• Journal of the Chungcheong Mathematical Society
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• v.29 no.3
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• pp.509-514
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• 2016

By constructing a canonical basis for the space $M_k^{\sharp}({\Gamma}_0(N))$ explicitly, we find a basis of the space of cusp forms for ${\Gamma}_0(N)$ consisting of $Poincar{\acute{e}}$ series.

• The Journal of the Korean dental association
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• v.10 no.11
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• pp.731-733
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• 1972

Occurence of different forms of Carabelli's cusp on the maxillary first permanent molar was studied in 282 Korean children. The results were as follows; 1. The Carabelli's cusp was absent in 36.2% of the teeth studied. 2. The percentages of various form appeared Carabelli's tubercle were as follows; a. Pronounced tubercle............9.9% b. Slight tubercle............24.5% c. Groove............25.5% d. Pit............3.9%

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.293-314
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• 2012

We prove that the Siegel modular form of D'Hoker and Phong that gives the chiral superstring measure in degree two is a lift. This gives a fast algorithm for computing its Fourier coefficients. We prove a general lifting from Jacobi cusp forms of half integral index t/2 over the theta group ${\Gamma}_1$(1, 2) to Siegel modular cusp forms over certain subgroups ${\Gamma}^{para}$(t; 1, 2) of paramodular groups. The theta group lift given here is a modification of the Gritsenko lift.

• Communications of the Korean Mathematical Society
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• v.16 no.4
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• pp.585-594
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• 2001

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)

• Honam Mathematical Journal
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• v.41 no.3
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• pp.569-579
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• 2019

Cusp (parabolic) points in the extended modular group ${\bar{\Gamma}}$ are basically the images of infinity under the group elements. This implies that the cusp points of ${\bar{\Gamma}}$ are just rational numbers and the set of cusp points is $Q_{\infty}=Q{\cup}\{{\infty}\}$.The Farey graph F is the graph whose set of vertices is $Q_{\infty}$ and whose edges join each pair of Farey neighbours. Each rational number x has an integer continued fraction expansion (ICF) $x=[b_1,{\cdots},b_n]$. We get a path from ${\infty}$ to x in F as $<{\infty},C_1,{\cdots},C_n>$ for each ICF. In this study, we investigate relationships between Fibonacci numbers, Farey graph, extended modular group and ICF. Also, we give a computer program that computes the geodesics, block forms and matrix represantations.

• Journal of the Korean Mathematical Society
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• v.52 no.2
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• pp.333-347
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• 2015

In this paper, by using the modular forms of weight nk ($2{\leq}n{\in}\mathbb{N}$ and $k{\in}\mathbb{Z}$), we construct a formula which generates modular forms of weight 2nk+4. This formula consist of some known results in [14] and [4]. Moreover, we obtain Fourier expansion of these modular forms. We also give some properties of an operator related to the derivative formula. Finally, by using the function $j_4$, we obtain the Fourier coefficients of modular forms with weight 4.

• Journal of the korean academy of Pediatric Dentistry
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• v.37 no.4
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• pp.488-496
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• 2010

Dens evaginatus is a tooth with cylindrical enamel projection which forms a nodule on occlusal surface. It could be explained as outward overgrowth of inner enamel epithelium or localized hyperplasia of pulpal mesenchymal tissue during tooth development. A problem is that it is likely to be worn out or fractured by mastication ensuing pulpal inflammation. It is occasionally found on the lingual surface of upper anterior teeth as well, called talon cusp. Dens invaginatus is a tooth with deep lingual pit made by invagination of lingual enamel epithelium during tooth development while it is considered normal in terms of size and shape. Radiographically, a part of cervical enamel shows inward growth forming cavity and it is reasonable to say that the base is possibly open to pulpal cavity since they are very close. Talon cusp and dens invaginatus are relatively common abnormality of shape. However it becomes the opposite if the two exist in the same tooth. Once the talon cusp is broken by occlusal force or fissure between cusps is decayed, the complicated structure of canals makes the pulpal treatment difficult. Preventive treatments such as occlusal equilibrium and sealant, and regular oral examination should be preceded and thorough understanding of canal shape, using radiography, is required when pulpal treatment is necessary. This report is about a 9- year-old boy(lower left central incisor), a 8-year-old girl(upper right central incisor), and a 7-year-old boy(upper right central incisor), who have dens invaginatus and talon cusp in the same teeth. The first and the second patients are under pulpal treatments, and the last one is being observed showing no pathologic impressions.

• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.445-464
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• 2013

We describe the one-dimensional and zero-dimensional cusps of the Satake compactification for the paramodular groups in degree two for arbitrary levels. We determine the crossings of the one-dimensional cusps. Applications to computing the dimensions of Siegel modular forms are given.

• Kyungpook Mathematical Journal
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• v.59 no.3
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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.

• Journal of the Korean Mathematical Society
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• v.61 no.5
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• pp.997-1033
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• 2024

We use Borcherds products to give a new construction of the weight 3 paramodular nonlift eigenform f_N for levels N = 61, 73, 79. We classify the congruences of f_N to Gritsenko lifts. We provide techniques that compute eigenvalues to support future modularity applications. Our method does not compute Hecke eigenvalues from Fourier coefficients but instead uses elliptic modular forms, specifically the restrictions of Gritsenko lifts and their images under the slash operator to modular curves.