Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 807176 | https://doi.org/10.1155/2010/807176

C. S. Ryoo, T. Kim, "A Note on the -Extension of Bernoulli Numbers and Bernoulli Polynomials", Discrete Dynamics in Nature and Society, vol. 2010, Article ID 807176, 11 pages, 2010. https://doi.org/10.1155/2010/807176

A Note on the ( ā„Ž , š‘ž ) -Extension of Bernoulli Numbers and Bernoulli Polynomials

Academic Editor: Leonid Shaikhet
Received13 May 2010
Accepted11 Jul 2010
Published23 Aug 2010

Abstract

We observe the behavior of roots of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the q-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). The main purpose of this paper is also to investigate the zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). Furthermore, we give a table for the zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„).

1. Introduction

Throughout this paper ā„¤,ā„¤š‘,ā„šš‘, and ā„‚š‘ will be denoted by the ring of rational integers, the ring of š‘-adic integers, the field of š‘-adic rational numbers, and the completion of algebraic closure of ā„šš‘, respectively, compare with [1ā€“6]. Let šœˆš‘ be the normalized exponential valuation of ā„‚š‘ with |š‘|š‘=š‘āˆ’šœˆš‘(š‘)=š‘āˆ’1. When one talks of š‘ž-extension, š‘ž is variously considered as an indeterminate, a complex number š‘žāˆˆā„‚, or š‘-adic number š‘žāˆˆā„‚š‘. If š‘žāˆˆā„‚š‘, then we normally assume that |š‘žāˆ’1|š‘<š‘āˆ’1/(š‘āˆ’1), so that š‘žš‘„=exp(š‘„logš‘ž) for |š‘„|š‘ā‰¤1. If š‘žāˆˆā„‚, then we normally assume that |š‘ž|<1. For š‘“āˆˆš‘ˆš·(ā„¤š‘,ā„‚š‘)={š‘“āˆ£š‘“āˆ¶ā„¤š‘ā†’ā„‚š‘isuniformlydiļ¬€erentiablefunction}, the š‘-adic š‘ž-integral (or š‘ž-Volkenborn integration) was defined as š¼š‘ž(ī€œš‘“)=ā„¤š‘š‘“(š‘„)š‘‘šœ‡š‘ž(š‘„)=limš‘ā†’āˆž1ī€ŗš‘š‘ī€»š‘žš‘š‘āˆ’1ī“š‘„=0š‘“(š‘„)š‘žš‘„,(1.1) where [š‘„]š‘ž=(1āˆ’š‘žš‘„)/(1āˆ’š‘ž), compare with [1ā€“8]. Thus, we note that š¼1(š‘“)=limš‘žā†’1š¼š‘ž(ī€œš‘“)=ā„¤š‘š‘“(š‘„)š‘‘šœ‡1(š‘„)=limš‘ā†’āˆž1š‘š‘ī“0ā‰¤š‘„<š‘š‘š‘“(š‘„),comparewith[].1,2,3,4,5,6(1.2) By (1.2), we easily see that š¼1ī€·š‘“1ī€ø=š¼1(š‘“)+š‘“ī…ž(0),comparewith[],1,2,3,4,5,6(1.3) where š‘“1(š‘„)=š‘“(š‘„+1),š‘“ī…ž(0)=(š‘‘/š‘‘š‘„)š‘“(š‘„)|š‘„=0.

In (1.3), if we take š‘“(š‘„)=š‘žā„Žš‘„š‘’š‘„š‘”, then we have ī€œā„¤š‘š‘žā„Žš‘„š‘’š‘„š‘”š‘‘šœ‡1(š‘„)=ā„Žlogš‘ž+š‘”š‘žā„Žš‘’š‘”,āˆ’1comparewith[6],(1.4) for |š‘”|ā‰¤š‘āˆ’1/(š‘āˆ’1),ā„Žāˆˆā„¤.

Recently, many mathematicians have studied Bernoulli numbers and Bernoulli polynomials. Bernoulli polynomials possess many interesting properties and arising in many areas of mathematics and physics. For more studies in this subject we may see references [1ā€“8]. The motivation for this study comes from the following papers. Some interesting analogues of the Bernoulli numbers and polynomials were investigated by Ryoo and Kim [6]. We begin by recalling here definitions of (ā„Ž,š‘ž)-extension of Bernoulli numbers and polynomials as follows.

