• 한국CDE학회논문집
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• 제11권4호
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• pp.246-255
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• 2006

Ribs and fans are interesting geometric entities that are derived from an ordinary $B\'{e}zier$ curve or surface. A rib itself is a $B\'{e}zier$ curve or surface with a lower degree than the given curve or surface. A fan is a vector field whose degree is also lower than its origin. First, we present methods to transform the control points of a $B\'{e}zier$ curve or surface into the control points and vectors of its ribs and fans. Then, we show that a $B\'{e}zier$ curve of degree n is decomposed into a rib of degree (n-1), a fan of degree (n-2), and a scalar function of degree 2. We also show that a $B\'{e}zier$ surface of degree (m, n) is decomposed into a rib of degree (m-1, n-1) and three fans of degrees (m-1, n-2), (m-2, n-1), and (m-2, n-2), respectively. In addition, the lengths of the fans are further controlled by scalar functions of degree 2 and (2, 2). We present relevant notations and definitions, introduce theories, and present some of design examples.

• Kyungpook Mathematical Journal
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• 제58권1호
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• pp.117-135
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• 2018

The orthogonal trajectories of the first tangents of the curve are called the involutes of x. The hyperspheres which have higher order contact with a curve x are known osculating hyperspheres of x. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve x in n-dimensional Euclidean space ${\mathbb{E}}^n$. In the present study, we give a characterization of involute curves of order k (resp. evolute curves) of the given curve x in n-dimensional Euclidean space ${\mathbb{E}}^n$. Further, we obtain some results on these type of curves in ${\mathbb{E}}^3$ and ${\mathbb{E}}^4$, respectively.

• 한국가축번식학회지
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• 제6권1호
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• pp.31-35
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• 1982

A study of 188 lactations of Holstein cows in Gwangju area was undertaken to establish the shape of lactation curve during the period from October in 1980 to January in 1982. The Gammafunction described by Wood(1967) was fitted to the lactations observed. The results obtained in the present study were summarized as follows; 1. The lactation curve of the 188 lactations was expressed by the equation based on Wood's model (1967) as follows; Yn=24.5m0.0762e-0.0944n(R2=0.99) 2. The lactation curves by parity were represented by the equations as follows; Yn=18.81n0.1486e-0.0741n(R2=0.98)……………parity 1 Yn=26.51n0.1161e-0.1200n(R2=0.96)……………parity 2 Yn=26.95n0.2804e-0.1703n(R2=0.99)……………parity 3 Yn=27.92n0.1429e-0.1427n(R2=0.98)……………parity 4 Yn=22.61n0.1985e-0.1211n(R2=0.94)……………parity 5 3. The lactation curves by calving seasons were represented by the equationes as follows; Yn=27.05n0.0739e-0.1005n(R2=0.98)……………spring Yn=23.08n0.2040e-0.1202n(R2=0.98)……………summer Yn=26.81n0.0460e-0.1134n(R2=0.98)……………autumn Yn=23.40n0.1299e-0.1101n(R2=0.95)……………winter

• 대한수학회지
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• 제60권5호
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• pp.1087-1107
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• 2023

Let E be an elliptic curve defined over ℚ of conductor N, c the Manin constant of E, and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K, P_K the Heegner point in E(K), and III(E/K) the Shafarevich-Tate group of E over K. Let 2u_K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P_K has infinite order in E(K), then the integer c · m · u_K · |III(E/K)| $\frac{1}{2}$ is divisible by |E(ℚ)_tor|. In this paper, we prove that this conjecture is true if E(ℚ)_tor ≅ ℤ/2ℤ or ℤ/4ℤ except for two explicit families of curves. Further, we show these exceptions can be removed under Stein-Watkins conjecture.

• 대한수학회보
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• 제33권4호
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• pp.537-544
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• 1996

Crofton's formula on Euclidean plane $E^2$ states: Let $\Gamma$ be a rectifiable curve of length L and let G be a straight line. Then $$ \int_{G \cap \Gamma \neq \phi} n dG = 2L $$ where n is the number of the intersection points of G with the curve $\Gamma$.

