• Communications of the Korean Mathematical Society
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• v.24 no.1
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• pp.127-144
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• 2009

Let f be a diffeomorphism of a compact $C^{\infty}$ manifold, and let p be a hyperbolic periodic point of f. In this paper we introduce the notion of $C^1$-stable inverse shadowing for a closed f-invariant set, and prove that (i) the chain recurrent set $\cal{R}(f)$ of f has $C^1$-stable inverse shadowing property if and only if f satisfies both Axiom A and no-cycle condition, (ii) $C^1$-generically, the chain component $C_f(p)$ of f associated to p is hyperbolic if and only if $C_f(p)$ has the $C^1$-stable inverse shadowing property.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.393-402
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• 2013

For 0 < $p$ < ${\infty}$, ${\alpha}$ > -1 and 0 < $r$ < 1, we show that if $f$ is in the space of Dirichlet type $\mathfrak{D}^p_{p-1}$, then ${\int}_{1}^{0}M_{p}^{p}(r,f^{\prime})(1-r)^{p-1}rdr$ < ${\infty}$ and ${\int}_{1}^{0}M_{(2+{\alpha})p}^{(2+{\alpha})p}(r,f^{\prime})(1-r)^{(2+{\alpha})p+{\alpha}}rdr$ < ${\infty}$ where $M_p(r,f)=\[\frac{1}{2{\pi}}{\int}_{0}^{2{\pi}}{\mid}f(re^{it}){\mid}^pdt\]^{1/p}$. For 1 < $p$ < $q$ < ${\infty}$ and ${\alpha}+1$ < $p$, we show that if there exists some positive constant $c$ such that ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathfrak{D}^p_{\alpha}}$ for all $f{\in}\mathfrak{D}^p_{\alpha}$, then ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathcal{B}_p(q)}$ where $\mathcal{B}_p(q)$ is the weighted Besov space. We also find the condition of measure ${\mu}$ such that ${\sup}_{a{\in}D}{\int}_D(k_a(z)(1-{\mid}a{\mid}^2)^{(p-a-1)})^{q/p}d{\mu}(z)$ < ${\infty}$.

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.539-545
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• 1994

Let $q = p^r$, where p is a prime. A polynomial $f(x) \in GF(q)[x]$ is called a permutation polynomial (PP) over GF(q) if the numbers f(a) where $a \in GF(Q)$ are a permutation of the a's. In other words, the equation f(x) = a has a unique solution in GF(q) for each $a \in GF(q)$. More generally, $f(x_1, \cdots, x_n)$ is a PP in n variables if $f(x_1,\cdots,x_n) = \alpha$ has exactly $q^{n-1}$ solutions in $GF(q)^n$ for each $\alpha \in GF(q)$. Mullen ([3], [4], [5]) has studied the concepts of local permutation polynomials (LPP's) over finite fields. A polynomial $f(x_i, x_2, \cdots, x_n) \in GF(q)[x_i, \codts,x_n]$ is called a LPP if for each i = 1,\cdots, n, f(a_i,\cdots,x_n]$ is a PP in $x_i$ for all $a_j \in GF(q), j \neq 1$.Mullen ([3],[4]) found a set of necessary and three variables over GF(q) in order that f be a LPP. As examples, there are 12 LPP's over GF(3) in two indeterminates ; $f(x_1, x_2) = a_{10}x_1 + a_{10}x_2 + a_{00}$ where $a_{10} = 1$ or 2, $a_{01} = 1$ or x, $a_{00} = 0,1$, or 2. There are 24 LPP's over GF(3) of three indeterminates ; $F(x_1, x_2, x_3) = ax_1 + bx_2 +cx_3 +d$ where a,b and c = 1 or 2, d = 0,1, or 2.

• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.247-257
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• 1999

In this article, we give an example of local derivation, that is not derivation, on the algebra F(x1,…, xn) of rational functions in x1, …, xn over an infinite field F, and show that if X is a set of symbols and {x1,…, xn} is a finite subset of X, n$\geq$1, then each local derivation of F[x1,…, xn] into F[X] is a F-derivation and each local derivation of F[X] into itself is also a F-derivation.