Definition 1.1 (see [6]). The (ā„Ž,š‘ž)-extension of Bernoulli numbers šµ(ā„Ž)š‘›,š‘ž and polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) is defined by means of the generating functions as follows: š¹š‘ž(ā„Ž)(š‘”)=ā„Žlogš‘ž+š‘”š‘žā„Žš‘’š‘”=āˆ’1āˆžī“š‘›=0šµ(ā„Ž)š‘›,š‘žš‘”š‘›,š¹š‘›!š‘ž(ā„Ž)(š‘”,š‘„)=ā„Žlogš‘ž+š‘”š‘žā„Žš‘’š‘”š‘’āˆ’1š‘„š‘”=āˆžī“š‘›=0šµ(ā„Ž)š‘›,š‘žš‘”(š‘„)š‘›.š‘›!(1.5)
Note that šµ(ā„Ž)š‘›,š‘ž(0)=šµ(ā„Ž)š‘›,š‘ž,limš‘žā†’1šµ(ā„Ž)š‘›,š‘ž(š‘„)=šµš‘›(š‘„), and šµ(0)š‘›,š‘ž(š‘„)=šµš‘›(š‘„), where šµš‘› are the š‘›th Bernoulli numbers.

By (1.4) and (1.5), we have the following Witt formula. For ā„Žāˆˆā„¤,š‘žāˆˆā„‚š‘ with |1āˆ’š‘ž|š‘ā‰¤š‘āˆ’1/(š‘āˆ’1), we have ī€œā„¤š‘š‘žā„Žš‘„š‘„š‘›š‘‘šœ‡1(š‘„)=šµ(ā„Ž)š‘›,š‘ž,ī€œā„¤š‘š‘žā„Žš‘¦(š‘„+š‘¦)š‘›š‘‘šœ‡1(š‘¦)=šµ(ā„Ž)š‘›,š‘ž(š‘„).(1.6) In this paper, we investigate the (ā„Ž,š‘ž)-extension of Bernoulli numbers and Bernoulli polynomials in order to obtain some interesting results and explicit relationships. The aim of this paper to observe an interesting phenomenon of ā€œscatteringā€ of the zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). The outline of this paper is as follows. In Section 2, we study the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). In Section 3, we describe the beautiful zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) using a numerical investigation. Also we display distribution and structure of the zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) by using computer. By using the results of our paper, the readers can observe the regular behaviour of the roots of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). Finally, we carried out computer experiments for demonstrating a remarkably regular structure of the complex roots of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„).

2. Basic Properties for the (ā„Ž,š‘ž)-Extension of Bernoulli Numbers and Bernoulli Polynomials

Let š‘ž be a complex number with |š‘ž|<1 and ā„Žāˆˆā„¤. By the meaning of (1.5), the (ā„Ž,š‘ž)-extension of Bernoulli numbers šµ(ā„Ž)š‘›,š‘ž and Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) is defined by means of the following generating function: š¹š‘ž(ā„Ž)(š‘”)=ā„Žlogš‘ž+š‘”š‘žā„Žš‘’š‘”=āˆ’1āˆžī“š‘›=0šµ(ā„Ž)š‘›,š‘žš‘”š‘›,š‘›!(2.1)š¹š‘ž(ā„Ž)(š‘„,š‘”)=ā„Žlogš‘ž+š‘”š‘žā„Žš‘’š‘”š‘’āˆ’1š‘„š‘”=āˆžī“š‘›=0šµ(ā„Ž)š‘›,š‘žš‘”(š‘„)š‘›,š‘›!(2.2) respectively.

Here is the list of the first (ā„Ž,š‘ž)-extension of Bernoulli numbers šµ(ā„Ž)š‘›,š‘ž. šµ(ā„Ž)0,š‘ž=ā„Žlogš‘žāˆ’1+š‘žā„Ž,šµ1,š‘ž=1āˆ’1+š‘žā„Žāˆ’ā„Žš‘žā„Žlogš‘žī€·āˆ’1+š‘žā„Žī€ø2,šµ(ā„Ž)2,š‘žā„Ž=āˆ’2š‘žā„Žī€·āˆ’1+š‘žā„Žī€ø2āˆ’ā„Žš‘žā„Žlogš‘žī€·āˆ’1+š‘žā„Žī€ø2+2ā„Žš‘ž2ā„Žlogš‘žī€·āˆ’1+š‘žā„Žī€ø3,šµ(ā„Ž)3,š‘žā„Ž=āˆ’3š‘žā„Žī€·āˆ’1+š‘žā„Žī€ø2+6š‘ž2ā„Žī€·āˆ’1+š‘žā„Žī€ø3āˆ’ā„Žš‘žā„Žlogš‘žī€·āˆ’1+š‘žā„Žī€ø2+6ā„Žš‘ž2ā„Žlogš‘žī€·āˆ’1+š‘žā„Žī€ø3āˆ’6ā„Žš‘ž3ā„Žlogš‘žī€·āˆ’1+š‘žā„Žī€ø4,ā€¦,(2.3)

