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
The numerical method for multi-dimensional integrals is of great importance, particularly in the uncertainty quantification of engineering structures. The key is to generate representative points as few as possible bu...The numerical method for multi-dimensional integrals is of great importance, particularly in the uncertainty quantification of engineering structures. The key is to generate representative points as few as possible but of acceptable accuracy. A generalized L2（GL2）-discrepancy is studied by taking unequal weights for the point set. The extended Koksma-Hlawka inequality is discussed. Thereby, a worst-case error estimate is provided by such defined GL2-discrepancy, whose dosed-form expression is available. The characteristic values of GL2-discrepancy are investigated. An optimal strategy for the selection of the representative point sets with a prescribed cardinal number is proposed by minimizing the GL2-discrepancy. The three typical examples of the multi-dimensional integrals are investigated. The stochastic dynamic response analysis of a nonlinear structure is then studied by incorporating the proposed method into the probability density evolution method. It is shown that the proposed method is advantageous in achieving tradeoffs between the efficiency and accuracy of the exemplified problems. Problems to be further studied are discussed.展开更多
The purpose of the present article is to introduce a class of mixed two- and three-level extended designs obtained by adding some new runs to an existing mixed two- and three-level design. A formulation of wrap-around...The purpose of the present article is to introduce a class of mixed two- and three-level extended designs obtained by adding some new runs to an existing mixed two- and three-level design. A formulation of wrap-around L2-discrepancy for the extended designs is developed. As a benchmark of obtaining （nearly） uniform asymmetrical extended designs, a lower bound to the wrap-around L2- discrepancy for our proposed designs is established. Thorough numerical results are displayed, which provide further corroboration to the derived theoretical results.展开更多
The objective of this paper is to study the issue of uniformity on three-level U-type designs in terms of the wrap-around L2-discrepancy.Based on the known formula,we present a new lower bound of wrap-around L2-discre...The objective of this paper is to study the issue of uniformity on three-level U-type designs in terms of the wrap-around L2-discrepancy.Based on the known formula,we present a new lower bound of wrap-around L2-discrepancy for three-level U-type designs and compare it with those existing ones through figures,numerical simulation and illustrative examples.展开更多
How to obtain an effective design is a major concern of scientific research. This topic always involves high-dimensional inputs with limited resources. The foldover is a quick and useful technique in construction of f...How to obtain an effective design is a major concern of scientific research. This topic always involves high-dimensional inputs with limited resources. The foldover is a quick and useful technique in construction of fractional designs, which typically releases aliased factors or interactions.This paper takes the wrap-around L_2-discrepancy as the optimality measure to assess the optimal three-level combined designs. New and efficient analytical expressions and lower bounds of the wraparound L_2-discrepancy for three-level combined designs are obtained. The new lower bound is useful and sharper than the existing lower bound. Using the new analytical expression and lower bound as the benchmarks, the authors may implement an effective algorithm for constructing optimal three-level combined designs.展开更多
Abstract The objective of this paper is to study the issue of employing the uniformity criterion measured by the wrap-around L2-discrepancy to assess the optimal foldover plans for three-level designs.For three-level ...Abstract The objective of this paper is to study the issue of employing the uniformity criterion measured by the wrap-around L2-discrepancy to assess the optimal foldover plans for three-level designs.For three-level fractional factorials as the original designs,the general foldover plan and combined design under a foldover plan are defined,some theoretical properties of the defined foldover plans are obtained,a tight lower bound of the wrap-around L2-discrepancy of combined designs under a general foldover plan is also obtained,which can be used as a benchmark for searching optimal foldover plans.For illustration of the usage of our theoretical results,a catalog of optimal foldover plans for uniform initial designs with s three-level factors is tabulated,where 2≤ s ≤11.展开更多
