• Journal of the Korean Mathematical Society
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• v.51 no.2
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• pp.383-402
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• 2014

We expose a pattern for establishing Friedman-Weiermann style independence results according to which there are thresholds of provability of some parameterized variants of well-partial-ordering. For this purpose, we investigate an ordinal notation system for ${\vartheta}{\Omega}^{\omega}$, the small Veblen ordinal, which is the proof-theoretic ordinal of the theory $({\prod}{\frac{1}{2}}-BI)_0$. We also show that it sometimes suffices to prove the independence w.r.t. PA in order to obtain the same kind of independence results w.r.t. a stronger theory such as $({\prod}{\frac{1}{2}}-BI)_0$.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.179-196
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• 2018

In this paper we extend the two q-additions with powers in the umbrae, define a q-multinomial-coefficient, which implies a vector version of the q-binomial theorem, and an arbitrary complex power of a JHC power series is shown to be equivalent to a special case of the first q-Lauricella function. We then present several q-analogues of hypergeometric integral formulas from the two books by Exton and the paper by Choi and Rathie. We also find multiple q-analogues of hypergeometric integral formulas from the recent paper by Kim. Finally, we prove several multiple q-hypergeometric integral formulas emanating from a paper by Koschmieder, which are special cases of more general formulas by Exton.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.4
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• pp.289-296
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• 2023

When G is an abelian group, we use the notation M(G, n) to denote the Moore space. The space X is the wedge product space of Moore spaces, given by X = M(ℤ_q, n+ 2) ∨ M(ℤ_q, n+ 1) ∨ M(ℤ_q, n). We determine the self-homotopy classes group [X, X] and the self-homotopy equivalence group 𝓔(X). We investigate the subgroups of [M_j , M_k] consisting of homotopy classes of maps that induce the trivial homomorphism up to (n + 2)-homotopy groups for j ≠ k. Using these results, we calculate the subgroup 𝓔^dim_#(X) of 𝓔(X) in which all elements induce the identity homomorphism up to (n + 2)-homotopy groups of X.

• Journal of The Korean Association of Information Education
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• v.7 no.1
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• pp.11-26
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• 2003

Although many mathematical teaching/learning materials are already developed in the web, diverse utilization of this materials such as calculation, searching, or reusing of expressions are limited since the expression is actually a figure. To cope with this, MathML which describing mathematical notation was developed. In the paper, we proposed the methods of developing teaching materials using MathML, making learning assistance tools which utilize MathML, and applying MathML to information exchange community for Mathematics courses in ICT environment. Using MathML to develop a teaching material makes easy to correct and reuse the mathematical notations conveniently. Furthermore, learning assistance tools made by placing MathML help teachers reorganize and utilize these materials in the classroom as well as enhancing the connection between mathematical notations and concepts. The web-board that can make a use the mathematical notations using MathML enables the teachers and students to exchange information actively. It also helps to fulfill different types of teaching using ICT such as "discussion on the web".

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.687-700
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• 1994

Throughout this paper we will let H denote the complete Heyting algebra ($H, \vee, \wedge, *$) with order reversing involution *. 0 and 1 denote the supermum and the infimum of $\emptyset$, respectively. Given any set X, any element of $H^X$ is called H-fuzzy set (or, simply f.set) in X and will be denoted by small Greek letters, such as $\mu, \nu, \rho, \sigma$. $H^X$ inherits a structure of H with order reversing involution in natural way, by definding $\vee, \wedge, *$ pointwise (sam notations of H are usual). If $f$ is a map from a set X to a set Y and $\mu \in H^Y$, then $f^{-1}(\mu)$ is the f.set in X defined by f^{-1}(\mu)(x) = \mu(f(x))$. Also for $\sigma \in H^X, f(\sigma)$ is the f.set in Y defined by $f(\sigma)(y) = sup{\sigma(x) : f(x) = y}$ ([4]). A preorder R on a set X is reflexive and transitive relation on X, the pair (X,R) is called preordered set. A map $f$ from a preordered set (X, R) to another one (Y,T) is said to be preorder preserving (inverting) if for $x,y \in X, xRy$ implies $f(x)T f(y) (resp. f(y)Tf(x))$. For the terminology and notation, we refer to [10, 11, 13] for category theory and [7] for H-fuzzy semitopogenous spaces.

