• 호남수학학술지
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• 제20권1호
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• pp.97-109
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• 1998

We first obtain the white noise calculus to the computation of Feynman integral for a generalized function, according to the definition of Feynman integrals by T. Hida and L. Streit. We next give the translation theorem for Feynman integral of a generalized function.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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• pp.317-324
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• 2008

This paper shows the degree of simultaneous neural network approximation for a target function in $C^r$[-1, 1] and its first derivative. We use the Jackson's theorem for differentiable functions to get a degree of approximation to a target function by algebraic polynomials and trigonometric polynomials. We also make use of the de La Vall$\grave{e}$e Poussin sum to get an approximation order by algebraic polynomials to the derivative of a target function. By showing that the divided difference with a generalized translation network can be arbitrarily closed to algebraic polynomials on [-1, 1], we obtain the degree of simultaneous approximation.

• 대한수학회지
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• 제57권2호
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• pp.451-470
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• 2020

Let C[0, T] denote an analogue of Wiener space, the space of real-valued continuous functions on the interval [0, T]. For a partition 0 = t₀ < t₁ < ⋯ < t_n < t_n+1 = T of [0, T], define X_n : C[0, T] → ℝⁿ⁺¹ by X_n(x) = (x(t₀), x(t₁), …, x(t_n)). In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on C[0, T] with the conditioning function X_n which has a drift and does not contain the present position of paths. As applications of the formula with X_n, we evaluate the Radon-Nikodym derivatives of the functions ∫₀^T[x(t)]^mdλ(t)(m∈ℕ) and [∫₀^Tx(t)dλ(t)]² on C[0, T], where λ is a complex-valued Borel measure on [0, T]. Finally we derive two translation theorems for the Radon-Nikodym derivatives of the functions on C[0, T].

• Korean Journal of Mathematics
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• 제27권1호
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• pp.221-267
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• 2019

In the five first sections of this paper we define and study the hypergeometric transmutation operators $V^W_k$ and $^tV^W_k$ called also the trigonometric Dunkl intertwining operator and its dual corresponding to the Heckman-Opdam's theory on ${\mathbb{R}}^d$. By using these operators we define the hypergeometric translation operator ${\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, and its dual $^t{\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, we express them in terms of the hypergeometric Fourier transform ${\mathcal{H}}^W$, we give their properties and we deduce simple proofs of the Plancherel formula and the Plancherel theorem for the transform ${\mathcal{H}}^W$. We study also the hypergeometric convolution product on W-invariant $L^p_{\mathcal{A}k}$-spaces, and we obtain some interesting results. In the sixth section we consider a some root system of type $BC_d$ (see [17]) of whom the corresponding hypergeometric translation operator is a positive integral operator. By using this positivity we improve the results of the previous sections and we prove others more general results.

• 대한수학회지
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• 제33권4호
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• pp.763-775
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• 1996

For $Q = [0,S] \times [0,T]$ let C(Q) denote Yeh-Wiener space, i.e., the space of all real-valued continuous functions x(s,t) on Q such that x(0,t) = x(s,0) = 0 for every (s,t) in Q. Yeh [10] defined a Gaussian measure $m_y$ on C(Q) (later modified in [13]) such that as a stochastic process ${x(s,t), (s,t) \epsilon Q}$ has mean $E[x(s,t)] = \smallint_{C(Q)} x(s,t)m_y(dx) = 0$ and covariance $E[x(s,t)x(u,\upsilon)] = min{s,u} min{t,\upsilon}$. Let $C_\omega \equiv C[0,T]$ denote the standard Wiener space on [0,T] with Wiener measure $m_\omega$. Yeh [12] introduced the concept of the conditional Wiener integral of F given X, E(F$\mid$X), and for case X(x) = x(T) obtained some very useful results including a Kac-Feynman integral equation.

