• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.137-143
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• 1992

The purpose of this paper is to provide the affirmative solution of the following conjecture due to Davis and Geramita. Conjecture; Let A=R[T] be a polynomial ring in one variable, where R is a regular local ring of dimension d. Then maximal ideals in A are complete intersection. Geramita has proved that the conjecture is true when R is a regular local ring of dimension 2. Whatwadekar has rpoved that conjecture is true when R is a formal power series ring over a field and also when R is a localization of an affine algebra over an infinite perfect field. Nashier also proved that conjecture is true when R is a local ring of D[ $X_{1}$,.., $X_{d-1}$] at the maximal ideal (.pi., $X_{1}$,.., $X_{d-1}$) where (D,(.pi.)) is a discrete valuation ring with infinite residue field. The methods to establish our results are following from Nashier's method. We divide this paper into three sections. In section 1 we state Theorems without proofs which are used in section 2 and 3. In section 2 we prove some lemmas and propositions which are used in proving our results. In section 3 we prove our main theorem.eorem.rem.

• Bulletin of the Korean Mathematical Society
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• v.36 no.1
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• pp.25-31
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• 1999

Let $\nu$ be a positive Borel measure on a space of homogeneous type (X, d, $\mu$), satisfying the doubling property. A condition on a weight $\omega$ for whixh a maximal operator $M\nu f$(x) defined by M$mu$f(x)=supr>0{{{{ { 1} over {ν(B(x,r)) } INT _{ B(x,r)} │f(y)│d mu (y)}}}}, is of weak type (p,p) with respect to (ν, $omega$), is that there exists a constant C such that C $omega$(y) for a.e. y$\in$B(x, r) if p=1, and {{{{( { 1} over { upsilon (B(x,r) } INT _{ B(x,r)}omega(y) ^ (-1/p-1) d mu (y))^(p-1)}}}} C, if 1$infty$.

• Bulletin of the Korean Mathematical Society
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• v.22 no.2
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• pp.117-119
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• 1985

Let R be a commutative noetherian ring with 1.neq.0, denoting by .nu.(I) the cardinality of a minimal basis of the ideal I. Let A be a polynomial ring in n>0 variables with coefficients in R, and let M be a maximal ideal of A. Generally it is shown that .nu.(M $A_{M}$).leq..nu.(M).leq..nu.(M $A_{M}$)+1. It is well known that the lower bound is not always satisfied, and the most classical examples occur in nonfactional Dedekind domains. But in many cases, (e.g., A is a polynomial ring whose coefficient ring is a field) the lower bound is attained. In [2] and [3], the conditions when the lower bound is satisfied is investigated. Especially in [3], it is shown that .nu.(M)=.nu.(M $A_{M}$) if M.cap.R=p is a maximal ideal or $A_{M}$ (equivalently $R_{p}$) is not regular or n>1. Hence the problem of determining whether .nu.(M)=.nu.(M $A_{M}$) can be studied when p is not maximal, $A_{M}$ is regular and n=1. The purpose of this note is to provide some conditions in which the lower bound is satisfied, when n=1 and R is a regular local ring (hence $A_{M}$ is regular)./ is regular).

• The Korean Journal of Physiology
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• v.2 no.2
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• pp.11-20
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• 1968

Maximal oxygen uptake was measured in thirty-three secondary school girls by means of the treadmill test. Eighteen middle school girls aged 14.0 (range: $13.0{\sim}15.9$) years and fifteen high school girls aged 16.9 (range: $16.0{\sim}18.0$) years served as subjects. Maximal treadmill run lasted for 2 minutes and 20 seconds and the expired air was collected in a Douglas bag through a J-valve during the last one minute period. In general, absolute values of various measurements in the high school girls were greater than those of the middle school girls. When values were expressed on the body weight or lean body weight basis, however, work capacity of middle school girls was superior to that of the high school girls. The detailed results are as follows: 1. In middle school girls maximal oxygen uptake was 1.78 l/min., 47.4 ml/kg body weight, 12.3 ml/cm body height, and 61.7ml/kg lean body mass. In high school girls maximal oxygen uptake was 1.93 l/min., 39.7ml/kg body weight, 12.3 ml/cm body height, and 51.2 ml/kg LBM. Although the absolute value of maximal oxygen uptake was greater in high school girls than in middle school girls, values expressed on the body weight basis showed the reverse trend, namely, values of the middle school girls was greater than those of the high school girls. 2. The ratio of maximal to resting oxygen uptake was 8.8 in the middle school girls and was 10.2 in the high school girls. 3. Maximal pulmonary ventilation in the middle school girls was 55.3 l/min. and 66.1 l/min. in the high school girls. The ratio of maximal to resting pulmonary ventilation was 10.2 in the middle school girls and 10.1 in the high school girls. 4. The correlation between body weight and maximal oxygen uptake was relatively high, namely, r=0.79 both in middle and high school girls. The correlation coefficient between body weight and maximal pulmonary ventilation was a little less that of between maximal oxygen uptake and showed a value of r=0.60 both in middle and high school girls. The lean body mass was a poor reference of maximal oxygen uptake or maximal pulmonary ventilation as compared to body weight. The correlation between maximal oxygen uptake and maximal pulmonary ventilation was high and the coefficient of correlation in middle school girls was 0.927 and in high school girls it was 0.856. 5. Maximal ventilation equivalent was 30.9 liters in middle school girls and 33.9 liters in high school girls. This indicated that no hyperventilation was induced during the maximal of oxygen uptake exercise period as related to the maximal oxygen uptake. 6. Heart rate reached to the peak value within 1.5 minutes after beginning of maximal oxygen uptake run and remained at the same peak plateau level throughout the entire running period. Heart rate decreased steeply on cessation of running and subsided slowly thereafter. The maximal heart rate was 184 beat/min. in middle school girls and 189 beat/min. in high school girls. 7. Maximal oxygen pulse was 9.4 in middle school girls and 9.9 ml/beat in high school girls.