• 대한수학회보
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• 제44권3호
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• pp.429-438
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• 2007

In this paper, we study the position vectors of a spacelike W-curve (or a helix), i.e., curve with constant curvatures, with spacelike, timelike and null principal normal in the Minkowski 3-space $\mathbb{E}_1^3$. We give some characterizations for spacelike W - curves whose image lies on the pseudohyperbolical space $\mathbb{H}_0^2$ and Lorentzian sphere $\mathbb{S}_1^2$ by using the positions vectors of the curve.

• 암석학회지
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• 제27권3호
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• pp.109-120
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• 2018

경상분지 남동부 일대의 백악기 및 제3기 암류에서 발달하는 단층분절에 대한 분포특성을 도출하였다. 선형을 보이는 267조의 단층분절은 광역 지질도 상에서 표시된 곡선의 단층선에서 추출하였다. 첫째, 단층분절에 대한 방향각(${\theta}$)-길이(L)의 도면을 작성하였다. 관계도에서 단층분절의 전반적인 분포형태를 도출하였다. 도면의 분포곡선은 전체 형태에 따라서 4개의 구간으로 구분하였다. 상기 구간의 정점에 해당하는 북북동, 북북서 및 서북서의 방향은 양산, 울산 및 가음 단층계의 방향을 시사한다. 단층분절의 집단은 최대 정점에 해당하는 $N19^{\circ}E$의 방향에 대하여 거의 대칭 분포를 보여 준다. 둘째, 방향각-빈도수(N), 평균 길이(Lm), 총 길이(Lt) 및 밀도(${\rho}$)의 도면을 작성하였다. 관계도에서 상기한 도면의 전 영역을 분포곡선의 분포상에 의하여 19개의 영역으로 구분하였다. 상기한 영역의 정점에 해당하는 방향은 암체에 가해진 대표적인 응력의 방향을 시사한다. 셋째, 18개의 부집단에 대한 길이-누적 빈도수 그래프를 작성하였다. 관계도에서 지수(${\lambda}$)는 시계방향($N10{\sim}20^{\circ}E{\rightarrow}N50{\sim}60^{\circ}E$)과 반시계방향($N10{\sim}20^{\circ}W{\rightarrow}N50{\sim}60^{\circ}W$)으로 갈수록 증가한다. 반면 길이의 분포 폭 및 평균 길이는 감소한다. 서로 다른 진화 특성을 갖는 상기한 부집단에 대한 도면은 진화과정의 한 단면을 나타내고 있다. 넷째, 18개의 그래프에 대한 종합 분포도를 작성하였다. 관계도에서 상기한 그래프를 분포 구역에 따라 5개의 그룹(A~E)으로 분류하였다. 단층분절의 길이는 그룹 E ($N80{\sim}90^{\circ}E{\cdot}N70{\sim}80^{\circ}E{\cdot}N80{\sim}90^{\circ}W{\cdot}N50{\sim}60^{\circ}W{\cdot}N30{\sim}40^{\circ}W{\cdot}N40{\sim}50^{\circ}W$) < D ($N70{\sim}80^{\circ}W{\cdot}N60{\sim}70^{\circ}W{\cdot}N60{\sim}70^{\circ}E{\cdot}N50{\sim}60^{\circ}E{\cdot}N40{\sim}50^{\circ}E{\cdot}N0{\sim}10^{\circ}W$) < C ($N20{\sim}30^{\circ}W{\cdot}N10{\sim}20^{\circ}W$) < B ($N0{\sim}10^{\circ}E{\cdot}N30{\sim}40^{\circ}E$) < A ($N20{\sim}30^{\circ}E{\cdot}N10{\sim}20^{\circ}E$)의 순으로 증가한다. 특히 그래프의 형태는 균등 분포에서 지수 분포로 점차 변화한다. 마지막으로, 단층분절의 길이에 대한 여섯 개 변수의 값을 5개 그룹으로 구분하였다. 여섯 개 변수 중, 평균 길이 및 가장 긴 단층분절의 길이는 그룹 III ($N10^{\circ}W{\sim}N20^{\circ}E$) > IV ($N20{\sim}60^{\circ}E$) > II ($N10{\sim}60^{\circ}W$) > I ($N60{\sim}90^{\circ}W$) > V ($N60{\sim}90^{\circ}E$)의 순으로 감소한다. 그룹 V에 속하는 단층분절의 빈도수, 최장 길이, 총 길이, 평균 길이 및 밀도가 가장 낮은 값을 보여 준다. 5개 그룹 사이의 상기 배열순은 단층분절의 상대적인 생성시기와의 상관성을 시사한다.

