• Journal of the Chungcheong Mathematical Society
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• v.16 no.1
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• pp.7-14
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• 2003

Let $P_c(z)=z^n+c$ be a complex polynomial with an integer $n{\geq}2$. We derive a criterion that the critical orbit of $P_c$ escapes to infinity and investigate its applications to the degree-n bifurcation set. The intersection of the degree-n bifurcation set with the real line as well as with a typical symmetric axis is explicitly written as a function of n. A well-defined escape-time algorithm is also included for the improved construction of the degree-n bifurcation set.

• Journal of Korean Neurosurgical Society
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• v.60 no.5
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• pp.504-510
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• 2017

Objective : To investigate the morphological and hemodynamic parameters associated with middle cerebral artery (MCA) bifurcation aneurysm rupture. Methods : A retrospective study of 67 consecutive patients was carried out based on 3D digital subtraction angiography data. Morphological and hemodynamic parameters including aneurysm size parameters (dome width, height, and perpendicular height), longest dimension from the aneurysm neck to the dome tip, neck width, aneurysm area, aspect ratio, Longest dimension from the aneurysm neck to the dome tip (Dmax) to dome width, and height-width, Bottleneck factor, as well as wall shear stress (WSS), low WSS area (LSA), percentage of LSA (LSA%) and energy loss (EL) were estimated. Parameters between ruptured and un-ruptured groups were analyzed. Receiver operating characteristics were generated to check prediction performance of all significant variables. Results : Sixty-seven patients with MCA bifurcation aneurysm were included (31 unruptured, 36 ruptured). Dmax (p=0.008) was greater in ruptured group than that in un-ruptured group. D/W (p<0.001) and the percentage of the low WSS area ($0.09{\pm}0.13$ vs. $0.01{\pm}0.03$, p<0.001) were also greater in the ruptured group. Moreover, the EL in ruptured group was higher than that in unruptured group ($6.39{\pm}5.04$ vs. $1.53{\pm}0.86$, p<0.001). Multivariate regression analysis suggested D/W and EL were significant predictors of rupture of MCA bifurcation aneurysms. Correlation analyses revealed the D/W value was positively associated with the EL (R=0.442, p<0.01). Conclusion : D/W and EL might be the most two favorable factors to predict rupture risk of MCA bifurcation aneurysms.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.7 no.6
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• pp.1421-1424
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• 2006

By extending the Mandelbrot set for the complex polynomial $$M={c\in C\;:\; _{k\rightarrow\infty}^{lim}P_c^k(0)\;{\neq}\;{\infty}$$ we define the degree-n bifurcation set. In this paper, we formulate the boundary equation of a period-2 component on the main component in the degree-9 bifurcation set by parameterizing its image. We establish an algorithm constructing a period-2 component in the degree-9 bifurcation set and the typical implementations show the satisfactory result with Mathematica codes grounded on the analysis.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1149-1167
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• 2015

We get a theorem which shows the existence of at least four $2{\pi}$-periodic weak solutions for the bifurcation problem of the Hamiltonian system with the superquadratic nonlinearity. We obtain this result by using the variational method, the critical point theory induced from the limit relative category theory.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.667-680
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• 2016

We get a theorem which shows the multiple weak solutions for the bifurcation problem of the superquadratic nonlinear Hamiltonian system. We obtain this result by using the variational method, the critical point theory in terms of the $S^1$-invariant functions and the $S^1$-invariant linear subspaces.

• Korean Journal of Mathematics
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• v.20 no.3
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• pp.333-341
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• 2012

We investigate the number of solutions for the superlinear elliptic bifurcation problem with Dirichlet boundary condition. We get a theorem which shows the existence of at least $k$ weak solutions for the superlinear elliptic bifurcation problem with boundary value condition. We obtain this result by using the critical point theory induced from invariant linear subspace and the invariant functional.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.402-418
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• 2022

Noise is a fundamental factor to increased validity and regularity of spike propagation and neuronal firing in the nervous system. In this paper, we examine the stochastic version of the Izhikevich-FitzHugh neuron dynamical model. This approach is based on techniques presented by Luo and Guo, which provide a general framework for the bifurcation and stability analysis of two dimensional stochastic dynamical system as an Itô averaging diffusion system. By using largest lyapunov exponent, local and global stability of the stochastic system at the equilibrium point are investigated. We focus on the two kinds of stochastic bifurcations: the P-bifurcation and the D-bifurcations. By use of polar coordinate, Taylor expansion and stochastic averaging method, it is shown that there exists choices of diffusion and drift parameters such that these bifurcations occurs. Finally, numerical simulations in various viewpoints, including phase portrait, evolution in time and probability density, are presented to show the effects of the diffusion and drift coefficients that illustrate our theoretical results.

• Restorative Dentistry and Endodontics
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• v.16 no.2
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• pp.174-181
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• 1991

A model system was used which enabled the same root canal system to be measured before and after coronal flaring of 51 extracted mandibular molars. The concavity of the distal surface of the mesial root was measured and the amount of reduction was compared after coronal flaring using step-back flared preparation, Gates-Glidden dirll or ultrasonic system(Quick-$\varepsilon$) at the furcation and apical 3mm from the furcation. The results were as follows: 1. The mean concavity of mesial root of manchbular molar was $0.73{\pm}0.27mm$ at the bifurcation and $0.65{\pm}0.23mm$ at the 3.0mm apical from the bifurcation. 2. The thickness of the root canal wall of the mesiobuccal canal was $1.08{\pm}0.26mm$ at the bifurcation and $1.00{\pm}0.23mm$ at the 3.0mm apical from the bifurcation. 3. The thickness of the root canal wall of the mesiolingual was $1.09{\pm}0.21mm$ at the bifurcation and $0.98{\pm}0.29mm$ at the 3.0mm apical from the bifurcation. 4. In the amount of reduction at the furcation and at the 3.0mm apical from the furcation there was no statistically significant difference between the step-back preparation and Gates-Glidden drill preparation, and ultrasonic preparation(P>0.05).

• Korean Journal of Mathematics
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• v.22 no.3
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• pp.471-489
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• 2014

We investigate the number of the weak periodic solutions for the bifurcation problem of the Hamiltonian system with the superquadratic nonlinearity. We get one theorem which shows the existence of at least two weak periodic solutions for this system. We obtain this result by using variational method, critical point theory induced from the limit relative category theory.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1185-1202
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• 2010

In this paper, we have studied the dynamical behaviour such as boundedness, local and global stabilities, bifurcation of a single species population affected by environmental toxicant and population toxicant. We have also studied the effect of discrete delay of the environmental toxicant on the instantaneous growth rates of the population biomass and population toxicant due to incubation period. The length of delay preserving the stability is also estimated. Computer simulations are carried out to illustrate our analytical findings.