• Kyungpook Mathematical Journal
• /
• 제49권3호
• /
• pp.435-450
• /
• 2009

On page 366 of his lost notebook [15], Ramanujan recorded a cubic continued fraction and several theorems analogous to Rogers-Ramanujan's continued fractions. In this paper, we derive several general formulas for explicit evaluations of Ramanujan's cubic continued fraction, several reciprocity theorems, two formulas connecting V (q) and V ($q^3$) and also establish some explicit evaluations using the values of remarkable product of theta-function.

• 대한수학회보
• /
• 제51권1호
• /
• pp.157-171
• /
• 2014

We show that the recurrence rates of Laurent series about continued fractions almost surely coincide with their pointwise dimensions of the Haar measure. Moreover, let $E_{{\alpha},{\beta}}$ be the set of points with lower and upper recurrence rates ${\alpha},{\beta}$, ($0{\leq}{\alpha}{\leq}{\beta}{\leq}{\infty}$), we prove that all the sets $E_{{\alpha},{\beta}}$, are of full Hausdorff dimension. Then the recurrence sets $E_{{\alpha},{\beta}}$ have constant multifractal spectra.

• 대한수학회보
• /
• 제59권3호
• /
• pp.697-707
• /
• 2022

It is well known that the continued fraction expansion of ${\sqrt{d}}$ has the form $[{\alpha}_0,\;{\bar{{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},\;2_{{\alpha}_0}}]$ and ${\alpha}_1,\;{\ldots},\;{\alpha}_{l-1}$ is a palindromic sequence of positive integers. For a given positive integer l and a palindromic sequence of positive integers ${\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},$ we define the set $S(l;{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1})\;:=\;\{d{\in}{\mathbb{Z}}{\mid}d>0,\;{\sqrt{d}}=[{\alpha}_0,\;{\bar{{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},\;2_{{\alpha}_0}}],\;where\;{\alpha}_0={\lfloor}{\sqrt{d}}{\rfloor}\}.$ In this paper, we completely determine when $S(l;{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1})$ is not empty in the case that l is 4, 5, 6, or 7. We also give similar results for $(1+{\sqrt{d}})/2.$ For the case that l is 4, 5, or 6, we explicitly describe the fundamental units of the real quadratic field ${\mathbb{Q}}({\sqrt{d}}).$ Finally, we apply our results to the Mordell conjecture for the fundamental units of ${\mathbb{Q}}({\sqrt{d}}).$

• 충청수학회지
• /
• 제25권3호
• /
• pp.475-489
• /
• 2012

A plateau in a Motzkin path is a sequence of three steps: an up step, a horizontal step, then a down step. We find three different forms for the bivariate generating function for plateaus in Motzkin paths, then generalize to longer plateaus. We conclude by describing a further generalization: a continued fraction form from which one can easily derive new multivariate generating functions for various kinds of path statistics. Several examples of generating functions are given using this technique.

• 대한수학회보
• /
• 제50권4호
• /
• pp.1315-1328
• /
• 2013

We first derive several modular equations of degree 5 and present their concise proofs based on algebraic computations. We then establish explicit relations and formulas for some parameterizations for the theta functions ${\varphi}$ and ${\psi}$ by using the derived modular equations. In addition, we find specific values of the parameterizations and evaluate some numerical values of the Rogers-Ramanujan continued fraction.

• 대한수학회보
• /
• 제50권4호
• /
• pp.1221-1233
• /
• 2013

We first derive some modular equations of degrees 3 and 9 and present their concise proofs based on algebraic computations. We then use these modular equations to establish explicit relations and formulas for the parameterizations for the theta functions ${\varphi}$ and ${\psi}$ In addition, we find specific values of the parameterizations to evaluate some numerical values of the cubic continued fraction.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제6권2호
• /
• pp.1-8
• /
• 2002

Let $T_{\phi}$ be a generalized Gauss transformation and $[a_1,\;a_2,\;{\cdots}]_{T_{\phi}}$ be a symbolic representation of $x{\in}[0,\;1)$ induced by $T_{\phi}$, i.e., generalized continued fraction expansion induced by $T_{\phi}$. It is shown that the distribution of relative frequency of [$k_1,\;{\cdots},\;k_n$] in $[a_1,\;a_2,\;{\cdots}]_{T_p}$ satisfies Central Limit Theorem where $k_i{\in}{\mathbb{N}}$ for $1{\leq}i{\leq}n$.

• 대한수학회보
• /
• 제52권5호
• /
• pp.1559-1568
• /
• 2015

The relative class number $H_d(f)$ of a real quadratic field $K=\mathbb{Q}(\sqrt{m})$ of discriminant d is the ratio of class numbers of $O_f$ and $O_K$, where $O_K$ denotes the ring of integers of K and $O_f$ is the order of conductor f given by $\mathbb{Z}+fO_K$. In a recent paper of A. Furness and E. A. Parker the relative class number of $\mathbb{Q}(\sqrt{m})$ has been investigated using continued fraction in the special case when $(\sqrt{m})$ has a diagonal form. Here, we extend their result and show that there exists a conductor f of relative class number 1 when the continued fraction of $(\sqrt{m})$ is non-diagonal of period 4 or 5. We also show that there exist infinitely many real quadratic fields with any power of 2 as relative class number if there are infinitely many Mersenne primes.

• Asian-Australasian Journal of Animal Sciences
• /
• 제15권7호
• /
• pp.1022-1025
• /
• 2002

Effect of chemical stress of daily administration of glucocorticoid (dexamethasone @0.125 mg./calf/day) injections on plasma zinc levels, Zn status of body tissues and its distribution in sub cellular fractions, was studied in neonatal buffalo calves. Daily i/m injections of dexamethasone, starting at the completion of 1 week of age and continued till 8th week, led to a significant decline in plasma Zn concentration from 3rd week onwards, which then persisted throughout the rest of the experimental period. In control group, liver had the highest concentration of zinc, followed by heart, muscle, spleen, kidney and testis. In all these tissues, cytosolic fractions had the highest (>60%) zinc levels followed by nuclear, mitochondrial and microsomal fractions. In dexamethasone treated calves, there was a significant increase in the Zn uptake by the tissues of liver and muscle. This increase in zinc concentration was observed in all the sub cellular fractions of liver and muscle, however about 80% of this increase was in cytosolic fraction. It was concluded that glucocorticoid-induced stress caused increase in Zn levels of liver/muscles and decrease in blood plasma zinc, thus indicating a redistribution of Zn in body.

• Structural Engineering and Mechanics
• /
• 제66권5호
• /
• pp.649-663
• /
• 2018

The scaled boundary finite element method is extended to evaluate the dynamic stress intensity factors and T-stress with a numerical procedure based on the improved continued-fraction. The improved continued-fraction approach for the dynamic stiffness matrix is introduced to represent the inertial effect at high frequencies, which leads to numerically better conditioned matrices. After separating the singular stress term from other high order terms, the internal displacements can be obtained by numerical integration and no mesh refinement is needed around the crack tip. The condition numbers of coefficient matrix of the improved method are much smaller than that of the original method, which shows that the improved algorithm can obtain well-conditioned coefficient matrices, and the efficiency of the solution process and its stability can be significantly improved. Several numerical examples are presented to demonstrate the increased robustness and efficiency of the proposed method in both homogeneous and bimaterial crack problems.