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A quantum algorithm for Toeplitz matrix-vector multiplication
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作者 高尚 杨宇光 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第10期248-253,共6页
Toeplitz matrix-vector multiplication is widely used in various fields,including optimal control,systolic finite field multipliers,multidimensional convolution,etc.In this paper,we first present a non-asymptotic quant... Toeplitz matrix-vector multiplication is widely used in various fields,including optimal control,systolic finite field multipliers,multidimensional convolution,etc.In this paper,we first present a non-asymptotic quantum algorithm for Toeplitz matrix-vector multiplication with time complexity O(κpolylogn),whereκand 2n are the condition number and the dimension of the circulant matrix extended from the Toeplitz matrix,respectively.For the case with an unknown generating function,we also give a corresponding non-asymptotic quantum version that eliminates the dependency on the L_(1)-normρof the displacement of the structured matrices.Due to the good use of the special properties of Toeplitz matrices,the proposed quantum algorithms are sufficiently accurate and efficient compared to the existing quantum algorithms under certain circumstances. 展开更多
关键词 quantum algorithm Toeplitz matrix-vector multiplication circulant matrix
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Performance Prediction Based on Statistics of Sparse Matrix-Vector Multiplication on GPUs 被引量:1
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作者 Ruixing Wang Tongxiang Gu Ming Li 《Journal of Computer and Communications》 2017年第6期65-83,共19页
As one of the most essential and important operations in linear algebra, the performance prediction of sparse matrix-vector multiplication (SpMV) on GPUs has got more and more attention in recent years. In 2012, Guo a... As one of the most essential and important operations in linear algebra, the performance prediction of sparse matrix-vector multiplication (SpMV) on GPUs has got more and more attention in recent years. In 2012, Guo and Wang put forward a new idea to predict the performance of SpMV on GPUs. However, they didn’t consider the matrix structure completely, so the execution time predicted by their model tends to be inaccurate for general sparse matrix. To address this problem, we proposed two new similar models, which take into account the structure of the matrices and make the performance prediction model more accurate. In addition, we predict the execution time of SpMV for CSR-V, CSR-S, ELL and JAD sparse matrix storage formats by the new models on the CUDA platform. Our experimental results show that the accuracy of prediction by our models is 1.69 times better than Guo and Wang’s model on average for most general matrices. 展开更多
关键词 SPARSE matrix-vector multiplication Performance Prediction GPU Normal DISTRIBUTION UNIFORM DISTRIBUTION
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A small microring array that performs large complex-valued matrix-vector multiplication 被引量:10
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作者 Junwei Cheng Yuhe Zhao +7 位作者 Wenkai Zhang Hailong Zhou Dongmei Huang Qing Zhu Yuhao Guo Bo Xu Jianji Dong Xinliang Zhang 《Frontiers of Optoelectronics》 EI CSCD 2022年第2期1-15,共15页
As an important computing operation,photonic matrix-vector multiplication is widely used in photonic neutral networks and signal processing.However,conventional incoherent matrix-vector multiplication focuses on real-... As an important computing operation,photonic matrix-vector multiplication is widely used in photonic neutral networks and signal processing.However,conventional incoherent matrix-vector multiplication focuses on real-valued operations,which cannot work well in complex-valued neural networks and discrete Fourier transform.In this paper,we propose a systematic solution to extend the matrix computation of microring arrays from the real-valued field to the complex-valued field,and from small-scale(i.e.,4×4)to large-scale matrix computation(i.e.,16×16).Combining matrix decomposition and matrix partition,our photonic complex matrix-vector multiplier chip can support arbitrary large-scale and complex-valued matrix computation.We further demonstrate Walsh-Hardmard transform,discrete cosine transform,discrete Fourier transform,and image convolutional processing.Our scheme provides a path towards breaking the limits of complex-valued computing accelerator in conventional incoherent optical architecture.More importantly,our results reveal that an integrated photonic platform is of huge potential for large-scale,complex-valued,artificial intelligence computing and signal processing. 展开更多
关键词 Photonic matrix-vector multiplication Complex-valued computing Microring array Signal/image processing
