• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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• pp.657-661
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• 2010

We study a singular function which we call a generalized cylinder convex(concave) function induced from different generalized dyadic expansion systems on the unit interval. We show that the generalized cylinder convex(concave)function is a singular function and the length of its graph is 2. Using a local dimension set in the unit interval, we give some characterization of the distribution set using its derivative, which leads to that this singular function is nowhere differentiable in the sense of topological magnitude.

• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.261-275
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• 2011

We give characterizations of the differentiability points and the non-differentiability points of the Riesz-N$\'{a}$gy-Tak$\'{a}$cs(RNT) singulr function using the distribution sets in the unit interval. Using characterizations, we show that the Hausdorff dimension of the non-differentiability points of the RNT singular function is greater than 0 and the packing dimension of the infinite derivative points of the RNT singular function is less than 1. Further the RNT singular function is nowhere differentiable in the sense of topological magnitude, which leads to that the packing dimension of the non-differentiability points of the RNT singular function is 1. Finally we show that our characterizations generalize a recent result from the ($\tau$, $\tau$ - 1)-expansion associated with the RNT singular function adding a new result for a sufficient condition for the non-differentiability points.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1173-1183
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• 2017

We give some sufficient conditions for the null and infinite derivatives of the $Riesz-N{\acute{a}}gy-Tak{\acute{a}}cs$ (RNT) singular function. Using these conditions, we show that the Hausdorff dimension of the set of the infinite derivative points of the RNT singular function coincides with its packing dimension which is positive and less than 1 while the Hausdorff dimension of the non-differentiability set of the RNT singular function does not coincide with its packing dimension 1.

• Communications of the Korean Mathematical Society
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• v.30 no.1
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• pp.7-21
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• 2015

We give the characterization of H$\ddot{o}$lder differentiability points and non-differentiability points of the Riesz-N$\acute{a}$gy-Tak$\acute{a}$cs (RNT) singular function ${\Psi}_{a,p}$ satisfying ${\Psi}_{a,p}(a)=p$. It generalizes recent multifractal and metric number theoretical results associated with the RNT function. Besides, we classify the singular functions using the singularity order deduced from the H$\ddot{o}$lder derivative giving the information that a strictly increasing smooth function having a positive derivative Lebesgue almost everywhere has the singularity order 1 and the RNT function ${\Psi}_{a,p}$ has the singularity order $g(a,p)=\frac{a{\log}p+(1-a){\log}(1-p)}{a{\log}a+(1-a){\log}(1-a)}{\geq}1$.

• Bulletin of the Korean Mathematical Society
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• v.52 no.6
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• pp.1797-1803
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• 2015

Let ($S_{\alpha}(n))_{n{\geq}1}$ be the lexicographically greatest Sturmian word of slope ${\alpha}$ > 0. For a fixed ${\sigma}$ > 1, we consider Dirichlet series of the form ${\nu}_{\sigma}({\alpha})$ := ${\Sigma}_{n=1}^{\infty}s_{\alpha}(n)n^{-{\sigma}}$. This paper studies the singular properties of the real function ${\nu}_{\sigma}$, and the Lebesgue-Stieltjes measure whose distribution is given by ${\nu}_{\sigma}$.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.421-426
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• 2013

We give a series of discrete random variables which converges to a random variable whose distribution function is the Riesz-N$\acute{a}$gy-Tak$\acute{a}$cs (RNT) distribution. We show this using the correspondence theorem that if the moments coincide then their corresponding distribution functions also coincide.

• Journal of Sensor Science and Technology
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• v.20 no.3
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• pp.151-155
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• 2011

A variety of pattern recognition algorithms including neural networks may be applicable to the identification of odors. In this paper, an identification technique for an electronic odor sensing system applicable to wound state monitoring is presented. The performance of the radial basis function(RBF) network is highly dependent on the choice of centers and widths in basis function. For the fine tuning of centers and widths, those parameters are initialized by an ill-conditioned genetic fuzzy c-means algorithm, and the distribution of input patterns in the very first stage, the stochastic gradient(SG), is adapted. The adaptive RBF network with singular value decomposition(SVD), which provides additional adaptation capabilities to the RBF network, is used to process data from array-based gas sensors for early detection of wound infection in burn patients. The primary results indicate that infected patients can be distinguished from uninfected patients.

• Journal of Integrative Natural Science
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• v.7 no.4
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• pp.273-277
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• 2014

In this article, we consider a singular and a degenerate elliptic operators in a divergence form. The singularities exist on a part of boundary, and comparable to the logarithmic distance function or its inverse. If we assume that the operator can be treated in a pointwise sense than distribution sense, with this operator we obtain a priori Harnack continuity near the boundary. In the proof we transform the singular elliptic operator to uniformly bounded elliptic operator with unbounded first order terms. We study this type of estimations considering a De Giorgi conjecture. In his conjecture, he proposed a certain ellipticity condition to guarantee a continuity of a solution.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.227-231
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• 2008

We give some relation between the Riemann-Stieltjes integrals with respect to the Riesz-N$\'{a}$gy-Tak$\'{a}$cs(RNT) distribution $H_{a,p}$ satisfying the equation 1 - a = $a^m$ over different intervals, which generalizes recent results([6]) for a = ${\frac{1}{2}}$.

• Honam Mathematical Journal
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• v.30 no.4
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• pp.671-676
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• 2008

We give the relation between the Riemann-Stieltjes integrals with respect to the Riesz-N$\'{a}$gy-Tak$\'{a}$cs(RNT) distribution $H_{a,p}$ satisfying the equation a = $(1-a)^m$ and those with respect to the dual RNT distribution $H_{1-a,1-p}$, which leads to a generalization of recent results for a = $\frac{1}{2}$.