• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.71-77
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• 2013

Consider a multiplicative group of integers modulo $n$, denoted by $\mathbb{Z}_n^*$. Any element $a{\in}\mathbb{Z}_n^*$ is said to be a semi-primitive root if the order of $a$ modulo $n$ is ${\phi}(n)/2$, where ${\phi}(n)$ is the Euler phi-function. In this paper, we discuss some interesting properties of the multiplicative groups of integers possessing semi-primitive roots and give its applications to solving certain congruences.

• Journal of Communications and Networks
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• v.12 no.5
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• pp.406-410
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• 2010

This paper presents an approach to the construction of non-binary quasi-cyclic (QC) low-density parity-check (LDPC) codes based on multiplicative groups over one Galois field GF(q) and Euclidean geometries over another Galois field GF($2^S$). Codes of this class are shown to be regular with girth $6{\leq}g{\leq}18$ and have low densities. Finally, simulation results show that the proposed codes perform very wel with the iterative decoding.

• Honam Mathematical Journal
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• v.33 no.2
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• pp.181-186
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• 2011

Consider a multiplicative group of integers modulo n, denoted by $\mathbb{Z}_n^*$. Any element $a{\in}\mathbb{Z}_n^*$ n is said to be a semi-primitive root if the order of a modulo n is $\phi$(n)/2, where $\phi$(n) is the Euler phi-function. In this paper, we classify the multiplicative groups of integers having semi-primitive roots and give interesting properties of such groups.

• The Mathematical Education
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• v.50 no.3
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• pp.337-354
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• 2011

The purpose of the study was to investigate how students understand multiplication and division of fractions and how their understanding influences the solutions of fractional word problems. Thirteen students from 5th to 6th grades were involved in the study. Students' understanding of operations with fractions was categorized into "a part of the parts", "multiplicative comparison", "equal groups", "area of a rectangular", and "computational procedures of fractional multiplication (e.g., multiply the numerators and denominators separately)" for multiplications, and "sharing", "measuring", "multiplicative inverse", and "computational procedures of fractional division (e.g., multiply by the reciprocal)" for divisions. Most students understood multiplications as a situation of multiplicative comparison, and divisions as a situation of measuring. In addition, some students understood operations of fractions as computational procedures without associating these operations with the particular situations (e.g., equal groups, sharing). Most students tended to solve the word problems based on their semantic structure of these operations. Students with the same understanding of multiplication and division of fractions showed some commonalities during solving word problems. Particularly, some students who understood operations on fractions as computational procedures without assigning meanings could not solve word problems with fractions successfully compared to other students.

• Asian Pacific Journal of Cancer Prevention
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• v.13 no.5
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• pp.1803-1808
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• 2012

This study aimed to investigate the effects of Ser/Cys polymorphism in hOGG1 gene, Arg/Pro polymorphism in p53 gene, smoking and their interactions on the development of lung cancer. Ser/Cys polymorphism in hOGG1 and Arg/Pro polymorphism in p53 among 124 patients with lung cancer and 128 normal people were detected using PCR-RFLP. At the same time, smoking status was investigated between the two groups. Logistic regression was used to estimate the effects of Ser/Cys polymorphism and Arg/Pro polymorphisms, smoking and their interactions on the development of lung cancer. ORs (95% CI) of smoking, hOGG1 Cys/Cys and p53 Pro/Pro genotypes were 2.34 (1.41-3.88), 2.12 (1.03-4.39), and 2.12 (1.15-3.94), respectively. The interaction model of smoking and Cys/Cys was super-multiplicative or multiplicative, and the OR (95% CI) for their interaction item was 1.67 (0.36 -7.78). The interaction model of smoking and Pro/Pro was super-multiplicative with an OR (95%CI) of their interaction item of 5.03 (1.26-20.1). The interaction model of Pro/Pro and Cys/Cys was multiplicative and the OR (95%CI) of their interaction item was 0.99 (0.19-5.28). Smoking, hOGG1 Cys/Cys, p53 Pro/Pro and their interactions may be the important factors leading to the development of lung cancer.

