• The Pure and Applied Mathematics
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• v.19 no.2
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• pp.199-209
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• 2012

For a locally compact higher rank graph ${\Lambda}$, we construct a two-sided path space ${\Lambda}^{\Delta}$ with shift homeomorphism ${\sigma}$ and its corresponding path groupoid ${\Gamma}$. Then we find equivalent conditions of aperiodicity, cofinality and irreducibility of ${\Lambda}$ in (${\Lambda}^{\Delta}$, ${\sigma}$), ${\Gamma}$, and the groupoid algebra $C^*({\Gamma})$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1699-1718
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• 2017

We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic K-theory and depends on the canonical ${\mathbb{Z}}-torsor$ of a locally linearly compact k-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.207-218
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• 2020

Some modular representations of reflection groups related to Weyl groups are considered. The rational cohomology of the classifying space of a compact connected Lie group G with a maximal torus T is expressed as the ring of invariants, H*(BG; ℚ) ≅ H*(BT; ℚ)^W(G), which is a polynomial ring. If such Lie groups are locally isomorphic, the rational representations of their Weyl groups are equivalent. However, the integral representations need not be equivalent. Under the mod p reductions, we consider the structure of the rings, particularly for the Weyl group of symplectic groups Sp(n) and for the alternating groups A_n as the subgroup of W(SU(n)). We will ask if such rings of invariants are polynomial rings, and if each of them can be realized as the mod p cohomology of a space. For n = 3, 4, the rings under a conjugate of W(Sp(n)) are shown to be polynomial, and for n = 6, 8, they are non-polynomial. The structures of H*(BT^n-1; 𝔽_p)^A_n will be also discussed for n = 3, 4.

• Honam Mathematical Journal
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• v.33 no.1
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• pp.73-84
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• 2011

Let f, g, h, k : $\mathbb{R}^n{\rightarrow}\mathbb{C}$ be locally integrable functions. We deal with the $L^{\infty}$-version of the Hyers-Ulam stability of the quadratic functional inequality and the Pexiderized quadratic functional inequality $${\parallel}f(x + y) + f(x - y) -2f(x) - 2f(y){\parallel}_{L^{\infty}(\mathbb{R}^n)}\leq\varepsilon$$ $${\parallel}f(x + y) + g(x - y) -2h(x) - 2f(y){\parallel}_{L^{\infty}(\mathbb{R}^n)}\leq\varepsilon$$ based on the concept of linear functionals on the space of smooth functions with compact support.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.3
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• pp.457-467
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• 2011

In this paper, we establish some comparison theorems about volumes of tubes in metric spaces with nonpositive curvature. First we compare the Hausdorff measure of tube about a metric ball contained in an (n-1)-dimensional totally geodesic subspace of an n-dimensional locally compact, geodesically complete Hadamard space with Lebesgue measure of its corresponding tube in Euclidean space ${\mathbb{R}}^n$, and then develop the result to the case of an m-dimensional totally geodesic subspace for 1 < m < n with an additional condition. Also, we estimate the Hausdorff measure of the tube about a shortest curve in a metric space of curvature bounded above and below.

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.81-87
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• 1992

Wellknown invariance of domain theorems are Brower's invariance of domain theorem for continuous mappings defined on a finite dimensional space and Schauder-Leray's invariance of domain theorem for the class of mappings I+C defined on a infinite dimensional Banach space with I the identity and C compact. The two classical invariance of domain theorems were proved by applying the homotopy invariance of Brower's degree and Leray-Schauder's degree respectively. Degree theory for some class of mappings is a useful tool for mapping theorems. And mapping theorems (or surjectivity theorems of mappings) are closely related with invariance of domain theorems for mappings. In[4, 5], Browder and Petryshyn constructed a multi-valued degree theory for A-proper mappings. From this degree Petryshyn [9] obtained some invariance of domain theorems for locally A-proper mappings. Recently Browder [6] has developed a degree theory for demicontinuous mapings of type ( $S_{+}$) from a reflexive Banach space X to its dual $X^{*}$. By applying this degree we obtain some invariance of domain theorems for demicontinuous mappings of type ( $S_{+}$). ( $S_{+}$).

