• Korean Journal of Mathematics
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• v.32 no.1
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• pp.43-57
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• 2024

Given a domain X and a collection H of functions h : X → {0, 1}, the Vapnik-Chervonenkis (VC) dimension of H measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions ℋ'²_t(E) : 𝔽²_q → {0, 1}, corresponding to indicator functions of circles centered at points in a subset E ⊆ 𝔽²_q. They showed that when |E| is large enough, the VC-dimension of ℋ'²_t(E) is the same as in the case that E = 𝔽²_q. We study a related hypothesis class, ℋ^d_t(E), corresponding to intersections of spheres in 𝔽^d_q, and ask how large E ⊆ 𝔽^d_q needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever |E| ≥ C_dq^d-1/(d-1) for d ≥ 3, the VC-dimension of ℋ^d_t(E) is as large as possible. We get a slightly stronger result if d = 3: this result holds as long as |E| ≥ C_3q^7/3. Furthermore, when d = 2 the result holds when |E| ≥ C_2q^7/4.

• Geomechanics and Engineering
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• v.36 no.5
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• pp.417-426
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• 2024

Shield tunneling method is widely used to build tunnels in complex geological environment. Stability control of tunnel face is the key to the safety of projects. To improve the excavation efficiency or perform equipment maintenance, the excavation chamber sometimes is not fully filled with support medium, which can reduce the load and increase tunneling speed while easily lead to ground collapse. Due to the high risk of the face failure under non-fully support mode, the tunnel face stability should be carefully evaluated. Whether compressive air is required for compensation and how much air pressure should be provided need to be determined accurately. Based on the upper bound theorem of limit analysis, a non-fully support rotational failure model is developed in this study. The failure mechanism of the model is verified by numerical simulation. It shows that increasing the density of supporting medium could significantly improve the stability of tunnel face while the increase of tunnel diameter would be unfavorable for the face stability. The critical support ratio is used to evaluate the face failure under the nonfully support mode, which could be an important index to determine whether the specific unsupported height could be allowed during shield tunneling. To avoid of face failure under the non-fully support mode, several charts are provided for the assessment of compressed air pressure, which could help engineers to determine the required air pressure for face stability.

• Journal of the Korea Society of Computer and Information
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• v.16 no.10
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• pp.101-109
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• 2011

The Riemann's zeta function $\zeta(s)$ has been known as answer for a number of primes $\pi$(x) less than given number x. In prime number theorem, there are another approximation function $\frac{x}{lnx}$,Li(x), and R(x). The error about $\pi$(x) is R(x) < Li(x) < $\frac{x}{lnx}$. The logarithmic integral function is Li(x) = $\int_{2}^{x}\frac{1}{lnt}dt$ ~ $\frac{x}{lnx}\sum\limits_{k=0}^{\infty}\frac{k!}{(lnx)^k}=\frac{x}{lnx}(1+\frac{1!}{(lnx)^1}+\frac{2!}{(lnx)^2}+\cdots)$. This paper shows that the $\pi$(x) can be represent with finite Li(x), and presents generalized prime counting function $\sqrt{{\alpha}x}{\pm}{\beta}$. Firstly, the $\pi$(x) can be represent to $Li_3(x)=\frac{x}{lnx}(\sum\limits_{t=0}^{{\alpha}}\frac{k!}{(lnx)^k}{\pm}{\beta})$ and $Li_4(x)=\lfloor\frac{x}{lnx}(1+{\alpha}\frac{k!}{(lnx)^k}{\pm}{\beta})}k\geq2$ such that $0{\leq}t{\leq}2k$. Then, $Li_3$(x) is adjusted by $\pi(x){\simeq}Li_3(x)$ with ${\alpha}$ and error compensation value ${\beta}$. As a results, this paper get the $Li_3(x)=Li_4(x)=\pi(x)$ for $x=10^k$. Then, this paper suggests a generalized function $\pi(x)=\sqrt{{\alpha}x}{\pm}{\beta}$. The $\pi(x)=\sqrt{{\alpha}x}{\pm}{\beta}$ function superior than Riemann's zeta function in representation of prime counting.

