• Bulletin of the Korean Mathematical Society
• /
• v.60 no.4
• /
• pp.1017-1024
• /
• 2023

A celebrated result in the study of integer partitions is the identity due to Lehmer whereby the number of partitions of n with an even number of even parts minus the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct odd parts. Inspired by Lehmer's identity, we prove explicit formulas for evaluating generating functions for sequences that enumerate integer partitions of fixed width with an even/odd number of even parts. We introduce a technique for decomposing the even entries of a partition in such a way so as to evaluate, using a finite sum over q-binomial coefficients, the generating function for the sequence of partitions with an even number of even parts of fixed, odd width, and similarly for the other families of fixed-width partitions that we introduce.

• Journal of Industrial Convergence
• /
• v.19 no.4
• /
• pp.23-28
• /
• 2021

This paper finds the regular pattern of n = [3, ∞] for Euler square game related with n = 6(6×6=36) thirty-six officer problem that is still unsolved problem. The solution of this problem is exists for n = [3, 10] without n = 6. Also, previous researchers finds the random assigned solution for specific number using computer programming. Therefore, the solution of n = [11, ∞] Euler squares are unsolved problem because of anything but easy. This paper attempts to find generalized patterns for domains that have been extended to n = [3, ∞], while existing studies have been limited to n = [3, 10]. This paper classify the n = [3, ∞] into n = odd, 4k even, 4k+2 even of three classes. Then we find the simple regular pattern solution for n = odd and 4k even(n/2 = even). But we can't find the regular pattern for 4k+2 even(n/2 = odd).

• Journal of the Korean Institute of Electrical and Electronic Material Engineers
• /
• v.12 no.1
• /
• pp.69-74
• /
• 1999

We have investigated that the high pretilt angle of the NLC, 4-n-pentyl-4-cyanobiphenyl (5CB), was observed on rubbed polythiophene (PTP) surfaces with alkyl chains with more than 10 carbon atoms; it is attributed to the surface-excluded volume effect by the alkyl chain lengths between the LCs and the PTP surfaces. Next, we investigated that the odd-even effect of the polar anchoring strength in 5CB on rubbed PTP surfaces with alkyl chain lengths has been successfully evaluated. The anchoring strength of 5CB for rubbed PTP surfaces with odd-number is weak compared with even-number up to the 6 carbon atoms in the alkyl chain; however, odd-number is strong compared with even-number above 7 carbon atoms. The weak anchoring strength of 5CB is approximately $1\times10^{-3} (J/m^2$) on rubbed PTP surface with 7 carbon atoms; it is relatively strong anchoring strength. Consequently, we conclude that the odd-even effects of the polar anchoring strength in NLCs are strongly related to the characteristics of the polymer and observed clearly for short alkyl chain lengths.

• Proceedings of the IEEK Conference
• /
• 1998.10a
• /
• pp.1137-1140
• /
• 1998

In this paper, we propose the new hardware architecture which implements the stereo matching algorithm using the dynamic programming method. The dynamic programming method is used in finding the corresponding pixels between the left image and the right image. The proposed MOEPE(Merged Odd-Even PE) architecture operates in the systolic manner and finds the disparities from the intensities of the pixels on the epipolar line. The number of PEs used in the MOEPE architecture is the number of the range constraint, which reduced the number of the necessary PEs dramatically compared to the traditional method which uses the PEs with the number of pixels on the epipolar line. For the normal method by 25 times. The proposed architecture is modeled with the VHDL code and simulated by the SYNOPSYS tool.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1357-1365
• /
• 2013

We find a lower bound on the number of real/inert imagi-nary/ramified imaginary quadratic extensions of the function field $\mathbb{F}_q(t)$ whose ideal class groups have an element of a fixed order, where $q$ is a power of 2.

• Journal of the Korean Society of Costume
• /
• v.50 no.7
• /
• pp.113-127
• /
• 2000

The purpose of this study was to examine the symbolic meaning of oriental numbers based on Yin-Yang(陰陽) theory. Based on the thought of Yin-Yang Wu-Hsing(陰陽五行), the number was divided the number of the heaven (positive number) 1, 3. 5, 7, 9 from the number cf the earth(negative number) 2. 4, 6, 8, 10. It was descrived very well in the dress and its ornaments and the folk customs. In the costume of the Court, there were 9, 7, 5, 3 patterns costume for the king and queen. Even though an even number, 12 patterns costume for the emperor symbolized 12 months and made it of the principal of the universe. Korean traditional costume Han-bock(韓服) was formed with the three dimentional principal of circle (圓.$\bigcirc$), square(方.$\square$) and triangularity(角.$\Delta$). In the middle of odd numbers, number 3 was regarded as a holy number of the heaven (天), the earth(地) and a man(人). Taken for a highest number. number 3 had the symbolism of wishes for good fortune. Number 10 and number 100, which meant the fullness and the long life, were used regularly. With Ten longevity patterns(十長生紋), the feast of a hundred-day-old baby, our race prayed for the healthy long life. As mentioned above. the symbolism of the number though the costume prefered the positive number to the negative one. Accommodating to the universal principal and the cycle. The deep meaning and the symbolism of the number has been implied the mental wishes.

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

• Kyungpook Mathematical Journal
• /
• v.61 no.1
• /
• pp.49-59
• /
• 2021

Recently, Andrews introduced partition functions ����(n) and ${\bar{\mathcal{EO}}}$(n) where the function ����(n) denotes the number of partitions of n in which every even part is less than each odd part and the function ${\bar{\mathcal{EO}}}$(n) denotes the number of partitions enumerated by ����(n) in which only the largest even part appears an odd number of times. In this paper we obtain some congruences modulo 2, 4, 10 and 20 for the partition function ${\bar{\mathcal{EO}}}$(n). We give a simple proof of the first Ramanujan-type congruences ${\bar{\mathcal{EO}}}$ (10n + 8) ≡ 0 (mod 5) given by Andrews.

• The Journal of the Institute of Internet, Broadcasting and Communication
• /
• v.18 no.4
• /
• pp.177-182
• /
• 2018

The home-and-way round-robin sports leagues scheduling problem with minimum brake is very hard to solve in polynomial time. This problem is NP-hard, the complexity status is not yet determined. This paper suggests round-robin sports leagues scheduling algorithm not computer-aided program but by hand with O(n) time complexity for arbitrary number of teams n with always same pattern. The algorithm makes a list of mathes using $n{\times}n$ canonical latin square for n=even teams. Then trying to get home(H) and away(A) with n-2 minimum number of brakes. Also, we get the n=odd scheduling with none brakes delete a team own maximum number of brakes from n=even scheduling.

• Journal of the Institute of Electronics Engineers of Korea SD
• /
• v.37 no.10
• /
• pp.57-64
• /
• 2000

In this paper, we propose the new hardware architecture which implements the stereo matching algorithm using the dynamprogrammethod. The proposed MOEPE(Merged Odd-Even PE) architecture operates in the systolic manner and finds the disparities form the intensities of the pixels on the epipolar line. The number of PEs used in the MOEPE architecture is the same number of the range constraint, which reduced the nuMber of the necessary PEs draMatically compared to the traditional method which uses the PEs with the same number of pixels on the epipolar line. For the normal sized images, the numof the MOEPE architecture is less than that of the PEs in the traditional method by 25${\times}$The proposed architecture is modeled with the VHDL code and simulated by the SYNOPSYS tool.