• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1079-1095
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• 2021

In this paper, we give new congruences with the generalized Catalan numbers and harmonic numbers modulo p². One of our results is as follows: for prime number p > 3, $${\sum\limits_{k=(p+1)/2}^{p-1}}\;k^2B_{p,k}B_{p,k-(p-1)/2}H_k{\equiv}(-1)^{(p-1)/2}\(-{\frac{521}{36}}p-{\frac{1}{p}}-{\frac{41}{12}}+pH^2_{3(p-1)/2}-10pq^2_p(2)+4\({\frac{10}{3}}p+1\)q_p(2)\)\;(mod\;p^2),$$ where q_p(2) is Fermat quotient.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.26 no.6
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• pp.1401-1411
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• 2016

The Multi-Prime RSA and the Prime Power RSA are the variants of the RSA cryptosystem, where the Multi-Prime RSA uses the modulus $N=p_1p_2{\cdots}p_r$ for distinct primes $p_1,p_2,{\cdots},p_r$ (r>2) and the Prime Power RSA uses the modulus $N=p^rq$ for two distinct primes p, q and a positive integer r(>1). This paper analyzes the security of these systems by using the technique given by Heninger and Shacham. More specifically, this paper shows that if the $2-2^{1/r}$ random portion of bits of $p_1,p_2,{\cdots},p_r$ is given, then $N=p_1p_2{\cdots}p_r$ can be factorized in the expected polynomial time and if the $2-{\sqrt{2}}$ random fraction of bits of p, q is given, then $N=p^rq$ can be factorized in the expected polynomial time. The analysis is then validated with experimental results for $N=p_1p_2p_3$, $N=p^2q$ and $N=p^3q$.

• Horticultural Science & Technology
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• v.35 no.3
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• pp.289-299
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• 2017

The objective of this study was to determine the effect of replacing perlite (PL) with pellets processed with poultry feather fiber as an inert material to prepare growing medium. The growth and flowering of Salvia splendens 'Vista Red' grown in individual growing medium $Biosangto^{(R)}$, peat moss (PM), PL, and two pellets (P45-1 and P45-2) were evaluated. Peat moss was mixed with PL, P45-1, or P45-2 at various ratios (1:0 to 1:3 or 3:1 by volume) to investigate the feasibility of replacing PL with pellets. Nutrient composition of the growing medium and leaf tissues was analyzed. The number of florets, inflorescence length, plant height, and fresh weight of plants grown in media containing P45-1 or P45-2 were reduced compared to those grown in individual growing medium PM or PL. As the mixing ratio of P45-1 or P45-2 to PM was higher, the growth of salvia, such as inflorescence length, plant height, number of leaves, and fresh weight was inhibited. Our results indicate that mixing three parts PM with one part of P45-1 (PM/P45-1/3:1) or P45-2 (PM/P45-2/3:1) accelerated flowering and increased the number of florets and leaves compared to other mixing ratios of PM and pellets media. The concentrations of phosphorus (P), calcium (Ca), boron (B), iron (Fe), and copper (Cu) in individual growing medium PL, P45-1, and P45-2 were significantly lower than those in PM. The concentration of N was the highest in leaves of plants grown in P45-1 or P45-2 amended media, and the concentrations of P, Ca, and zinc (Zn) in leaves were lower in individual growing medium P45-1 or P45-2 than in PM and PL. The pH of PM/P45-1/3:1 or PM/P45-2/3:1 media was maintained at optimal level (5.8-5.9) and the concentrations of macro- and micro-elements in the media and leaves were considered to be optimal levels. Therefore, mixing three parts PM with one part P45-1(PM:P45-1/3:1) or P45-2 (PM:P45-2/3:1) is recommended for improved growth and flowering in salvia. This suggests that P45-1 or P45-2 can replace PL as an inert material to prepare growing medium.

