• Journal of applied mathematics & informatics
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• 제33권3_4호
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• pp.365-377
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• 2015

For a (molecular) graph, the first and second Zagreb indices (M₁ and M₂) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajstić. The first Zagreb index M₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let $K_{n_1,n_2}^{P}$ with n₁ $\leq$ n₂, n₁ + n₂ = n and p < n₁ be the set of bipartite graphs obtained by deleting p edges from complete bipartite graph K_n1,n2. In this paper, we determine sharp upper and lower bounds on Zagreb indices of graphs from $K_{n_1,n_2}^{P}$ and characterize the corresponding extremal graphs at which the upper and lower bounds on Zagreb indices are attained. As a corollary, we determine the extremal graph from $K_{n_1,n_2}^{P}$ with respect to Zagreb coindices. Moreover a problem has been proposed on the first and second Zagreb indices.

• 대한치과의사협회지
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• 제10권11호
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• pp.725-729
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• 1972

Ist permanent molars of 3340(1717;1623) primary school children in Kyung Nam and Kang Won province were observed. The results were as follows: 1. In dental health index of permanent 1st molars, male was 98.89, female 97.75, both sexes 95.95 in Kang Won and male was 97.94, female 96.25, both sexes 97.09 in Kyung Nam province. 2. In D.M.F. index, male was 1.11, female 2.25, both sexes 1.68 in Kang Won and male was 2.06, female 3.75, both sexes 2.91 in Kyung Nam province. 3. Female was generally lower than male in dental health index and male was higher than female in D.M.F. index. 4. Epidemiologically, Kang Won was higher than Kyung Nam province in dental health index. 5. In D.M.F. index, Kang Won province was lowest than other districts; city and farm and fishing village. In dental health index, it was increased in Kyung Nam and next was Kang Won province.

• Kyungpook Mathematical Journal
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• 제53권1호
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• pp.117-124
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• 2013

We show that for 1 < $p$ < ${\infty}$, $k$, $m{\in}\mathbb{N}$, $n^{(k)}(l_p)=inf\{n^{(k)}(l^m_p):m{\in}\mathbb{N}\}$ and that for any positive measure ${\mu}$, $n^{(k)}(L_p({\mu})){\geq}n^{(k)}(l_p)$. We also prove that for every $Q{\in}P(^kl_p:l_p)$ (1 < $p$ < ${\infty}$), if $v(Q)=0$, then ${\parallel}Q{\parallel}=0$.

• Korean Journal of Mathematics
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• 제28권3호
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• pp.405-419
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• 2020

The zeroth-order general Randić index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where d(u) is the degree of vertex u and α is an arbitrary real number. In this paper, the maximum value of zeroth-order general Randić index on the graphs of order n with a given clique number is presented for any α ≠ 0, 1 and α ∉ (2, 2n-1], where n = |V (G)|. The minimum value of zeroth-order general Randić index on the graphs with a given clique number is also obtained for any α ≠ 0, 1. Furthermore, the corresponding extremal graphs are characterized.

• Current Optics and Photonics
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• 제4권4호
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• pp.326-329
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• 2020

The Eaton lens, with spherical symmetry to its refractive index, was described by Eaton in 1952 and was found recently in the design of an invisible sphere for cloaking. In this paper, an Eaton lens for rotating 90° was designed using Luneburg theory, by which we found it was a fourth-order equation in the refractive index n. Therefore, the refractive index n has four roots. The equation in n was solved and studied using mathematical technology. The unsuitable complex roots of the equation should be dropped; consequently, only one of the four roots remained. To verify the refractive-index profile, the only root was solved for, before a simulation using finite-element analysis (FEA) was performed. The simulation showed that all rays will bend 90° to the right. The result of the simulation is identical to our expectation. This treatment provides a possible method for rotating light at many other angles.

• 대한수학회보
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• 제59권3호
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• pp.709-723
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• 2022

In this note, we revise some vanishing and finiteness results on hypersurfaces with finite index in ℝⁿ⁺¹. When the hypersurface is stable minimal, we show that there is no nontrivial L^2p harmonic 1-form for some p. The our range of p is better than those in [7]. With the same range of p, we also give finiteness results on minimal hypersurfaces with finite index.

