• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.377-392
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• 2008

Let X be a real reflexive Banach space with a uniformly $G\hat{a}teaux$ differentiable norm, C a nonempty closed convex subset of X, T : C $\rightarrow$ X a continuous pseudocontractive mapping, and A : C $\rightarrow$ C a continuous strongly pseudocontractive mapping. We show the existence of a path ${x_t}$ satisfying $x_t=tAx_t+(1- t)Tx_t$, t $\in$ (0,1) and prove that ${x_t}$ converges strongly to a fixed point of T, which solves the variational inequality involving the mapping A. As an application, we give strong convergence of the path ${x_t}$ defined by $x_t=tAx_t+(1-t)(2I-T)x_t$ to a fixed point of firmly pseudocontractive mapping T.

• Communications of the Korean Mathematical Society
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• v.12 no.1
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• pp.69-73
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• 1997

We show that if T is an $\varepsilon$-approximate Jordan functional such that T(a) = 0 implies $T(a^2) = 0 (a \in A)$ then T is continuous and $\Vert T \Vert \leq 1 + \varepsilon$. Also we prove that every $\varepsilon$-near Jordan mapping is an $g(\varepsilon)$-approximate Jordan mapping where $g(\varepsilon) \to 0$ as $\varepsilon \to 0$ and for every $\varepsilon > 0$ there is an integer m such that if T is an $\frac {\varepsilon}{m}$-approximate Jordan mapping on a finite dimensional Banach algebra then T is an $\varepsilon$-near Jordan mapping.

• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.37-51
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• 2002

Let X be a reflexive Banach space with a uniformly Gateaux differentiable norm, C a nonempty bounded open subset of X, and T a continuous mapping from the closure of C into X which is locally pseudo-contractive mapping on C. We show that if the closed unit ball of X has the fixed point property for nonexpansive self-mappings and T satisfies the following condition: there exists z $\in$ C such that ∥z-T(z)∥<∥x-T(x)∥ for all x on the boundary of C, then the trajectory tlongrightarrowz$_{t}$$\in$C, t$\in$[0, 1) defined by the equation z$_{t}$ = tT(z$_{t}$)+(1-t)z is continuous and strongly converges to a fixed point of T as t longrightarrow 1￣.ow 1￣.

• Investigative Magnetic Resonance Imaging
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• v.24 no.3
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• pp.141-153
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• 2020

Purpose: Myocardial T1 and T2 relaxation times are affected by technical factors such as cardiovascular magnetic resonance platform/vendor. We aimed to validate T1 and T2 mapping sequences using a phantom; establish reference T1, T2, and extracellular volume (ECV) measurements using two sequences at 3T in normal Koreans; and compare the protocols and evaluate the differences from previously reported measurements. Materials and Methods: Eleven healthy subjects underwent cardiac magnetic resonance imaging (MRI) using 3T MRI equipment (Verio, Siemens, Erlangen, Germany). We did phantom validation before volunteer scanning: T1 mapping with modified look locker inversion recovery (MOLLI) with 5(3)3 and 4(1)3(1)2 sequences, and T2 mapping with gradient echo (GRE) and TrueFISP sequences. We did T1 and T2 mappings on the volunteers with the same sequences. ECV was also calculated with both sequences after gadolinium enhancement. Results: The phantom study showed no significant differences from the gold standard T1 and T2 values in either sequence. Pre-contrast T1 relaxation times of the 4(1)3(1)2 protocol was 1142.27 ± 36.64 ms and of the 5(3)3 was 1266.03 ± 32.86 ms on the volunteer study. T2 relaxation times of GRE were 40.09 ± 2.45 ms and T2 relaxation times of TrueFISP were 38.20 ± 1.64 ms in each. ECV calculation was 24.42% ± 2.41% and 26.11% ± 2.39% in the 4(1)3(1)2 and 5(3)3 protocols, respectively, and showed no differences at any segment or slice between the sequences. We also calculated ECV from the pre-enhancement T1 relaxation time of MOLLI 5(3)3 and the post-enhancement T1 relaxation time of MOLLI 4(1)3(1)2, with no significant differences between the combinations. Conclusion: Using phantom-validated sequences, we reported the normal myocardial T1, T2, and ECV reference values of healthy Koreans at 3T. There were no statistically significant differences between the sequences, although it has limited statistical value due to the small number of subjects studied. ECV showed no significant differences between calculations based on various pre- and post-mapping combinations.

• Journal of Cardiovascular Imaging
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• v.31 no.2
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• pp.71-82
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• 2023

BACKGROUND: Cardiac magnetic resonance fingerprinting (cMRF) enables simultaneous mapping of myocardial T1 and T2 with very short acquisition times. Breathing maneuvers have been utilized as a vasoactive stress test to dynamically characterize myocardial tissue in vivo. We tested the feasibility of sequential, rapid cMRF acquisitions during breathing maneuvers to quantify myocardial T1 and T2 changes. METHODS: We measured T1 and T2 values using conventional T1 and T2-mapping techniques (modified look locker inversion [MOLLI] and T2-prepared balanced-steady state free precession), and a 15 heartbeat (15-hb) and rapid 5-hb cMRF sequence in a phantom and in 9 healthy volunteers. The cMRF_5-hb sequence was also used to dynamically assess T1 and T2 changes over the course of a vasoactive combined breathing maneuver. RESULTS: In healthy volunteers, the mean myocardial T1 of the different mapping methodologies were: MOLLI 1,224 ± 81 ms, cMRF_15-hb 1,359 ± 97 ms, and cMRF_5-hb 1,357 ± 76 ms. The mean myocardial T2 measured with the conventional mapping technique was 41.7 ± 6.7 ms, while for cMRF_15-hb 29.6 ± 5.8 ms and cMRF_5-hb 30.5 ± 5.8 ms. T2 was reduced with vasoconstriction (post-hyperventilation compared to a baseline resting state) (30.15 ± 1.53 ms vs. 27.99 ± 2.07 ms, p = 0.02), while T1 did not change with hyperventilation. During the vasodilatory breath-hold, no significant change of myocardial T1 and T2 was observed. CONCLUSIONS: cMRF_5-hb enables simultaneous mapping of myocardial T1 and T2, and may be used to track dynamic changes of myocardial T1 and T2 during vasoactive combined breathing maneuvers.

