• 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).

• School Mathematics
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• v.16 no.3
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• pp.613-631
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• 2014

The aim of this study was to investigate how the small group activity system influences individual to form concepts of prime number and composite number through activity theory on learning process of mathematically gifted 5th-grade students. Student's worksheets, recorded video, and interview were gathered and transcribed for analyzing data. Process of concept formation and using symbol behavior were used to derive the stage of mathematical concept from students, and the activity system and stage of concept formation process were schematized through analysis of whole class activity system and small group activity system based on activity theory. According to the results of this study, two students who were in different activity groups separated into the state of semi-concept and the stage of complex thinking respectively, and therefore, social context and the activity system had effects on process of concept formation among the students.

• The Korean Journal of Ecology
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• v.23 no.2
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• pp.107-112
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• 2000

Comparison of oribatid mite community structures among Namsan, Kwangreung, and Mt, Jumbong, which receive different levels of environmental stress from severe to almost none, was made in coniferous and deciduous forests, respectively. The number of species of oribatid mites was significantly lower in Namsan and Mt. Jumbong than in Kwangreung in the coniferous forest (p<0.05). In the deciduous forests, the number of species of oribatid mites was significantly lower in Namsan than in Kwangreung and Mt. Jumbong. Dominant species in 3 regions were remarkably different. Similarity of the oribatid community between Namsan and Kwangreung was much higher (ca. 2 times) than similarities between Namsan and Mt. Jumbong. and Kwangreung and Mt Jumbong. Diversity index (H$^{\prime}$) value of oribatid communities in deciduous forests in Namsan, Kwangreung and Mt. Jumbong was 2.74, 2.78, and 2.87, respectively. Diversity (H$^{\prime}$) value of oribatid communities in coniferous forests in Namsan, Kwangreung and Mt. Jumbong was 2.83, 2.62, and 2.38, respectively. Namsan and Kwangreung were characterized as O-type in both coniferous and deciduous forests On the contrary, Mt. Jumbong was characterized as MG-type in MGP-I analysis.

• International Journal of Computer Science & Network Security
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• v.23 no.7
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• pp.49-60
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• 2023

The amount of data generated by electronic systems through e-commerce, social networks, and data computation has risen. However, the security of data has always been a challenge. The problem is not with the quantity of data but how to secure the data by ensuring its confidentiality and privacy. Though there are several research on cloud data security, this study proposes a security scheme with the lowest execution time. The approach employs a non-linear time complexity to achieve data confidentiality and privacy. A symmetric algorithm dubbed the Non-Deterministic Cryptographic Scheme (NCS) is proposed to address the increased execution time of existing cryptographic schemes. NCS has linear time complexity with a low and unpredicted trend of execution times. It achieves confidentiality and privacy of data on the cloud by converting the plaintext into Ciphertext with a small number of iterations thereby decreasing the execution time but with high security. The algorithm is based on Good Prime Numbers, Linear Congruential Generator (LGC), Sliding Window Algorithm (SWA), and XOR gate. For the implementation in C, thirty different execution times were performed and their average was taken. A comparative analysis of the NCS was performed against AES, DES, and RSA algorithms based on key sizes of 128kb, 256kb, and 512kb using the dataset from Kaggle. The results showed the proposed NCS execution times were lower in comparison to AES, which had better execution time than DES with RSA having the longest. Contrary, to existing knowledge that execution time is relative to data size, the results obtained from the experiment indicated otherwise for the proposed NCS algorithm. With data sizes of 128kb, 256kb, and 512kb, the execution times in milliseconds were 38, 711, and 378 respectively. This validates the NCS as a Non-Deterministic Cryptographic Algorithm. The study findings hence are in support of the argument that data size does not determine the execution.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.83-96
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• 2013

Let K be a number field and fix a prime number $p$. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set $B_{K,p}$ of primes of K satisfying that any elliptic curve over K with $B_{K,p}$-reduction has no $p$-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with $B_{K,p}$-reduction and a $p$-torsion point. The action of the absolute Galois group on the $p$-torsion subgroup of E gives its associated Galois representation $\bar{\rho}_{E,p}$ modulo $p$. We also study the irreducibility and surjectivity of $\bar{\rho}_{E,p}$ for semistable elliptic curves with $B_{K,p}$-reduction.