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Technological Exploration of RRAM Crossbar Array for Matrix-Vector Multiplication 被引量:5
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作者 Lixue Xia Peng Gu +7 位作者 Boxun Li Tianqi Tang Xiling Yin Wenqin Huangfu Shimeng Yu Yu Cao Yu Wang Huazhong Yang 《Journal of Computer Science & Technology》 SCIE EI CSCD 2016年第1期3-19,共17页
Matrix-vector multiplication is the key operation for many computationally intensive algorithms.The emerging metal oxide resistive switching random access memory(RRAM)device and RRAM crossbar array have demonstrated a... Matrix-vector multiplication is the key operation for many computationally intensive algorithms.The emerging metal oxide resistive switching random access memory(RRAM)device and RRAM crossbar array have demonstrated a promising hardware realization of the analog matrix-vector multiplication with ultra-high energy efficiency.In this paper,we analyze the impact of both device level and circuit level non-ideal factors,including the nonlinear current-voltage relationship of RRAM devices,the variation of device fabrication and write operation,and the interconnect resistance as well as other crossbar array parameters.On top of that,we propose a technological exploration flow for device parameter configuration to overcome the impact of non-ideal factors and achieve a better trade-off among performance,energy,and reliability for each specific application.Our simulation results of a support vector machine(SVM)and Mixed National Institute of Standards and Technology(MNIST)pattern recognition dataset show that RRAM crossbar array based SVM is robust to input signal fluctuation but sensitive to tunneling gap deviation.A further resistance resolution test presents that a 6-bit RRAM device is able to realize a recognition accuracy around 90%,indicating the physical feasibility of RRAM crossbar array based SVM.In addition,the proposed technological exploration flow is able to achieve 10.98%improvement of recognition accuracy on the MNIST dataset and 26.4%energy savings compared with previous work.Experimental results also show that more than 84.4%power saving can be achieved at the cost of little accuracy reduction. 展开更多
关键词 resistive switching random access memory(RRAM) machine learning electronic design automation matrixvector multiplication non-ideal factor
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Products of multiplication,composition and differentiation on weighted Bergman spaces in the unit ball
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作者 ZHANG Chao QIN Yuehai 《中山大学学报(自然科学版)(中英文)》 北大核心 2025年第2期160-169,共10页
The boundness and compactness of products of multiplication,composition and differentiation on weighted Bergman spaces in the unit ball are studied.We define the differentiation operator on the space of holomorphic fu... The boundness and compactness of products of multiplication,composition and differentiation on weighted Bergman spaces in the unit ball are studied.We define the differentiation operator on the space of holomorphic functions in the unit ball by radial derivative.Then we extend the Sharma's results. 展开更多
关键词 composition operator multiplication operator differentiation operator weighted Bergman space
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Illusion of inactivity:Revisiting progressive multiple sclerosis treatment paradigms
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作者 Tal Ganz Tamir Ben-Hur 《Neural Regeneration Research》 2026年第3期1134-1135,共2页
Active inflammation in“inactive”progressive multiple sclerosis:Traditionally,the distinction between relapsing-remitting multiple sclerosis and progressive multiple sclerosis(PMS)has been framed as an inflammatory v... Active inflammation in“inactive”progressive multiple sclerosis:Traditionally,the distinction between relapsing-remitting multiple sclerosis and progressive multiple sclerosis(PMS)has been framed as an inflammatory versus degenerative dichotomy.This was based on a broad misconception regarding essentially all neurodegenerative conditions,depicting the degenerative process as passive and immune-independent occurring as a late byproduct of active inflammation in the central nervous system(CNS),which is(solely)systemically driven. 展开更多
关键词 inflammatory versus degenerative dichotomythis treatment paradigms illusion inactivity multiple sclerosis traditionallythe central nervous system cns which progressive multiple sclerosis active inflammation progressive multiple sclerosis pms
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Predicting lymph node metastasis in colorectal cancer using caselevel multiple instance learning
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作者 Ling-Feng Zou Xuan-Bing Wang +4 位作者 Jing-Wen Li Xin Ouyang Yi-Ying Luo Yan Luo Cheng-Long Wang 《World Journal of Gastroenterology》 2026年第1期110-125,共16页