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.237-242
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• 1994

One of the most well-known geometric lattices is a partition lattice. Every upper interval of a partition lattice is a partition lattice. The whitney numbers of a partition lattices are the Stirling numbers, and the characteristic polynomial is a falling factorial. The set of partitions with a single non-trivial block containing a fixed element is a Boolean sublattice of modular elements, so the partition lattice is supersolvable in the sense of Stanley [6]. In this paper, we rephrase four results due to Heller[1] and Murty [4] in terms of matroids and give several characterizations of partition lattices. Our notation and terminology follow those in [8,9]. To clarify our terminology, let G, be a finte geometric lattice. If S is the set of points (or rank-one flats) in G, the lattice structure of G induces the structure of a (combinatorial) geometry, also denoted by G, on S. The size vertical bar G vertical bar of the geometry G is the number of points in G. Let T be subset of S. The deletion of T from G is the geometry on the point set S/T obtained by restricting G to the subset S/T. The contraction G/T of G by T is the geometry induced by the geometric lattice [cl(T), over ^1] on the set S' of all flats in G covering cl(T). (Here, cl(T) is the closure of T, and over ^ 1 is the maximum of the lattice G.) Thus, by definition, the contraction of a geometry is always a geometry. A geometry which can be obtained from G by deletions and contractions is called a minor of G.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1031-1054
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• 2010

Let $\mathbb{Z}^n$ be the Cartesian product of the set of integers $\mathbb{Z}$ and let ($\mathbb{Z}$, T) and ($\mathbb{Z}^n$, $T^n$) be the Khalimsky line topology on $\mathbb{Z}$ and the Khalimsky product topology on $\mathbb{Z}^n$, respectively. Then for a set $X\;{\subset}\;\mathbb{Z}^n$, consider the subspace (X, $T^n_X$) induced from ($\mathbb{Z}^n$, $T^n$). Considering a k-adjacency on (X, $T^n_X$), we call it a (computer topological) space with k-adjacency and use the notation (X, k, $T^n_X$) := $X_{n,k}$. In this paper we introduce the notions of KD-($k_0$, $k_1$)-homotopy equivalence and KD-k-deformation retract and investigate a classification of (computer topological) spaces $X_{n,k}$ in terms of a KD-($k_0$, $k_1$)-homotopy equivalence.

• Journal for History of Mathematics
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• v.25 no.1
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• pp.15-27
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• 2012

Thomas Harriot(1560-1621) introduced a simplified notation for algebra. His fundamental research on the theory of equations was far ahead of that time. He invented certain symbols which are used today. Harriot treated all answers to solve equations equally whether positive or negative, real or imaginary. He did outstanding work on the solution of equations, recognizing negative roots and complex roots in a way that makes his solutions look like a present day solution. Since he published no mathematical work in his lifetime, his achievements were not recognized in mathematical history and mathematics education. In this paper, by comparing his works with Viete and Descartes those are mathematicians in the same age, I show his achievements in mathematics.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.15 no.3
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• pp.161-175
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• 2011

An n-string tangle is a three dimensional ball with n strings properly embedded in it. A tangle model of a DNA-protein complex is first introduced by C. Ernst and D. Sumners in 1980's. They assumed the protein bound DNA as strings and the protein as a three dimensional ball. By using a tangle analysis, one can predict the topology of DNA within the complex. S.Kim and I. Darcy developed the biologically reasonable 4-string tangle equations and decided a solution tangle, called R-standard tangle. The author discussed more about the simple solution tangles of the equations and found a generalized R-standard tangle solution.

• Proceedings of the KSR Conference
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• 1999.05a
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• pp.283-290
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• 1999

Recently, the ${\mu}$-processor based-controlled systems instead of conventional relays circuitry are widely used to industrial applications, and also those technology is available to railway signalings which are safety-critical systems. However, the safety and reliability of software for those systems are harder to demonstrate than in traditional relays circuitry because the faults or errors can not be analyzed and predicted to those systems. So, the safety problems are crucial more and more in ${\mu}$-processor based-controlled system. In this paper, the Grafcet language, the graphical and mathematical form, is used to obtain the high-level safety and reliability of software control logic. The general description for Grafcet notation are provided. And some partial of interlocking logic are formally modeled and simulated by Grafcet language and graphical windows.