• Journal of the Korean Magnetics Society
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• v.25 no.6
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• pp.175-179
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• 2015

The half-metallicity and magnetism of the (001) surfaces of $(AlP)_1/(CrP)_1$ superlattice were investigated by means of FLAPW (Full-potential Liniarized Augmented Plane Wave) method. We considered four types of (001) surface termination, i.e., Al(S)-, Cr(S)-, P(S)Al(S-1)- and P(S)Cr(S-1)-term systems. We found that only Cr(S)-term system maintains the half-metallicity at the surface as only this system has the calculated magnetic moment of integer number of bohr magnetons. The magnetic moment of Cr(S) atom in the system was $3.02{\mu}_B$ which was increased from the bulk value by the effects of band narrowing and increased spin-splitting at the surface. The electronic density of states of the P(S) atom in the P(S)Al(S-1)-term showed very sharp surface states due to the broken $p_z$ bonds at the surface. We found there is still a strong p-d hybridization between the P(S) and Cr(S-1) layers in the P(S)Cr(S-1)-term which causes a considerable increase of magnetic moment of P(S) atom.

• Kyungpook Mathematical Journal
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• v.28 no.2
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• pp.129-139
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• 1988

Let $T^{\alpha}_{\lambda}$(p, A, B) denote the class of functions $$f(z)=z^p-{\sum\limits^{\infty}_{k=1}}{\mid}a_{p+k}{\mid}z^{p+k}$$ which are regular and p valent in the unit disc U = {z: |z| <1} and satisfying the condition $\left|{\frac{{e^{ia}}\{{\frac{f^{\prime}(z)}{z^{p-1}}-p}\}}{(A-B){\lambda}p{\cos}{\alpha}-Be^{i{\alpha}}\{\frac{f^{\prime}(z)}{z^{p-1}}-p\}}}\right|$<1, $z{\in}U$, where 0<${\lambda}{\leq}1$, $-\frac{\pi}{2}$<${\alpha}$<$\frac{\pi}{2}$, $-1{\leq}A$<$B{\leq}1$, 0<$B{\leq}1$ and $p{\in}N=\{1,2,3,{\cdots}\}$. In this paper, we obtain sharp results concerning coefficient estimates, distortion theorem and radius of convexity for the class $T^{\alpha}_{\lambda}$(p, A, B). It is further shown that the class $T^{\alpha}_{\lambda}$(p, A, B) is closed under "arithmetic mean" and "convex linear combinations". We also obtain class preserving integral operators of the form $F(z)=\frac{p+c}{z^c}{\int^z_0t^{c-1}}f(t)dt$, c>-p, for the class $T^{\alpha}_{\lambda}$(p, A, B). Conversely when $F(z){\in}T^{\alpha}_{\lambda}$(p, A, B), radius of p valence of f(z) has also determined.

• East Asian mathematical journal
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• v.31 no.1
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• pp.103-107
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• 2015

We use some known contiguous function relations for $_2F_1$ to show how simply the following three recurrence relations for Jacobi polynomials $P_n^{({\alpha},{\beta)}(x)$: (i) $({\alpha}+{\beta}+n)P_n^{({\alpha},{\beta})}(x)=({\beta}+n)P_n^{({\alpha},{\beta}-1)}(x)+({\alpha}+n)P_n^{({\alpha}-1,{\beta})}(x);$ (ii) $2P_n^{({\alpha},{\beta})}(x)=(1+x)P_n^{({\alpha},{\beta}+1)}(x)+(1-x)P_n^{({\alpha}+1,{\beta})}(x);$ (iii) $P_{n-1}^{({\alpha},{\beta})}(x)=P_n^{({\alpha},{\beta}-1)}(x)+P_n^{({\alpha}-1,{\beta})}(x)$ can be established.

• Communications of the Korean Mathematical Society
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• v.24 no.3
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• pp.367-379
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• 2009

