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

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.747-759
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• 2018

A graph G = (V (G), E(G) of order n = |V (G)| and size m = |E(G)| is said to be odd harmonious if there exists an injection $f:V(G){\rightarrow}\{0,\;1,\;2,\;{\ldots},\;2m-1\}$ such that the induced function $f^*:E(G){\rightarrow}\{1,\;3,\;5,\;{\ldots},\;2m-1\}$ defined by $f^*(uv)=f(u)+f(v)$ is bijection. While a bipartite graph G with partite sets A and B is said to be bigraceful if there exist a pair of injective functions $f_A:A{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ and $f_B:B{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ such that the induced labeling on the edges $f_{E(G)}:E(G){\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ defined by $f_{E(G)}(uv)=f_A(u)-f_B(v)$ (with respect to the ordered partition (A, B)), is also injective. In this paper we prove that odd harmonious graphs and bigraceful graphs are equivalent. We also prove that the number of distinct odd harmonious labeled graphs on m edges is m! and the number of distinct strongly odd harmonious labeled graphs on m edges is [m/2]![m/2]!. We prove that the Cartesian product of strongly odd harmonious trees is strongly odd harmonious. We find some new disconnected odd harmonious graphs.

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

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

• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.607-617
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• 2014

Let S = C(r,m), the finite cyclic semigroup with index r and period m. Each subsemigroup of S is cyclic if and only if either r = 1; r = 2; or r = 3 with m odd. For $r{\neq}1$, the maximum value of the minimum number of elements in a (minimal) generating set of a subsemigroup of S is 1 if r = 3 and m is odd; 2 if r = 3 and m is even; (r-1)/2 if r is odd and unequal to 3; and r/2 if r is even. The number of cyclic subsemigroups of S is $r-1+{\tau}(m)$. Formulas are also given for the number of 2-generated subsemigroups of S and the total number of subsemigroups of S. The minimal generating sets of subsemigroups of S are characterized, and the problem of counting them is analyzed.

• Proceedings of the IEEK Conference
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• 1999.11a
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• pp.495-498
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• 1999

In this paper, we propose the adaptive Odd/Even Multi-shell Median Filter(adaptive O/E MMF) to improve the defect that Modified Multi-shell Median Filter(MMMF) can not recover missing lines of vertical and cross direction. This filter uses odd/even multi-shells and new proposed threshold strategy The performance of the proposed filter is evaluated over image 'airfield 'by using MATLAB. As the proposed threshold strategy eliminate the number of redundant replacement, it suppresses impulse noise and recovers missing lines.

• JSTS:Journal of Semiconductor Technology and Science
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• v.14 no.4
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• pp.376-382
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• 2014

In this paper, a 9-bit 1GS/s high precision folding A/D converter with a 45 nm CMOS technology is proposed. In order to improve the asymmetrical boundary condition error of a conventional folding ADC, a novel scheme with an odd number of folding blocks is proposed. Further, a new digital encoding technique is described to implement the odd number of folding technique. The proposed ADC employs a digital error correction circuit to minimize device mismatch and external noise. The chip has been fabricated with 1.1V 45nm Samsung CMOS technology. The effective chip area is $2.99mm^2$ and the power dissipation is about 120 mW. The measured result of SNDR is 45.35 dB, when the input frequency is 150 MHz at the sampling frequency of 1 GHz. The measured INL is within +7 LSB/-3 LSB and DNL is within +1.5 LSB/-1 LSB.

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

• KSBB Journal
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• v.8 no.1
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• pp.49-54
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• 1993

The contents of odd number fatty acid in walnut, peach seed and apricot seed were analyzed by HPLC. The results were as follows; the contents of crude oil in walnut, peach seed and apricot seed were 61.7%, 48.21%, and 52.34%, respectively. The contents of neutral lipid in walnut peach seed, and apricot seed were contained 62.32%, 91.50%, and 88.00%, respectively. The contents of glycolipid in walnut peach seed and apricot seed were contained 34.14%, 3.25%, and 4.78%, respectively. The contents of phospholipid in walnut peach seed and apricot seed were contained 3.54%, 5.25%, and 7.27%, respectively. The contents of odd number fatty acid and even number fatty acid in neutralipid were walnut 37.07% and 36.72%, peach seed 5.4% and 94.60%, and apricot seed 6.85% and 93.92%, respectively. The contents of odd number fatty acid and even number fatty acid in glycolipid were walnut 5.37% and 46.21%, peach seed 6.97% and 89.42%, and apricot seed 10.50% and 89.42%, respectively. The contents of odd number fatty acid and even number fatty acid in phospholipld were walnut 3.57% and 96.43%, peach speed 6.51% and 93.49%, and apricot seed 15.00% and 84.20%, respectively. The contents of odd number fatty and even number fatty acid in total fatty acid were walnut 14.47% and 85.53%, peach seed 1.17% and 98.83%, and apricot seed 4.3% and 95.97%, respectively.

• Clinical and Experimental Pediatrics
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• v.51 no.10
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• pp.1038-1041
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• 2008

Aminoglycosides are frequently used antibiotics in children. The multiple daily dosing (MDD) in infants and children is twice or three times daily depending on age. Recent studies in adults have shown that once daily dosing (ODD) maximizes the bactericidal activity and might minimize the toxicity of antibiotics. So, I reviewed many studies about efficacy, toxicity and cost effectiveness of ODD of aminoglycosides in children. Most studies suggest that ODD compared with MDD of aminoglycosides is theoretically more efficacious and has no higher toxicity in infants and children. But, the total number of patients included in the studies is not large. Multi-center, controlled prospective studies are required in larger numbers of infants and children to determine the efficacy and safety of the ODD regimen in children before ODD of aminoglycosides can be recommended for routine use.