• Structural Engineering and Mechanics
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• 제24권5호
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• pp.585-599
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• 2006

This paper describes a series of laboratory tests carried out to evaluate the influence of bed joint orientation on interlocking grouted stabilised mud-flyash brick masonry under uniaxial cyclic compressive loading. Five cases of loading at $0^{\circ}$, $22.5^{\circ}$, $45^{\circ}$, $67.5^{\circ}$ and $90^{\circ}$ with the bed joints were considered. The brick units and masonry system developed by Prof. S.N. Sinha were used in present investigation. Eighteen specimens of size $500mm{\times}100mm{\times}700mm$ and twenty seven specimens of size $500mm{\times}100mm{\times}500mm$ were tested. The envelope stress-strain curve, common point curve and stability point curve were established for all five cases of loading with respect to bed joints. A general analytical expression is proposed for these curves which fit reasonably well with the experimental data. Also, the stability point curve has been used to define the permissible stress level in the brick masonry.

• 한국의류학회지
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• 제23권3호
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• pp.353-360
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• 1999

An investigation made of the variations of angle of bias on the top of the sleeve cap curve line and calculated easing contraction ratio by capheights(A ; a,h$\times$5,/6) B: A, H/4 +4cm C:A.H/3 D: A.H/ 4+3cm E:AH/4+2cm, F: A,H/4+1cm, G: A,H/4, H:A,H/6, I:A,H/8) and the efects of easing contraction on the cap curve lines of sleeve A, D, G by easing stitch density with the gathering foot: sewing condition-lockstitch industrial machine stitch density(N1.0 ; 38stitches/3cm N1.5: 26stitches/3cm, N2.0 ; 19stitches/3cm, N2.5 ; 14stitches/ 3cm) The results obtained were as follows; 1) The variations of the angle of bias on the top of the sleeve cap curve line by cap heights can be done according to the angle balance (front; $\alpha$-$\beta$ back ; $\alpha$'- $\beta$') between the angle (front ;$\alpha$, $\beta$, back ; $\alpha$'- $\beta$') of bias of the two base-lines. 2) The higher cap height the more higher the calculated easing contraction ratio. 3) The lower the stitch density the higher easing contraction ratio. 4) The effects of easing contraction was that sleeve G was more than sleeve A, D.

• 한국CDE학회논문집
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• 제3권2호
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• pp.135-139
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• 1998

The hodograph, which are usually defined as the derivative of parametric curve or surface, is useful far various geometric operations. It is known that the hodographs of Bezier curves and surfaces can be represented in the closed form. However, the counterparts of rational Bezier curves and surface have not been discussed yet. In this paper, the equations are derived, which are the closed form of rational Bezier curves and surfaces. The hodograph of rational Bezier curves of degree n can be represented in another rational Bezier curve of degree 2n. The hodograph of a rational Hazier surface of degree m×n with respect to a parameter can be also represented in rational Bezier surface of degree 2m×2n. The control points and corresponding weight of the hodographs are directly computed using the control points and weights of the given rational curves or surfaces.