The objective of this paper is to discuss the issue of the projection uniformity of asymmetric fractional factorials.On the basis of Lee discrepancy,the authors define the projection Lee discrepancy to measure the uni...The objective of this paper is to discuss the issue of the projection uniformity of asymmetric fractional factorials.On the basis of Lee discrepancy,the authors define the projection Lee discrepancy to measure the uniformity for low-dimensional projection designs.Moreover,the concepts of uniformity pattern and minimum projection uniformity criterion are proposed,which can be used to assess and compare two and three mixed levels factorials.Statistical justification of uniformity pattern is also investigated.展开更多
In this paper,the study of projection properties of two-level factorials in view of geometry is reported.The concept of uniformity pattern is defined.Based on this new concept,criteria of uniformity resolution and min...In this paper,the study of projection properties of two-level factorials in view of geometry is reported.The concept of uniformity pattern is defined.Based on this new concept,criteria of uniformity resolution and minimum projection uniformity are proposed for comparing two-level factorials.Relationship between minimum projection uniformity and other criteria such as minimum aberration,generalized minimum aberration and orthogonality is made explict.This close relationship raises the hope of improving the connection between uniform design theory and factorial design theory.Our results provide a justification of orthogonality,minimum aberration,and generalized minimum aberration from a natural geometrical interpretation.展开更多
We report the fabrication of 4-inch nano patterned wafer by two-beam laser interference lithography and analyze the uniformity in detail. The profile of the dots array with a period of 800 nm divided into five regions...We report the fabrication of 4-inch nano patterned wafer by two-beam laser interference lithography and analyze the uniformity in detail. The profile of the dots array with a period of 800 nm divided into five regions is characterized by a scanning electron microscope. The average size in each region ranges from 270 nm to 320 nm,and the deviation is almost 4%, which is approaching the applicable value of 3% in the industrial process. We simulate the two-beam laser interference lithography system with MATLAB software and then calculate the distribution of light intensity around the 4 inch area. The experimental data fit very well with the calculated results. Analysis of the experimental data and calculated data indicates that laser beam quality and space filter play important roles in achieving a periodical nanoscale pattern with high uniformity and large area. There is the potential to obtain more practical applications.展开更多
In this article, we consider the characterization problem in design theory. The objective is to characterize minimum projection uniformity for two-level designs in terms of their complementary designs. Here, the compl...In this article, we consider the characterization problem in design theory. The objective is to characterize minimum projection uniformity for two-level designs in terms of their complementary designs. Here, the complementary design means a design in which all the Hamming distances of any two runs are the same, which generalizes the concept of a pair of complementary designs in the literature. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity （MPU） rule to assess and compare two-level factorials.展开更多
In constructing two-level fractional factorial designs, the so-called doubling method has been employed. In this paper, we study the problem of uniformity in double designs. The centered L2-discrepancy is employed as ...In constructing two-level fractional factorial designs, the so-called doubling method has been employed. In this paper, we study the problem of uniformity in double designs. The centered L2-discrepancy is employed as a measure of uniformity. We derive results connecting the centered L2-discrepancy value of D（X） and generalized wordlength pattern of X, which show the uniformity relationship between D（X） and X. In addition, we also obtain lower bounds of centered L2-discrepancy value of D（X）, which can be used to assess uniformity of D（X）.展开更多
We study the complementary design problem, which is to express the uniformity pattern of a q-level design in terms of that of its complementary design. Here, a pair of complementary designs form a design in which all ...We study the complementary design problem, which is to express the uniformity pattern of a q-level design in terms of that of its complementary design. Here, a pair of complementary designs form a design in which all the Hamming distances of any two distinct runs are the same, and the uniformity pattern proposed by H. Qin, Z. Wang, and K. Chatterjee [J. Statist. Plann. Inference, 2012, 142： 1170-11771 comes from discrete discrepancy for q-level designs. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity rule to assess and compare q-level factorials.展开更多
This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered Lu-discrepancy values of combined designs is obtained, which can be used as a benchmar...This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered Lu-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design. Keywords Centered L2-discrepancy, optimal foldover plan, uniformity, uniformity pattern展开更多
基金supported by the National Natural Science Foundations of China under Grant Nos.11271147 and 11401596
文摘The objective of this paper is to discuss the issue of the projection uniformity of asymmetric fractional factorials.On the basis of Lee discrepancy,the authors define the projection Lee discrepancy to measure the uniformity for low-dimensional projection designs.Moreover,the concepts of uniformity pattern and minimum projection uniformity criterion are proposed,which can be used to assess and compare two and three mixed levels factorials.Statistical justification of uniformity pattern is also investigated.
