• 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.

• Journal of the Korean Statistical Society
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• v.25 no.2
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• pp.175-183
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• 1996

Let $S_n,n$ = 1, 2,... denote the partial sums of not necessarily in-dependent random variables. Let N(c) = min${ n ; S_n > c}$, c $\geq$ 0. Theorem 2 states that N (c), (suitably normalized), tends to 0 in p-mean, 1 $\leq$ p < 2, as c longrightarrow $\infty$ under mild conditions, which generalizes earlier result by Gut(1974). The proof follows by applying Theorem 1, which generalizes the known result $E$\mid$S_n$\mid$^p$ = o(n), 0 < p< 2, as n .rarw..inf. to randomly indexed partial sums.

• Journal of Environmental Science International
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• v.17 no.5
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• pp.509-516
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• 2008

New functional surfactant, N,N-dimethyl-N-dodecyl-N-(2-methyl benzimidazoyl) ammonium chloride(DDBAC) having benzimidazole(BI) functional group have been synthesized and the critical micellar concentration of DDBAC measured by surface tentiometry and electric conductivity method was $8.9{\times}10^{-4}M$. Micellar effects in DDBAC functional surfactant solution on the hydrolysis of p-nitrophenylacetate(p-NPA), p-nitro-phenylpropionate(p-NPP) and p-nitrophenylvalerate(p-NPV) were observed with change of various pH (Tris-buffer). The pseudo first rate constants of hydrolysis of p-NPA, p-NPP and p-NPV in optimum concentration of DDBAC solution increase to about 160, 280 and 600 times, respectively, as compared with those of aqueous solution at pH 8.00(Tris-buffer). It is considered that benzimidazole functional moiety accelerates the reaction rates of hydrolysis because they act as nucleophile or general base. In optimum concentration of DDBAC solution, the rate constants of hydrolysis of p-NPP and p-NPV increase to about 1.5 and 3.0 times, respectively, as compared with that of p-NPA. It means that the more the carbon numbers of alkyl group of substrates, the larger the binding constants between DDBAC micelle and substrates are. To know the hydrolysis mechanism of p-NPCE(p-NPA, p-NPP and p-NPV), the deuterium kinetic isotope effects were measured in $D_2O$ solutions. Consequently the pseudo first order rate constant ratios in $H_2O$ and $D_2O$ solution, $k_{H_2O}/k_{D_2O}$, were about $2.8{\sim}3.0$ range. It means that the mechanism of hydrolysis were proceeded by nucleophile and general base attack in approximately same value.

• Bulletin of the Korean Chemical Society
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• v.16 no.10
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• pp.938-945
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• 1995

Hydrolysis of p-nitrophenyl acetate (PNPA) and p-nitrophenyldiphenyl phosphate (PNPDPP) by N-chloro compounds in micellar solution were studied. N,N'-dichloroisocyanuric acid sodium salt (DCI) in cetyltrimethylammonium chloride (CTACl) micellar solution gave pseudo first-order kinetics. But, DCI in cetyltrimethylammonium bromide (CTABr) micellar solution showed typical series first-order kinetics - fast hydrolysis of the esters and concomitant slow decay of the hydrolyzed product, p-nitrophenolate. The hydrolysis rate was decreased as the hydrophobicity of N-chloro compounds was increased, which is the opposite trend to the usual bimolecular micellar reaction. This curious behavior of the N-chloro compounds in the catalytic hydrolysis of PNPA and PNPDPP in a cationic micellar system can be best explained by participation of counter ions of the surfactants during hydrolysis.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.23-34
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• 2019

In this paper, we first define generalizations of Pell and Pell-Lucas sequences by the recurrence relations $$p_n=2ap_{n-1}+(b-a^2)p_{n-2}\;and\;q_n=2aq_{n-1}+(b-a^2)q_{n-2}$$ with initial conditions $p_0=0$, $p_1=1$, and $p_0=2$, $p_1=2a$, respectively. We give generating functions and Binet's formulas for these sequences. Also, we obtain some identities of these sequences.

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.155-166
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• 2023

In 2018, Andrews introduced the partition functions ${\mathfrak{EO}}$(n) and ${\bar{\mathfrak{EO}}}$(n). The first of these denotes the number of partitions of n in which every even part is less than each odd part, and the second counts the number of partitions enumerated by the first in which only the largest even part appears an odd number of times. In 2021, Pore and Fathima introduced a new partition function ${\mathfrak{EO}}_e$(n) which counts the number of partitions of n which are enumerated by ${\bar{\mathfrak{EO}}}$(n) together with the partitions enumerated by ${\bar{\mathfrak{EO}}}$(n) where all parts are odd and the number of parts is even. They also proved some particular congruences for ${\bar{\mathfrak{EO}}}$(n) and ${\mathfrak{EO}}_e$(n). In this paper, we establish infinitely many families of congruences modulo 2, 4, 5 and 8 for ${\bar{\mathfrak{EO}}}$(n) and modulo 4 for ${\mathfrak{EO}}_e$(n). For example, if p ≥ 5 is a prime with Legendre symbol $({\frac{-3}{p}})=-1$, then for all integers n ≥ 0 and α ≥ 0, we have ${\bar{\mathfrak{EO}}}(8{\cdot}p^{2{\alpha}+1}(pn+j)+{\frac{19{\cdot}p^{2{\alpha}+2}-1}{3}}){\equiv}0$ (mod 8); 1 ≤ j ≤ (p - 1).