• Communications for Statistical Applications and Methods
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• v.1 no.1
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• pp.149-156
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• 1994

For an r > 2 and a finite B, $E\mid max \;1\leq k\leq n \;\sum\limits_{j=m+1}^{m+k}X_j\mid^r\leq Bn^ {\frac{r}{2}}$ (all $n\geq 1$) is obtained for a negatively associated sequence $\{X_j \;:\; j\in N\}$. We also derive the maximal inequelity for a negatively associated sequence. Stationarity is not required.

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.210-214
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• 1994

The purpose of this paper is to investigate the relationship between the depths of the Rees algebra R[It] and the associated graded ring g $r_{I}$(R) of an ideal I in a local ring (R,m) of dim(R) > 0. The relationship between the Cohen-Macaulayness of these two rings has been studied extensively. Let (R, m) be a local ring and I an ideal of R. An ideal J contained in I is called a reduction of I if J $I^{n}$ = $I^{n+1}$ for some integer n.geq.0. A reduction J of I is called a minimal reduction of I. The reduction number of I with respect to J is defined by (Fig.) S. Goto and Y.Shimoda characterized the Cohen-Macaulay property of the Rees algebra of the maximal ideal of a Cohen-Macaulay local ring in terms of the Cohen-Macaulay property of the associated graded ring of the maximal ideal and the reduction number of that maximal ideal. Let us state their theorem.m.m.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.155-162
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• 1995

Let $\mu$ be a positive Borel measure on $R^n$ which is positive on cubes. For any cube $Q \subset R^n$, a Borel measurable nonnegative function $\varphi_Q$, supported and positive a.e. with respect to $\mu$ in Q, is given. We consider a maximal function $$ M_{\mu}f(x) = sup \int \varphi Q$\mid$f$\mid$d_{\mu} $$ where the supremum is taken over all $\varphi Q$ such that $x \in Q$.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.373-386
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• 2014

Let (R,m) be a commutative Noetherian local ring. In this paper we show that a finitely generated R-module M of dimension d is Cohen-Macaulay if and only if there exists a proper ideal I of R such that depth($M/I^nM$) = d for $n{\gg}0$. Also we show that, if dim(R) = d and $I_1{\subset}\;{\cdots}\;{\subset}I_n$ is a chain of ideals of R such that $R/I_k$ is maximal Cohen-Macaulay for all k, then $n{\leq}{\ell}_R(R/(a_1,{\ldots},a_d)R)$ for every system of parameters $a1,{\ldots},a_d$ of R. Also, in the case where dim(R) = 2, we prove that the ideal transform $D_m(R/p)$ is minimax balanced big Cohen-Macaulay, for every $p{\in}Assh_R$(R), and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.

• The Korean Journal of Physiology
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• v.2 no.2
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• pp.1-10
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• 1968

Maximal oxygen consumption measurements were performed on 15 middle school boys (age: mean 14.0, range: $13{\sim}16$ years) and 14 high school boys (age: mean 17.4, range: $16{\sim}19$ years). General body build was greater in the high school boys and absolute values of body height, body weight, skinfold thicknesses, maximal oxygen uptake, and maximal pulmonary ventilation followed the same trend. Considered on the basis of body build, however, the values of high school boys were not always greater than those of middle school boys. The following results were obtained. 1. Maximal oxygen consumption in middle school boys was 2.11 l/min., 53.7ml/kg b. weight, 13.9 ml/cm body height, and 63.7 ml/kg LBM. In high school boys the values were: 2.86 l/min., 52.7 ml/kg b.wt., 17.5 ml/cm b. height, and 57.9 ml/kg LBM. Thus, middle school boys were superior to high school boys on body weight and lean body mass basis. They were also superior to the European boys of the same age. 2. The ratio of maximal oxygen uptake to resting value was 9.7 in middle school boys, and 10.8 in high school boys. 3. Maximal pulmonary ventilation in middle school boys was 58.0 l/min., and 84.0 l/min. in high school boys. The ratio of maximal ventilation to resting value was the same as oxygen uptake, namely, 9.7 in middle school boys and 10.7 in high school boys. 4. Ventilation equivalent in middle school boys was 27.5 and 29.3 in high school boys. These values represent values of untrained male subjects. 5. Maximal heart rate in high school boys reached to 193 beat/min. and is 2.9 times that of resting heart rate. 6. Maximal oxygen pulse in high school boys was 16.6 ml/beat and was same as that of untrained subject. 7. Correlation between body weight and maximal oxygen consumption in middle school boys was r=0.570, and r=0.162 in high school boys. Correlation between lean body mass in middle school boys was r=0.499, and r=0.158 in high school boys. Interrelation between body weight and maximal pulmonary ventilation was poor. 8. The differences between trained and untrained subjects were discussed.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.277-303
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• 2021

Under the rough kernels Ω belonging to the block spaces B0,qr (Sn-1) or the radial Grafakos-Stefanov kernels W����(Sn-1) for some r, �� > 1 and q ≤ 0, the boundedness and continuity were proved for two classes of rough maximal singular integrals and maximal operators associated to polynomial mappings on the Triebel-Lizorkin spaces and Besov spaces, complementing some recent boundedness and continuity results in [27, 28], in which the authors established the corresponding results under the conditions that the rough kernels belong to the function class L(log L)α(Sn-1) or the Grafakos-Stefanov class ����(Sn-1) for some α ∈ [0, 1] and �� ∈ (2, ∞).