• Journal of applied mathematics & informatics
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• v.37 no.3_4
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• pp.177-182
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• 2019

This study is about the number expressed and the number not expressed in terms of the sum of consecutive natural numbers with a difference of 2k. Since it is difficult to generalize in cases of onsecutive positive integers with a difference of 2k, the table of cases of 4, 6, 8, 10, and 12 was examined to find the normality and to prove the hypothesis through the results. Generalized guesswork through tables was made to establish and prove the hypothesis of the number of possible and impossible numbers that are to all consecutive natural numbers with a difference of 2k. Finally, it was possible to verify the possibility and impossibility of the sum of consecutive cases of 124 and 2010. It is expected to be investigated the sum of consecutive natural numbers with a difference of 2k + 1, as a future research task.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.37-51
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• 2003

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5${\times}$10$\^$10/, yielding the counting function ,r2,4(5${\times}$10$\^$10/) = l18905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computed B$_4$(5 ${\times}$ 10$\^$10/) = 1.1970s4473029 and B$_4$ = 1.197054 ${\pm}$ 7 ${\times}$ 10$\^$-6/. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5${\times}$10$\^$10/, yielding the counting function $\pi$$\_$2.6/(5${\times}$10$\^$10/) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computed B$\_$6/(5${\times}$10$\^$10/) = 0.93087506039231 and B$\_$6/ = 1.135835 ${\pm}$ 1.2${\times}$10$\^$-6/.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.441-452
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• 2006

We introduce the counting function ${\pi}^*_{2.8}(x)$ of the primes with difference 8 between consecutive primes ($p_n,\;p_{n+l}=p_n+8$) can be approximated by logarithm integral $Li^*_{2.8}$. We calculate the values of ${\pi}^*_{2.8}(x)$ and the sum $C_{2,8}(x)$ of reciprocals of primes with difference 8 between consecutive primes $p_n,\;p_{n+l}=p_n+8$ where x is counted up to $7{\times}10^{10}$. From the results of these calculations. we obtain ${\pi}^*_{2.8}(7{\times}10^{10}$)= 133295081 and $C_{2.8}(7{\times}10^{10}) = 0.3374{\pm}2.6{\times}10^{-4}$.

• Journal for History of Mathematics
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• v.19 no.1
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• pp.1-16
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• 2006

In 1713, J. Bernoulli first discovered the method which one can produce those formulae for the sum $\sum\limits_{\iota=1}^{n}\;\iota^k$ for any natural numbers k ([5],[6]). In this paper, we investigate for the historical background and motivation of the sums of powers of consecutive integers due to J. Bernoulli. Finally, we introduce and discuss for the subjects which are studying related to these areas in the recent.

• Journal of Internet of Things and Convergence
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• v.6 no.4
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• pp.49-57
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• 2020

In this paper, I provided an approximation function Li*2,10(x) using logarithm integral for the counting function π*2,10(x) of consecutive deca primes. Several personal computers and Mathematica were used to validate the approximation function Li*2,10(x). I found the real value of π*2,10(x) and approximate value of Li*2,10(x) for various x ≤ 1011. By the result of theses calculations, most of the error rates are margins of error of 0.005%. Also, I proved that the sum C2,10(∞) of reciprocals of all primes with difference 10 between primes is finite. To find C2,10(∞), I computed the sum C2,10(x) of reciprocals of all consecutive deca primes for various x ≤ 1011 and I estimate that C2,10(∞) probably lies in the range C2,10(∞)=0.4176±2.1×10-3.

• Journal for History of Mathematics
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• v.20 no.4
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• pp.71-84
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• 2007

J. Bernoulli first discovered the method which one can produce those formulae for the sum $S_n(k)=\sum_{{\iota}=1}^n\;{\iota}^k$ for any natural numbers k. After then, there has been increasing interest in Bernoulli and Euler numbers associated with Riemann zeta functions. Recently, Kim have been studied extended q-Bernoulli numbers and q-Euler numbers associated with p-adic q-integral on $\mathbb{Z}_p$, and sums of powers of consecutive q-integers, etc. In this paper, we investigate for the historical background and evolution process of the sums of powers of consecutive q-integers and discuss for Euler zeta functions subjects which are studying related to these areas in the recent.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.66 no.12
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• pp.1772-1781
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• 2017

In this study, the design methodology for alleviating the overfitting problem of Polynomial Neural Networks(PNN) is realized with the aid of two kinds techniques such as L2 regularization and Sum of Squared Coefficients (SSC). The PNN is widely used as a kind of mathematical modeling methods such as the identification of linear system by input/output data and the regression analysis modeling method for prediction problem. PNN is an algorithm that obtains preferred network structure by generating consecutive layers as well as nodes by using a multivariate polynomial subexpression. It has much fewer nodes and more flexible adaptability than existing neural network algorithms. However, such algorithms lead to overfitting problems due to noise sensitivity as well as excessive trainning while generation of successive network layers. To alleviate such overfitting problem and also effectively design its ensuing deep network structure, two techniques are introduced. That is we use the two techniques of both SSC(Sum of Squared Coefficients) and $L_2$ regularization for consecutive generation of each layer's nodes as well as each layer in order to construct the deep PNN structure. The technique of $L_2$ regularization is used for the minimum coefficient estimation by adding penalty term to cost function. $L_2$ regularization is a kind of representative methods of reducing the influence of noise by flattening the solution space and also lessening coefficient size. The technique for the SSC is implemented for the minimization of Sum of Squared Coefficients of polynomial instead of using the square of errors. In the sequel, the overfitting problem of the deep PNN structure is stabilized by the proposed method. This study leads to the possibility of deep network structure design as well as big data processing and also the superiority of the network performance through experiments is shown.