because šœ•š¹šœ•š‘„š‘ž(ā„Ž)(š‘„,š‘”)=š‘”š¹š‘ž(ā„Ž)(š‘„,š‘”)=āˆžī“š‘›=0š‘‘šµš‘‘š‘„(ā„Ž)š‘›,š‘ž(š‘”š‘„)š‘›,š‘›!(2.4) it follows the important relation š‘‘šµš‘‘š‘„(ā„Ž)š‘›,š‘ž(š‘„)=š‘›šµ(ā„Ž)š‘›āˆ’1,š‘ž(š‘„).(2.5) We have the integral formula as follows: ī€œš‘š‘Žšµ(ā„Ž)š‘›āˆ’1,š‘ž1(š‘„)š‘‘š‘„=š‘›ī‚€šµ(ā„Ž)š‘›,š‘ž(š‘)āˆ’šµ(ā„Ž)š‘›,š‘žī‚.(š‘Ž)(2.6)

Here is the list of the first (ā„Ž,š‘ž)-extension of Bernoulli Polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). šµ(ā„Ž)0,š‘ž=ā„Žlogš‘žāˆ’1+š‘žā„Ž,šµ(ā„Ž)1,š‘ž=1ī€·āˆ’1+š‘žā„Žī€øāˆ’ā„Žš‘žā„Žlogš‘žī€·āˆ’1+š‘žā„Žī€ø2+ā„Žš‘„logš‘žī€·āˆ’1+š‘žā„Žī€ø,šµ(ā„Ž)2,š‘ž=2š‘žā„Žī€·āˆ’1+š‘žā„Žī€ø2+2š‘„ī€·āˆ’1+š‘žā„Žī€ø+2ā„Žš‘ž2ā„Žlogš‘žī€·āˆ’1+š‘žā„Žī€ø3āˆ’ā„Žš‘žā„Žlogš‘žī€·āˆ’1+š‘žā„Žī€ø2āˆ’2ā„Žš‘žā„Žš‘„logš‘žī€·āˆ’1+š‘žā„Žī€ø2+ā„Žš‘„2logš‘žī€·āˆ’1+š‘žā„Žī€ø,ā€¦.(2.7)

Since āˆžī“š‘™=0šµ(ā„Ž)š‘™,š‘ž(š‘”š‘„+š‘¦)š‘™=š‘™!ā„Žlogš‘ž+š‘”š‘žā„Žš‘’š‘”š‘’āˆ’1(š‘„+š‘¦)š‘”=āˆžī“š‘›=0šµ(ā„Ž)š‘›,š‘ž(š‘”š‘„)š‘›š‘›!āˆžī“š‘š=0š‘¦š‘šš‘”š‘š=š‘š!āˆžī“š‘™=0īƒ©š‘™ī“š‘›=0šµ(ā„Ž)š‘›,š‘žš‘”(š‘„)š‘›š‘¦š‘›!š‘™āˆ’š‘›š‘”š‘™āˆ’š‘›īƒŖ=(š‘™āˆ’š‘›)!āˆžī“š‘™=0īƒ©š‘™ī“š‘›=0ī‚µš‘™š‘›ī‚¶šµ(ā„Ž)š‘›,š‘ž(š‘„)š‘¦š‘™āˆ’š‘›īƒŖš‘”š‘™,š‘™!(2.8) we have the following theorem.