BACKGROUND The accurate prediction of lymph node metastasis(LNM)is crucial for managing locally advanced(T3/T4)colorectal cancer(CRC).However,both traditional histopathology and standard slide-level deep learning ofte... BACKGROUND The accurate prediction of lymph node metastasis(LNM)is crucial for managing locally advanced(T3/T4)colorectal cancer(CRC).However,both traditional histopathology and standard slide-level deep learning often fail to capture the sparse and diagnostically critical features of metastatic potential.AIM To develop and validate a case-level multiple-instance learning(MIL)framework mimicking a pathologist's comprehensive review and improve T3/T4 CRC LNM prediction.METHODS The whole-slide images of 130 patients with T3/T4 CRC were retrospectively collected.A case-level MIL framework utilising the CONCH v1.5 and UNI2-h deep learning models was trained on features from all haematoxylin and eosinstained primary tumour slides for each patient.These pathological features were subsequently integrated with clinical data,and model performance was evaluated using the area under the curve(AUC).RESULTS The case-level framework demonstrated superior LNM prediction over slide-level training,with the CONCH v1.5 model achieving a mean AUC(±SD)of 0.899±0.033 vs 0.814±0.083,respectively.Integrating pathology features with clinical data further enhanced performance,yielding a top model with a mean AUC of 0.904±0.047,in sharp contrast to a clinical-only model(mean AUC 0.584±0.084).Crucially,a pathologist’s review confirmed that the model-identified high-attention regions correspond to known high-risk histopathological features.CONCLUSION A case-level MIL framework provides a superior approach for predicting LNM in advanced CRC.This method shows promise for risk stratification and therapy decisions,requiring further validation. 展开更多
关键词 Colorectal cancer Lymph node metastasis Deep learning multiple instance learning HISTOPATHOLOGY
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Effects of Exogenous Hormones on Subculture Multiplication and Root Induction of Tissuecultured Seedlings of Mulberry 被引量:2
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作者 朱宏 韦鹏霄 +2 位作者 岑秀芬 植菊芳 周芳芳 《Agricultural Science & Technology》 CAS 2014年第5期760-764,共5页
Objective The purpose of this experiment was to investigate the effects of exogenous hormone factors on subculture multiplication and root induction of tis-suecultured seedlings in mulberry. [Method] The aseptic seedl... Objective The purpose of this experiment was to investigate the effects of exogenous hormone factors on subculture multiplication and root induction of tis-suecultured seedlings in mulberry. [Method] The aseptic seedlings of Guisangyou 12 were used as the materials and the comparative experiment on subculture multi-plication and root induction used different kinds,concentrations and combinations of exogenous hormones. [Result] The hormone combination of 0.1 mg/L IBA+1.5 mg/L 6-BA+0.03 mg/L TDZ showed the best effects to bud multiplication, and the multiple of bud multiplication was 5.28. The effects of NAA was better than IBA and IAA on root induction. The hormone combination of 2.0 mg/L NAA+1.0 mg/L PP333 was most suitable to the root induction, in which the rooting rate arrived to 100%, the root number was 7.01 and the root length was 1.38 cm on average. [Conclusion] The results wil provide some technical reference for large-scale propagation of mul-berry seedlings in vitro. 展开更多
关键词 MULBERRY Aseptic seedlings Exogenous hormone Bud multiplication Root induction
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Study on Multiplication, Rooting and Transplanting of Tissue Culture Plantlets of Rhododendron chrysanthum Pall 被引量:1
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作者 王蕾 巴春影 +2 位作者 曹后男 宗成文 姚航 《Agricultural Science & Technology》 CAS 2015年第7期1348-1351,1373,共5页
[Objective] This study aimed to shorten the multiplication culture and root- ing culture periods of Rh. chrysanthum Pall. [Method] The Rh. chrysanthum Pall tis- sue culture plantlets collected from Changbai Mountain w... [Objective] This study aimed to shorten the multiplication culture and root- ing culture periods of Rh. chrysanthum Pall. [Method] The Rh. chrysanthum Pall tis- sue culture plantlets collected from Changbai Mountain were used as material, and the effects of different hormone combinations and coconut milk on the proliferation and differentiation of Rh. chrysanthum Pall tissue culture plantlets were investigated. In addition, the rooting medium and transplanting matrix for Rh. chrysanthum Pall tissue culture plantlets were explored. [Result] The medium composed of modified MS, iBA (3 mg/L) and ZT (1.5 mg/L) was the optimum medium for subculture mul- tiplication of Rh. chrysanthum Pall tissue culture plantlets. The multiplication multiple and average plant height were significantly improved by adding coconut milk into the medium (150 mg/L). [Conclusion] For Rh. chrysanthum Pall tissue culture plantlets, the optimum rooting culture medium was composed of modified MS (1/4) and IBA (5.0 mg/L), and the tissue culture plantlets began to root 8 d after the inoculation. The root induction treatment was carried out after a 15-d sand culture, and the suitable matrix was composed of tufty soil, humus soil and perlite (2:1:1) with a survival rate of 95.66%. 展开更多
