• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1423-1433
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• 2019

Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.

• East Asian mathematical journal
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• v.32 no.2
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• pp.253-289
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• 2016

This study attempted to figure out new teaching method of mathematics teaching-learning by applying Polya's 4-level strategy to mild disability students at the H Special-education high school where the research works for. In particular, epilogue and suggestion, which Polya stressed were selected and reconstructed for mild disability students. Prior test and post test were carried by putting the Polya's problem solving strategy as independent variable, and problem solving ability as dependent variable. As a result, by continual use of Polya's program in mathematics teaching course, it suggested necessary strategies to solve mathematics problems for mild disability students and was proven that Polya's heuristic training was of help to improve problem solving in mathematics.

• The Korean Journal of Applied Statistics
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• v.18 no.3
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• pp.607-615
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• 2005

Recently the interval estimation of a binomial proportion is revisited in various literatures. This is mainly due to the erratic behavior of the coverage probability of the will-known Wald confidence interval. Various alternatives have been proposed. Among them, Agresti-Coull confidence interval has been recommended by Brown et al. (2001) with other confidence intervals for large sample, say n $\ge$ 40. On the other hand, a noninformative Bayesian approach called Polya posterior often produces statistics with good frequentist's properties. In this note, an interval estimator is developed using weighted Polya posterior. The resulting interval estimator is essentially the Agresti-Coull confidence interval with some improved features. It is shown that the weighted Polys posterior produce an effective interval estimator for small sample size and a severely skewed binomial distribution.

• Journal of Elementary Mathematics Education in Korea
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• v.17 no.1
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• pp.87-103
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• 2013

This study was investigated to examine the effects of writing activities based on Polya's Problem Solving Stages on Learning Accomplishment and Attitudes. A total of 54 students were selected from two Grade 6 classes of P Elementary School in G City to form an experimental group(n=27) and a control group (n=27). The experimental group was applied to a class which was creating writing activities according to Polya's Problem Solving Stages to problem solving and inquiry activities. The control group was taught by the traditional method to the same activities. The five questions for each area were selected as a descriptive assessment of the second semester of Grade 5 in the area of the Academic Achievement pre-test, developed by the G Education and Science Research. The post-test was selected by a descriptive assessment of the content of the first semester in Grade 6. The same questions were posed for both the pre-test and the post-test of the Mathematical Attitudes assessment. We examined the pre-test at the beginning of the school term, then the students were re-examined after one semester, using the same questions as the pre-test. This research showed that there was a meaningful difference in Learning Accomplishment as a result of T-test in the 5% level of significance. Secondly, there was a meaningful difference in the Mathematical Attitudes as a result of T-tests. It shows that writing activities based on Polya's Problem Solving Stages have an influence on improving Learning Accomplishment and Attitudes.

• East Asian mathematical journal
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• v.34 no.4
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• pp.359-369
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• 2018

This note, we first show that the famous Carleman's inequality can be improved if we find a positive sequence $\{c_n\}$ such that $c_n{\sum\limits_{j=n}^{\infty}}{\frac{1}{j\(\prod_{k=1}^{j}ck\)^{\frac{1}{j}}}}$ < e. Then we list a lot of known results in the literature improving Carleman's inequality by this method. These results can be a good source to a further research for interested students. We next consider about similar improvement of Polya-Knopp's inequality, which is a continuous version of Carleman's inequality. We show by a manner parallel to the case of Carleman's inequality that Polya-Knopp's inequality can be improved if we find a positive function c(x) such that $c(x){\int}_{x}^{\infty}\frac{1}{t\;{\exp}\(\frac{1}{t}{\int}_{0}^{t}{\ln}\;c(s)\;ds\)}dt$ < e. But there are no known results improving Polya-Knopp's inequality by this method. Suggesting to find a new method, we lastly show that there is no nice continuous function c(x) that satisfies the inequality.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.267-271
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• 2011

Let 0 < s < 1. For $f^{\ast}$ representing the symmetric radial decreasing rearrangement of f, we build up a fractional version of Polya-$Szeg{\ddot{o}}$ inequality: $${\int}_{\mathbb{R}^n}{\mid}(-\Delta)^{s/2}f^{\ast}(x){\mid}^2dx{\leq}{\int}_{\mathbb{R}^n}{\mid}(-\Delta)^{s/2}f(x){\mid}^2dx$$.

• Journal of Korean Institute of Industrial Engineers
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• v.28 no.4
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• pp.351-359
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• 2002

In this paper, stochastically dependent single and sequential acceptance sampling plans are dealt when the process follows a Polya process model. A Monte-Cairo algorithm is used to find the acceptance and rejection probabilities of a lot. The number of defectives for the test to be accepted and rejected in a probability ratio sequential test can be found by using these probabilities. The formula to measure performance of these sampling plans is developed. Type I and II error probabilities are estimated by simulation. Dependent multiple acceptance sampling plans can be derived by extending the sequential sampling procedure. In numerical examples, single and sequential sampling plans of a Polya dependent process are examined and the characteristics are compared according to the change of the dependency factor.

• The Korean Journal of Applied Statistics
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• v.19 no.1
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• pp.171-181
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• 2006

The Wald confidence interval has been considered as a standard method for the difference of proportions. However, the erratic behavior of the coverage probability of the Wald confidence interval is recognized in various literatures. Various alternatives have been proposed. Among them, Agresti-Caffo confidence interval has gained the reputation because of its simplicity and fairly good performance in terms of coverage probability. It is known however, that the Agresti-Caffo confidence interval is conservative. In this note, a confidence interval is developed using the weighted Polya posterior which was employed to obtain a confidence interval for the binomial proportion in Lee(2005). The resulting confidence interval is simple and effective in various respects such as the closeness of the average coverage probability to the nominal confidence level, the average expected length and the mean absolute error of the coverage probability. Practically it can be used for the interval estimation of the difference of proportions for any sample sizes and parameter values.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.657-667
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• 1996

This paper is concerned with the zeros of successive derivatives of real entire functions. In order to state the background to our results, as well as the results themselves, let us introduce some terminologies.

• Communications of the Korean Mathematical Society
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• v.16 no.1
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• pp.67-73
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• 2001

We classify self-converse oriented trees into two types, namely, bicentral self-converse oriented trees and central ones, according to their centers and characterize these tow types. Using characterizations and Polya enumeration theorem, we derive the ordinary generating function for self-converse oriented trees.