• Journal of the Korean Society of Radiology
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• v.81 no.2
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• pp.365-378
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• 2020

Purpose This study was performed to determine whether the T1 relaxation time of gadoxetic acid-enhanced liver MR imaging is useful for detecting and staging liver fibrosis in patients with chronic liver disease. Materials and Methods One hundred and three patients with suspected focal liver lesion underwent MR imaging and Fibroscan. Fibroscan was chosen as the reference standard for classifying liver fibrosis. T1 relaxation times were acquired before (preT1), 20 minutes after (postT1) contrast administration, and reduction rate of T1 relaxation time (rrT1) on transverse 3D VIBE (volumetric interpolated breath-hold examination) sequence using 3T MR imaging. The optimal cut-off values for the fibrosis staging were determined with ROC analysis. Results PreT1 and postT1 increased and rrT1 decreased constantly with increasing severity of liver fibrosis according to the METAVIR score (F0-F4). There were statistically significant differences between F2 and F3 in preT1 (F2, 836.0 ± 74.7 ms; F3, 888.6 ± 77.5 ms, p < 0.05) and between F3 and F4 in postT1 (F3, 309.0 ± 80.2 ms; F4, 406.6 ± 147.7 ms, p < 0.05) and rrT1 (F3, 65.4 ± 7.7%; F4, 57.3 ± 11.4%, p < 0.05). ROC analysis revealed that combination test (preT1 + postT1) was the best test for predicting liver fibrosis. Conclusion PreT1 and postT1 increased constantly with increasing severity of liver fibrosis. T1 mapping in gadoxetic acid-enhanced liver MR imaging could be a helpful complementary sequence to determine the liver fibrosis stage.

• The Pure and Applied Mathematics
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• v.24 no.4
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• pp.243-251
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• 2017

In this paper, we solve the additive ${\rho}-functional$ equations (0.1) $f(x+y)+f(x-y)-2f(x)={\rho}(2f(\frac{x+y}{2})+f(x-y)-2f(x))$, and (0.2) $2f(\frac{x+y}{2})+f(x-y)-2f(x)={\rho}(f(x+y)+f(x-y)-2f(x))$, where ${\rho}$ is a fixed (complex) number with ${\rho}{\neq}1$, Using the direct method, we prove the Hyers-Ulam stability of the additive ${\rho}-functional$ equations (0.1) and (0.2) in ${\beta}-homogeneous$ (complex) F-spaces.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.91-101
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• 2015

In this paper, we investigate the stability of the functional equation f(-x + y + z + w) + f(x - y + z + w) + f(x + y - z + w) + f(x + y + z - w) = 3f(x) + f(-x) + 3f(y) + f(-y) + 3f(z) + f(-z) + 3f(w) + f(-w) by using the direct method in the sense of Hyers.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.27-36
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• 2014

In this paper, we introduce the notion of f-derivations from a semilattice S to a lattice L, as a generalization of derivation and f-derivation of lattices. Also, we define the simple f-derivation from S to L, and research the properties of them and the conditions for a lattice L to be distributive. Finally, we prove that a distributive lattice L is isomorphic to the class $SD_f(S,L)$ of all simple f-derivations on S to L for every ${\wedge}$-homomorphism $f:S{\rightarrow}L$ such that $f(x_0){\vee}f(y_0)=1$ for some $x_0,y_0{\in}S$, in particular, $$L{\simeq_-}=SD_f(S,L)$$ for every ${\wedge}$-homomorphism $f:S{\rightarrow}L$ such that $f(x_0)=1$ for some $x_0{\in}S$.

• Proceedings of the Korean Radioactive Waste Society Conference
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• 2003.11a
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• pp.426-430
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• 2003

Fissionable uranium will be separated from long-lived nuclear materials in pyroprocess for transmutation. This study was measured decomposition voltage and deposition rate on cathode of uranium in $LiF-BeF_2$ molten salt by electrowinning. The result of experimental is that decomposition voltage of $UF_4$ and $LiF-BeF_2$ molten salt is -1.4 and -1.5 volt at $500^{\circ}C$ Deposition rate of uranium on cathode increases with increase of uranium concentration in molten salt.

• The Pure and Applied Mathematics
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• v.23 no.2
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• pp.155-162
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• 2016

In this paper, we solve the additive ρ-functional inequalities (0.1)${\parallel}f(x+y)+f(x-y)-2f(x){\parallel}$ $\leq$ ${\parallel}{\rho}(2f(\frac{x+y}{2})+f(x-y)-2f(x)){\parallel}$, where ρ is a fixed complex number with |ρ| < 1, and (0.2) ${\parallel}2f(\frac{x+y}{2})+f(x-y)-2f(x)){\parallel}$ $\leq$ ${\parallel}{\rho}f(x+y)+f(x-y)-2f(x){\parallel}$, where ρ is a fixed complex number with |ρ| < 1. Furthermore, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces.