in the sense of Hyers, Ulam, Rassias, Gavruta, for some well-known equations such as the gamma, beta and G-function type equations.

• BMB Reports
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• v.40 no.3
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• pp.419-425
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• 2007

In vitro directed evolution through DNA shuffling is a powerful molecular tool for creation of new biological phenotypes. E. coli $\beta$-galactosidase and $\beta$-glucuronidase are widely used, and their biological function, catalytic mechanism, and molecular structures are well characterized. We applied an in vitro directed evolution strategy through DNA shuffling and obtained five mutants named YG6764, YG6768, YG6769, YG6770 and YG6771 after two rounds of DNA shuffling and screening, which exhibited more $\beta$-glucuronidase activity than wild-type $\beta$-galactosidase. These variants had mutations at fourteen nucleic acid sites, resulting in changes in ten amino acids: S193N, T266A, Q267R, V411A, D448G, G466A, L527I, M543I, Q626R and Q951R. We expressed and purified those mutant proteins. Compared to the wild-type protein, five mutant proteins exhibited high $\beta$-glucuronidase activity. The comparison of molecular models of the mutated and wildtype enzymes revealed the relationship between protein function and structural modification.