• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.255-260
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• 2006

Let $m$ and $k$ be any positive integers, let $\mathbb{Z}_k$ the ring of integers of modulo $k$, let $G_m(\mathbb{Z}_k)$ the group of all $m$ by $m$ nonsingular matrices over $\mathbb{Z}_k$ and let ${\phi}_m(k)$ the order of $G_m(\mathbb{Z}_k)$. In this paper, ${\phi}_m(k)$ can be computed by the following investigation: First, for any relatively prime positive integers $s$ and $t$, $G_m(\mathbb{Z}_{st})$ is isomorphic to $G_m(\mathbb{Z}_s){\times}G_m(\mathbb{Z}_t)$. Secondly, for any positive integer $n$ and any prime $p$, ${\phi}_m(p^n)=p^{m^2}{\cdot}{\phi}_m(p^{n-1})=p{^{2m}}^2{\cdot}{\phi}_m(p^{n-2})={\cdots}=p^{{(n-1)m}^2}{\cdot}{\phi}_m(p)$, and so ${\phi}_m(k)={\phi}_m(p_1^n1){\cdot}{\phi}_m(p_2^{n2}){\cdots}{\phi}_m(p_s^{ns})$ for the prime factorization of $k$, $k=p_1^{n1}{\cdot}p_2^{n2}{\cdots}p_s^{ns}$.

• Journal of KIISE:Computer Systems and Theory
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• v.29 no.12
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• pp.665-679
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• 2002

The fault diameter of a graph G is the maximum of lengths of the shortest paths between all two vertices when there are $\chi$(G) - 1 or less faulty vertices, where $\chi$(G) is connectivity of G. In this paper, we analyze the fault diameter of recursive circulant $G(2^m,2^k)$ with $k{\geq}3$. Let $ dia_{m.k}$ denote the diameter of $G(2^m,2^k)$. We show that if $2{\leq}m,2{\leq}k, the fault diameter of $G(2{\leq}m,2{\leq}k)$ is equal to $2^m-2$, and if m=k+1, it is equal to $2^k-1$. It is also shown that for m>k+1, the fault diameter is equal to di a_$m{\neq}1$(mod 2k); otherwise, it is less than or equal to$dia_{m.k+2}$.

• Journal of Plant Biology
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• v.39 no.1
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• pp.41-48
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• 1996

Two forms of homo serine dehydrogenase have been isolated from 8-day-old cotyledons of Canavalin lineata by a heat denaturation, ammonium sulfate fractionation, DEAE-8ephacel ion exchange and Sephacryl 8-300 gel filtration chromatographies, and Pro cion red dye, Cibacron blue dye and Resource Q column chromatographies. The molecular weights of T -form (threonine-sensitive) and K-form(threonine- insensitive) were estimated to 230 kD and 135 kD, respectively. In the presence of 10 mM threonine, the activity of T-form was inhibited with almost 70%, but that of K-form was not at all. The Km values tor homo serine of T- and Kform were 1.6 mM and 0.3 mM, respectively. The Km values for NAD of T- and K-form were 2.34 mM and 0.03 mM, respectively. And Km values for NADP of two isozymes were the same as 0.01 mM. The activities of T- and K-form were markedly stimulated up to 4.9and 2.8-fold, respectively, by 400 mM KCI. The partial purified(gel filtration) enzymes(Tform and K-form) can be reversibly converted.verted.

• Journal of Applied Biological Chemistry
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• v.42 no.1
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• pp.12-18
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• 1999

Sucrose synthase (EC 2.4.1.13) has been purified from the plant cytosolic fraction of chickpea (Cicer arietinum L. cv. Amethyst) nodules. The native enzyme had a molecular mass of $356{\pm}15kD$. The subunit molecular mass was $87{\pm}2kD$, and a tetrameric structure is proposed for sucrose synthase of chickpea nodule. Optimum activities in the sucrose cleavage and synthesis directions were at pH 6.5 and 9.0, respectively. The purified enzyme displayed typical hyperbolic kinetics with substrates in cleavage and synthesis reactions. Chickpea nodules sucrose synthase had a high affinity for UDP ($K_m$, $8.0{\mu}M$) and relatively low affinities for ADP ($K_m$, 0.23 mM), CDP ($K_m$, 0.87 mM), and GDP ($K_m$, 1.51 mM). The $K_m$ for sucrose was 29.4 mM. In the synthesis reaction, UDP-glucose ($K_m$, $24.1{\mu}M$) was a more effective glucosyl donor than ADP-glucose ($K_m$, 2.7 mM), and the $K_m$ for fructose was 5.4 mM. Divalent cations, such as $Ca^{2+}$, $Mg^{2+}$, and $Mn^{2+}$, stimulated the enzyme activity in both the cleavage and synthesis directions, and the enzyme was very sensitive to inhibition by $HgCl_2$ and $CuSO_4$.

