• 품질경영학회지
• /
• 제32권4호
• /
• pp.220-228
• /
• 2004

The orthogonality is an important property in the experimental designs. When we use nearly orthogonal arrays, we need evaluate the degree of the orthogonality of given experimental designs. Graphical methods for evaluating the degree of the orthogonality of nearly orthogonal arrays are suggested.

• 응용통계연구
• /
• 제23권1호
• /
• pp.167-178
• /
• 2010

직교성은 실험계획에서 중요한 개념이다. 실험계획에서 실험점의 개수보다 인자의 개수가 많은 상황에서 우리는 초포화계획을 사용한다. 이러한 초포화계획은 직교성을 만족하지 못하게 되는 데 얼마나 직교성을 만족하는 지를 평가하는 데 우리는 주로 수치적인 측도들을 사용한다. 우리는 초포화계획의 직교성의 정도를 평가하는 방법으로서 그래픽방법을 사용할 수 있다.

• 품질경영학회지
• /
• 제36권3호
• /
• pp.13-21
• /
• 2008

The orthogonality is an important property in the experimental designs. When we use nearly orthogonal arrays(for example, supersaturated designs), we need evaluate the degree of the orthogonality of given nearly orthogonal arrays. We can use the mutual information as a new criterion for evaluating and testing the degree of the orthogonality of given nearly orthogonal arrays.

• 한국정밀공학회지
• /
• 제15권12호
• /
• pp.169-179
• /
• 1998

Plane mirror type laser interferometers are popularly being used in many modern ultraprecision machines, as they can perform simultaneous measurements of multiple axis positions with nanometer resolution capabilities. One important issue in this application of laser interferometers is to provide a good level of alignment between the reflecting mirrors and the laser beams so that measurement errors due to undesirable coupling effects can be avoided in multiple axis measurements In this investigation, a thorough metrological analysis is given to develop an suitable mathematical model for a precision x-y stage in which the orthogonality misalignment between the reflecting mirrors significantly affects overall x-y mea-surement results. Then a noble calibration method is suggested in which two-dimensional displacement sensors of moire gratings of concentric circles are used to realize the reversal principle of orthogonality evaluation in situ. Finally, actual experimental results are discussed to verify that the suggested method can effectively calibrate the orthogonality error with an uncertainty of 0.2667 arcsec.

• Korean Journal of Mathematics
• /
• 제22권2호
• /
• pp.325-337
• /
• 2014

In 1911 Schur[6] derived degree and character formulas for projective representations of the symmetric groups remarkably similar to the corresponding formulas for ordinary representations. Morris[3] derived a recurrence for evaluation of spin characters and Stembridge[8] gave a combinatorial reformulation for Morris' recurrence. In this paper we give combinatorial interpretations for the orthogonality relations of spin characters based on Stembridge's combinatorial reformulation for Morris' rule.

• 한국정밀공학회지
• /
• 제24권9호
• /
• pp.68-75
• /
• 2007

The pitch and orthogonality of two-dimensional (2D) gratings have been measured by using a metrological atomic force microscope (MAFM) and measurement uncertainty has been analyzed. Gratings are typical standard artifacts for the calibration of precision microscopes. Since the magnification and orthogonality in two perpendicular axes of microscopes can be calibrated simultaneously using 2D gratings, it is important to certify the pitch and orthogonality of 2D gratings accurately for nano-metrology using precision microscopes. In the measurement of 2D gratings, the MAFM can be used effectively for its nanometric resolution and uncertainty, but a new measurement scheme was required to overcome some limitations of current MAFM such as nonnegligible thermal drift and slow scan speed. Two kinds of 2D gratings, each with the nominal pitch of 300 nm and 1000 nm, were measured using line scans for the pitch measurement of each direction. The expanded uncertainties (k = 2) of measured pitch values were less than 0.2 nm and 0.4 nm for each specimen, and those of measured orthogonality were less than 0.09 degree and 0.05 degree respectively. The experimental results measured using the MAFM and optical diffractometer were coincident with each other within the expanded uncertainty of the MAFM. As a future work, we also proposed another scheme for the measurements of 2D gratings to increase the accuracy of calculated peak positions.

• International Journal of Precision Engineering and Manufacturing
• /
• 제9권2호
• /
• pp.18-22
• /
• 2008

The pitch and orthogonality of two-dimensional (2-D) gratings were measured using a metrological atomic force microscope (MAFM), and the measurement uncertainty was analyzed. Gratings are typical standard devices for the calibration of precision microscopes, Since the magnification and orthogonality in two perpendicular axes of microscopes can be calibrated simultaneously using 2-D gratings, it is important to certify the pitch and orthogonality of such gratings accurately for nanometrology. In the measurement of 2-D gratings, the MAFM can be used effectively for its nanometric resolution and uncertainty, but a new measurement scheme is required to overcome limitations such as thermal drift and slow scan speed. Two types of 2-D gratings with nominal pitches of 300 and 1000 nm were measured using line scans to determine the pitch measurement in each direction. The expanded uncertainties (k = 2) of the measured pitch values were less than 0.2 and 0.4 nm for each specimen, and the measured orthogonality values were less than $0.09^{\circ}$ and $0.05^{\circ}$, respectively. The experimental results measured using the MAFM and optical diffractometer agreed closely within the expanded uncertainty of the MAFM. We also propose an additional scheme for measuring 2-D gratings to increase the accuracy of calculated peak positions, which will be the subject of future study.

• Communications for Statistical Applications and Methods
• /
• 제15권2호
• /
• pp.235-244
• /
• 2008

실험계획의 요인배치법에서 부분실험을 설계할 때, 직교배열을 이용한 실험설계방법이 널리 사용된다. 그러나 부분실험의 해상도(resolution)가 큰 경우, 직교배열을 일반화한 균형배열이 효과적으로 사용될 수 있다. 특히 2-수준 요인실험법에서는 강도(strength)가 4인 균형배 열은 resolution-V인 부분실험법과 동일하다는 것이 알려져 있다. 본 연구에서는 강도(strength)가 4인 균형배열의 직교성을 평가하는 측도를 제시하고자 한다. 그리고 본 연구에서 제시된 평가측도를 응용하여 실험횟수가 가장 적고 직교성에 가까운 최소 균형 resolution-V 부분실험법의 설계방법을 제시하고자 한다.

• Journal of applied mathematics & informatics
• /
• 제20권1_2호
• /
• pp.623-635
• /
• 2006

In this paper, we derive an elementary formula for the linearization coefficients of the product of two Legendre polynomials. Our main purpose is to study the method which can be used to derive the formula, which is straightforward from the integration and their orthogonality.

• 한국품질경영학회:학술대회논문집
• /
• 한국품질경영학회 2009년도 추계학술대회
• /
• pp.23-29
• /
• 2009

직교성은 실험계획에서 중요한 개념이다. 실험계획에서 실험점의 개수보다 인자의 개수가 많은 상황에서 우리는 초포화계획을 사용한다. 이러한 초포화계획은 직교성을 만족하지 못하게 되는 데 얼마나 직교성을 만족하는 지를 평가하는 데 우리는 주로 수치적인 측도들을 사용한다. 우리는 초포화계획의 직교성의 정도를 평가하는 또 다른 탐색적 방법으로서 그래픽방법을 사용할 수 있다. 또한 초포화계획의 예측 능력을 평가하는 방법으로서 우리는 그래픽 방법을 사용할 수 있다.