It is our pedagogical challenge to introduce various mathematical concept in an educationally acceptable way and to prepare course materials that make students understand that deeply. As described in this paper, we present some of our attempts to verify the effects of using carefully prepared course materials with high-quality graphs in collegiate education of mathematics. Through our experiment, we detected the change of students’ brain activity by conducting behavioral observation and neuroimaging simultaneously. In an experiment aimed at helping students understand the concept of an exponential growth comparing exponential and power functions, we prepared several graphs for that purpose. Seven students observed the graphs while we tracked their responses. Simultaneously, we monitored their brain activities using electroencephalography (EEG). Three students altered their judgments, we found, on viewing the triggering graph. Some changes in the trend of their EEG signal were recognized while they were viewing the graph. These results of our experiments show that the use of favorable graphs as course materials might promote learners’ reasoning processes.

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The notion of soft BCI-positive implicative ideals and BCI-positive implicative idealistic soft BCI-algebras is introduced and their basic properties are discussed. Relations between soft ideals and soft BCIpositive implicative ideals of soft BCI-algebras are provided. Also idealistic soft BCI-algebras and BCI-positive implicative idealistic soft BCI-algebras are being related. The intersection, union, “AND” operation and “OR” operation of soft BCI-positive implicative ideals and BCI-positive implicative idealistic soft BCI-algebras are established. The characterizations of (fuzzy) BCI-positive implicative ideals in BCI-algebras are given by using the concept of soft sets. Relations between fuzzy BCI-positive implicative ideals and BCI-positive implicative idealistic soft BCI-algebras are discussed.

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We consider classification of homeomorphism classes of connected sums of closed surfaces by the Euler characteristic and the K-theory ranks of C*-algebras. We next consider classification of those of connected sums of higher dimensional, closed topological manifolds by the Euler characteristic and the K-theory ranks of C*-algebras.

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It is proved (with given different proofs) that the von Neumann- Wold and the Nagy-Foias-Langer decompositions are valid in more general classes than the classical W*-algebras.

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