• The Korean Journal of Physiology and Pharmacology
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• v.3 no.4
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• pp.415-425
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• 1999

$[K^+]_o$ can be increased under a variety of conditions including subarachnoid hemorrhage. The increase of $[K^+]_o$ in the range of $5{\sim}15$ mM may affect tensions of blood vessels and cause relaxation of agonist-induced precontracted vascular smooth muscle $(K^+-induced$ relaxation). In this study, effect of the increase in extracellular $K^+$ concentration on the agonist-induced contractions of various arteries including resistant arteries of rabbit was examined, using home-made Mulvany-type myograph. Extracellular $K^+$ was increased in three different ways; from initial 1 to 3 mM, from initial 3 to 6 mM, or from initial 6 to 12 mM. In superior mesenteric arteries, the relaxation induced by extracellular $K^+$ elevation from initial 6 to 12 mM was the most prominent among the relaxations induced by the elevations in three different ways. In cerebral arteries, the most prominent relaxation was produced by the elevation of extracellular $K^+$ from initial 1 to 3 mM and a slight relaxation was provoked by the elevation from initial 6 to 12 mM. In superior mesenteric arteries, $K^+-induced$ relaxation by the elevation from initial 6 to 12 mM was blocked by $Ba^{2+}\;(30\;{\mu}M)$ and the relaxation by the elevation from 1 to 3 mM or from 3 to 6 mM was not blocked by $Ba^{2+}.$ In cerebral arteries, however, $K^+-induced$ relaxation by the elevation from initial 3 to 6 mM was blocked by $Ba^{2+},$ whereas the relaxation by the elevation from 1 to 3 mM was not blocked by $Ba^{2+}.$ Ouabain inhibited all of the relaxations induced by the extracellular $K^+$ elevations in three different ways. In cerebral arteries, when extracellular $K^+$ was increased to 14 mM with 2 or 3 mM increments, almost complete relaxation was induced at 1 or 3 mM of initial $K^+$ concentration and slight relaxation occurred at 6 mM. TEA did not inhibit $Ba^{2+}-sensitive$ relaxation at all and NMMA or endothelial removal did not inhibit $K^+-induced$ relaxation. Most conduit arteries such as aorta, carotid artery, and renal artery were not relaxed by the elevation of extracellular $K^+.$ Among conduit arteries, trunk of superior mesenteric artery and basilar artery were relaxed by the elevations of $[K^+]_o.$ These data suggest that $K^+-induced$ relaxation has two independent components, $Ba^{2+}-sensitive$ and $Ba^{2+}-insensitive$ one and there are different mechanisms for $K^+-induced$ relaxation in various arteries.

• The Korean Journal of Physiology and Pharmacology
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• v.2 no.6
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• pp.695-703
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• 1998