Tandem asymmetric double Michael addition/internal nucleophilic substitution of the novel chiral source, 5-(l-menthyloxy)-3-bromo-2(5H)-furanone with nucleophilic alcohol compounds has been investigated. The tandem as...Tandem asymmetric double Michael addition/internal nucleophilic substitution of the novel chiral source, 5-(l-menthyloxy)-3-bromo-2(5H)-furanone with nucleophilic alcohol compounds has been investigated. The tandem asymmetric reaction can afford four new stereogenic centers with one reaction and give optically pure spiro-cyclopropane derivatives 5a--5d which are difficult to obtain by routine methods. The synthetic method for 5a--5d was studied in detail and the new compounds were identified on the basis of their analytical data and spectroscopic data, such as [α]^(20),IR,~1H NMR,^(13)C NMR, MS and elementary analysis. The absolute configuration of the sprio [5-l-menthyloxy-3-bromo butyrolactocyclopropane-3″, 3′(4′-methyloxy-5′-menthyloxybutyrolactone)] (5a) was established by X-ray crystallography. The work can provide important synthetic strategy in synthesis of some new optically active spiro-cyclopropane analogues and some biologically active molecules with complex structure.展开更多
The synthetic method of the novel chiral synthon, 5-/-menthyloxy-3-chloro-2-(5H)-furanone 5a and its application in asymmetric reactions were investigated. 5a is easily obtained in highly optical purity, and acts as a...The synthetic method of the novel chiral synthon, 5-/-menthyloxy-3-chloro-2-(5H)-furanone 5a and its application in asymmetric reactions were investigated. 5a is easily obtained in highly optical purity, and acts as a stable acceptor of Michael addition with oxygen nucleophiles in tandem double Michael addition / internal nucieophilic substitution to offer the spiro-cyclopropane derivative containing four stereogenic centers 8, which it is difficult to obtain by routine methods. The synthetic methods for 5a and 8 are reported in detail and the new compounds are identified on the basis of their analytical data and spectroscopic data, such as UV, IR, H NMR,13 C NMR, MS and elementary analysis. The absolute configuration of the interesting spiro-cyclopropanes, spiro [1-chloro-4-(/-menthyloxy)-5-oxo-6-oxa-biscyclo[3.1.0]hexane-2,3i(4’-/-menthyloxy-5’-/-menthyloxy- butyrolactone)] 8 was established by X-ray crystallography. This result can provide important synthetic strategy in synthesis of some complex展开更多
The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the c...The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the centered L2-discrepancy to assess the optimal foldover plans was studied for four-level design. A new analytical expression and a new lower bound of the centered L2-discrepancy for fourlevel combined design under a general foldover plan are respectively obtained. A necessary condition for the existence of an optimal foldover plan meeting this lower bound was described. An algorithm for searching the optimal four-level foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient Mgorithms for searching the optimal four-level foldover plans.展开更多
Space-filling designs are widely used in various fields because of their nice space-filling properties.Uniform designs are one of space-filling designs,which desires the experimental points to scatter uniformly over t...Space-filling designs are widely used in various fields because of their nice space-filling properties.Uniform designs are one of space-filling designs,which desires the experimental points to scatter uniformly over the experimental area.For practical need,the construction and their properties of nine-level uniform designs are discussed via two code mappings in this paper.Firstly,the algorithm of constructing nine-level uniform designs is presented from an initial three-level design by the Type-I code mapping and tripling technique.Secondly,the algorithm of constructing nine-level uniform designs is presented from a three-level base design by the Type-II code mapping and generalized orthogonal arrays.Moreover,relative properties are discussed based on the two code mappings.Finally,some numerical examples are given out for supporting our theoretical results.展开更多
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
基金supported by the National Natural Science Foundation of China(Grant Nos.51538010&51261120374)the State Key Laboratory of Disaster Reduction in Civil Engineering(Grant No.SLDRCE14-B-17)the Fundamental Funding for Central Universities
文摘The numerical method for multi-dimensional integrals is of great importance, particularly in the uncertainty quantification of engineering structures. The key is to generate representative points as few as possible but of acceptable accuracy. A generalized L2（GL2）-discrepancy is studied by taking unequal weights for the point set. The extended Koksma-Hlawka inequality is discussed. Thereby, a worst-case error estimate is provided by such defined GL2-discrepancy, whose dosed-form expression is available. The characteristic values of GL2-discrepancy are investigated. An optimal strategy for the selection of the representative point sets with a prescribed cardinal number is proposed by minimizing the GL2-discrepancy. The three typical examples of the multi-dimensional integrals are investigated. The stochastic dynamic response analysis of a nonlinear structure is then studied by incorporating the proposed method into the probability density evolution method. It is shown that the proposed method is advantageous in achieving tradeoffs between the efficiency and accuracy of the exemplified problems. Problems to be further studied are discussed.