• Korean Journal of Mathematics
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• v.5 no.1
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• pp.17-27
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• 1997

In this paper we define a torsion function ${\log}\;T(M,{\varphi})(t)$) for $t{\gg}0$ and show that it has an asymptotic expansion $\frac{1}{2}{\chi}(M)t$ as $t{\rightarrow}{\infty}$.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1151-1164
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• 2009

Let E be a reflexive Banach space with a weakly sequentially continuous duality mapping, C be a nonempty closed convex subset of E, f : C $\rightarrow$C a contractive mapping (or a weakly contractive mapping), and T : C $\rightarrow$ C a nonexpansive mapping with the fixed point set F(T) ${\neq}{\emptyset}$. Let {$x_n$} be generated by a new composite iterative scheme: $y_n={\lambda}_nf(x_n)+(1-{\lambda}_n)Tx_n$, $x_{n+1}=(1-{\beta}_n)y_n+{\beta}_nTy_n$, ($n{\geq}0$). It is proved that {$x_n$} converges strongly to a point in F(T), which is a solution of certain variational inequality provided the sequence {$\lambda_n$} $\subset$ (0, 1) satisfies $lim_{n{\rightarrow}{\infty}}{\lambda}_n$ = 0 and $\sum_{n=0}^{\infty}{\lambda}_n={\infty}$, {$\beta_n$} $\subset$ [0, a) for some 0 < a < 1 and the sequence {$x_n$} is asymptotically regular.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.71-79
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• 1996

In this paper, we prove for a nonexpansive mapping T that under certain conditions the trajectory $t \to G_t(x), t \in [0,1]$, defined by the equation $G_t(x) = (1 - t)x + tTG_t(x)$ strongly converges to a fixed point of T as $t \to 1^{-1}$.

• Korean Journal of Radiology
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• v.21 no.8
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• pp.978-986
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• 2020

Objective: To compare native and post-contrast T1 mapping with late gadolinium enhancement (LGE) imaging for detecting and measuring the microvascular obstruction (MVO) area in reperfused acute myocardial infarction (MI). Materials and Methods: This study included 20 patients with acute MI who had undergone 1.5T cardiovascular magnetic resonance imaging (CMR) after reperfusion therapy. CMR included cine imaging, LGE, and T1 mapping (modified look-locker inversion recovery). MI size was calculated from LGE by full-width at half-maximum technique. MVO was defined as an area with low signal intensity (LGE) or as a region of visually distinguishable T1 values (T1 maps) within infarcted myocardium. Regional T1 values were measured in MVO, infarcted, and remote myocardium on T1 maps. MVO area was measured on and compared among LGE, native, and post-contrast T1 maps. Results: The mean MI size was 27.1 ± 9.7% of the left ventricular mass. Of the 20 identified MVOs, 18 (90%) were detected on native T1 maps, while 10 (50%) were recognized on post-contrast T1 maps. The mean native T1 values of MVO, infarcted, and remote myocardium were 1013.5 ± 58.5, 1240.9 ± 55.8 (p < 0.001), and 1062.2 ± 55.8 ms (p = 0.169), respectively, while the mean post-contrast T1 values were 466.7 ± 26.8, 399.1 ± 21.3, and 585.2 ± 21.3 ms, respectively (p < 0.001). The mean MVO areas on LGE, native, and post-contrast T1 maps were 134.1 ± 81.2, 133.7 ± 80.4, and 117.1 ± 53.3 mm², respectively. The median (interquartile range) MVO areas on LGE, native, and post-contrast T1 maps were 128.0 (58.1-215.4), 110.5 (67.7-227.9), and 143.0 (76.7-155.3) mm², respectively (p = 0.002). Concordance correlation coefficients for the MVO area between LGE and native T1 maps, LGE and post-contrast T1 maps, and native and post-contrast T1 maps were 0.770, 0.375, and 0.565, respectively. Conclusion: MVO areas were accurately delineated on native T1 maps and showed high concordance with the areas measured on LGE. However, post-contrast T1 maps had low detection rates and underestimated MVO areas. Collectively, native T1 mapping is a useful tool for detecting MVO within the infarcted myocardium.

• Journal of the Chungcheong Mathematical Society
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• v.3 no.1
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• pp.103-110
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• 1990

Let K be a nonempty weakly compact convex subset of a Banach space X and T : K ${\rightarrow}$ C(X) a nonexpansive mapping satisfying $P_T(x){\cap}clI_K(x){\neq}{\emptyset}$. We first show that if I - T is semiconvex type then T has a fixed point. Also we obtain the same result without the condition that I - T is semiconvex type in a Banach space satisfying Opial's condition. Lastly we extend the result of [5] to the case, that T is an 1-set contraction mapping.