• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1159-1172
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• 2019

A semigroup S is called a weakly abundant semigroup if its every $\tilde{\mathcal{L}}$-class and every $\tilde{\mathcal{R}}$-class contains an idempotent. Our purpose is to study an analogue of orthodox semigroups in the class of weakly abundant semigroups. Such an analogue is called a left quasi-abundant semigroup, which is a weakly abundant semigroup with a left quasi-normal band of idempotents and having the congruence condition (C). To build our main structure theorem for left quasi-abundant semigroups, we first give a sufficient and necessary condition of the idempotent set E(S) of a weakly abundant semigroup S being a left quasi-normal band. And then we construct a left quasi-abundant semigroup in terms of weak spined products. Such a result is a generalisation of that of Guo and Shum for left semi-perfect abundant semigroups. In addition, we consider a type Q semigroup which is a left quasi-abundant semigroup having the PC condition.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.537-558
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• 2008

Consider a weak solution u of the Navier-Stokes equations in the class $L^2((0,T);X_1(\mathbb{R}^d)^d)$. We establish a new approach to treat the regularity problem for the Navier-Stokes equation in term of the multiplier space $X_1(\mathbb{R}^d)$.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.921-936
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• 2011

In this paper, our main purpose is to establish the existence of weak solutions of a weak solutions of a class of p-q-Laplacian system involving concave-convex nonlinearities: $$\{\array{-{\Delta}_pu-{\Delta}_qu={\lambda}V(x)|u|^{r-2}u+\frac{2{\alpha}}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\;x{\in}{\Omega}\\-{\Delta}p^v-{\Delta}q^v={\theta}V(x)|v|^{r-2}v+\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\;x{\in}{\Omega}\\u=v=0,\;x{\in}{\partial}{\Omega}}$$ where ${\Omega}$ is a bounded domain in $R^N$, ${\lambda}$, ${\theta}$ > 0, and 1 < ${\alpha}$, ${\beta}$, ${\alpha}+{\beta}=p^*=\frac{N_p}{N_{-p}}$ is the critical Sobolev exponent, ${\Delta}_su=div(|{\nabla}u|^{s-2}{\nabla}u)$ is the s-Laplacian of u. when 1 < r < q < p < N, we prove that there exist infinitely many weak solutions. We also obtain some results for the case 1 < q < p < r < $p^*$. The existence results of solutions are obtained by variational methods.

• 대한공업교육학회지
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• v.30 no.1
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• pp.106-119
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• 2005

Many problems with the offline class, which is the traditional education type in corporations or universities, were indicated and people hoped that e-learning, which is web-based instruction, would solve these problems. However, e-learning also has weak points in that it should be self-paced and media-based in many ways. Therefore, when considering the good and weak points of offline classes and e-learning, blended learning seems to be necessary. Until now, blended learning has usually been used in corporations, and there have been almost no studies on the effectiveness or management of blended learning in universities. Thus, in this study, I would like to design blended classes, manage them at the level of university classes, and verify the effectiveness of blended classes, by comparing academic achievement, student participation, and student satisfaction. The subject students who signed up for Computer & Technology at C University in 2005 were divided into three study groups: offline class, online class, and blended class. The offline class was taught using the traditional class teaching method. For the online class and the blended class, multimedia contents were developed and a different LMS was used. The results of 13 weeks of teaching are as follows. For the academic achievement in the offline, online and blended classes, there was no statistically significant difference (f=2.387, p=.096). But when comparing the average achievement, the average of the blended class was higher than that of the other classes, so that it can be said that the blended class has positive effects on academic achievement. Second, when comparing the learners' participation in the online class and the blended class, the total posts were 85 and 138 respectively, which shows a considerable difference. The hit counts for each post in the online class and the blended class are 10 and 20, respectively. Moreover, the login counts for subjects are 3 in the online class and 4 in the blended class. In the questionnaire for the students' academic satisfaction in the online class and the blended class, all of the 15 items showed higher satisfaction in the blended class. Considering all these results, if adequate media are properly combined, the blended class is better than either the pure online class or the pure offline class.