关键词 Rh. chrysanthum Pall multiplication culture ROOTING Transplanting matrix
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Study on Designing Standards System Framework of Hainan National Breeding and Multiplication
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作者 陈冠铭 杨静 +2 位作者 林亚琼 汪李平 李劲松 《Agricultural Science & Technology》 CAS 2012年第7期1437-1442,共6页
A preliminary feasibility study was set up for the HNBM standard system. By introducing the set theory and basic methods of the solid geometry and adopting the Verman three-dimensional structure ideology innovatively,... A preliminary feasibility study was set up for the HNBM standard system. By introducing the set theory and basic methods of the solid geometry and adopting the Verman three-dimensional structure ideology innovatively, the standards system of Hainan National Breeding and Multiplication (HNBM) was studied with the four-dimensional structure and integration by combining with the time dimension. The four-dimensional structure standards system will be useful for guiding the construction of the HNBM standards system, carrying on the division of labor and realizing the dynamic development of the standards system. The standards set of HNBM helps to computer programming and information management. 展开更多
关键词 Hainan National Breeding and multiplication Four-dimensional standards system ASSEMBLING Dynamic system stardards set
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Research on a Method for Purification and Multiplication of PTGMS Lines
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作者 何强 庞镇宇 +4 位作者 邢俊杰 舒服 张武汉 杨益善 邓华凤 《Agricultural Science & Technology》 CAS 2014年第3期352-354,366,共4页
[Objective] The aims was to explore PTGMS line purification and multipli- cation technology. [Method] Three commercial PTGMS lines, Annong 810S, P88S and C815S, were used in this study. In the first year, two sterile ... [Objective] The aims was to explore PTGMS line purification and multipli- cation technology. [Method] Three commercial PTGMS lines, Annong 810S, P88S and C815S, were used in this study. In the first year, two sterile line populations with same genetic background were multiplied by dividing seedling; meanwhile, se- lection and multiplication of the core individuals were conducted; within the year, the core seeds harvested were multiplied under the isolated condition in Hainan, and at the same time, superior individuals were selected and used as the seeds for two populations with same genetic background in the next year. In the second year, seeds harvested in Hainan were planted in Changsha, while the core individuals were selected and multiplied; the core seeds harvested were continued to multiply in Hainan. The above procedures were repeated in the following years. [Result] The test results showed that in two years the proportion of core individuals in the popu- lation of Annong 810S, P88S and C815S increased from 40.3%, 0.7% and 0.3% in the first year to 71.67%, 66.00% and 68.67% in the second year respectively. [Conclusion] This suggested that this method of purification and multiplication of the PTGMS lines for consecutive years can effectively control the drift of the critical sterility inducing temperature, and at the same time, core seeds with high purity and controlled inducing temperature can be harvested. 展开更多
关键词 PTGMS PURIFICATION multiplication RICE
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Fast and secure elliptic curve scalar multiplication algorithm based on special addition chains
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作者 刘双根 胡予濮 《Journal of Southeast University(English Edition)》 EI CAS 2008年第1期29-32,共4页
To resist the side chaimel attacks of elliptic curve cryptography, a new fast and secure point multiplication algorithm is proposed. The algorithm is based on a particular kind of addition chains involving only additi... To resist the side chaimel attacks of elliptic curve cryptography, a new fast and secure point multiplication algorithm is proposed. The algorithm is based on a particular kind of addition chains involving only additions, providing a natural protection against side channel attacks. Moreover, the new addition formulae that take into account the specific structure of those chains making point multiplication very efficient are proposed. The point multiplication algorithm only needs 1 719 multiplications for the SAC260 of 160-bit integers. For chains of length from 280 to 260, the proposed method outperforms all the previous methods with a gain of 26% to 31% over double-and add, 16% to22% over NAF, 7% to 13% over4-NAF and 1% to 8% over the present best algorithm--double-base chain. 展开更多
