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INTEGRABILITY AS VALUES OF CUSP FORMS IN IMAGINARY QUADRATIC
Kim, Dae-Yeoul,Koo, Ja-Kyung Korean Mathematical Society 2001 대한수학회논문집 Vol.16 No.4
Let η be the complex upper half plane, let h($\tau$) be a cusp form, and let $\tau$ be an imaginary quadratic in η. If h($\tau$)$\in$$\Omega$( $g_{2}$($\tau$)$^{m}$ $g_{3}$ ($\tau$)$^{ι}$with $\Omega$the field of algebraic numbers and m. l positive integers, then we show that h($\tau$) is integral over the ring Q[h/$\tau$/n/)…h($\tau$+n-1/n)] (No Abstract.see full/text)
ARITHMETIC OF INFINITE PRODUCTS AND ROGERS-RAMANUJAN CONTINUED FRACTIONS
Kim, Dae-Yeoul,Koo, Ja-Kyung,Simsek, Yilmaz Korean Mathematical Society 2007 대한수학회논문집 Vol.22 No.3
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}$. We find a lot of algebraic properties derived from theta functions, and by using this we explore some new algebraic numbers from Rogers-Ramanujan continued fractions.
ONVOLUTION SUM Σ<sub>m</sub><sub><</sub><sub>n/8</sub>σ<sub>1</sub>(2m)σ<sub>1</sub>(n-8m)
Kim, Dae-Yeoul,Kim, Ae-Ran,Park, Hwa-Sin The Honam Mathematical Society 2012 호남수학학술지 Vol.34 No.1
In this paper, we present the convolution sum ${\sum}_{m<n/8}{\sigma}_1(2m){\sigma}_1(n-8m)$ evaluated for all $n{\in}\mathbb{N}$.
ALGEBRAIC NUMBERS, TRANSCENDENTAL NUMBERS AND ELLIPTIC CURVES DERIVED FROM INFINITE PRODUCTS
Kim, Dae-Yeoul,Koo, Ja-Kyung Korean Mathematical Society 2003 대한수학회지 Vol.40 No.6
Let k be an imaginary quadratic field, η the complex upper half plane, and let $\tau$ $\in$ η $textsc{k}$, p = $e^{{\pi}i{\tau}}$. In this article, using the infinite product formulas for g2 and g3, we prove that values of certain infinite products are transcendental whenever $\tau$ are imaginary quadratic. And we derive analogous results of Berndt-Chan-Zhang ([4]). Also we find the values of (equation omitted) when we know j($\tau$). And we construct an elliptic curve E : $y^2$ = $x^3$ + 3 $x^2$ + {3-(j/256)}x + 1 with j = j($\tau$) $\neq$ 0 and P = (equation omitted) $\in$ E.
Combinatoric Convolution Sums Containing σ and ð of the Form 2m p
( Dae Yeoul Kim ),( Joong Soo Park ) 호남수학회 2014 호남수학학술지 Vol.36 No.3
In this paper, we study combinatoric convolution sums of divisor functions and get values of this sum when n=2m p. We find that the value of this convolution sum is represented by a sum of powers of 2 and Bernoulli or Euler number.
A REMARK OF EISENSTEIN SERIES AND THETA SERIES
Kim, Dae-Yeoul,Koo, Ja-Kyung Korean Mathematical Society 2002 대한수학회보 Vol.39 No.2
As a by-product of [5], we produce algebraic integers of certain values of quotients of Eisenstein series. And we consider the relation of $\Theta_3(0,\tau)$ and $\Theta_3(0,\tau^n)$. That is,we show that $$\mid$\Theta_3(0,\tau^n)$\mid$=$\mid$\Theta_3(0,\tau)$\mid$,\bigtriangleup(0,\tau)=\bigtriangleup(0,\tau^n)$ and $J(\tau)=J(\tau^n)$ for some $\tau\in\eta$.
REMARK OF P<sub>i,k</sub> ON ELLIPTIC CURVES AND APPLICATION FOR MANCHESTER CODING
Kim, Dae-Yeoul,Kim, Min-Soo The Honam Mathematical Society 2011 호남수학학술지 Vol.33 No.2
Greg([Greg]) considered that $$N_k= \sum\limits_{i=1}^k(-1)^{i+1}P_{i,k}(p)N_1^i$$ where the $P_{i,k}$'s were polynomials with positive integer coefficients. In this paper, we will give the equations for $\sum\limits{P_{i,k}$ modulo 3. Using this, if we send a information for elliptic curve to sender, we can make a new checksum method for Manchester coding in IEEE 802.3 or IEEE 802.4.
Convolution Sum ∑(m<n/8)/σ1(2m)σ1(n-8m)
( Dae Yeoul Kim ),( Ae Ran Kim ),( Hwa Sin Park ) 호남수학회 2012 호남수학학술지 Vol.34 No.1
In this paper, we present the convolution sum ∑(m<n/8)/ σ1(2m)σ1(n-8m) evaluated for all n □ N.
ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES II
Kim, Dae-Yeoul,Koo, Ja-Kyung Korean Mathematical Society 2008 대한수학회지 Vol.45 No.5
Let k be an imaginary quadratic field, ${\eta}$ the complex upper half plane, and let ${\tau}{\in}{\eta}{\cap}k,\;q=e^{{\pi}{i}{\tau}}$. For n, t ${\in}{\mathbb{Z}}^+$ with $1{\leq}t{\leq}n-1$, set n=${\delta}{\cdot}2^{\iota}$(${\delta}$=2, 3, 5, 7, 9, 13, 15) with ${\iota}{\geq}0$ integer. Then we show that $q{\frac}{n}{12}-{\frac}{t}{2}+{\frac}{t^2}{2n}{\prod}_{m=1}^{\infty}(1-q^{nm-t})(1-q^{{nm}-(n-t)})$ are algebraic numbers.