• 한국학교수학회논문집
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• 제2권1호
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• pp.181-195
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• 1999

This research is produced as a applet of learning materials, and is made with the internet languages HTML, Java, and NamoWebeditor. It contains "Greatest Common Divisor and Least Common Multiple", "Parallel translation of function of second order", "Pythagoras Theorem", which is the current middle school mathmatics textbook for third graders. The keynote of this research is that the students can study individually through logging into the internet on their own computers; the program is made using graphics and animation on order to develop the learners′ interest in mathematics. I hope that this research can supplement our currently insufficient internet educational data.

• 대한수학회지
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• 제41권1호
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• pp.65-76
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• 2004

An entire function $f\;{\in}\;H(\mathbb{C})$ is called universal with respect to translations if for any $g\;{\in}\;H(\mathbb{C}),\;R\;>\;0,\;and\;{\epsilon}\;>\;0$, there is $n\;{\in}\;{\mathbb{N}}$ such that $$\mid$f(z\;+\;n)\;-\;g(z)$\mid$\;<\;{\epsilon}$ whenever $$\mid$z$\mid$\;{\leq}\;R$. Similarly, it is universal with respect to differentiation if for any g, R, and $\epsilon$, there is n such that $$\mid$f^{(n)}(z)\;-\;g(z)$\mid$\;<\;{\epsilon}\;for\;$\mid$z$\mid$\;{\leq}\;R$. In this note, we review G. MacLane's proof of the existence of universal functions with respect to differentiation, and we give a simplified proof of G. D. Birkhoff's theorem showing the existence of universal functions with respect to translation. We also discuss Godefroy and Shapiro's extension of these results to convolution operators as well as some new, related results and problems.

• Kyungpook Mathematical Journal
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• 제61권2호
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• pp.223-237
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• 2021

Let n be a fixed natural number. Menger algebras of rank n can be regarded as a canonical generalization of arbitrary semigroups. This paper is concerned with studying algebraic properties of Menger algebras of rank n by first defining a special class of full n-place functions, the so-called a left translation, which possess necessary and sufficient conditions for an (n + 1)-groupoid to be a Menger algebra of rank n. The isomorphism parts begin with introducing the concept of homomorphisms, and congruences in Menger algebras of rank n. These lead us to establish a quotient structure consisting a nonempty set factored by such congruences together with an operation defined on its equivalence classes. Finally, the fundamental homomorphism theorem and isomorphism theorems for Menger algebras of rank n are given. As a consequence, our results are significant in the study of algebraic theoretical Menger algebras of rank n. Furthermore, we extend the usual notions of ordinary semigroups in a natural way.

• 대한조선학회논문집
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• 제29권4호
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• pp.114-131
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• 1992

본 논문은 2차원 자유표면파문제에서 시간영역해법을 이용하여 2차원 운동문제에 적용할 수 있는 수치해석을 하였다. 경계조건으로는 엄밀한 물체표면 경계조건과 비선형 자유표면경계 조건을 부과했다. 수치해를 구하는데는 코시이론을 이용하여 제2종 프레드흘름 경계적분방정식을 도출하고 이를 이산화시켜 처리하였다. 수치계산을 위해 전영역을 유한한 영역으로 제한하여야 한다. 제한된 영역에서 방사해의 부과를 위해 전영역을 수치해석영역과 외부영역으로 나누고, 외부영역의 해는 그린 제2정리를 이용하고, 선형자유표면조건을 만족하는 과도그린함수를 사용한다. 위의 그린 제2정리를 이용한 식으로 부터 초기조건, 선형 자유표면조건, 무한원방조건을 이용하여 단순화시킨 다음 포텐셜과 유동함수의 관계식으로 치환하면 비선형해와 정합할 수 있는 정합행렬을 구할 수 있다. 본 논문에서 개발한 정합방법을 이용하여 적용할 문제로서 첫째는 주상체가 상하동요, 수평동요하는 경우 계산이고 두번째로는 수면하에서 타원형실린더가 일정속도로 항진할 때 계산을 수행한 결과를 고차스펙트럴방법과 비교하였다.