基金supported by the National Natural Science Foundation of China under Grant Nos.11271147,11471135,11471136support of Excellent Doctoral Dissertation to Cultivate Project of Central China Normal University under Grant No.2017YBZZ057
文摘The purpose of the present article is to introduce a class of mixed two- and three-level extended designs obtained by adding some new runs to an existing mixed two- and three-level design. A formulation of wrap-around L2-discrepancy for the extended designs is developed. As a benchmark of obtaining （nearly） uniform asymmetrical extended designs, a lower bound to the wrap-around L2- discrepancy for our proposed designs is established. Thorough numerical results are displayed, which provide further corroboration to the derived theoretical results.
基金Supported in part by the National Natural Science Foundation of China(Nos.11871237,11801576,11271147,11401596)the Fundamental Research Funds for the Central Universities(South-Central University for Nationalities(Nos.CZQ18017,CZQ19010))
文摘The objective of this paper is to study the issue of uniformity on three-level U-type designs in terms of the wrap-around L2-discrepancy.Based on the known formula,we present a new lower bound of wrap-around L2-discrepancy for three-level U-type designs and compare it with those existing ones through figures,numerical simulation and illustrative examples.
基金supported by the National Natural Science Foundation of China under Grant Nos.11271147,11471135,11471136the UIC Grant R201409+1 种基金the Zhuhai Premier Discipline Grantthe Self-Determined Research Funds of CCNU from the Colleges Basic Research and Operation of MOE under Grant Nos.CCNU14A05041,CCNU16A02012
文摘How to obtain an effective design is a major concern of scientific research. This topic always involves high-dimensional inputs with limited resources. The foldover is a quick and useful technique in construction of fractional designs, which typically releases aliased factors or interactions.This paper takes the wrap-around L_2-discrepancy as the optimality measure to assess the optimal three-level combined designs. New and efficient analytical expressions and lower bounds of the wraparound L_2-discrepancy for three-level combined designs are obtained. The new lower bound is useful and sharper than the existing lower bound. Using the new analytical expression and lower bound as the benchmarks, the authors may implement an effective algorithm for constructing optimal three-level combined designs.
基金supported by National Natural Science Foundation of China(Grant Nos.11201177 and 11271147)China Postdoctoral Science Foundation(Grant No.2013M531716)+2 种基金Scientific Research Plan Item of Hunan Provincial Department of Education(Grant No.12C0287)Jishou University Doctor Science Foundation(Grant No.jsdxxcfxbskyxm201113)Scientific Research Plan Item of Jishou University(Grant No.13JDY041)
文摘Abstract The objective of this paper is to study the issue of employing the uniformity criterion measured by the wrap-around L2-discrepancy to assess the optimal foldover plans for three-level designs.For three-level fractional factorials as the original designs,the general foldover plan and combined design under a foldover plan are defined,some theoretical properties of the defined foldover plans are obtained,a tight lower bound of the wrap-around L2-discrepancy of combined designs under a general foldover plan is also obtained,which can be used as a benchmark for searching optimal foldover plans.For illustration of the usage of our theoretical results,a catalog of optimal foldover plans for uniform initial designs with s three-level factors is tabulated,where 2≤ s ≤11.