• Journal of KIISE:Databases
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• v.33 no.6
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• pp.600-619
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• 2006

According to recent technical advances on sensors and mobile devices, processing of data streams generated by the devices is becoming an important research issue. The data stream of real values obtained at continuous time points is called streaming time-series. Due to the unique features of streaming time-series that are different from those of traditional time-series, similarity matching problem on the streaming time-series should be solved in a new way. In this paper, we propose an efficient algorithm for streaming time- series matching problem that supports normalization transform. While the existing algorithms compare streaming time-series without any transform, the algorithm proposed in the paper compares them after they are normalization-transformed. The normalization transform is useful for finding time-series that have similar fluctuation trends even though they consist of distant element values. The major contributions of this paper are as follows. (1) By using a theorem presented in the context of subsequence matching that supports normalization transform[4], we propose a simple algorithm for solving the problem. (2) For improving search performance, we extend the simple algorithm to use $k\;({\geq}\;1)$ indexes. (3) For a given k, for achieving optimal search performance of the extended algorithm, we present an approximation method for choosing k window sizes to construct k indexes. (4) Based on the notion of continuity[8] on streaming time-series, we further extend our algorithm so that it can simultaneously obtain the search results for $m\;({\geq}\;1)$ time points from present $t_0$ to a time point $(t_0+m-1)$ in the near future by retrieving the index only once. (5) Through a series of experiments, we compare search performances of the algorithms proposed in this paper, and show their performance trends according to k and m values. To the best of our knowledge, since there has been no algorithm that solves the same problem presented in this paper, we compare search performances of our algorithms with the sequential scan algorithm. The experiment result showed that our algorithms outperformed the sequential scan algorithm by up to 13.2 times. The performances of our algorithms should be more improved, as k is increased.

• Journal of KIISE:Computer Systems and Theory
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• v.30 no.5_6
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• pp.324-329
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• 2003

The public-key cryptosystems such as Diffie-Hellman Key Distribution and Elliptical Curve Cryptosystems are built on the basis of the operations defined in GF(2$^{m}$ ):addition, subtraction, multiplication and multiplicative inversion. It is important that these operations should be computed at high speed in order to implement these cryptosystems efficiently. Among those operations, as being the most time-consuming, multiplicative inversion has become the object of lots of investigation Formant's theorem says $\beta$$^{-1}$ =$\beta$$^{2}$sup m/-2/, where $\beta$$^{-1}$ is the multiplicative inverse of $\beta$$\in$GF(2$^{m}$ ). Therefore, to compute the multiplicative inverse of arbitrary elements of GF(2$^{m}$ ), it is most important to reduce the number of times of multiplication by decomposing 2$^{m}$ -2 efficiently. Among many algorithms relevant to the subject, the algorithm proposed by Itoh and Tsujii[2] has reduced the required number of times of multiplication to O(log m) by using normal basis. Furthermore, a few papers have presented algorithms improving the Itoh and Tsujii's. However they have some demerits such as complicated decomposition processes[3,5]. In this paper, in the case of 2$^{m}$ -2, which is mainly used in practical applications, an efficient algorithm is proposed for computing the multiplicative inverse at high speed by using both the factorization formula x$^3$-y$^3$=(x-y)(x$^2$＋xy＋y$^2$) and normal basis. The number of times of multiplication of the algorithm is smaller than that of the algorithm proposed by Itoh and Tsujii. Also the algorithm decomposes 2$^{m}$ -2 more simply than other proposed algorithms.

• Nuclear Medicine and Molecular Imaging
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• v.42 no.1
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• pp.52-60
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• 2008

Purpose: To establish the methods for sinogram formation and correction in order to appropriately apply the filtered backprojection (FBP) reconstruction algorithm to the data acquired using PET scanner with multiple scintillation crystal layers. Materials and Methods: Formation for raw PET data storage and conversion methods from listmode data to histogram and sinogram were optimized. To solve the various problems occurred while the raw histogram was converted into sinogram, optimal sampling strategy and sampling efficiency correction method were investigated. Gap compensation methods that is unique in this system were also investigated. All the sinogram data were reconstructed using 20 filtered backprojection algorithm and compared to estimate the improvements by the correction algorithms. Results: Optimal radial sampling interval and number of angular samples in terms of the sampling theorem and sampling efficiency correction algorithm were pitch/2 and 120, respectively. By applying the sampling efficiency correction and gap compensation, artifacts and background noise on the reconstructed image could be reduced. Conclusion: Conversion method from the histogram to sinogram was investigated for the FBP reconstruction of data acquired using multiple scintillation crystal layers. This method will be useful for the fast 20 reconstruction of multiple crystal layer PET data.

