• Journal of applied mathematics & informatics
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• v.33 no.3_4
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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.

• The Journal of the Korean dental association
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• v.10 no.11
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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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• v.53 no.1
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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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• v.28 no.3
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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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• v.4 no.4
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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.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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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.

• Journal of the Korean Crystal Growth and Crystal Technology
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• v.17 no.2
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• pp.47-51
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• 2007

F-doped SiOC : H thin films with low refractive index were deposited on Si wafer and glass substrate by plasma enhanced chemical vapor deposition (PECVD) as a function of rf powers, substrate temperatures, gas rates and their composition flow ratios ($SiH_4,\;CF_4$ and $N_2O$). The refractive index of the F-doped SiOC : H film continuously decreased with increasing deposition temperature and rf power. As $N_2O$ gas flow rate decreased, the refractive index of the deposited films decreased down to 1.3778, reaching a minimum value at rf power of 180W and $100^{\circ}C$ without $N_2O$ gas. The fluorine content of F-doped SiOC : H film increased from 1.9 at% to 2.4 at% as the rf power was increased from 60 W to 180 W, which results in the decrease of refractive index.

• Journal of the Korean Society of Radiology
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• v.12 no.2
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• pp.185-191
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• 2018

The purpose of this study was to evaluate the usefulness of BMI application to Urography CT by applying different tube voltages in accordance with body mass index. Group A (n = 38) with body mass index of lower than 25 was examined with tube voltage of 100 kVp while Group B (n = 45) with a BMI of 25 and higher was examined with tube voltage of 120 kVp. C group (n = 37) with body mass index (BMI) of lower than 25 was examined with tube voltage of 120kVp. Although the difference in average dose between group A (100 kVp) and group C (120 kVp) with low body mass index (BMI) of lower than 25 was $214.8mGy{\cdot}cm$, there was no significant difference in qualitative evaluation and, compared with patient group with body mass index of 25 and higher, results obtained were rather good. Therefore, this study verified that the tube voltage of lower than 100 kVp does not have adverse effect on the quality of image for patients with body mass index (BMI) of lower than 25.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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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.

• Horticultural Science & Technology
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• v.33 no.6
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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.