Theorem 2.1. (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) satisfies the following relation: šµ(ā„Ž)š‘™,š‘ž(š‘„+š‘¦)=š‘™ī“š‘›=0ī‚µš‘™š‘›ī‚¶šµ(ā„Ž)š‘›,š‘ž(š‘„)š‘¦š‘™āˆ’š‘›.(2.9)
From (2.2), we can derive the following equality: āˆžī“š‘›=0ī‚€š‘žā„Žšµ(ā„Ž)š‘›,š‘ž(š‘„+1)āˆ’šµ(ā„Ž)š‘›,š‘ž(ī‚š‘”š‘„)š‘›=š‘›!āˆžī“š‘›=0ī€·š‘„š‘›ā„Žlogš‘ž+š‘›š‘„š‘›āˆ’1ī€øš‘”š‘›.š‘›!(2.10)
Hence, we obtain the following difference equation.

Theorem 2.2. For any positive integer š‘›, we obtain š‘žā„Žšµ(ā„Ž)š‘›,š‘ž(š‘„+1)āˆ’šµ(ā„Ž)š‘›,š‘ž(š‘„)=š‘„š‘›ā„Žlogš‘ž+š‘›š‘„š‘›āˆ’1.(2.11)

3. Distribution and Structure of the Zeros

In this section, we assume that š‘žāˆˆā„‚, with |š‘ž|<1. We observed the behavior of real roots of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). We display the shapes of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµš‘›,š‘ž(š‘„) and we investigate the zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). For š‘›=1,ā€¦,10, we can draw a plot of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„), respectively. This shows the ten plots combined into one. We display the shape of šµ(ā„Ž)š‘›,š‘ž(š‘„),āˆ’1ā‰¤š‘„ā‰¤1,š‘ž=1/2 (Figure 1). We investigate the beautiful zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) by using a computer. We plot the zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(3)š‘›,š‘ž(š‘„) for š‘›=15,20,25,30 and š‘„āˆˆā„‚ (Figure 2).

Our numerical results for approximate solutions of real zeros of šµ(ā„Ž)š‘›,1/2(š‘„) are displayed (Tables 1 and 2).


degree š‘› ā„Ž = 3 ā„Ž = 5
real zeros complex zeros real zeros complex zeros

1 1 0 1 0
2 2 0 2 0
3 3 0 3 0
4 4 0 2 2
5 32 32
6 4 2 4 2
7 5 2 5 2
8 6 2 4 4
9 3 6 5 4
10 4 6 6 4
11 5 6 5 6
12 6 6 4 8


degree š‘› š‘„

1 0 . 3 3 8 0 4 1 2 0 4
2 0 . 0 7 7 2 7 7 1 0 8 , 0 . 5 9 8 8 0 5 3 0 1
3 āˆ’ 0 . 0 7 9 0 7 8 4 0 1 , 0 . 2 7 5 4 8 8 1 7 , 0 . 8 1 7 7 1 3 8 4
4 āˆ’ 0 . 1 4 8 5 9 6 4 9 , 0 . 0 0 6 3 9 1 6 4 , 0 . 4 8 7 9 8 3 8 7 , 1 . 0 0 6 3 8 5 7 9
5 0 . 1 9 2 2 8 0 4 , 0 . 6 9 6 0 3 2 0 , 1 . 1 6 9 9 1 8 3 2
6 āˆ’ 0 . 1 0 1 9 3 3 5 , 0 . 3 9 0 8 6 0 1 , 0 . 8 9 7 2 2 9 4 , 1 . 3 1 0 0 2 5 3
7 āˆ’ 0 . 3 0 0 7 3 7 , 0 . 0 9 4 1 3 2 , 0 . 5 9 2 0 6 0 , 1 . 0 9 4 1 6 7 , 1 . 4 2 5 0 4 4 4
ā‹® ā‹®

We plot the zeros of (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) for š‘›=30,š‘ž=1/2,ā„Ž=5,7,9,11, and š‘„āˆˆā„‚ (Figure 3). We plot the zeros of (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) for š‘›=30,š‘ž=9/10,99/100, and š‘„āˆˆā„‚ (Figure 4).

We observe a remarkably regular structure of the complex roots of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). We hope to verify a remarkably regular structure of the complex roots of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) (Table 1). This numerical investigation is especially exciting because we can obtain an interesting phenomenon of scattering of the zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). These results are used not only in pure mathematics and applied mathematics, but also used in mathematical physics and other areas. Next, we calculated an approximate solution satisfying the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). The results are given in Table 2.