• Nuclear Engineering and Technology
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• v.49 no.7
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• pp.1405-1413
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• 2017

To investigate the heat loads imposed on a reactor vessel through the natural convection of core melts in severe accidents, mass transfer experiments were performed based on the heat transfer/mass transfer analogy, using two- (2-D) and three-dimensional (3-D) facilities of various heights. The modified Rayleigh numbers ranged from $10^{12}$ to $10^{15}$, with a fixed Prandtl number of 2,014. The measured Nusselt numbers showed a trend similar to those of existing studies, but the absolute values showed discrepancies owing to the high Prandtl number of this system. The measured angle-dependent Nusselt numbers were analyzed for 2-D and 3-D geometries, and a multiplier was developed that enables the extrapolation of 2-D data into 3-D data. The definition of $Ra^{\prime}_H$ was specified for 2-D geometries, so that results could be extrapolated for 3-D geometries; also, heat transfer correlations were developed.

• Journal for History of Mathematics
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• v.17 no.1
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• pp.33-42
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• 2004

In this paper, we identify the essential ideas of Fundamental Theorem of Arithmetic(FTA). Then, we compare these ideas with several theorems of Euclid's Elements to investigate whether the essential ideas of FTA are contained in Elements or not. From this, we have the following conclusion: Even though Elements doesn't contain FTA explicitly, it contains all of the essential ideas of FTA. Finally, we assert two reasons why Greeks couldn't mention FTA explicitly. First, they oriented geometrically, and so they understood the concept of 'divide' as 'metric'. So they might have difficulty to find the divisor of the given number and the divisor of the divisor continuously. Second, they have limit to use notation in Mathematics. So they couldn't represent the given composite number as multiplication of all of its prime divisors.

• Journal of the Korean Data and Information Science Society
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• v.3 no.1
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• pp.47-63
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• 1992

Bechhofer and Tamhane(1981) proposed Balanced Treatment Incomplete Block (BTIB) desings for comparing p test treatments with a control treatment in blocks of size ${\kappa}$. Notz and Tamhane(1983) solved the problem about determination of the minimal complete class for ${\kappa}=3$. However there are a number of design parameters for which BTIB designs do not exist. We suggest a new class of designs called Group Divisible Treatment Desings(GDTD's) that is a larger class including BTIB designs as a subclass. In this paper we give the minimal complete classes of generator designs for GDTD's with ${\kappa}=2,\;p{\geq}4(except\;prime\;number)\;and\;{\kappa}=3,\;p=4(2)6$.

• Korean Journal of Mathematics
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• v.21 no.4
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• pp.365-374
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• 2013

Let $k=\mathbb{F}_q(T)$ be a rational function field over the finite field $\mathbb{F}_q$, where q is a power of an odd prime number, and $\mathbb{A}=\mathbb{F}_q[T]$. Let ${\gamma}$ be a generator of $\mathbb{F}^*_q$. Let $\mathcal{H}_n$ be the subset of $\mathbb{A}$ consisting of monic square-free polynomials of degree n. In this paper we obtain an asymptotic formula for the mean value of $L(1,{\chi}_{\gamma}{\small{D}})$ and calculate the average value of the ideal class number $h_{\gamma}\small{D}$ when the average is taken over $D{\in}\mathcal{H}_{2g+2}$.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.39-44
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• 1996

Throughout this paper $Z^p, Q_p$ and C_p$ will denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of $Q_p$, respectively. Let $v_p$ be the normalized exponential valuation of $C_p$ with $$\mid$p$\mid$_p = p^{-v_p (p)} = p^{-1}$. We set $p^* = p$ for any prime p > 2 $p^* = 4 for p = 2$.