• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.643-702
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• 2021

Fix ℤ/2 is the prime field of two elements and write 𝒜₂ for the mod 2 Steenrod algebra. Denote by GL_d := GL(d, ℤ/2) the general linear group of rank d over ℤ/2 and by ${\mathfrak{P}}_d$ the polynomial algebra ℤ/2[x₁, x₂, …, x_d] as a connected unstable 𝒜₂-module on d generators of degree one. We study the Peterson "hit problem" of finding the minimal set of 𝒜₂-generators for ${\mathfrak{P}}_d$. Equivalently, we need to determine a basis for the ℤ/2-vector space $$Q{\mathfrak{P}}_d:={\mathbb{Z}}/2{\otimes}_{\mathcal{A}_2}\;{\mathfrak{P}}_d{\sim_=}{\mathfrak{P}}_d/{\mathcal{A}}^+_2{\mathfrak{P}}_d$$ in each degree n ≥ 1. Note that this space is a representation of GL_d over ℤ/2. The problem for d = 5 is not yet completely solved, and unknown in general. In this work, we give an explicit solution to the hit problem of five variables in the generic degree n = r(2^t - 1) + 2^ts with r = d = 5, s = 8 and t an arbitrary non-negative integer. An application of this study to the cases t = 0 and t = 1 shows that the Singer algebraic transfer of rank 5 is an isomorphism in the bidegrees (5, 5 + (13.2⁰ - 5)) and (5, 5 + (13.2¹ - 5)). Moreover, the result when t ≥ 2 was also discussed. Here, the Singer transfer of rank d is a ℤ/2-algebra homomorphism from GL_d-coinvariants of certain subspaces of $Q{\mathfrak{P}}_d$ to the cohomology groups of the Steenrod algebra, $Ext^{d,d+*}_{\mathcal{A}_2}$ (ℤ/2, ℤ/2). It is one of the useful tools for studying these mysterious Ext groups.

• Journal of the Korean Geographical Society
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• v.43 no.3
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• pp.432-447
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• 2008

This study is to examine and analyze the the connection types between location knowledge and understanding the characteristics of world regions which 8th grade students are recognizing. In the analysis of the relation between the location knowledge and region understanding(clothing, food, housing and living style and so on), I analyzed the relation types between location knowledge and region living figures. Absolutely understanding the region needed the efact location knowledge on the world map, the knowledge of the longitude and latitude, symbolic area and the higher ranking region and the information about natural surroundings in which I grasp the region living figures. In addition, it needed the knowledge of locational attributes for the connection formation between region location knowledge on maps and understanding the region. The relative location knowledge related to the other regions and those natural surroundings and the linkage of those region locational attributes are needed in order that the location knowledge on a map could be connected to the knowledge about the actual region living figures. So I call it 'the structured location knowledge'. In order to form the structured location knowledge, the location knowledge of region on a map have to be connected to the multiple knowledge about absolute and relative locations with the knowledge of diverse location attributes. Therefore, it would be necessary that the location teaming for getting the characteristics of region have to be systematically accomplished departments of location-climate$^{\circ}{\mathbb{S}}$geographical features-region living figures.