• Journal of the Korean Mathematical Society
• /
• v.61 no.4
• /
• pp.779-796
• /
• 2024

In this paper, under the ABC-conjecture, we show that there exist infinitely many real quadratic fields with odd period of minimal type.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.4
• /
• pp.875-883
• /
• 2012

Let $\mathbb{F}_q$ be a finite field with q elements and $\mathbb{F}_q((X^{-1}))$ be the field of all formal Laurent series with coefficients lying in $\mathbb{F}_q$. This paper concerns with the size of the set of points $x{\in}\mathbb{F}_q((X^{-1}))$ with their partial quotients $A_n(x)$ both lying in a given subset $\mathbb{B}$ of polynomials in $\mathbb{F}_q[X]$ ($\mathbb{F}_q[X]$ denotes the ring of polynomials with coefficients in $\mathbb{F}_q$) and deg $A_n(x)$ tends to infinity at least with some given speed. Write $E_{\mathbb{B}}=\{x:A_n(x){\in}\mathbb{B},\;deg\;A_n(x){\rightarrow}{\infty}\;as\;n{\rightarrow}{\infty}\}$. It was shown in [8] that the Hausdorff dimension of $E_{\mathbb{B}}$ is inf{$s:{\sum}_{b{\in}\mathbb{B}}(q^{-2\;deg\;b})^s$ < ${\infty}$}. In this note, we will show that the above result is sharp. Moreover, we also attempt to give conditions under which the above dimensional formula still valid if we require the given speed of deg $A_n(x)$ tends to infinity.

• Honam Mathematical Journal
• /
• v.31 no.4
• /
• pp.463-478
• /
• 2009

Let k be an imaginary quadratic field, ℌ the complex upper half plane, and let ${\tau}{\in}k{\cap}$ℌ, q = exp(${\pi}i{\tau}$). And let n, t be positive integers with $1{\leq}t{\leq}n-1$. Then $q^{{\frac{n}{12}}-{\frac{t}{2}}+{\frac{t^2}{2n}}}{\prod}^{\infty}_{m=1}(1-q^{nm-t})(1-q^{nm-(n-t)})$ is an algebraic number [10]. As a generalization of this result, we find several infinite series and products giving algebraic numbers using Ramanujan's $_{1{\psi}1}$ summation. These are also related to Rogers-Ramanujan continued fractions.

• The Pure and Applied Mathematics
• /
• v.31 no.2
• /
• pp.189-200
• /
• 2024

We derive modular equations of degree 3 to find corresponding theta-function identities. We use them to find some new evaluations of $G(e^{-{\pi}{\sqrt{n}}})$ and $G(-e^{-{\pi}{\sqrt{n}}})$ for $n\,=\,\frac{25}{3{\cdot}4^{m-1}}$ and $\frac{4^{1-m}}{3{\cdot}25}$, where m = 0, 1, 2.

• Asian-Australasian Journal of Animal Sciences
• /
• v.12 no.3
• /
• pp.366-370
• /
• 1999

Samples of whole plant, leaf and stem of Akbar, Neelum, UM-81 and lZ-31 cultivars of maize fodder harvested up to 14 weeks at different growth stages were drawn and analysed for dry matter contents and various cell wall constituents such as NDF, ADF, hemicellulose, cellulose, lignin, cutin and silica. The dry matter contents of whole maize plant, leaf and stem increased significantly (p<0.01) with advancing plant age. Maximum dry matter was found in the leaf fraction of the plant. The cell wall components continued to increase significantly (p<0.001) in whole maize plant and its morphological fractions as the age advanced. Maximum values for NDF, ADF, cellulose and lignin were observed in stem followed by whole plant and leaf, whereas hemicellulose, cutin and silica contents were higher in leaf fraction of the plant. The cultivars were observed to have some effects on chemical composition of all plant fraction. The results indicated that maturity had a much greater effect on the concentration of all the structural components than did the cultivars. It was concluded that maize fodder should be cut preferably between 8th to 9th week of age (flowering stage) to obtain more nutritious and digestible feed for livestock. Among the maize cultivars, Neelum proved to be the best, due to its higher dry matter contents and lower lignin concentration.

• Communications of the Korean Mathematical Society
• /
• v.24 no.4
• /
• pp.629-640
• /
• 2009

The aim of the present paper is to establish certain relations for partial mock theta functions and mock theta functions of order eight with other partial mock theta functions and mock theta functions of order two, six, eight and ten respectively.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.2
• /
• pp.535-547
• /
• 2019

In this paper, by using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and $Peth{\ddot{o}}$, we find all generalized Fibonacci numbers which are Pell numbers. This paper continues a previous work that searched for Pell numbers in the Fibonacci sequence.

• East Asian mathematical journal
• /
• v.40 no.1
• /
• pp.25-36
• /
• 2024

Full univariate moment problems have been studied using continued fractions, orthogonal polynomials, spectral measures, and so on. On the other hand, the truncated moment problem has been mainly studied through confirming the existence of the extension of the moment matrix. A few articles on the multivariate moment problem implicitly presented about some results of this note, but we would like to rearrange the important results for the existence of a representing measure of a moment sequence. In addition, new techniques with orthogonal polynomials will be introduced to expand the means of studying truncated moment problems.

• Journal of the Korean Mathematical Society
• /
• v.44 no.1
• /
• pp.55-107
• /
• 2007

Let k be an imaginary quadratic field, h the complex upper half plane, and let $\tau{\in}h{\cap}k,\;q=e^{{\pi}i\tau}$. In this article, we obtain algebraic numbers from the 130 identities of Rogers-Ramanujan continued fractions investigated in [28] and [29] by using Berndt's idea ([3]). Using this, we get special transcendental numbers. For example, $\frac{q^{1/8}}{1}+\frac{-q}{1+q}+\frac{-q^2}{1+q^2}+\cdots$ ([1]) is transcendental.

• Journal of the Korean Mathematical Society
• /
• v.54 no.3
• /
• pp.835-845
• /
• 2017

The $GCF_{\epsilon}$ expansion is a new class of continued fractions induced by the transformation $T_{\epsilon}:(0, 1]{\rightarrow}(0, 1]$: $T_{\epsilon}(x)={\frac{-1+(k+1)x}{1+k-k{\epsilon}x}}$ for $x{\in}(1/(k+1),1/k]$. Under the algorithm $T_{\epsilon}$, every $x{\in}(0,1]$ corresponds to an increasing digits sequences $\{k_n,n{\geq}1\}$. Their basic properties, including the ergodic properties, law of large number and central limit theorem have been discussed in [4], [5] and [7]. In this paper, we study the large deviation for the $GCF_{\epsilon}$ expansion and show that: $\{{\frac{1}{n}}{\log}k_n,n{\geq}1\}$ satisfies the different large deviation principles when the parameter ${\epsilon}$ changes in [-1, 1], which generalizes a result of L. J. Zhu [9] who considered a case when ${\epsilon}(k){\equiv}0$ (i.e., Engel series).