• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.59-68
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• 2022

For a finite group G ⊂ GL(n, ℂ), the G-Hilbert scheme is a fine moduli space of G-clusters, which are 0-dimensional G-invariant subschemes Z with H⁰(𝒪_Z ) isomorphic to ℂ[G]. In many cases, the G-Hilbert scheme provides a good resolution of the quotient singularity ℂⁿ/G, but in general it can be very singular. In this note, we prove that for a cyclic group A ⊂ GL(n, ℂ) of type ${\frac{1}{r}}$(1, …, 1, a) with r coprime to a, A-Hilbert Scheme is smooth and irreducible.

• Korean Journal of Mathematics
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• v.32 no.1
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• pp.59-72
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• 2024

In the presented paper, the first section contains strong convergence and demiclosedness property of a sequence generated by Karakaya et al. iteration scheme in a Hilbert space for quasi-nonexpansive mappings and also the comparison between the iteration scheme given by Karakaya et al. with well-known iteration schemes for the convergence rate. The second section contains some applications of the fixed point theory in solution of different mathematical problems.

• Journal of KIISE
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• v.41 no.10
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• pp.792-798
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• 2014

In spatial databases, R-tree is one of the most widely used indexing structures and many variants have been proposed for its performance improvement. Among these variants, Hilbert R-tree is a representative method using Hilbert curve to process large amounts of data without high cost split techniques to construct the R-tree. This Hilbert R-tree, however, is hardly applicable to large-scale applications in practice mainly due to high pre-processing costs and slow bulk-load time. To overcome the limitations of Hilbert R-tree, we propose a novel approach for parallelizing Hilbert mapping and thus accelerating bulk-loading of Hilbert R-tree on GPU memory. Hilbert R-tree based on GPU improves bulk-loading performance by applying the inversed-cell method and exploiting parallelism for packing the R-tree structure. Our experimental results show that the proposed scheme is up to 45 times faster compared to the traditional CPU-based bulk-loading schemes.

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.767-786
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• 2018

In this paper, we introduce new iterative schemes by using the modified Ishikawa iteration for two hybrid multivalued mappings in a Hilbert space. We then obtain weak convergence theorem under suitable conditions. We use CQ and shrinking projection methods with Ishikawa iteration for obtaining strong convergence theorems. Furthermore, we give examples and numerical results for supporting our main results.

• International Journal of Internet, Broadcasting and Communication
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• v.15 no.2
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• pp.22-30
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• 2023

The portable mobile devices with high performance and high speed 5G network activate and explode the demands for ubiquitous information services that remove the limitations of time for the communication and places to request for the information. NN (Nearest Neighbor) search is one of the most important types of queries to be processed efficiently in the information services. Various indexes have been proposed to support efficient NN search in the wireless broadcast system. The indexes adopting Hilbert curve, grid partition or Voronoi diagram enable the clients to search for NN quickly in the wireless broadcast channel. It is necessary that an efficient means to evaluate the performances of various indexes. In this paper, we propose an open framework that can adopt a variety of indexing schemes and evaluate and compare the performances of them. The proposed framework is organized with open and flexible structure that can adopt hybrid indexing schemes extensible to Voronoi diagram as well as simple indexing schemes. With the implemented framework, we demonstrate the efficiency and scalability and flexibility of the proposed framework by evaluating various indexing schemes for NN query.

• Journal of KIISE:Computing Practices and Letters
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• v.16 no.11
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• pp.1071-1075
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• 2010

Because a sensor node in wireless sensor networks(WSNs) has limited resources, such as battery capacity and memory, data aggregation techniques have been studied to manage the limited resources efficiently. Because sensor network uses wireless communication, a data can be disclosed by attacker. Thus, the study on data protection schemes for data aggregation is essential in WSNs. But the existing data aggregation methods require both a large number of computation and communication, in case of network construction and data aggregation processing. To solve the problem, we propose a data protection scheme based on Hilbert-curve for data aggregation. Our scheme can minimizes communications among neighboring sensor nodes by using tree-based routing. Moreover, it can protect the data from attacker by doing encryption through a Hilbert-curve technique based on a private seed, Finally, we show that our scheme outperforms the existing methods in terms of message transmission and average sensor node lifetime.

• Korean Journal of Mathematics
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• v.25 no.4
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• pp.469-481
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• 2017

In this paper, we established a general nonconvex split variational inequality problem, this is, an extension of general convex split variational inequality problems in two different Hilbert spaces. By using the concepts of prox-regularity, we proved the convergence of the iterative schemes for the general nonconvex split variational inequality problems. Further, we also discussed the iterative method for the general convex split variational inequality problems.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.515-531
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• 1997

In this paper, finite difference method is applied to approximate the generalized solutions of Sobolev equations. Using the Steklov mollifier and Bramble-Hilbert Lemma, a priori error estimates in discrete $L^2$ as well as in discrete $H^1$ norms are derived frist for the semidiscrete methods. For the fully discrete schemes, both backward Euler and Crank-Nicolson methods are discussed and related error analyses are also presented.

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.381-403
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• 2022

The main goal of this research is to solve variational inequalities involving quasimonotone operators in infinite-dimensional real Hilbert spaces numerically. The main advantage of these iterative schemes is the ease with which step size rules can be designed based on an operator explanation rather than the Lipschitz constant or another line search method. The proposed iterative schemes use a monotone and non-monotone step size strategy based on mapping (operator) knowledge as a replacement for the Lipschitz constant or another line search method. The strong convergences have been demonstrated to correspond well to the proposed methods and to settle certain control specification conditions. Finally, we propose some numerical experiments to assess the effectiveness and influence of iterative methods.

• Journal of Korea Multimedia Society
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• v.17 no.10
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• pp.1182-1188
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• 2014

Recently, the research on database outsourcing has been actively done with the popularity of cloud computing. However, because users' data may contain sensitive personal information, such as health, financial and location information, the data encryption methods have attracted much interest. Existing data encryption schemes process a query without decrypting the encrypted databases in order to support user privacy protection. On the other hand, to efficiently handle the large amount of data in cloud computing, it is necessary to study the distributed index structure. However, existing index structure and query processing algorithms have a limitation that they only consider single-column query processing. In this paper, we propose a grid-based multi column indexing scheme and an encrypted query processing algorithm. In order to support multi-column query processing, the multi-dimensional index keys are generated by using a space decomposition method, i.e. grid index. To support encrypted query processing over encrypted data, we adopt the Hilbert curve when generating a index key. Finally, we prove that the proposed scheme is more efficient than existing scheme for processing the exact and range query.