• Journal for History of Mathematics
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• v.25 no.3
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• pp.15-27
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• 2012

This paper aims to examine Alan Turing's life at the centenary of his birth and to discuss a philosophical implication of his work by concentrating on halting theorem particularly. Turing negatively solved Hilbert's decision problem by proving impossibility of solving halting problem. In this paper I claim that the impossibility implies limits of reason, and accordingly that the marginality in cognition and/or in action should be recognized.

• Journal of KIISE
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• v.44 no.4
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• pp.399-410
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• 2017

We present a system that automatically annotates unverified Web sentences with information from credible sources. The system turns to neural theorem proving for an annotating task for cancer related Wikipedia data (1,486 propositions) with Korean National Cancer Center data (19,304 propositions). By switching the recursive module in a neural theorem prover to a word embedding module, we overcome the fundamental problem of tremendous learning time. Within the identical environment, the original neural theorem prover was estimated to spend 233.9 days of learning time. In contrast, the revised neural theorem prover took only 102.1 minutes of learning time. We demonstrated that a neural theorem prover, which encodes a proposition in a tensor, includes a classic theorem prover for exact match and enables end-to-end differentiable logic for analogous words.

• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.85-91
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• 1995

For a locally compact Hausdorff space, we denote by $C_0(X)$ the Banach space of all continuous complex valued functions defined on X which vanish at infinity, equipped with the usual sup norm. In case X is compact, we write C(X) instead of $C_0(X)$. A well-known Banach-Stone theorem states that the existence of an isometry between the function spaces $C_0(X)$ and $C_0(Y)$ implies X and Y are homemorphic. D. Amir [1] and M. Cambern [2] independently generalized this theorem by proving that if $C_0(X)$ and $C_0(Y)$ are isomorphic under an isomorphism T satisfying $\left\$\mid$ T \right\$\mid$ \left\$\mid$ T^1 \right\$\mid$ < 2$, then X and Y must also be homeomorphic.

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.667-680
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• 2004

We generalize a theorem of W. Benz by proving the following result: Let $H_{\theta}$ be a half space of a real Hilbert space with dimension $\geq$ 3 and let Y be a real normed space which is strictly convex. If a distance $\rho$ > 0 is contractive and another distance N$\rho$ (N $\geq$ 2) is extensive by a mapping f : $H_{\theta}$ \longrightarrow Y, then the restriction f│$_{\theta}$ $H_{+}$$\rho$/2// is an isometry, where $H_{\theta}$＋$\rho$/2/ is also a half space which is a proper subset of $H_{\theta}$. Applying the above result, we also generalize a classical theorem of Beckman and Quarles.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.221-233
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• 2016

The present paper proposes a new monotone iteration method for existence as well as approximation of the coupled solutions for a coupled periodic boundary value problem of first order ordinary nonlinear differential equations. A new hybrid coupled fixed point theorem involving the Dhage iteration principle is proved in a partially ordered normed linear space and applied to the coupled periodic boundary value problems for proving the main existence and approximation results of this paper. An algorithm for the coupled solutions is developed and it is shown that the sequences of successive approximations defined in a certain way converge monotonically to the coupled solutions of the related differential equations under some suitable mixed hybrid conditions. A numerical example is also indicated to illustrate the abstract theory developed in the paper.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.281-292
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• 2005

In this paper a nonlinear alternative of Leray-Schauder type is proved in a Banach algebra involving three operators and it is further applied to a functional nonlinear integral equation of mixed type $$x(t)=k(t,x({\mu}(t)))+[f(t,x({\theta}(t)))]\(q(t)+{\int}_0{^{\sigma}^{(t)}}v(t,s)g(s,x({\eta}\(s)))ds\)$$ for proving the existence results in Banach algebras under generalized Lipschitz and $Carath{\acute{e}}odory$ conditions.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.399-422
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• 1998

A useful method of proving the finite decidability of an equationally definable class V of algebras (i.e., variety) is to prove the decidability of the class of finite directly indecomposable members of V. The validity of this method relies on the well-known result of Feferman-Vaught: if a class K of first-order structures is decidable, then so is the class {$\prod$$_{i}$＜n/ $A_{i}$│ $A_{i}$ $\in$ X (i < n), n $\in$ $\omega$}. In this paper we show that the converse of this does not necessarily hold.d.d.

• Proceedings of the Korean Information Science Society Conference
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• 2003.10a
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• pp.58-60
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• 2003

원숭이와 바나나 문제는 논리적 추론에 의한 문제 해결 과정을 설명하기 위해 사용되는 대표적 예제이다. 본 논문에서는 전통적인 접근 방식과는 달리, 이 문제를 그래프 탈색의 그것으로 이해한 후 DNA 컴퓨팅에 근거한 너비 우선 탐색(breadth-first search, BFS)을 통해 해들을 발견하고자 한다. 그 결과, 최적해 (optimal solution)를 포함한 최소 4개 이상의 다양한 해들이 실제 DNA 생화학 실험을 통해 확인되었다.

• The Pure and Applied Mathematics
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• v.10 no.3
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• pp.193-198
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• 2003

Let X and Y be n-dimensional Euclidean spaces with $n\;{\geq}\;3$. In this paper, we generalize a classical theorem of Bookman and Quarles by proving that if a mapping, from a half space of X into Y, preserves a distance $\rho$, then the restriction of f to a subset of the half space is an isometry.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.52 no.1
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• pp.51-54
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• 2003

The z-transform method is a basic mathematical tool in analyzing and designing digital signal processing systems for discrete input and output signals. There are may cases where the output signal is in the form of a discrete convolution sum of an input function and a designed digital processing algorithm function. It is well known that the z-transform of the convolution sum becomes the product of the two z-transforms of the input function and the digital processing function, whose proofs require the causality of the digital signal processing function in the almost all the available references. However, not all of the convolution sum functions are based on the causality. Many digital signal processing systems such as image processing system may depend not on the time information but on the spatial information, which has nothing to do with causality requirement. Thus, the application of the causality-based z-transform theorem on the convolution sum cannot be used without difficulty in this case. This paper proves the z-transform theorem on the discrete convolution sum without causality requirement, and make it possible for the theorem to be used in analysis and desing for any cases.