关键词 scalar multiplication algorithm special addition chains side channel attacks double base chain
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Analysis of the Key Issues about the Regularized Construction of Hainan National Breeding and Multiplication Base
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作者 陈冠铭 林亚琼 +1 位作者 李劲松 汪李平 《Agricultural Science & Technology》 CAS 2014年第9期1450-1456,1465,共8页
Based on winter nursery and shuttle breeding, Hainan National Breeding and Multiplication (HNBM), with off-season breeding and seed production as the essence, has become a national strategic resource. The way of HNB... Based on winter nursery and shuttle breeding, Hainan National Breeding and Multiplication (HNBM), with off-season breeding and seed production as the essence, has become a national strategic resource. The way of HNBM rising to the national strategy was analyzed through the aspects of technology features, historical characteristics, irreplaceability, function and value hierarchy, and the key problems. As the major achievement of the practice of scientific research and historical selec- tion, HNBM played an irreplaceable role in the climatic conditions, scientific breed- ing, and seed production in disaster relief, protecting the safety of the seed industry and Iocational advantages. Based on the analysis on the functions, values and ex- isted problems of HNBM, the function and value hierarchical structure of HNBM was established through introducing concepts of value engineering, industrial clusters and regional economy. But to fully achieve the industrial clustering development and functional values of HNBM, it needed to solve the key problems existed in top-level planning, experimental bases, system design, hardware construction and ecological security. Some suggestions were put forward in this paper, including setting up key scientific research protection zones, improving the regulations and policies safeguard mechanism and management system of HNBM, drawing up and carrying out the base development planning of HNBM, and striving for financial support from national policies. 展开更多
关键词 Hainan National Breeding Basic functions: Technoloov functions: and multiplication (HNBM) Value engineering Social value: Location of HNBM
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POINTWISE MULTIPLICATION OPERATORS FROM HARDY SPACES TO WEIGHTED BERGMAN SPACES IN THE UNIT BALL OF ■ 被引量:2
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作者 Ru PENG Xiaolei XING Liangying JIANG 《Acta Mathematica Scientia》 SCIE CSCD 2019年第4期1003-1016,共14页
This article is devoted to characterizing the boundedness and compactness of multiplication operators from Hardy spaces to weighted Bergman spaces in the unit ball of ■.
关键词 Poin twise multiplication opera TORS Hardy SPACES BERGMAN SPACES CARLESON measure
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ON MULTIPLICATION AND COMULTIPLICATION MODULES 被引量:2
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作者 H. Ansari-Toroghy F. Farshadifar 《Acta Mathematica Scientia》 SCIE CSCD 2011年第2期694-700,共7页
This article deals with some results concerning multiplication and comultipli- cation modules over a commutative ring.
关键词 multiplication modules comultiplication modules cocyclic modules
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Studies on Rhizome Multiplication and Differentiation of Cymbidium sinense × Cymbidium goeringii 被引量:2
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作者 Lijuan ZUO Ling YU Bo CHEN 《Agricultural Biotechnology》 CAS 2017年第6期22-25,共4页
With the rhizomes obtained from the germination of Cymbidium sinense ~ Cymbidium goeringii seeds as a material, the effects of different basic media and different plant growth regulators on the multiplication and diff... With the rhizomes obtained from the germination of Cymbidium sinense ~ Cymbidium goeringii seeds as a material, the effects of different basic media and different plant growth regulators on the multiplication and differentiation of rhizomes and rooting and transplanting of seedlings were investigated. The results showed that the basal medium more suitable for the multiplication and differentiation of rhizomes was 1/2MS; the medium more suitable for the multiplication and differentiation of rhizomes of the hybrid orchid was 1/2MS + 6 - BA 2.0 mg/L + KT 0.5 mg/L + NAA 0.05 mg/L; the medium more suitable for the rooting of the hybrid orchid was 1/2MS + IBA 0.5 mg/L; and after training of seedlings, transplanting was performed onto moss, and after 25 d, the survival rate was 95.6%. 展开更多
关键词 RHIZOME multiplication DIFFERENTIATION ROOTING