• Journal of The Korean Astronomical Society
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• v.47 no.1
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• pp.1-13
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• 2014

We analyze the dependence of disk morphology (arm class, Hubble type, bar type) of nearby spiral galaxies on the galaxy environment by using local background density (${\Sigma}_n$), projected distance ($r_p$), and tidal index (T I) as measures of the environment. There is a strong dependence of arm class and Hubble type on the galaxy environment, while the bar type exhibits a weak dependence with a high frequency of SB galaxies in high density regions. Grand design fractions and early-type fractions increase with increasing ${\Sigma}_n$, $1/r_p$, and T I, while fractions of flocculent spirals and late-type spirals decrease. Multiple-arm and intermediate-type spirals exhibit nearly constant fractions with weak trends similar to grand design and early-type spirals. While bar types show only a marginal dependence on ${\Sigma}_n$, they show a fairly clear dependence on $r_p$ with a high frequency of SB galaxies at small $r_p$. The arm class also exhibits a stronger correlation with $r_p$ than ${\Sigma}_n$ and T I, whereas the Hubble type exhibits similar correlations with ${\Sigma}_n$ and $r_p$. This suggests that the arm class is mostly affected by the nearest neighbor while the Hubble type is affected by the local densities contributed by neighboring galaxies as well as the nearest neighbor.

• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.333-344
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• 2023

In this paper, we generalize the notions uniformly S-flat, briefly u-S-flat, modules and dimensions. We introduce and study the notions of weak u-S-flat modules. An R-module M is said to be weak u-S-flat if Tor^R₁ (R/I, M) is u-S-torsion for any ideal I of R. This new class of modules will be used to characterize u-S-von Neumann regular rings. Hence, we introduce the weak u-S-flat dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed.

• Bulletin of the Korean Mathematical Society
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• v.20 no.2
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• pp.87-93
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• 1983

Let X and Y be normed linear spaces. An operator T defined on X with the range in Y is continuous in the sense that if a sequence {x$_{n}$} in X converges to x for the weak topology .sigma.(X.X') then {Tx$_{n}$} converges to Tx for the norm topology in Y. We shall denote the class of such operators by WC(X, Y). For example, if T is a compact operator then T.mem.WC(X, Y). In this note we discuss relationships between WC(X, Y) and the class of weakly of bounded linear operators B(X, Y). In the last section, we will consider some characters for an operator in WC(X, Y).).

• Korean Journal of Mathematics
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• v.21 no.4
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• pp.483-493
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• 2013

We investigate the multiplicity of the solutions for a class of the system of the biharmonic equations with some singular potential nonlinearity. We obtain a theorem which shows the existence of the nontrivial weak solution for a class of the system of the biharmonic equations with singular potential nonlinearity and Dirichlet boundary condition. We obtain this result by using variational method and the generalized mountain pass theorem.

• Imaging Science in Dentistry
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• v.46 no.3
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• pp.211-216
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• 2016

Purpose: The use of electronic tools in teaching is growing rapidly in all fields, and there are many options to choose from. We present one such platform, Learning Catalytics$^{TM}$ (LC) (Pearson, New York, NY, USA), which we utilized in our oral and maxillofacial radiology course for second-year dental students. Materials and Methods: The aim of our study was to assess the correlation between students' performance on course exams and self-assessment LC quizzes. The performance of 354 predoctoral dental students from 2 consecutive classes on the course exams and LC quizzes was assessed to identify correlations using the Spearman rank correlation test. The first class was given in-class LC quizzes that were graded for accuracy. The second class was given out-of-class quizzes that were treated as online self-assessment exercises. The grading in the self-assessment exercises was for participation only and not accuracy. All quizzes were scheduled 1-2 weeks before the course examinations. Results: A positive but weak correlation was found between the overall quiz scores and exam scores when the two classes were combined (P<0.0001). A positive but weak correlation was likewise found between students' performance on exams and on in-class LC quizzes (class of 2016) (P<0.0001) as well as on exams and online LC quizzes (class of 2017) (P<0.0001). Conclusion: It is not just the introduction of technological tools that impacts learning, but also their use in enabling an interactive learning environment. The LC platform provides an excellent technological tool for enhancing learning by improving bidirectional communication in a learning environment.

• East Asian mathematical journal
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• v.33 no.1
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• pp.83-102
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• 2017

The main purpose of this paper is to introduce a pair new class of primal and dual problem associated with an operator. We prove the sufficient optimality theorem, weak duality theorem and strong duality theorem for these problems. The equivalence between the generalized optimization problems and the generalized variational inequality problems is studied in ordered topological vector space modeled in Hilbert spaces. We introduce the concept of partial differential associated (PDA)-operator, PDA-vector function and PDA-antisymmetric function to show the existence of a new class of function called, ($T_{\eta};{\xi}_{\theta}$)-invex functions. We discuss first and second kind of ($T_{\eta};{\xi}_{\theta}$)-invex functions and establish their existence theorems in ordered topological vector spaces.