• 한국결정성장학회지
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• 제17권2호
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• pp.47-51
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• 2007

저굴절 재료인 F-doped SiOC : H 박막을 Si 웨이퍼와 유리기판위에 rf power, 기판온도, 그리고 가스유량($SiH_4,\;CF_4$ and $N_2O$)을 변수로 하여 PECVD법으로 증착하였다. 기판 온도와 rf power증가에 따라 F-doped SiOC : H 박막의 굴절률은 감소하는 경향을 보였다. $N_2O$ 가스 유량이 감소함에 따라 증착된 박막의 굴절률은 감소하였으며, rf power가 180W 기판온도 $100^{\circ}C$, 그리고 $N_2O$ 가스를 첨가하지 않은 조건에서 증착한 박막은 최소 굴절률인 1.3778을 갖는 것을 알 수 있었다. Rf power 60W에서 180W로 증가시킴에 따라 증착된 박막의 불소 함량은 1.9at%에서 2.4at%로 증가하였으며 이러한 이유로 박막의 굴절률은 감소하는 경향을 나타냈다.

• 한국방사선학회논문지
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• 제12권2호
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• pp.185-191
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• 2018

본 연구에서는 체질량 지수에 따라 각기 다른 관전압을 적용하여 비뇨기계 CT 검사 시 체질량 지수 적용의 유용성을 평가하고자 하였다. A그룹(n=38)은 체질량 지수가 25 이하이며 관전압 100 kVp로 검사하였고 B그룹(n=45)은 체질량 지수가 25 이상이며 관전압 120kVp로 검사하였다. C그룹(n=37)은 체질량 지수가 25 이하이며 관전압 120kVp로 검사하였다. 체질량 지수가 25 이하로 낮은 A그룹(100 kVp)과 C그룹(120kVp)의 두 그룹간 평균 선량의 차이는 $214.8mGy{\cdot}cm$의 차이를 나타내었으나 정성적 평가에서는 큰 차이가 없음을 확인할 수 있었으며 체질량 지수가 25이상인 환자군과 비교하여서는 오히려 좋은 결과를 확인할 수 있었다. 따라서 체질량 지수가 25 이하로 낮은 환자는 관전압을 100 kVp 로 낮추어 촬영하여도 영상의 질에는 부정적인 영향을 주지 않음을 알 수 있다.

• 대한수학회보
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• 제44권1호
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• pp.83-86
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• 2007

We determine the largest integer i such that $0 and the coefficient of $t^{i}$ is odd in the polynomial $(1+t+t^{2}+{\cdots}+t^{n})^{n+1}$. We apply this to prove that the co-index of the tangent bundle over $FP^{n}$ is stable if $2^{r}{\leq}n<2^{r}+\frac{1}{3}(2^{r}-2)$ for some integer r.

• 원예과학기술지
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• 제33권6호
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• pp.845-853
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• 2015

The objective of this study was to predict the timing of nitrogen (N) demand through analyzing chlorophyll fluorescence (ChlF), soil-plant analysis development (SPAD), and normalized difference vegetation index (NDVI), which are positively correlated with foliar N concentration in star cluster (Pentas lanceolata). The plants were grown in potting soil under optimal conditions for 30 d, followed by weekly irrigation with five concentrations (0, 4, 8, 16, and 24 mM) of N for an additional 30 d. These five N application levels corresponded to leaf N concentrations of 2.62, 3.48, 4.00, 4.23, and 4.69%, respectively. We measured 13 morphological and physiological parameters, as well as the responses of these parameters to various N-fertilizer treatments. The general increases in Dickson's quality index (DQI), above-ground dry weight (DW), total DW, flowering rate, ${\Delta}F/Fm$', and qP in response to treatment with 0 to 8 mM N were similar to those of SPAD, NDVI, and Fv/Fm. Consistent and strong correlations ($R^2$= 0.60 to 0.85) were observed between leaf N concentration (%) and SPAD, NDVI, ${\Delta}F/Fm$', and above-ground DW. Validation of leaf S PAD, NDVI, and ${\Delta}F/Fm$' revealed that these vegetation indices are accurate predictors of leaf N concentration that can be used for non-destructive estimation of the proper timing for N-solution irrigation of P. lanceolata. Moreover, irrigation with 8 mM N-fertilizer i s recommended w hen leaf N concentration, SPAD, NVDI, and ${\Delta}F/Fm$' ratios are reduced from their saturation values of 4.00, 50.68, 0.64, and 0.137%, respectively.