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Injective dimension of Rees Rings

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Science China Mathematics 1993年第12期36卷 1417-1421页

作者：王志玺 Department of Mathematics University of Science and Technology of China

Let A be a left and right Noetherian ring and let x be a central regular element of A. A_x denotes the localization of A at the central multiplicatively closed subset {1,x,x^2,…}. Suppose M is an A-module such that x is a nonzero divisor in. M. It is shown that there is an equality relation among three iniective dimensions ld_A(M), IdA_x(M_x), and I_dA/xA(M/xM). Then the result is applied to the case of Rees rings of filtered rings and an improved and uniform form of the two results of E. K. Ekstrm is obtained. Moreover, the results in this paper generalize the relevant results of Li Huishi, M. Van den Bergh and F. Van Oystaeyen.

关键词： Noetherian ring injective dimension filtered ring Rees ring

Category and dimension of compact subsets of R^n

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Chinese Science Bulletin 1997年第20期42卷 1680-1683页

作者： Feng, DJ Wu, J Wuhan Univ Dept Math Wuhan 430072 Peoples R China

IN fractal geometry, two classes of sets play important roles. One is the regular set (the setHausdorff and packing dimension coincide), the other is the set whose Bouligand dimensionexists. A natural question is how to measure 'the size' of these sets mentioned above. In thisnote, by using category, we answer this question. The main result is Theorem 1.

关键词： fractal dimension category.

Microsphere-assisted,nanospot,non-destructive metrology for semiconductor devices

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Light(Science & Applications) 2022年第2期11卷 262-275页

作者： Soonyang Kwon Jangryul Park Kwangrak Kim Yunje Cho Myungjun Lee Equipment R&D Team 4 Mechatronics ResearchSamsung Electronics Co.Ltd.1-1 Samsungjeonja-roHwaseong-siGyeonggi-do 18848Republic of Korea

As smaller structures are being increasingly adopted in the semiconductor industry,the performance of memory and logic devices is being continuously improved with innovative 3D integration schemes as well as shrinking and stacking strategies.Owing to the increasing complexity of the design architectures,optical metrology techniques including spectroscopic ellipsometry(SE)and reflectometry have been widely used for efficient process development and yield ramp-up due to the capability of 3D structure measurements.However,there has been an increasing demand for a significant reduction in the physical spot diameter used in the SE technique;the spot diameter should be at least 10 times smaller than the cell dimension(~30×40μm2)of typical dynamic random-access memory to be able to measure in-cell critical dimension(CD)variations.To this end,this study demonstrates a novel spectrum measurement system that utilizes the microsphere-assisted super-resolution effect,achieving extremely small spot spectral metrology by reducing the spot diameter to~210 nm,while maintaining a sufficiently high signal-to-noise ratio.In addition,a geometric model is introduced for the microsphere-based spectral metrology system that can calculate the virtual image plane magnification and depth of focus,providing the optimal distance between the objective lens,microsphere,and sample to achieve the best possible imaging quality.The proof of concept was fully verified through both simulations and experiments for various samples.Thus,owing to its ultra-small spot metrology capability,this technique has great potential for solving the current metrology challenge of monitoring in-cell CD variations in advanced logic and memory devices.

关键词： metrology dimension utilize

The pointwise dimension of self-similar measures

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Chinese Science Bulletin 1999年第23期44卷 2136-2140页

作者： CHEN Ercai and XIONG Jincheng Department of Mathematics , Nanjing Normal University , Nanjing 210097, China Department of Mathematics , South China Normal University, Guangzhou 510631 , China Department of Mathematics Nanjing Normal University Nanjing China Department of Mathematics South China Normal University Guangzhou China

The problem about the existence of pointwise dimension of self-similar measure has been discussed. The Hausdorff dimension of the set of those points for which the pointwise dimension does not exist equals the dimenti... 详细信息

关键词： self-similar measure point wise dimension Hausdorff dimension multifractal symbolic space.

Homological dimension of Finitely Generated Module Having Simple Module as Direct Factor

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Chinese Science Bulletin 1994年第18期39卷 1503-1506页

作者：王明生 Department of Mathematics Beijing Normal University Beijing 100875 PRC

In Ref. [1], Yao demonstrates that the projective dimension of a simple module over a commutative Noetherian ring equals its injective dimension and characterizes the commutative ring possessing an injective simple mo... 详细信息

关键词： simple module projective dimension injective dimension.

On the dimension of an Arbitrary Ascending Chain

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Chinese Science Bulletin 1993年第10期38卷 799-804页

作者：高小山 Institute of Systems Science Academia Sinica Beijing 100080 PRC

We know that the dimension for an irreducible ascending chain ASC is a crucial concept in Ritt-Wu’s constructive theory of algebraic geometry. We can also define the dimension for an arbitrary ascending chain similarly. But one may say that this definition has no geometric meaning. In this note, we shall show that the dimension of an

关键词： ascending chain dimension of a variety Ritt-Wu’s decomposition unmixed decomposition.

dimension AND DIFFERENTIABILIIY OF A CLASS OF FRACTAL INTERPOLATION FUNCTIONS

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Applied Mathematics(A Journal of Chinese Universities) 1996年第1期11卷 85-100页

作者： WANG GUOZHONG Department of Mathematics, Zhejiang University Hangzhou 310027

In this paper, a new iterated function system consisting of non-linear affinemaps is constructed. We investigate the fractal interpolation functions generated bysuch a system and get its differentiability, its box dim... 详细信息

关键词： Fractal interpolation function dimension differentiability

RELATIONSHIP BETWEEN GAUGE dimension, GRAIN SIZE AND SUPERPLASTIC ELONGATION

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Chinese Science Bulletin 1991年第12期36卷 1046-1049页

作者：蒋兴钢崔建忠马龙翔 Department of Metal Forming Northeast University of Technology Shenyang 110006 PRC

Although several formulas for predicting the total elongation of the superplastic deformation have been reported, up to now, there is still no one which directly shows the relationship between gauge dimension, grain size and superplastic elongation of a specimen. In consideration of the irregularity of the fracture surface of a superplastic tensile specimen which is approximately self-similar, a fractal model for the fracture of materials during

关键词： elongation gauge dimension grain size.

DETERMINATION OF THE SHAPE AND dimension OF PRECIPITATE FROM AGED Lc4 ALUMINIUM ALLOY BY SMALL ANGLE X-RAY SCATTERING

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Chinese Science Bulletin 1989年第17期34卷 1489-1493页

作者：孟昭富 Materials Science Institute Jilin University Changchun

Small angle X-ray scattering (SAXS) is extensively used for determining the dimension of a particle, but rarely used for determining its shape. The difficulty is that it needs to do extensive calculations. In addition, it can only be carried out for the mono-dispersion system. In re cent years, Reuther et al. succeeded in determining the shape and dimension of the particle of a polymer gel.

关键词： small angle X-ray scattering aluminium alloy shape of particle dimension of particle