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Shoot multiplication of Syringa reticulata var.mandshurica from in vitro cultured seedlings 被引量:1
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作者 Chun-ping Liu Hua-ying Liu +3 位作者 Bo-wen Yang Ling Yang Peng Zhang Hai-long Shen 《Journal of Forestry Research》 SCIE CAS CSCD 2017年第1期41-46,共6页
We developed a shoot multiplication protocol for Syringa reticulata Blume vat. mandshurica Hara from in vitro cultured seedlings that derived from in vitro germinated seeds. The shoots could be induced on Murashige an... We developed a shoot multiplication protocol for Syringa reticulata Blume vat. mandshurica Hara from in vitro cultured seedlings that derived from in vitro germinated seeds. The shoots could be induced on Murashige and Skoog (MS) medium with proper plant growth regulator combinations of 6-benzylaminopurine (BA) and indole-3-butyric acid (IBA). The better medium for shoot multiplication and growth was MS + 5 mg L-1 BA + 0.5 mg L-1 IBA + 20 g L-1 sucrose + 7 g L-1 agar, and the corresponding shoot induction rate was 75 %. The plantlets grew well after rooting on I/2MS medium (macro-elements of MS medium are at half-strength) sup- plemented with 1 mg L-1 IBA, and the survival percentage was 〉80 % at 16 weeks after transplanting. 展开更多
关键词 Manchurian lilac MICROPROPAGATION Plantgrowth regulator Shoot multiplication Sucroseconcentration
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Electro dislocation multiplication and strain effect in 2091 Al-Li alloy 被引量:1
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作者 刘志义 雷毅 李山东 《中国有色金属学会会刊:英文版》 CSCD 2000年第1期39-43,共5页
Electro dislocation multiplication and strain effect were investigated. TEM observation showed that when statically held for 15 min, the microstructure of dislocation cell in the specimens without current pulse turned... Electro dislocation multiplication and strain effect were investigated. TEM observation showed that when statically held for 15 min, the microstructure of dislocation cell in the specimens without current pulse turned to subgrains with few dislocations in cold rolled state, but when current pulse charged, there were a lot of dislocations although a more rapid recrystallization occurred concurrently; when deformed, the interior dislocations in the specimens without current pulse were annihilated due to dynamic recrystallization, but were not decreased in the current pulse charged specimens in dynamic recrystallization, instead increased. It is indicated that “electron wind” force created by current pulse produced a shear stress for dislocation slip, and caused a dislocation multiplication in static holding of 2091 Al Li alloy; when strain exerted, an additional shear stress was provided to increase the effect of current pulse on dislocation multiplication. 展开更多
关键词 Al Li alloy DEFORMATION DISLOCATION multiplication current PULSE
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Elliptic Curve Scalar Multiplication with x-Coordinate 被引量:1
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作者 ZHANG Ning PEI Qingqi XIAO Guozhen 《Wuhan University Journal of Natural Sciences》 CAS 2007年第1期163-166,共4页
This paper gives a comprehensive method to do Elliptic Curve Scalar Multiplication with only x-coordinate. Explicit point operation formulae for all types of defining equations of the curves are derived. For each type... This paper gives a comprehensive method to do Elliptic Curve Scalar Multiplication with only x-coordinate. Explicit point operation formulae for all types of defining equations of the curves are derived. For each type of curve, the performance is analyzed. The formulae are applied in Montgomery Ladder to get scalar multiplication algorithm operated with only x-coordinate. The new scalar multiplication has the same security level and computation amount with protected binary scalar multiplication (PBSM) against side channel attack, and has the advantages of higher security and little memory needed. 展开更多
关键词 elliptic curve scalar multiplication Montgomery ladder x-coordinate
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MULTIPLICATION OPERATORS ON INVARIANT SUBSPACES OF FUNCTION SPACES 被引量:1
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作者 B.YOUSEFI Sh.KHOSHDEL Y.JAHANSHAHI 《Acta Mathematica Scientia》 SCIE CSCD 2013年第5期1463-1470,共8页
Let Mφ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator Mz, we derive some spectral properties of the... Let Mφ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator Mz, we derive some spectral properties of the multiplication operator Mφ : F→F. We characterize norm, spectrum, essential norm and essential spectrum of such operators when F has the codimension n property with n∈{1,2,...,+∞}. 展开更多
关键词 invariant subspace Hilbert space of analytic functions essential spectrum essential norm Fredholm operator multiplication operator
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