• 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.

• 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.

• 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: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.

• Journal of Korea Spatial Information System Society
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• v.11 no.1
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• pp.115-126
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• 2009

Recent development in wireless communication and mobile positioning technologies makes Location-Based Services (LBSs) popular. However, because, in the LBSs, users request a query to database servers by using their exact locations, the location information of the users can be misused by adversaries. Therefore, a mechanism for users' privacy protection is required for the safe use of LBSs by mobile users. For this, we, in this paper, propose a efficient cloaking region creation scheme using Hilbert curves in distributed grid environment, so as to protect users' privacy in LBSs. The proposed scheme generates a minimum cloaking region by analyzing the characteristic of a Hilbert curve and computing the Hilbert curve values of neighboring cells based on it, so that we may create a cloaking region to satisfy K-anonymity. In addition, to reduce network communication cost, we make use of a distributed hash table structure, called Chord. Finally, we show from our performance analysis that the proposed scheme outperforms the existing grid-based cloaking method.

• East Asian mathematical journal
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• v.29 no.3
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• pp.303-314
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• 2013

In this paper, we introduce an iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem and the set of common fixed points of a finite family of asymptotically ${\kappa}$-strict pseudo-contractions in Hilbert spaces. Weak and strong convergence theorems are established for the iterative scheme.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1263-1275
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• 2010

In this paper, we introduce a new iterative scheme for finding a common element of the set of common fixed points of infinite family of nonexpansive mappings and the set of solutions to a generalized equilibrium problem and the set of solutions to a variational inequality problem in a real Hilbert space. Then strong convergence of the scheme to a common element of the three sets is proved. As applications, three new strong convergence theorems are obtained. Our theorems extend important recent results.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.59-82
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• 2022

In this paper, we present a modified (improved) generalized M-iteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we obtain a weak convergence result under suitable conditions and the strong convergence result is achieved using the hybrid projection method with our modified generalized M-iteration. Finally, we apply our convergence results to certain optimization problem, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other improved iterative methods (modified SP-iterative scheme) in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.16 no.9
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• pp.3087-3103
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• 2022

Nearest neighbor query in location-based services has become a popular application. Aiming at the shortcomings of the privacy protection algorithms of traditional ciphertext nearest neighbor query having the high system overhead because of the usage of the double Hilbert curves and having the inaccurate query results in some special circumstances, a privacy protection algorithm of ciphertext nearest neighbor query which is based on the single Hilbert curve has been proposed. This algorithm uses a single Hilbert curve to transform the two-dimensional coordinates of the points of interest into Hilbert values, and then encrypts them by the order preserving encryption scheme to obtain the one-dimensional ciphertext data which can be compared in numerical size. Then stores the points of interest as elements composed of index value and the ciphertext of the other information about the points of interest on the server-side database. When the user needs to use the nearest neighbor query, firstly calls the approximate nearest neighbor query algorithm proposed in this paper to query on the server-side database, and then obtains the approximate nearest neighbor query results. After that, the accurate nearest neighbor query result can be obtained by calling the precision processing algorithm proposed in this paper. The experimental results show that this privacy protection algorithm of ciphertext nearest neighbor query which is based on the single Hilbert curve is not only feasible, but also optimizes the system overhead and the accuracy of ciphertext nearest neighbor query result.

• Communications of the Korean Mathematical Society
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• v.5 no.2
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• pp.79-86
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• 1990