$[K^+]_o$ can be increased under a variety of conditions including subarachnoid hemorrhage. The increase of $[K^+]_o$ in the range of $5{\sim}15$ mM may affect tensions of blood vessels and can change their sensitivity to various vasoactive substances. Therefore, it was examined in the present study whether the sensitivity of cerebral arteries to vasoactive substances can be changed with the moderate increase of $[K^+]_o$, using Mulvany-type myograph and $[Ca^{2+}]_c$ measurement. The contractions of basilar artery and branch of middle cerebral artery induced by histamine were not increased with the elevation of $[K^+]_o$ from 6 mM to 9 mM or 12 mM. On the contrary, the contractions induced by serotonin were significantly increased with the elevation of $[K^+]_o$. The contractions were also significantly increased by the treatment with nitro-L-arginine $(10^{-4}$ M for 20 minutes). In the nitro-L-arginine treated arteries, the contractions induced by serotonin were significantly increased with the elevation of $[K^+]_o$ from 6 mM to 12 mM. $K^+-induced$ relaxation was evoked with the stepwise increment of extracellular $K^+$ from 0 or 2 mM to 12 mM by 2 mM in basilar arterial rings, which were contracted by histamine. But $[K^+]_o$ elevation from 4 or 6 mM to 12 mM by the stepwise increment evoked no significant relaxation. Basal tension of basilar artery was increased with $[K^+]_o$ elevation from 6 mM to 12 mM by 2 mM steps or by the treatment with ouabain and the increase of basal tension was blocked by verapamil. The cytosolic free $Ca^{2+}$ level was not increased by the single treatment with serotonin or with the elevation of $[K^+]_o$ from 4 mM to 8 or 12 mM. In contrast to the single treatment, the $Ca^{2+}$ level was increased by the combined treatment with serotonin and the elevation of $[K^+]_o$. The increase of free $Ca^{2+}$ concentration was blocked by the treatment with verapamil. These data suggest that the sensitivity of cerebral artery to serotonin is increased with the moderate increase of $[K^+]_o$ and the increased sensitivity to serotonin is due to the increased $[Ca^{2+}]_i$ induced by extracellular $Ca^{2+}$ influx.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1075-1090
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• 2023

In this paper, we study new classes of operators k-quasi (m, n)-paranormal operator, k-quasi (m, n)^*-paranormal operator, k-quasi (m, n)-class 𝒬 operator and k-quasi (m, n)-class 𝒬^* operator which are the generalization of (m, n)-paranormal and (m, n)^*-paranormal operators. We give matrix characterizations for k-quasi (m, n)-paranormal and k-quasi (m, n)^*-paranormal operators. Also we study some properties of k-quasi (m, n)-class 𝒬 operator and k-quasi (m, n)-class 𝒬^* operators. Moreover, these classes of composition operators on L² spaces are characterized.

• Communications of the Korean Mathematical Society
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• v.12 no.1
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• pp.59-67
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• 1997

Let $K'_M$ be the space of distributions on $R^m$ which grow no faster than $e^{M(kx)}$ for some k > 0 and an index function M(x) and $K'_M$ be the Fourier transform of $K'_M$. We establish the characterizations of the space $O_M(K'_m;K'_M)$ of multipliers in $K'_M$.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.435-447
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• 2012

Let G be a group. Let R be a G-graded commutative ring with identity and M be a G-graded multiplication module over R. A proper graded submodule Q of M is semiprime if whenever $I^nK{\subseteq}Q$, where $I{\subseteq}h(R)$, n is a positive integer, and $K{\subseteq}h(M)$, then $IK{\subseteq}Q$. We characterize semiprime submodules of M. For example, we show that a proper graded submodule Q of M is semiprime if and only if grad$(Q){\cap}h(M)=Q+{\cap}h(M)$. Furthermore if M is finitely generated then we prove that every proper graded submodule of M is contained in a graded semiprime submodule of M. A proper graded submodule Q of M is said to be almost semiprime if (grad(Q)$\cap$h(M))n(grad$(0_M){\cap}h(M)$) = (Q$\cap$h(M))n(grad$(0_M){\cap}Q{\cap}h(M)$). Let K, Q be graded submodules of M. If K and Q are almost semiprime in M such that Q + K $\neq$ M and $Q{\cap}K{\subseteq}M_g$ for all $g{\in}G$, then we prove that Q + K is almost semiprime in M.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.32 no.10C
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• pp.929-934
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• 2007

In this paper, we enumerate the number of distinct autocorrelation distributions that M-ary Sidel'nikov sequences can have, while we change the primitive element for generating the sequence. Let p be a prime and $M|p^n-1$. For M=2, there is a unique autocorrelation disuibution. If M>2 and $M|p^k+1$ for some k, $1{\leq}k, then the autocorrelatin distribution of M-ary Sidel'nikov sequences is unique. If M>2 and $M{\nmid}p^k+1$ for any k, $1{\leq}k, then the autocorrelation distribution of M-ary Sidel'nikov sequences is less than or equal to ${\phi}(M)/k'(or\;{\phi}(M)/2k')$, where k' is the smallest integer satisfying $M|p^{k'}-1$.