• Journal of the Korean Mathematical Society
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• v.43 no.3
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• pp.461-489
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• 2006

In this article, we establish the existence theorem for measure-valued Dyson series and show that it satisfies the Volterra-type integral equation. Furthermore, we prove the stability theorems for measure-valued Dyson series.

• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.137-145
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• 2004

In [13], we developed a theory of complex Finsler manifolds to investigate the global geometry of complex Finsler manifolds. There we proved a version of Bonnet-Myers' theorem for complex Finsler manifolds with a certain condition on the Finsler metric which is a generalization of the Kahler condition for the Hermitian metric. In this paper, we show that if the holomorphic sectional curvature of M is ${\geq}\;c^2\;>\;0$, then M is simply connected. This is a generalization of the Synge's theorem in the Riemannian geometry and the Tsukamoto's theorem for Kahler manifolds. The main point of the proof lies in how we can circumvent the convex neighborhood theorem in the Riemannian geometry. A second variation formula of arc length for complex Finsler manifolds is also derived.

• Communications of Mathematical Education
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• v.22 no.3
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• pp.311-335
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• 2008

The importance of students' precise understanding of mathematical definitions, formulas, and theorems can not be underestimated. In this survey research, we attempted to evaluate students' understanding of the concepts of five topics -limit, continuity and intermediate theorem, derivative, application of derivative and integral. On the basis of the research result, this paper suggests that we need to 1) be more inventive and speculative in making test problems, 2) explain the examples and counter-examples more concretely, 3) stress and repeat the basic concepts on the stage of introducing new concepts, 4) develop more effective problems for the measure of students' understanding of mathematical concepts, 5) use developed problems in actual teaching.

• Communications of the Korean Mathematical Society
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• v.30 no.2
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• pp.103-108
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• 2015

Fox [2] presented an interesting identity for $_pF_q$ which is expressed in terms of a finite summation of $_pF_q$'s whose involved numerator and denominator parameters are different from those in the starting one. Moreover Fox [2] found a very interesting and general summation formula for $_3F_2(1/2)$ as a special case of his above-mentioned general identity with the help of Kummer's second summation theorem for $_2F_1(1/2)$. Here, in this paper, we show how two general summation formulas for $$_3F_2\[\array{\hspace{110}{\alpha},{\beta},{\gamma};\\{\alpha}-m,\;\frac{1}{2}({\beta}+{\gamma}+i+1);}\;{\frac{1}{2}}\]$$, m being a nonnegative integer and i any integer, can be easily established by suitably specializing the above-mentioned Fox's general identity with, here, the aid of generalizations of Kummer's second summation theorem for $_2F_1(1/2)$ obtained recently by Rakha and Rathie [7]. Several known results are also seen to be certain special cases of our main identities.

• Honam Mathematical Journal
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• v.34 no.3
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• pp.311-326
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• 2012

The purpose of this paper is to introduce and investigate two new classes of generalized Bernoulli and Apostol-Bernoulli polynomials based on the definition given recently by the authors [29]. In particular, we obtain a new addition formula for the new class of the generalized Bernoulli polynomials. We also give an extension and some analogues of the Srivastava-Pint$\acute{e}$r addition theorem [28] for both classes. Finally, by making use of the new adition formula, we exhibit several interesting relationships between generalized Bernoulli polynomials and other polynomials or special functions.

• Journal for History of Mathematics
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• v.27 no.2
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• pp.127-138
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• 2014

In this paper we investigate how Newton discovered the generalized binomial theorem. Newton's binomial theorem, or binomial series can be found in Calculus text books as a special case of Taylor series. It can also be understood as a formal power series which was first conceived by Euler if convergence does not matter much. Discovered before Taylor or Euler, Newton's binomial theorem must have a good explanation of its birth and validity. Newton learned the interpolation method from Wallis' famous book ${\ll}$Arithmetica Infinitorum${\gg}$ and employed it to get the theorem. The interpolation method, which Wallis devised to find the areas under a family of curves, was by nature arithmetrical but not geometrical. Newton himself used the method as a way of finding areas under curves. He noticed certain patterns hidden in the integer binomial sequence appeared in relation with curves and then applied them to rationals, finally obtained the generalized binomial sequence and the generalized binomial theorem.

• Journal of the Korean Society of Clothing and Textiles
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• v.27 no.6
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• pp.665-674
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• 2003

The purpose of this study was to develop a grading increment at armhole area by apparel CAD(Computer Aided Design) system. In developing a grading increment at armhole area, we analyzed ease values of armhole area in bodice and sleeve by manual drafting patterns of five sizes. We suggested grading increments applied Pythagorean theorem to development the grading increment of the armhole of sleeve. The results and discussions of this study were as follows: 1. In drafting each size, the ease values were not identical. It was difficult to draft perfectly the same armhole line shape between sizes. 2. According to our developed grading increments applied Pythagorean theorem, the ease values were identical between sizes and difference of the armhole length between sizes was also identical. 3. The grading formulas were made out for apparel CAD system. Once grading increment or formula is set in the computer, it can be easily altered to various clothing items at any time. The efficiency of grading work will be also improved and grading time will be reduced.

• East Asian mathematical journal
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• v.25 no.4
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• pp.481-486
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• 2009

The first object of this note is to show that a summation formula due to Padmanabham for three-variables hypergeometric function $X_8$ introduced by Exton can be proved in a different (from Padmanabham's and his observation) yet, in a sense, conventional method, which has been employed in obtaining a variety of identities associated with hypergeometric series. The second purpose is to point out that one of two seemingly new hypergeometric identities due to Exton was already recorded and the other one is easily derivable from the first one. A corrected and a little more compact form of a general transform involving hypergeometric functions due to Exton is also given.

• Kyungpook Mathematical Journal
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• v.43 no.4
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• pp.489-494
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• 2003

In this paper, we first establish an interesting theorem involving the $\bar{H}$-function introduced by Inayat-Hussain ([7], [8]). The convergence and existence condition, basic properties of this function were given by Buschman and Srivastava ([2]). Next, we obtain certain new integrals and an expansion formula by the application of our theorem. On account of the most general nature of the functions involved herein, our main findings are capable of yielding a large number of new, interesting and useful integrals, expansion formulae involving simple special functions and polynomials as their special cases. A known special case of our main theorem in also given ([11]).

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.449-454
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• 2007

InIn this paper we consider generalizations of the classical Dixon's theorem and the classical Whipple's theorem on the sum of a $_3F_2$. The results are derived with the help of generalized Watson's theorem obtained earlier by Mitra. A large number of results contiguous to Dixon's and Whipple's theorems obtained earlier by Lavoie, Grondin and Rathie, and Lavoie, Grondin, Rathie and Arora follow special cases of our main findings.