• The Journal of the Korea Contents Association
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• v.22 no.6
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• pp.619-628
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• 2022

In general, as a method to confirm a urinary tract infection (UTI) in a medical institutions, urine culture including a urinalysis and an antimicrobial susceptibility test is performed. It is important to disinfect the area around the urethra and perineum before collecting urine samples, and it is important to collect it intermediate urine, not the first-void urine. We invented a patent urine cup (Patent No. 10-1732843) that can automatically and easily separate first-void urine and midstream urine and using this, the patent cup and the general cup were compared and evaluated using this. Nitrite (P<0.001), WBC (P=0.005), Bacterial colony count (P=0.001), colony positivity rate (P=0.004) in first-void urine (N=24), midstream urine (N=24) separated by patent cup to obtain a significantly higher value. This can be seen from the fact that the first-void urine and midstream urine separated using the patent cup were well separated. Also, the number of Bacterial colonies was statistically significantly higher in the midstream urine isolated using a patent cup (N=24) than in the midstream urine collected using a general cup (N=24) (average 7.9 vs. 4.0 on average, P= 0.002). Which means that the midstream urine separated using the patent cup is more sensitive to the UTI test than the midstream urine collected using a general cup.

• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.783-801
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• 2006

Let A be the mod p Steenrod algebra for p an arbitrary odd prime and S the sphere spectrum localized at p. In this paper, some useful propositions about the May spectral sequence are first given, and then, two new nontrivial homotopy elements ${\alpha}_1{\jmath}{\xi}_n\;(p{\geq}5,n\;{\geq}\;3)\;and\;{\gamma}_s{\alpha}_1{\jmath}{\xi}_n\;(p\;{\geq}\;7,\;n\;{\geq}\;4)$ are detected in the stable homotopy groups of spheres, where ${\xi}_n\;{\in}\;{\pi}_{p^nq+pq-2}M$ is obtained in [2]. The new ones are of degree 2(p - 1)($p^n+p+1$) - 4 and 2(p - 1)($p^n+sp^2$ + sp + (s - 1)) - 7 and are represented up to nonzero scalar by $b_0h_0h_n,\;b_0h_0h_n\tilde{\gamma}_s\;{\neq}\;0\;{\in}\;Ext^{*,*}_A^(Z_p,\;Z_p)$ in the Adams spectral sequence respectively, where $3\;{\leq}\;s\;<\;p-2$.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.423-432
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• 1998

Let p, q be integers such that 2 $\leq$ p, q $\leq$ n, and let $D_{p, q}$ denote the matrix obtained from $I_{n}$, the identity matrix of order n, by replacing each of the first p columns by an all 1's vector and by replacing each of the first two rows and each of the last q-2 rows by an all 1's vector. In this paper the permanent minimization problem over the face, determined by the matrix $D_{p, q}$, of the polytope of all n $\times$ n doubly stochastic matrices is treated.d.

• Journal of The Korean Society of Grassland and Forage Science
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• v.15 no.3
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• pp.164-168
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• 1995

A field experiment was carried out to determine the effects of nitrogen (N) fertilization levels after first harvest (0, 30, 60, 90 and 120 kglha) during spring season on the gms growth, dry matter (DM) yield, nutritive value, and efficiency of DM production per N kg applied in orchardgrass meadow. Additional N was applicated at early spring (70 kg), after second harvest (50 kg), and after fourth harvest (30 kg/ha) in all treatments, 1990. Growth and DM yield of orchardgrass at second harvest were significantly increased with increasing level of N after first harvest (P<0.05), but no yield differences were observed between application of N 60, N 90, and N 120 kg. The efficiency of DM increment per N kg applied was rapidly decreased with increasing level of N; that was 14.8, 12.0, 10.7 and 4.7 DM kg/N kgha in the plot of N 30, N 60, N 90 and N 120 kg, respectively. The content and yield of crude protein of orchardgrass at second harvest were significantly increased with increasing of N (P<0.05), ranged h r n 11.0% (N 0 kg) to 16.8% (N 120 kg), and from 195.0 kg (N 0 kg) to 508.2 kglha (N 120 kg), respectively. However, the content of neutral detergent fiber was not affected by N level after first harvest. From the above results, it is suggested that application of N fertilizer after first harvest was very effective for regrowth and nutritive value of orchardgrass during spring season, and the proper amount of N after first harvest was 60-90 kglha in this experiment.