• Bulletin of the Korean Mathematical Society
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• v.26 no.1
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• pp.1-9
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• 1989

In 1980 and 1983, it was proved that P $D^{2}$-groups are surface groups ([2], [3]). Since then, topologists have been positively studying about P $D^{n}$ -groups (or $D^{n}$ -groups). For example, let a topological space X have a right .pi.-action, where .pi. is a multiplicative group. If each x.memX has an open neighborhood U such that for each u.mem..pi., u.neq.1, U.cap. $U_{u}$ =.phi., this right .pi.-action is said to be proper. In this case, if X/.pi. is compact then (1) .pi.$_{1}$(X/.pi).iden..pi.(X:connected, .pi.$_{1}$: fundamental group) ([4]), (2) if X is a differentiable orientable manifold with demension n and .rho.X (the boundary of X)=.phi. then $H^{k}$ (X;Z).iden. $H_{n-k}$(X;Z), ([6]), where Z is the set of all integers.s.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.3
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• pp.309-319
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• 2018

Galois polynomials are defined as a generalization of the cyclotomic polynomials. The definition of Galois polynomials (and cyclotomic polynomials) is based on the multiplicative group of integers modulo n, i.e. ${\mathbb{Z}}_n^*$. In this paper, we define Galois polynomials which are based on the quotient group ${\mathbb{Z}}_n^*/H$.

• Journal of the Korean Mathematical Society
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• v.54 no.6
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• pp.1841-1851
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• 2017

Motivated by the Bloch-Beilinson conjectures, Voisin has made a conjecture concerning zero-cycles on self-products of Calabi-Yau varieties. We reformulate Voisin's conjecture in the setting of $hyperk{\ddot{a}}hler$ varieties, and we prove this reformulated conjecture for one family of $hyperk{\ddot{a}}hler$ fourfolds.

• Kyungpook Mathematical Journal
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• v.61 no.2
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• pp.371-381
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• 2021

In this paper, we attempt to obtain sufficient conditions for the existence of tight wavelet frame sets in locally compact abelian groups. The condition is generated by modulating a collection of characteristic functions that correspond to a generalized shift-invariant system via the Fourier transform. We present two approaches (for stationary and non-stationary wavelets) to construct the scaling function for L²(G) and, using the scaling function, we construct an orthonormal wavelet basis for L²(G). We propose an open problem related to the extension principle for Riesz wavelets in locally compact abelian groups.

• The Plant Pathology Journal
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• v.24 no.3
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• pp.337-351
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• 2008

Host-pathogen interaction in rice bacterial blight pathosystem was analyzed for a better understanding of their relationship and recognition of stable pathogenicity among the populations of Xanthomonas oryzae pv. oryzae. A total number of 52 bacterial strains isolated from diseased leaf samples collected from 12 rice growing states and one Union Territory of India, were inoculated on 16 rice varieties, each possessing known genes for resistance. Analysis of variance revealed that the host genotypes(G) accounted for largest(78.4%) proportion of the total sum of squares(SS), followed by 16.5% due to the pathogen isolates(I) and 5.1% due to the $I{\times}G$ interactions. Application of the Additive Main effects and Multiplicative Interaction(AMMI) model revealed that the first two interaction principal component axes(IPCA) accounted for 66.8% and 21.5% of the interaction SS, respectively. The biplot generated using the isolate and genotypic scores of the first two IPCAs revealed groups of host genotypes and pathogen isolates falling into four sectors. A group of five isolates with high virulence, high absolute IPCA-1 scores, moderate IPCA-2 scores, low AMMI stability index '$D_i$' values and minimal deviations from additive main effects displayed in AMMI biplot as well as response plot, were identified as possessing stable pathogenicity across 16 host genotypes. The largest group of 27 isolates with low virulence, small IPCA-1 as well as IPCA-2 scores, low $D_i$ values and minimal deviations from additive main effect predictions, possessed stable pathogenicity for low virulence. The AMMI analysis and biplot display facilitated in a better understanding of the host-pathogen interaction, adaptability of pathogen isolates to specific host genotypes, identification of isolates showing stable pathogenicity and most discriminating host genotypes, which could be useful in location specific breeding programs aiming at deployment of resistant host genotypes in bacterial blight disease control strategies.