• Journal of Korea Water Resources Association
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• v.42 no.12
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• pp.1079-1089
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• 2009

The objective of this study is to develop the accurate, robust and high resolution two-dimensional numerical model that solves the computationally difficult hydraulic problems, including the wave front propagation over dry bed and abrupt change in bathymetry. The developed model in this study solves the conservative form of the two-dimensional shallow water equations using an unsplit finite volume scheme and HLLC approximate Riemann solvers to compute the interface fluxes. Bed-slope term is discretized by the divergence theorem in the framework of FVM for application of unsplit scheme. Accurate and stable SGM, in conjunction with the MUSCL which is second-order-accurate both in space and time, is adopted to balance with fluxes and source terms. The exact C-property is shown to be satisfied for balancing the fluxes and the source terms. Since the spurious oscillations in second-order schemes are inherent, an efficient slope limiting technique is used to supply TVD property. The accuracy, conservation property and application of developed model are verified by comparing numerical solution with analytical solution and experimental data through the simulations of one-dimensional dam break flow without bed slope, steady transcritical flow over a hump and two-dimensional dam break flow with a constriction.

• Journal of fish pathology
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• v.26 no.2
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• pp.65-75
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• 2013

Population growth equation of scuticociliate Miamiensis avidus was obtained from the experimental results of in vitro culture condition to estimate the growth rate and carrying capacity from the growth equation. In addition, intraperitoneal infections into olive flounder Paralichthys olivaceus were carried out into 2 different conditions: different concentrations of M. avidus in same water temperature and same concentration of M. avidus in different water temperatures. Olive flounder mortality was threshold dependent with both the temperature and M. avidus density parameters. In this paper, we propose a mathematical model to study M. avidus growth in olive flounder based upon the interactions between parasite and host. The mathematical model was logistic growth differential equation (1.2). The parameters were found with Matlab program through the Levenberge-Marquardt method. In theorem, equilibrium values between the infected fish population and dead population could found. Our equilibrium points were a stable equilibrium and an unstable equilibrium. From the equation (1.6), it was possible to predict the amount of cumulative mortality of olive flounder along with the time after M. avidus infection.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.36 no.11A
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• pp.878-891
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• 2011

We consider the throughput-maximization problem for both the up- and downlink in a wireless network with interference channels. For this purpose, we design an iterative and distributive uplink algorithm based on Lagrangian relaxation. Using the uplink power prices and network duality, we achieve throughput-maximization in the dual downlink that has a symmetric channel and an equal power budget compared to the uplink. The network duality we prove here is a generalized version of previous research [10], [11]. Computational tests show that the performance of the up- and downlink throughput for our algorithms is close to the optimal value for the channel orthogonality factor, ${\theta}{\in}$(0.5, 1]. On the other hand, when the channels are slightly orthogonal (${\theta}{\in}$(0, 0.5]), we observe some throughput degradation in the downlink. We have extended our analysis to the real downlink that has a nonsymmetric channel and an unequal power budget compared to the uplink. It is shown that the modified duality-based approach is thoroughly applied to the real downlink. Considering the complexity of the algorithms in [6] and [18], we conclude that these results are quite encouraging in terms of both performance and practical applicability of the generalized duality theorem.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.11 no.1
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• pp.45-53
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• 1991

The underground structure, which has infinite or semi-infinite boundary conditions, is subjected by body forces and in-situ stresses. It also has stress concentration, which causes material nonlinear behavior, in the vicinity of the excavated surface. In this paper, some methods which can be used to transform domain integrals into boundary integrals are reviewed in order to analyze the effect of the body forces and the in-situ stresses. First, the domain integral of the body force is transformed into boundary integral by using the Galerkin tensor and divergence theorem. Second, it is transformed by writing the domain integral in cylindrical coordinates and using direct integration. The domain integral of the in-situ stress is transformed into boundary integral applying the direct integral method in cylindrical coordinates. The methodology is verified by comparing the results from the boundary element analysis with those of the finite element analysis. Coupling the above boundary elements with finite elements, the nonlinear behavior that occurs locally in the vicinity of the excavation is analyzed and the results are verified. Thus, it is concluded that the domain integrals of body forces and in-situ stresses could be performed effectively by transforming them into the boundary integrals, and the nonlinear behavior can be reasonably analyzed by coupled nonlinear finite element and boundary element method. The result of this research is expected to he used for the analysis of the underground structures in the effective manner.