For a prime number p, let $\mathbb{Q}_p$ denote the p-adic field and let $\mathbb{Q}_p^d$ denote a vector space over $\mathbb{Q}_p$ which consists of all d-tuples of $\mathbb{Q}_p$. For a function f ${\in}L_{loc}^1(\mathbb{Q}_p^d)$, we define the Hardy-Littlewood maximal function of f on $\mathbb{Q}_p^d$ by $$M_pf(x)=sup\frac{1}{\gamma{\in}\mathbb{Z}|B_{\gamma}(x)|H}{\int}_{B\gamma(x)}|f(y)|dy$$, where |E|$_H$ denotes the Haar measure of a measurable subset E of $\mathbb{Q}_p^d$ and $B_\gamma(x)$ denotes the p-adic ball with center x ${\in}\;\mathbb{Q}_p^d$ and radius $p^\gamma$. If 1 < q $\leq\;\infty$, then we prove that $M_p$ is a bounded operator of $L^q(\mathbb{Q}_p^d)$ into $L^q(\mathbb{Q}_p^d)$; moreover, $M_p$ is of weak type (1, 1) on $L^1(\mathbb{Q}_p^d)$, that is to say, |{$x{\in}\mathbb{Q}_p^d:|M_pf(x)|$>$\lambda$}|$_H{\leq}\frac{p^d}{\lambda}||f||_{L^1(\mathbb{Q}_p^d)},\;\lambda$ > 0 for any f ${\in}L^1(\mathbb{Q}_p^d)$.

• Korean Journal of Microbiology
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• v.21 no.2
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• pp.95-102
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• 1983

The penicillin G acylase(pga) gene was cloned in the vector plasmid pKM $300(Ar^r,\;Tc^r,\;6.33kb)$ for the study of the structure and expression of the pga gene. This recombinant plasmid pPAKS-1 DNA(24.5 Kb) was cleaved into 2 fragments by restriction enzyme Eco R1.1fragment by BamH1, 4fragments by Hind III, and 2 fragments by Pst I. The pga gene was located on the Eco R1.Hind III-C fragement of pPAKS-1. The recombinant plasmids pPAKS-1 and pPAKS-2, in which the Hind III-B and Hind III-D fragments pPAKS-1 are deleted, are characterized. The results are summarized as follows : 1. Doubling times of bacterial strain bearing pPAKS-1 and pPAKS-2 are 90 and 60 minutes, respectively. 2. pPAKS-1 and pPAKS-2 are present at about 16-32 and 70 copies per cell, respectively, are 0.66 and 5.5 units, respectively, which represent 2-fold and 20-fold higher enzyme 4. pPAKS-1 is very unstable, but pPAKS-2 is stable.

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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• pp.623-632
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• 2010

A holomorphic function F defined on the unit disc belongs to $A^{p,{\alpha}}$ (0 < p < $\infty$, 1 < ${\alpha}$ < $\infty$) if $\int\limits_U|F(z)|^p \frac{1}{1-|z|}(1+log)\frac{1}{1-|z|})^{-\alpha}$ dxdy < $\infty$. For boundedness of the composition operator defined by $C_{fg}=g{\circ}f$ mapping Blochs into $A^{p,{\alpha}$ the following (1) is a sufficient condition while (2) is a necessary condition. (1) $\int\limits_o^1\frac{1}{1-r}(1+log\frac{1}{1-r})^{-\alpha}M_p(r,\lambda{\circ}f)^p\;dr$ < $\infty$ (2) $\int\limits_o^1\frac{1}{1-r}(1+log\frac{1}{1-r})^{-\alpha+p}(1-r)^pM_p(r,f^#)^p\;dr$ < $\infty$.

• Honam Mathematical Journal
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• v.31 no.3
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• pp.369-380
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• 2009

Let $X,X_1,X_2,\;{\cdots}$ be identically distributed and negatively associated random variables with mean zeros and positive, finite variances. We prove that, if $E{\mid}X_1{\mid}^r$ < ${\infty}$, for 1 < p < 2 and r > $1+{\frac{p}{2}}$, and $lim_{n{\rightarrow}{\infty}}n^{-1}ES^2_n={\sigma}^2$ < ${\infty}$, then $lim_{{\epsilon}{\downarrow}0}{\epsilon}^{{2(r-p}/(2-p)-1}{\sum}^{\infty}_{n=1}n^{{\frac{r}{p}}-2-{\frac{1}{p}}}E\{{{\mid}S_n{\mid}}-{\epsilon}n^{\frac{1}{p}}\}+={\frac{p(2-p)}{(r-p)(2r-p-2)}}E{\mid}Z{\mid}^{\frac{2(r-p)}{2-p}}$, where $S_n\;=\;X_1\;+\;X_2\;+\;{\cdots}\;+\;X_n$ and Z has a normal distribution with mean 0 and variance ${\sigma}^2$.