• Journal of applied mathematics & informatics
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• 제39권3_4호
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• pp.359-380
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• 2021

The main purpose and motivation of this work is to investigate and provide some new results for coefficients derived from eta quotients related to 3. The result of this paper involve some restricted divisor numbers and their convolution sums. Also, our results give relation between the coefficients derived from infinite product, infinite sum and the convolution sum of restricted divisor functions.

• 대한수학회지
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• 제55권1호
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• pp.131-146
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• 2018

We generate ring class fields of imaginary quadratic fields in terms of the special values of certain eta-quotients, which are related to the relative norms of Siegel-Ramachandra invariants. These give us minimal polynomials with relatively small coefficients from which we are able to solve certain quadratic Diophantine equations concerning non-convenient numbers.

• Kyungpook Mathematical Journal
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• 제59권3호
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• pp.377-389
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• 2019

We evaluate the convolution sum $W_{a,b}(n):=\sum_{al+bm=n}{\sigma}(l){\sigma}(m)$ for (a, b) = (1, 28),(4, 7),(2, 7) for all positive integers n. We use a modular form approach. We also re-evaluate the known sums $W_{1,14}(n)$ and $W_{1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of n by the octonary quadratic form $x^2_1+x^2_2+x^2_3+x^2_4+7(x^2_5+x^2_6+x^2_7+x^2_8)$. Finally we express the modular forms ${\Delta}_{4,7}(z)$, ${\Delta}_{4,14,1}(z)$ and ${\Delta}_{4,14,2}(z)$ (given in [10, 14]) as linear combinations of eta quotients.

• 대한수학회논문집
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• 제34권1호
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• pp.27-41
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• 2019

We determine explicit formulas for the number of representations of a positive integer n by quaternary quadratic forms with coefficients 1, 2, 5 or 10. We use a modular forms approach.

• 대한수학회보
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• 제60권1호
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• pp.237-255
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• 2023

In this article, we find bases for the spaces of modular forms $M_2({\Gamma}_0(88),\;({\frac{d}{\cdot}}))$ for d = 1, 8, 44 and 88. We then derive formulas for the number of representations of a positive integer by the diagonal quaternary quadratic forms with coefficients 1, 2, 11 and 22.

• 대한수학회보
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• 제39권2호
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• pp.299-307
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• 2002

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 $\tau\in\eta$.

• 한국염색가공학회지
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• 제19권3호
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• pp.26-36
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• 2007

This study was carried out to treat the aqueous solutions containing reactive dyes(RB19, RR120 and RY179) by the Ozone demand flask method and adsorption process using activated carbon fiber(ACF) which are one of the main pollutants in dye wastewater. Ozone oxidation of three kinds of the reactive dyes was examined to investigate the reactivity of dyes with ozone, competition reaction and ozone utilization on various conditions for single- and multi-solute dye solution. Concentration of dyes was decreased continuously with increasing ozone dosage in the single-solute dye solutions. Competition quotient values were calculated to investigate the preferential oxidation of individual dyes in multi-solute dye solutions. Competition quotients(CQi) and values of the overall utilization efficiency, ${\eta}O_3$, were increased at 40mg/l of ozone dosage in multi-solute dye solutions. ACF(A-15) has much larger specific surface area$(1,584m^2/g-ACF)$ in comparison with granular activated carbon adsorbent (F400, $1,125m^2/g-GAC$), which is commonly used, and most of pores were found to be micropores with pore radius of 2nm and below. It was found that RB19 was most easily adsorbed among the dyes in this study. In the case of PCP (p-chlorophenol) and sucrose, which are single component adsorbate, adsorption capacities of ACF(A-15) were in good agreement with the batch adsorption measurement, and saturation time predicted of ACF columns for these components was also well agreed with practically measured time. But in the case of reactive dyes, which have relatively high molecular weight and aggregated with multi-components, adsorption capacities or saturation time predicted were not agreed with practically measured values.

• 대한수학회지
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• 제60권5호
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• pp.1073-1085
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• 2023

A partition of n is called a t-core partition if none of its hook number is divisible by t. In 2019, Hirschhorn and Sellers [5] obtained a parity result for 3-core partition function a3(n). Motivated by this result, both the authors [8] recently proved that for a non-negative integer α, a3αm(n) is almost always divisible by an arbitrary power of 2 and 3 and at(n) is almost always divisible by an arbitrary power of pji, where j is a fixed positive integer and t = pa11pa22···pamm with primes pi ≥ 5. In this article, by using Hecke eigenform theory, we obtain infinite families of congruences and multiplicative identities for a2(n) and a13(n) modulo 2 which generalizes some results of Das [2].