• 대한수학회보
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• 제56권4호
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• pp.1077-1097
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• 2019

Let M be a finitely generated module over a regular local ring (R, n). We will fix an n-stable filtration for M and show that the minimal free resolution of M can be obtained from any filtered free resolution of M by zero and negative consecutive cancellations. This result is analogous to [10, Theorem 3.1] in the more general context of filtered free resolutions. Taking advantage of this generality, we will study resolutions obtained by the mapping cone technique and find a sufficient condition for the minimality of such resolutions. Next, we give another application in the graded setting. We show that for a monomial order ${\sigma}$, Betti numbers of I are obtained from those of $LT_{\sigma}(I)$ by so-called zero ${\sigma}$-consecutive cancellations. This provides a stronger version of the well-known cancellation "cancellation principle" between the resolution of a graded ideal and that of its leading term ideal, in terms of filtrations defined by monomial orders.

• 충청수학회지
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• 제34권1호
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• pp.1-15
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• 2021

In the recent papers, S'anchez-Reyes [Appl. Math. Model. 40 (2016), 1676-1682] described the method for finding a minimal surface through a geodesic, and Li et al. [Appl. Math. Model. 37 (2013), 6415-6424] studied the approximation of minimal surfaces with a geodesic from Dirichlet function. In the present article, we consider an isoparametric surface generated by Frenet frame of a curve introduced by Wang et al. [Comput. Aided Des. 36 (2004), 447-459], and give the necessary and sufficient condition to satisfy both geodesic of the curve and minimality of the surface. From this, we construct minimal surfaces in terms of constant curvature and torsion of the curve. As a result, we present a new approach for constructions of the minimal surfaces from a prescribed closed geodesic and unclosed geodesic, and show some new examples of minimal surfaces with a circle and a helix as a geodesic. Our approach can be used in design of minimal surfaces from geodesics.

• 대한수학회보
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• 제60권5호
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• pp.1299-1320
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• 2023

In the present paper, firstly we obtain the general expression of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hyper-cones whose centers lie on a non-null curve with non-null Frenet vector fields in E⁴₁ and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also, we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in E⁴₁ by taking constant radius function and finally, we construct some examples and visualize them with the aid of Mathematica.

• 한국통신학회논문지
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• 제15권2호
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• pp.124-136
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• 1990

본 논문에서는 4변수 NP 同値類 대표함수의 AND-EXOR형 최소 논리식의 표를 제시한다. 여기에서 최소 논리식이란 우선, 첫째로 적항수가 최소이고 다음에 적항수 중에 Literal 수의 총화가 최소식이며 또한 그 최소 논리식의 성질에 대해서 검토한다. 이것을 기초로 해서 AND-OR 형 2단 논리 회로의 최소 논리식과 본 논문의 알고리즘을 이용한 AND-EXOR형 2단 논리회로의 최소 논리식을 비교한다. 그 결과, AND-OR형 최소 논리식의 경우는 적항수가 모두 8이하에서, AND-EXOR 형 2단 최소 논리식의 경우는 적항수가 6이하에서 모든 함수가 생성되고 있으며 전반적으로 AND-EXOR형 최소 논리회로 쪽이 4변수 함수를 실현하는데 훨씬 적항수가 적다는 것을 알았다. 본 논문에서 제시한 알고리즘들은 SUN 3/50 상에서 실현했으며 이것을 통해서 4변수 이하의 논리함수는 본 논문에서 제시한 표에 의해서 즉시 최소형을 얻을 수가 있다. 5변수 함수에 대해서는 그 일부의 함수를 적당한 변수로 Shanon 전개해서 이것에 본 논문의 4변수 최소형을 적용함으로서 단시간내에 최소형을 얻을 수 있는 것이 가능하며 이 방법은 6변수 이상의 함수에도 적용하는 것이 가능하다고 생각된다.