• Journal of the Korean Institute of Rural Architecture
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• v.20 no.1
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• pp.93-101
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• 2018

The area to be surveyed is the three-seat of Myeon office in Hwasun-gun, Jeollanam-do, where population is declining. Satisfaction with the use of local residents of rural central areas providing life services in the back area was identified, focusing on sustainable development and the compact cities. Based on the results of the study, the recommendations are as follows. First, the local spaces to be surveyed need a shift to a walk-centered choice and social structure, not a must-oriented, activity-centric structure. This not only improves the neighborhood of rural areas, but also increases the frequency of pedestrian exchange and helps keep the community afloat. Second, it could enhance the convenience of residents by reviving stocks in existing commercial areas and integrating public services. Finally, it has the effect of taking care of the socially disadvantaged and immigrants in terms of space welfare, and securing jobs through locally customized community businesses.

• Progress in Medical Physics
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• v.31 no.4
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• pp.189-193
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• 2020

An MR-based attenuation correction (MRAC) map plays an important role in quantitative positron emission tomography (PET) image evaluation in PET/magnetic resonance imaging (MRI) systems. However, the MRAC map is affected by the magnetic field inhomogeneity of MRIs. This study aims to evaluate the characteristics of MRAC maps of physical phantoms on PET/MRI images. Phantom measurements were performed using the Siemens Biograph mMR. The modular type physical phantoms that provide assembly versatility for phantom construction were scanned in a four-channel Body Matrix coil. The MRAC map was generated using the two-point Dixon-based segmentation method for whole-body imaging. The modular phantoms were scanned in compact and non-compact assembly configurations. In addition, the phantoms were scanned repeatedly to generate MRAC maps. The acquired MRAC maps show differently assigned values for void areas. An incorrect assignment of a void area was shown on a locally compact space between phantoms. The assigned MRAC values were distorted using a wide field-of-view (FOV). The MRAC values also differed after repeated scans. However, the erroneous MRAC values appeared outside of phantom, except for a large FOV. The MRAC map of the phantom was affected by phantom configuration and the number of scans. A quantitative study using a phantom in a PET/MRI system should be performed after evaluation of the MRAC map characteristics.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2008.10a
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• pp.406-409
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• 2008

We consider the problem of exploiting a taxonomy of propositionalized attributes in order to learn compact and robust classifiers. We introduce Propositionalized Attribute Taxonomy guided Naive Bayes Learner (PAT-NBL), an inductive learning algorithm that exploits a taxonomy of propositionalized attributes as prior knowledge to generate compact and accurate classifiers. PAT-NBL uses top-down and bottom-up search to find a locally optimal cut that corresponds to the instance space from propositionalized attribute taxonomy and data. Our experimental results on University of California-Irvine (UCI) repository data sets show that the proposed algorithm can generate a classifier that is sometimes comparably compact and accurate to those produced by standard Naive Bayes learners.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.12 no.12
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• pp.2357-2364
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• 2008

In this paper, we consider the problem of exploiting a taxonomy of propositionalized attributes in order to generate compact and robust classifiers. We introduce Propositionalized Attribute Taxonomy guided Naive Bayes Learner (PAT-NBL), an inductive learning algorithm that exploits a taxonomy of propositionalized attributes as prior knowledge to generate compact and accurate classifiers. PAT-NBL uses top-down and bottom-up search to find a locally optimal cut that corresponds to the instance space from propositionalized attribute taxonomy and data. Our experimental results on University of California-Irvine (UCI) repository data set, show that the proposed algorithm can generate a classifier that is sometimes comparably compact and accurate to those produced by standard Naive Bayes learners.