• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.73-93
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• 2004

In [10], Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we define the conditional generalized Fourier-Feynman transform and conditional generalized convolution product on function space. We then establish some relationships between the conditional generalized Fourier-Feynman transform and conditional generalized convolution product for functionals on function space that belonging to a Banach algebra.

• The Pure and Applied Mathematics
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• v.16 no.1
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• pp.107-119
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• 2009

In this paper we define a Banach algebra on very general function space induced by a generalized Brownian motion process rather than on Wiener space, but the Banach algebra can be considered as a generalization of Fresnel class defined on Wiener space. We then show that several interesting functions in quantum mechanic are elements of the class.

• Communications of the Korean Mathematical Society
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• v.38 no.2
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• pp.355-364
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• 2023

In this paper, we investigate the sums of the elements in the finite set $\{x^k:1{\leq}x{\leq}{\frac{n}{m}},\;gcd_u(x,n)=1\}$, where k, m and n are positive integers and gcd_u(x, n) is the unitary greatest common divisor of x and n. Moreover, for some cases of k and m, we can give the explicit formulae for the sums involving some well-known arithmetic functions.

• Journal for History of Mathematics
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• v.20 no.3
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• pp.167-180
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• 2007

In this study, we explored the role of analysis in the algebra development. For this, we classified ancient geometric analysis into an analysis by reduction and a Pappusian problematic analysis. this shows that both analyses have the function of reduction. Pappus' analysis consists of four steps; transformation, resolution, construction, demonstration. The transformation, by which conditions of given problem is transformed into other conditions which suggest a problem-solving, seems to be a kind of reduction. Mathematicians created new problems as a result of the reductional function of analysis, and became to see mathematics in the different view. An analytical thinking was a background at the birth of symbolic algebra, the reductional function of analysis played an important role in the development of symbolic algebra.

• The Journal of Korea Institute of Information, Electronics, and Communication Technology
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• v.16 no.1
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• pp.33-40
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• 2023

With the development of supercomputing technology and hardware technology, numerical computation methods are also being advanced. Accordingly, improved weather prediction becomes possible. In this paper, we propose to apply the LAPACK(Linear Algebra PACKage) BLAS(Basic Linear Algebra Subprograms) library to the linear algebraic numerical computation part within the source code to improve the performance of the cumulative parametric code, Unicon(A Unified Convection Scheme), which is included in SCAM(Single-Columns Atmospheric Model, simplified version of CESM(Community Earth System Model)) and performs standby operations. In order to analyze this, an overall execution structure diagram of SCAM was presented and a test was conducted in the relevant execution environment. Compared to the existing source code, the SCOPY function achieved 0.4053% performance improvement, the DSCAL function 0.7812%, and the DDOT function 0.0469%, and all of them showed a 0.8537% performance improvement. This means that the LAPACK BLAS application method, a library for high-density linear algebra operations proposed in this paper, can improve performance without additional hardware intervention in the same CPU environment.

• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.617-629
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• 2019

A term is called linear if each variable which occurs in the term, occurs only once. A hypersubstitution is said to be linear if it maps any operation symbol to a linear term of the same arity. Linear hypersubstitutions have some importance in Theoretical Computer Science since they preserve recognizability [7]. We show that the collection of all linear hypersubstitutions forms a monoid. Linear hypersubstitutions are used to define linear hyperidentities. The set of all linear term operations of a given algebra forms with respect to certain superposition operations a function algebra. Hypersubstitutions define endomorphisms on this function algebra.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.517-523
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• 2007

Let R/I be an Artinian algebra of codimension 3 with Hilbert function H such that $h_{d-1}>h_d=h_{d+1}$. Ahn and Shin showed that A cannot be level if ${\beta}_{1,d+2}(Gin(I))={\beta}_{2,d+2}(Gin(I))$ where Gin(I) is a generic initial ideal of I. We prove that some certain graded Artinian algebra R/I cannot be level if either ${\beta}_{1,d}(I^{lex})={\beta}_{2,d}(I^{lex})+1\;or\;{\beta}_{1,d+1}(I^{lex})={\beta}_{2,d+1}(I^{lex})\;where\;I^{lex}$ is a lex-segment ideal associated to I.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.1
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• pp.73-82
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• 2016

The utility of exponential generating functions is that they are relevant for combinatorial problems involving sets and subsets. Sequences of polynomials play a fundamental role in applied mathematics, such sequences can be described using the exponential generating functions. The actuarial polynomials ${\alpha}^{({\beta})}_n(x)$, n = 0, 1, 2, ${\cdots}$, which was suggested by Toscano, have the following exponential generating function: $${\limits\sum^{\infty}_{n=0}}{\frac{{\alpha}^{({\beta})}_n(x)}{n!}}t^n={\exp}({\beta}t+x(1-e^t))$$. A linear functional on polynomial space can be identified with a formal power series. The set of formal power series is usually given the structure of an algebra under formal addition and multiplication. This algebra structure, the additive part of which agree with the vector space structure on the space of linear functionals, which is transferred from the space of the linear functionals. The algebra so obtained is called the umbral algebra, and the umbral calculus is the study of this algebra. In this paper, we investigate some umbral representations in the actuarial polynomials.

• Bulletin of the Korean Mathematical Society
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• v.37 no.4
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• pp.669-674
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• 2000

We define the terminologies for the special type of functions defined on the field $\mathbb{C}$ of complex numbers and consider the solutions of function equations containing such functions in $\mathbb{C}$.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.397-420
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• 2011

In this paper we study the structure of closed weakly dense ideals in Privalov spaces $N^p$ (1 < p < $\infty$) of holomorphic functions on the disk $\mathbb{D}$ : |z| < 1. The space $N^p$ with the topology given by Stoll's metric [21] becomes an F-algebra. N. Mochizuki [16] proved that a closed ideal in $N^p$ is a principal ideal generated by an inner function. Consequently, a closed subspace E of $N^p$ is invariant under multiplication by z if and only if it has the form $IN^p$ for some inner function I. We prove that if $\cal{M}$ is a closed ideal in $N^p$ that is dense in the weak topology of $N^p$, then $\cal{M}$ is generated by a singular inner function. On the other hand, if $S_{\mu}$ is a singular inner function whose associated singular measure $\mu$ has the modulus of continuity $O(t^{(p-1)/p})$, then we prove that the ideal $S_{\mu}N^p$ is weakly dense in $N^p$. Consequently, for such singular inner function $S_{\mu}$, the quotient space $N^p/S_{\mu}N^p$ is an F-space with trivial dual, and hence $N^p$ does not have the separation property.