• Journal of applied mathematics & informatics
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• v.4 no.1
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• pp.109-116
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• 1997

In the present paper we consider an abstract partial dif-ferential equation of the form $\frac{\partial^2u}{{\partial}t^2}-\frac{\partial^2u}{{\partial}x^2}+A(x.t)u=f(x, t)$, where ${A(x, t):(x, t){\epsilon}\bar{G} }$ is a family of linear closed operators and $G=GU{\partial}G$, G is a suitable bounded region in the (x, t)-plane with bound-are ${\partial}G$. It is assumed that u is given on the boundary ${\partial}G$. The objective of this paper is to study the considered Dirichlet problem for a wide class of operators $A(x, t)$. A Dirichlet problem for non-elliptic partial differential equations of higher orders is also considerde.

• 한국데이터정보과학회:학술대회논문집
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• 2006.11a
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• pp.71-80
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• 2006

A nonparametric Bayesian multiple comparisons problem (MCP) for dependence parameters in I bivariate exponential populations is studied here. A simple method for pairwise comparisons of these parameters is also suggested. Here we extend the methodology studied by Gopalan and Berry (1998) using Dirichlet process priors. The family of Dirichlet process priors is applied in the form of baseline prior and likelihood combination to provide the comparisons. Computation of the posterior probabilities of all possible hypotheses are carried out through Markov Chain Monte Carlo method, namely, Gibbs sampling, due to the intractability of analytic evaluation. The whole process of MCP for the dependent parameters of bivariate exponential populations is illustrated through a numerical example.

• The Pure and Applied Mathematics
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• v.3 no.1
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• pp.67-71
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• 1996

We will show how bilinear form $\varepsilon_{\mu}$ related with some smooth measures can be extended to the $L^2$(\mathbb{R}^n,\mathbb {C})$ setting.

• East Asian mathematical journal
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• v.33 no.5
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• pp.495-509
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• 2017

With the accuracy of the nonlinearity guaranteed, plenty of time and large memory space are needed when we solve the finite element numerical solution of nonlinear partial differential equations. In this paper, we use the Group Element Method (GEM) to deal with the non-linearity of the BBM-Burgers Equation with Conservation form and perform a numerical analysis for two particular initial-boundary value (the Dirichlet boundary conditions and Neumann-Dirichlet boundary conditions) problems with the Finite Element Method (FEM). Some numerical experiments are performed to analyze the error between the exact solution and the FEM solution in MATLAB.

• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.19-23
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• 1995

We will estimate the lower bound of the first nonzero Neumann and Dirichlet eigenvalue of Laplacian equation on compact Riemannian manifold M with boundary. In case that the boundary of M has positive second fundamental form elements, Ly-Yau[3] gave the lower bound of the first nonzero neumann eigenvalue $\eta_1$. In case that the second fundamental form elements of $\partial$M is bounded below by negative constant, Roger Chen[4] investigated the lower bound of $\eta_1$. In [1], [2], we obtained the lower bound of the first nonzero Neumann eigenvalue is estimated under the condtion that the second fundamental form elements of boundary is bounded below by zero. Moreover, I realize that "the interior rolling $\varepsilon$ - ball condition" is not necessary when the first Dirichlet eigenvalue was estimated in [1].ed in [1].

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1123-1148
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• 2016

The aim of this note is to provide detailed proofs for local estimates near the boundary for weak solutions of second order parabolic equations in divergence form with time-dependent measurable coefficients subject to Neumann boundary condition. The corresponding parabolic equations with Dirichlet boundary condition are also considered.

• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.59-71
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• 2018

We consider the boundary value problem with a Dirichlet condition for a second order linear uniformly elliptic operator in a non-divergence form. We study some properties of a barrier at infinity which was introduced by Meyers and Serrin to investigate a solution in an exterior domains. Also, we construct a modified barrier for more general domain than an exterior domain.

• Communications of the Korean Mathematical Society
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• v.19 no.2
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• pp.205-210
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• 2004

According to Diriclet's Theorem, there are infinitely many primes of the form An+ 1 for any fixed positive integer A. For this, there already exists a classical simple proof using cyclotomic polynomials. In this paper, we develop another elementary proof of this statement.

• Honam Mathematical Journal
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• v.39 no.2
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• pp.297-305
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• 2017

Let ${\alpha}$ > 0 be a real number and $(s_{\alpha}(n))_{n{\geq}1}$ be the lexicographically greatest Sturmian word of slope ${\alpha}$. We investigate Dirichlet series of the form ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-s}$. To do this, a generalization of Euler's totient function is required. For a real ${\alpha}$ > 0 and a positive integer n, an arithmetic function ${\varphi}{\alpha}(n)$ is defined to be the number of positive integers m for which gcd(m, n) = 1 and 0 < m/n < ${\alpha}$. Under a condition Re(s) > 1, this paper establishes an identity ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-S}=1+{\sum}^{\infty}_{n=1}{\varphi}_{\alpha}(n)({\zeta}(s)-{\zeta}(s,1+n^{-1}))n^{-s}$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1201-1207
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• 2013

Let N be a complete Riemannian manifold with nonnegative Ricci curvature and let M be a complete noncompact oriented stable minimal hypersurface in N. We prove that if M has at least two ends and ${\int}_M{\mid}A{\mid}^2\;dv={\infty}$, then M admits a nonconstant harmonic function with finite Dirichlet integral, where A is the second fundamental form of M. We also show that the space of $L^2$ harmonic 1-forms on such a stable minimal hypersurface is not trivial. Our result is a generalization of one of main results in [12] because if N has nonnegative sectional curvature, then M admits no nonconstant harmonic functions with finite Dirichlet integral. And our result recovers a main theorem in [3] as a corollary.