基金partially supported by the Hong Kong RGC grant,RGC/HKBU 2044/02Pthe National Natural Science Foundation of China(Grant No.10071029)+1 种基金the Project-sponsored by SRF for ROCS(SEM)the NSF of Hubei Province for the second author.
文摘In this paper,the study of projection properties of two-level factorials in view of geometry is reported.The concept of uniformity pattern is defined.Based on this new concept,criteria of uniformity resolution and minimum projection uniformity are proposed for comparing two-level factorials.Relationship between minimum projection uniformity and other criteria such as minimum aberration,generalized minimum aberration and orthogonality is made explict.This close relationship raises the hope of improving the connection between uniform design theory and factorial design theory.Our results provide a justification of orthogonality,minimum aberration,and generalized minimum aberration from a natural geometrical interpretation.
基金Supported by the Scientific Equipment Research Program of Chinese Academy of Sciences under Grant No 2014Y4201449
文摘We report the fabrication of 4-inch nano patterned wafer by two-beam laser interference lithography and analyze the uniformity in detail. The profile of the dots array with a period of 800 nm divided into five regions is characterized by a scanning electron microscope. The average size in each region ranges from 270 nm to 320 nm,and the deviation is almost 4%, which is approaching the applicable value of 3% in the industrial process. We simulate the two-beam laser interference lithography system with MATLAB software and then calculate the distribution of light intensity around the 4 inch area. The experimental data fit very well with the calculated results. Analysis of the experimental data and calculated data indicates that laser beam quality and space filter play important roles in achieving a periodical nanoscale pattern with high uniformity and large area. There is the potential to obtain more practical applications.
基金supported by the NSF of China (10671080)NCET (06-672)the Key Project of Chinese Ministry of Education (105119)
文摘In this article, we consider the characterization problem in design theory. The objective is to characterize minimum projection uniformity for two-level designs in terms of their complementary designs. Here, the complementary design means a design in which all the Hamming distances of any two runs are the same, which generalizes the concept of a pair of complementary designs in the literature. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity （MPU） rule to assess and compare two-level factorials.
基金Supported by the National Natural Science Foundation of China(11271147)SRFDP(20090144110002)
文摘In constructing two-level fractional factorial designs, the so-called doubling method has been employed. In this paper, we study the problem of uniformity in double designs. The centered L2-discrepancy is employed as a measure of uniformity. We derive results connecting the centered L2-discrepancy value of D（X） and generalized wordlength pattern of X, which show the uniformity relationship between D（X） and X. In addition, we also obtain lower bounds of centered L2-discrepancy value of D（X）, which can be used to assess uniformity of D（X）.
基金Acknowledgements The authors greatly appreciate helpful suggestions of the referees that greatly improved the paper. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11271147, 11401596）.
文摘We study the complementary design problem, which is to express the uniformity pattern of a q-level design in terms of that of its complementary design. Here, a pair of complementary designs form a design in which all the Hamming distances of any two distinct runs are the same, and the uniformity pattern proposed by H. Qin, Z. Wang, and K. Chatterjee [J. Statist. Plann. Inference, 2012, 142： 1170-11771 comes from discrete discrepancy for q-level designs. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity rule to assess and compare q-level factorials.
基金Supported by SRFDP(Grant No.20090144110002)National Natural Science Foundation of China(Grant No.10671080)+3 种基金Key Project of Chinese Ministry of Education(Grant No.105119)NCET(Grant No.06-672)Scientific Research Plan Item of Hunan Provincial Department of Education(Grant No.10C1091)Innovation Program and Independent Research Project Funded by CCNU
文摘This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered Lu-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design. Keywords Centered L2-discrepancy, optimal foldover plan, uniformity, uniformity pattern