Stacks of zeros of šµ(ā„Ž)š‘›,š‘ž(š‘„) for š‘ž=1/3,1ā‰¤š‘›ā‰¤30 from a 3D structure are presented (in Figure 5).

Figure 6 presents the distribution of real zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(3)š‘›,š‘ž(š‘„) for š‘ž=1/2,1ā‰¤š‘›ā‰¤30.

Figure 7 presents the distribution of real zeros of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(3)š‘›,š‘ž(š‘„) for š‘ž=9/10,1ā‰¤š‘›ā‰¤30.

Figure 8 presents the distribution of real zeros of the Bernoulli polynomials šµš‘›(š‘„) for 1ā‰¤š‘›ā‰¤30.

4. Direction for Further Research

In [7], we observed the behavior of complex roots of the Bernoulli polynomials šµš‘›(š‘„), using numerical investigation. Prove that šµš‘›(š‘„),š‘„āˆˆā„‚, has Re(š‘„)=1/2 reflection symmetry in addition to the usual Im(š‘„)=0 reflection symmetry analytic complex functions. The obvious corollary is that the zeros of šµš‘›(š‘„) will also inherit these symmetries. Ifšµš‘›ī€·š‘„0ī€ø=0,thenšµš‘›ī€·1āˆ’š‘„0ī€ø=0=šµš‘›ī€·š‘„āˆ—0ī€ø=šµš‘›ī€·1āˆ’š‘„āˆ—0ī€ø,(4.1) where āˆ— denotes complex conjugation (see [7]). Finally, we shall consider the more general problems. Prove that šµš‘›(š‘„)=0 has š‘› distinct solutions. If šµ2š‘›+1(š‘„) has Re(š‘„)=1/2 and Im(š‘„)=0 reflection symmetries, and 2š‘›+1 nondegenerate zeros, then 2š‘› of the distinct zeros will satisfy (4.1). If the remaining one zero is to satisfy (4.1) too, it must reflect into itself, and therefore it must lie at 1/2, the center of the structure of the zeros, that is, šµš‘›ī‚€12ī‚=0āˆ€oddš‘›.(4.2) Prove that šµ(ā„Ž)š‘›,š‘ž(š‘„)=0 has š‘› distinct solutions, that is, all the zeros are nondegenerate. Find the numbers of complex zeros š¶šµ(ā„Ž)š‘›,š‘ž(š‘„) of šµ(ā„Ž)š‘›,š‘ž(š‘„),Im(š‘„)ā‰ 0. Since š‘› is the degree of the polynomial šµ(ā„Ž)š‘›,š‘ž(š‘„), the number of real zeros š‘…šµ(ā„Ž)š‘›,š‘ž(š‘„) lying on the real plane Im(š‘„)=0 is then š‘…šµ(ā„Ž)š‘›,š‘ž(š‘„)=š‘›āˆ’š¶šµ(ā„Ž)š‘›,š‘ž(š‘„), where š¶šµ(ā„Ž)š‘›,š‘ž(š‘„) denotes complex zeros. See Table 1 for tabulated values of š‘…šµ(ā„Ž)š‘›,š‘ž(š‘„) and š¶šµ(ā„Ž)š‘›,š‘ž(š‘„). Find the equation of envelope curves bounding the real zeros lying on the plane. We prove that šµ(ā„Ž)š‘›,š‘ž(š‘„),š‘„āˆˆā„‚, has Im(š‘„)=0 reflection symmetry analytic complex functions. If šµ(ā„Ž)š‘›,š‘ž(š‘„)=0, then šµ(ā„Ž)š‘›,š‘ž(š‘„āˆ—)=0, where āˆ— denotes complex conjugate (see Figures 2, 3, and 4). Observe that the structure of the zeros of the Bernoulli polynomials šµš‘›(š‘„) resembles the structure of the zeros of the š‘ž-Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) as š‘žā†’1 (see Figures 3, 4, and 5). In order to study the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„), we must understand the structure of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„). Therefore, using computer, a realistic study for the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) plays an important part. The author has no doubt that investigation along this line will lead to a new approach employing numerical method in the field of research of the (ā„Ž,š‘ž)-extension of Bernoulli polynomials šµ(ā„Ž)š‘›,š‘ž(š‘„) to appear in mathematics and physics. For related topics, the interested reader is referred to [3ā€“8].

References

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Copyright © 2010 C. S. Ryoo and T. Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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