• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.835-845
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• 2017

The $GCF_{\epsilon}$ expansion is a new class of continued fractions induced by the transformation $T_{\epsilon}:(0, 1]{\rightarrow}(0, 1]$: $T_{\epsilon}(x)={\frac{-1+(k+1)x}{1+k-k{\epsilon}x}}$ for $x{\in}(1/(k+1),1/k]$. Under the algorithm $T_{\epsilon}$, every $x{\in}(0,1]$ corresponds to an increasing digits sequences $\{k_n,n{\geq}1\}$. Their basic properties, including the ergodic properties, law of large number and central limit theorem have been discussed in [4], [5] and [7]. In this paper, we study the large deviation for the $GCF_{\epsilon}$ expansion and show that: $\{{\frac{1}{n}}{\log}k_n,n{\geq}1\}$ satisfies the different large deviation principles when the parameter ${\epsilon}$ changes in [-1, 1], which generalizes a result of L. J. Zhu [9] who considered a case when ${\epsilon}(k){\equiv}0$ (i.e., Engel series).

• Journal of the Korean Mathematical Society
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• v.52 no.3
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• pp.637-647
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• 2015

For generalized continued fraction (GCF) with parameter ${\epsilon}(k)$, we consider the size of the set whose partial quotients increase rapidly, namely the set $$E_{\epsilon}({\alpha}):=\{x{\in}(0,1]:k_{n+1}(x){\geq}k_n(x)^{\alpha}\;for\;all\;n{\geq}1\}$$, where ${\alpha}$ > 1. We in [6] have obtained the Hausdorff dimension of $E_{\epsilon}({\alpha})$ when ${\epsilon}(k)$ is constant or ${\epsilon}(k){\sim}k^{\beta}$ for any ${\beta}{\geq}1$. As its supplement, now we show that: $$dim_H\;E_{\epsilon}({\alpha})=\{\frac{1}{\alpha},\;when\;-k^{\delta}{\leq}{\epsilon}(k){\leq}k\;with\;0{\leq}{\delta}<1;\\\;\frac{1}{{\alpha}+1},\;when\;-k-{\rho}<{\epsilon}(k){\leq}-k\;with\;0<{\rho}<1;\\\;\frac{1}{{\alpha}+2},\;when\;{\epsilon}(k)=-k-1+\frac{1}{k}$$. So the bigger the parameter function ${\epsilon}(k_n)$ is, the larger the size of $E_{\epsilon}({\alpha})$ becomes.

• Journal of Korean Neurosurgical Society
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• v.56 no.3
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• pp.248-253
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• 2014

Objective : Cortical dysplasia (CD) is one of the common causes of epilepsy surgery. However, surgical outcome still remains poor, especially with frontal lobe epilepsy (FLE), despite the advancement of neuroimaging techniques and expansion of surgical indications. The aim of this study was to focus on surgical strategies in terms of extent of resection to improve surgical outcome in the cases of FLE with CD. Methods : A total of 11 patients of FLE were selected among 67 patients who were proven pathologically as CD, out of a total of 726 epilepsy surgery series since 1992. This study categorized surgical groups into three according to the extent of resection : 1) focal corticectomy, 2) regional corticectomy, and 3) partial functional lobectomy, based on the preoperative evaluation, in particular, ictal scalp EEG onset and/or intracranial recordings, and the lesions in high-resolution MRI. Surgical outcome was assessed following Engel's classification system. Results : Focal corticectomy was performed in 5 patients and regional corticectomy in another set of 5 patients. Only 1 patient underwent partial functional lobectomy. Types I and II CD were detected with the same frequency (45.45% each) and postoperative outcome was fully satisfactory (91%). Conclusion : The strategy of epilepsy surgery is to focus on the different characteristics of each individual, considering the extent of real resection, which is based on the focal ictal onset consistent with neuroimaging, especially in the practical point of view of neurosurgery.