• Honam Mathematical Journal
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• v.31 no.4
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• pp.579-590
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• 2009

The precise form of singular functions, singular function representation and the extraction form for the stress intensity factor play an important role in the singular function methods to deal with the domain singularities for the Poisson problems with most common boundary conditions, e.q. Dirichlet or Mixed boundary condition [2, 4]. In this paper we give an elementary step to get the singular functions of the solution for Poisson problem with Neumann boundary condition or Robin boundary condition. We also give singular function representation and the extraction form for the stress intensity with a result showing the number of singular functions depending on the boundary conditions.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.159-172
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• 2004

A map is singular if each edge is on the same face on a surface (i.e., those have only one face on a surface). Because any map with loop is not colorable, all maps here are assumed to be loopless. In this paper po-vides the explicit expression of chromatic sum functions for rooted singular maps on the projective plane, the torus and the Klein bottle. From the explicit expression of chromatic sum functions of such maps, the explicit expression of enumerating functions of such maps are also derived.

• The Pure and Applied Mathematics
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• v.23 no.1
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• pp.21-34
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• 2016

In this paper, we construct a strictly increasing continuous singular function which has a simple algebraic expression.

• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.425-443
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• 2006

The aim of this paper is to obtain several results in approximation by Jackson-type generalizations of complex Picard, Poisson-Cauchy and Gauss-Weierstrass singular integrals in terms of higher order moduli of smoothness. In addition, these generalized integrals preserve some sufficient conditions for starlikeness and univalence of analytic functions. Also approximation results for vector-valued functions defined on the unit disk are given.

• Communications for Statistical Applications and Methods
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• v.15 no.6
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• pp.837-842
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• 2008

Students taking linear model courses have difficulty in determining which parametric functions are estimable when the design matrix of a linear model is rank deficient. In this note a special form of estimable functions is presented with a linear combination of some orthogonal estimable functions. Here, the orthogonality means the least squares estimators of the estimable functions are uncorrelated and have the same variance. The number of the orthogonal estimable functions composing the special form is equal to the rank of the design matrix. The orthogonal estimable functions can be easily obtained through the singular value decomposition of the design matrix.

• Honam Mathematical Journal
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• v.29 no.4
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• pp.701-721
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• 2007

Recently, a new singular function(NSF) method was posed to get accurate numerical solution on quasi-uniform grids for two-dimensional Poisson and interface problems with domain singularities by the first author and his coworkers. Using the singular function representation of the solution, dual singular functions, and an extraction formula for stress intensity factors, the method poses a weak problem whose solution is in $H^2({\Omega})$ or $H^2({\Omega}_i)$. In this paper, we show that the singular functions, which are not in $H^2({\Omega})$, also satisfy the integration by parts and note that this fact suggests the possibility of different choice of the weak formulations. We show that the original choice of weak formulation of NSF method is critical.

• East Asian mathematical journal
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• v.38 no.5
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• pp.603-610
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• 2022

In [6, 7] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous boundary conditions, compute the finite element solutions using standard FEM and use the extraction formula to compute the stress intensity factor(s), then they posed new PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor(s), which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. They considered a partial differential equation with the input function f ∈ L²(Ω). In this paper we consider a PDE with the input function f ∈ H¹(Ω) and find the corresponding singular and dual singular functions. We also induce the corresponding extraction formula which are the basic element for the approach.

• 제어로봇시스템학회:학술대회논문집
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• 1993.10a
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• pp.482-485
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• 1993

This paper deals with structured singular value and mixed sensitivity problem for robust performance. We derive the sufficient condition that mixed sensitivity problem satisfies structured singular value in robust performance problem. And we show the bound of perturbation between structured singular value and norm of mixed sensitivity functions.

• Honam Mathematical Journal
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• v.40 no.4
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• pp.785-794
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• 2018

In [8] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous Dirichlet boundary condition with one corner singularity at the origin, and compute the finite element solution using standard FEM and use the extraction formula to compute the stress intensity factor, then pose a PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor, which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. This approach uses the polar coordinate and the cut-off function to control the singularity and the boundary condition. In this paper we consider Poisson equations with multiple singular points, which involves different cut-off functions which might overlaps together and shows the way of cording in FreeFEM++ to control the singular functions and cut-off functions with numerical experiments.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.713-732
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• 2017

Let ${\Omega}{\in}L^s(S^{n-1})$ for s > 1 be a homogeneous function of degree zero and b be BMO functions or Lipschitz functions. In this paper, we obtain some boundedness of the $Calder{\acute{o}}n$-Zygmund singular integral operator $T_{\Omega}$ and its commutator [b, $T_{\Omega}$] on Herz-type Hardy spaces with variable exponent.