基金Project supported by the National Natural Science Foundation of China (Grant No. 29672004)
文摘Tandem asymmetric double Michael addition/internal nucleophilic substitution of the novel chiral source, 5-(l-menthyloxy)-3-bromo-2(5H)-furanone with nucleophilic alcohol compounds has been investigated. The tandem asymmetric reaction can afford four new stereogenic centers with one reaction and give optically pure spiro-cyclopropane derivatives 5a--5d which are difficult to obtain by routine methods. The synthetic method for 5a--5d was studied in detail and the new compounds were identified on the basis of their analytical data and spectroscopic data, such as [α]^(20),IR,~1H NMR,^(13)C NMR, MS and elementary analysis. The absolute configuration of the sprio [5-l-menthyloxy-3-bromo butyrolactocyclopropane-3″, 3′(4′-methyloxy-5′-menthyloxybutyrolactone)] (5a) was established by X-ray crystallography. The work can provide important synthetic strategy in synthesis of some new optically active spiro-cyclopropane analogues and some biologically active molecules with complex structure.
文摘The synthetic method of the novel chiral synthon, 5-/-menthyloxy-3-chloro-2-(5H)-furanone 5a and its application in asymmetric reactions were investigated. 5a is easily obtained in highly optical purity, and acts as a stable acceptor of Michael addition with oxygen nucleophiles in tandem double Michael addition / internal nucieophilic substitution to offer the spiro-cyclopropane derivative containing four stereogenic centers 8, which it is difficult to obtain by routine methods. The synthetic methods for 5a and 8 are reported in detail and the new compounds are identified on the basis of their analytical data and spectroscopic data, such as UV, IR, H NMR,13 C NMR, MS and elementary analysis. The absolute configuration of the interesting spiro-cyclopropanes, spiro [1-chloro-4-(/-menthyloxy)-5-oxo-6-oxa-biscyclo[3.1.0]hexane-2,3i(4’-/-menthyloxy-5’-/-menthyloxy- butyrolactone)] 8 was established by X-ray crystallography. This result can provide important synthetic strategy in synthesis of some complex
基金Supported by National Natural Science Foundation of China(Grant Nos.11271147,11471135 and 11471136)
文摘The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the centered L2-discrepancy to assess the optimal foldover plans was studied for four-level design. A new analytical expression and a new lower bound of the centered L2-discrepancy for fourlevel combined design under a general foldover plan are respectively obtained. A necessary condition for the existence of an optimal foldover plan meeting this lower bound was described. An algorithm for searching the optimal four-level foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient Mgorithms for searching the optimal four-level foldover plans.
基金supported by the National Natural Science Foundation of China(Nos.12161040,119610271,1701213,11871237)Natural Science Foundation of Hunan Province(Nos.2020JJ4497,2021JJ30550)+2 种基金Scientific Research Plan Item of Hunan Provincial Department of Education(No.19A403)Graduate Scientific Research Innovation Item of Hunan Province(No.CX20211504)the Scientific Research Item of Jishou University(No.Jdy20057)。
文摘Space-filling designs are widely used in various fields because of their nice space-filling properties.Uniform designs are one of space-filling designs,which desires the experimental points to scatter uniformly over the experimental area.For practical need,the construction and their properties of nine-level uniform designs are discussed via two code mappings in this paper.Firstly,the algorithm of constructing nine-level uniform designs is presented from an initial three-level design by the Type-I code mapping and tripling technique.Secondly,the algorithm of constructing nine-level uniform designs is presented from a three-level base design by the Type-II code mapping and generalized orthogonal arrays.Moreover,relative properties are discussed based on the two code mappings.Finally,some numerical examples are given out for supporting our theoretical results.
