• Communications of the Korean Mathematical Society
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• v.31 no.4
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• pp.779-790
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• 2016

Using the extended generalized integral transform given by Luo et al. [6], we introduce some new generalized integral transforms to investigate such their (potentially) useful properties as inversion formulas and Parseval-Goldstein type relations. Classical integral transforms including (for example) Laplace, Stieltjes, and Widder-Potential transforms are seen to follow as special cases of the newly-introduced integral transforms.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.657-668
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• 2016

Given a partition ${\lambda}=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_l)$ of a positive integer n, let Tab(${\lambda}$, k) be the set of all tabloids of shape ${\lambda}$ whose weights range over the set of all k-compositions of n and ${\mathcal{OP}}^k_{\lambda}_{rev}$ the set of all ordered partitions into k blocks of the multiset $\{1^{{\lambda}_l}2^{{\lambda}_{l-1}}{\cdots}l^{{\lambda}_1}\}$. In [2], Butler introduced an inversion-like statistic on Tab(${\lambda}$, k) to show that the rank-selected $M{\ddot{o}}bius$ invariant arising from the subgroup lattice of a finite abelian p-group of type ${\lambda}$ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(${\lambda}$, k) and ${\mathcal{OP}}^k_{\hat{\lambda}}$. When k = 2, we also introduce a major-like statistic on Tab(${\lambda}$, 2) and study its connection to the inversion statistic due to Butler.

• Honam Mathematical Journal
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• v.27 no.2
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• pp.183-194
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• 2005

A permutation graph is the graph of inversions in a permutation. Here we determine whether a given labelled graph is a permutation graph or not and when a graph is a permutation graph we find the associated permutation. We also characterize all the 2-regular permutation graphs.

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.755-772
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• 2004

We show that the Fourier-Laplace series of a distribution on the sphere is uniformly Cesaro-summable to zero on a neighborhood of a point if and only if this point does not belong to the support of the distribution. Similar results on the ball and on the real projective space are also proved.

• Communications of the Korean Mathematical Society
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• v.16 no.3
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• pp.371-383
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• 2001

In this survey article, we shall introduce the applications of the theory of reproducing kernels to inverse problems. At the same time, we shall present some operator versions of our fundamental general theory for linear transforms in the framework of Hilbert spaces.

• Journal of the Korean Mathematical Society
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• v.43 no.3
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• pp.553-561
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• 2006

In this paper, we give an alternative proof that the number of doubly alternating Baxter permutations is Catalan.

• Communications of Mathematical Education
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• v.28 no.2
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• pp.219-234
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• 2014

Music and mathematics have a lot of structural similarities. Major and minor triads used importantly in music are in a relationship of inversion in which the sequence of the intervals is reversed, which is equivalent to reflection in mathematics. Geometrical expressions help understand structures in music as well as mathematics, and a diagram that shows tonal relationships in music is called Tonnetz. Relationships of reflection between major and minor triads can easily be understood by using Tonnetz, and also, transpositions can be expressed in translation. This study looks into existing Tonnetz and introduces S-Tonnetz newly formed by a mathematical principle.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.417-423
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• 2019

In the paper, by virtue of the $Fa{\grave{a}}$ di Bruno formula, properties of the Bell polynomials of the second kind, and the Lah inversion formula, the author simplifies coefficients in a family of ordinary differential equations related to the generating function of the Mittag-Leffler polynomials.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1773-1781
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• 2018

The Radon transform introduced by J. Radon in 1917 is the integral transform which is widely applicable to tomography. Here we study the discrete version of the Radon transform. More precisely, when $C({\mathbb{Z}}^n_p)$ is the set of complex-valued functions on ${\mathbb{Z}}^n_p$. We completely determine the subset of $C({\mathbb{Z}}^n_p)$ whose elements can be recovered from its Radon transform on ${\mathbb{Z}}^n_p$.

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.999-1014
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• 2003

For a fixed function h we deal with a class of convolution transforms $f\;{\rightarrow}\;f\;*\;h$, where $(f\;*\;h)(x)\;=\frac{1}{2x}\;{\int_{{R_{+}}^2}}^{e^1{\frac{1}{2}}(x\frac{u^2+y^2}{uy}+\frac{yu}{x})}\;f(u)h(y)dudy,\;x\;\in\;R_{+}$ as integral operators $L_p(R_{+};xdx)\;\rightarrow\;L_r(R_{+};xdx),\;p,\;r\;{\geq}\;1$. The Young type inequality is proved. Boundedness properties are investigated. Certain examples of these operators are considered and inversion formulas in $L_2(R_{+};xdx)$ are obtained.