• Journal of the Chungcheong Mathematical Society
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• v.20 no.3
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• pp.203-210
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• 2007

Let X be a hyperbolic region in the complex plane C such that the hyperbolic metrix ${\lambda}_X(w){\mid}dw{\mid}$ exists. Let $R(X)=sup\{{\delta}_X(w):w{\in}X\}$ where ${\delta}_X(w)$ is the euclidean distance from w to ${\partial}X$. Here ${\partial}X$ is the boundary of X. A hyperbolic region X is called a Bloch region if R(X) < ${\infty}$. In this paper, we obtain lower bounds for the hyperbolic metric on Bloch regions in terms of the distance to the boundary.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1335-1364
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• 2023

The aim of this paper is to investigate the spectral instability of roll waves bifurcating from an equilibrium in the 2-dimensional generalized Swift-Hohenberg equation. We characterize unstable Bloch wave vectors to prove that the rolls are spectrally unstable in the whole parameter region where the rolls exist, while they are Eckhaus stable in 1 dimension [13]. As compared to [18], showing that the stability of the rolls in the 2-dimensional Swift-Hohenberg equation without a quadratic nonlinearity is determined by Eckhaus and zigzag curves, our result says that the quadratic nonlinearity of the equation is the cause of such instability of the rolls.

• Investigative Magnetic Resonance Imaging
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• v.20 no.1
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• pp.27-35
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• 2016

Purpose: Among RF pulses, a sinc pulse is typically used for slice selection due to its frequency-selective feature. When a sinc pulse is implemented in practice, it needs to be apodized to avoid truncation artifacts at the expense of broadening the transition region of the excited-band profile. Here a sinc pulse tailored by a new apodization function is proposed that produces a sharper transition region with well suppression of truncation artifacts in comparison with conventional tailored sinc pulses. A multiband pulse designed using this newly apodized sinc pulse is also suggested inheriting the better performance of the newly apodized sinc pulse. Materials and Methods: A new apodization function is introduced to taper a sinc pulse, playing a role to slightly shift the first zero-crossing of a tailored sinc pulse from the peak of the main lobe and thereby producing a narrower bandwidth as well as a sharper pass-band in the excitation profile. The newly apodized sinc pulse was also utilized to design a multiband pulse which inherits the performance of its constituent. Performances of the proposed sinc pulse and the multiband pulse generated with it were demonstrated by Bloch simulation and phantom imaging. Results: In both simulations and experiments, the newly apodized sinc pulse yielded a narrower bandwidth and a sharper transition of the pass-band profile with a desirable degree of side-lobe suppression than the commonly used Hanning-windowed sinc pulse. The multiband pulse designed using the newly apodized sinc pulse also showed the better performance in multi-slice excitation than the one designed with the Hanning-windowed sinc pulse. Conclusion: The new tailored sinc pulse proposed here provides a better performance in slice (or slab) selection than conventional tailored sinc pulses. Thanks to the availability of analytical expression, it can also be utilized for multiband pulse design with great flexibility and readiness in implementation, transferring its better performance.