• Proceedings of the Korean Environmental Health Society Conference
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• 2004.12a
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• pp.45-50
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• 2004

Indoor air quality can be affected by indoor sources, ventilation, decay and outdoor levels. Alt hough technologies exist to measure these factors, direct measurements are often difficult. The purpose of this study was to develop an alternative method to characterize indoor environmental factors by multiple indoor and outdoor measurements. Daily indoor and outdoor NO2 concentrations were measured for 30 consecutive days in 28 houses in Brisbane, Australia, and for 21 consecutive days in 37 houses in Seoul, Korea. Using a mass balance model and regression analysis, penetration factor (ventilation rate divided by the sum of ventilation rate and deposition constant) and source strength factor (source strength divided by the sum of ventilation rate and deposition constant) were calculated using multiple indoor and outdoor measurements. Subsequently, the ventilation rate and NO2 source strength were estimated. Geometric means of ventilation rate were 1.44 ACH in Brisbane, assuming a residential NO2 deposition constant of 1.05 hr-1, and 1.36 ACH in Seoul, with the measured residential NO2 deposition constant of 0.94 hr-1. Source strengths of N02 were 15.8 ${\pm}$ 18.2 ${\mu}$g/m3${\cdot}$hr and 44.7 ${\pm}$ 38.1 ${\mu}$g/m3${\cdot}$hr in Brisbane and Seoul, respectively. In conclusion, indoor environmental factors were effectively characterized by this method using multiple indoor and outdoor measurements.

• Proceedings of the Korean Environmental Sciences Society Conference
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• 2003.05a
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• pp.155-160
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• 2003

Indoor air quality can be affected by indoor sources, ventilation, decay and outdoor levels. Although technologies exist to measure these factors, direct measurements are often difficult. The purpose of this study was to develop an alternative method to characterize indoor environmental factors by multiple indoor and outdoor measurements. Daily indoor and outdoor $NO_2$ concentrations were measured for 30 consecutive days in 28 houses in Brisbane, Australia, and for 21 consecutive days in 37 houses in Seoul, Korea. Using a mass balance model and regression analysis, penetration factor (ventilation rate divided by the sum of ventilation rate and deposition constant) and source strength factor (source strength divided by the sum of ventilation rate and deposition constant) were calculated using multiple indoor and outdoor measurements. Subsequently, the ventilation rate and $NO_2$ source strength were estimated. Geometric means of ventilation rate were 1.44 ACH in Brisbane, assuming a residential $NO_2$ deposition constant of 1.05 $hr^{-1}$, and 1.36 ACH in Seoul, with the measured residential $NO_2$ deposition constant of 0.94 $hr^{-1}$. Source strengths of $NO_2$ were 15.8 $\pm$ 18.2 ${\mu}g$/$m^3$.hr and 44.7 $\pm$ 38.1${\mu}g$/$m^3$.hr in Brisbane and Seoul, respectively. In conclusion, indoor environmental factors were effectively characterized by this method using multiple indoor and outdoor measurements.

• Journal of Korean Society for Quality Management
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• v.33 no.3
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• pp.126-134
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• 2005

This paper proposes a selectively cumulative sum(S-CUSUM) control chart for detecting shifts in the process mean. The basic idea of the S-CUSUM chart is to accumulate previous samples selectively in order to increase the sensitivity. The S-CUSUM chart employs a threshold limit to determine whether to accumulate previous samples or not. Consecutive samples with control statistics out of the threshold limit are to be accumulated to calculate a standardized control statistic. If the control statistic falls within the threshold limit, only the next sample is to be used. During the whole sampling process, the S-CUSUM chart produces an 'out-of-control' signal either when any control statistic falls outside the control limit or when L -consecutive control statistics fall outside the threshold limit. The number L is a decision variable and is called a 'control length'. A Markov chain approach is employed to describe the S-CUSUM sampling process. Formulae for the steady state probabilities and the Average Run Length(ARL) during an in-control state are derived in closed forms. Some properties useful for designing statistical parameters are also derived and a statistical design procedure for the S-CUSUM chart is proposed. Comparative studies show that the proposed S-CUSUM chart is uniformly superior to the CUSUM chart or the Exponentially Weighted Moving Average(EWMA) chart with respect to the ARL performance.