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Power forecasting method of ultra-short-term wind power cluster based on the convergence cross mapping algorithm
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作者 Yuzhe Yang Weiye Song +5 位作者 Shuang Han Jie Yan Han Wang Qiangsheng Dai Xuesong Huo Yongqian Liu 《Global Energy Interconnection》 2025年第1期28-42,共15页
The development of wind power clusters has scaled in terms of both scale and coverage,and the impact of weather fluctuations on cluster output changes has become increasingly complex.Accurately identifying the forward... The development of wind power clusters has scaled in terms of both scale and coverage,and the impact of weather fluctuations on cluster output changes has become increasingly complex.Accurately identifying the forward-looking information of key wind farms in a cluster under different weather conditions is an effective method to improve the accuracy of ultrashort-term cluster power forecasting.To this end,this paper proposes a refined modeling method for ultrashort-term wind power cluster forecasting based on a convergent cross-mapping algorithm.From the perspective of causality,key meteorological forecasting factors under different cluster power fluctuation processes were screened,and refined training modeling was performed for different fluctuation processes.First,a wind process description index system and classification model at the wind power cluster level are established to realize the classification of typical fluctuation processes.A meteorological-cluster power causal relationship evaluation model based on the convergent cross-mapping algorithm is pro-posed to screen meteorological forecasting factors under multiple types of typical fluctuation processes.Finally,a refined modeling meth-od for a variety of different typical fluctuation processes is proposed,and the strong causal meteorological forecasting factors of each scenario are used as inputs to realize high-precision modeling and forecasting of ultra-short-term wind cluster power.An example anal-ysis shows that the short-term wind power cluster power forecasting accuracy of the proposed method can reach 88.55%,which is 1.57-7.32%higher than that of traditional methods. 展开更多
关键词 Ultra-short-term wind power forecasting Wind power cluster Causality analysis convergence cross mapping algorithm
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Complete Convergenceand Complete Moment Convergence for Weighted Sums of ANA Random Variables
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作者 MENG Bing WU Qunying 《应用概率统计》 CSCD 北大核心 2024年第5期710-724,共15页
In this paper,we investigate the complete convergence and complete moment conver-gence for weighted sums of arrays of rowwise asymptotically negatively associated(ANA)random variables,without assuming identical distri... In this paper,we investigate the complete convergence and complete moment conver-gence for weighted sums of arrays of rowwise asymptotically negatively associated(ANA)random variables,without assuming identical distribution.The obtained results not only extend those of An and Yuan[1]and Shen et al.[2]to the case of ANA random variables,but also partially improve them. 展开更多
关键词 ANA random variables complete convergence complete moment convergence weighted sums
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Diagnostic values of questionnaires of Convergence Insufficiency Symptom Survey and College of Optometrists Vision Development Quality of Life in the screening of convergence insufficiency
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作者 Ling Xiong Qian Chen Ye Wu 《International Journal of Ophthalmology(English edition)》 SCIE CAS 2024年第5期904-908,共5页
AIM:To compare and analyse the diagnostic efficacy of the College of Optometrists Vision Development Quality of Life Questionnaire(COVD-QOL)and the Convergence Insufficiency Symptom Survey(CISS)in detecting convergenc... AIM:To compare and analyse the diagnostic efficacy of the College of Optometrists Vision Development Quality of Life Questionnaire(COVD-QOL)and the Convergence Insufficiency Symptom Survey(CISS)in detecting convergence insufficiency and to compare their diagnostic value in clinical applications.METHODS:Using the diagnostic test method,62 adult patients with convergence insufficiency(age:24.74±3.75y)and 62 normal participants(age:23.61±3.13y)who visited the Optometry Clinic of West China Hospital of Sichuan University from April 2021 to January 2023 were included.All subjects completed the CISS and COVD-QOL.Statistical analysis of the sensitivity and specificity of the CISS and COVD-QOL and comparison and joint experimental analysis of their diagnostic efficacy were performed.RESULTS:The sensitivity of the CISS and COVD-QOL for convergence insufficiency was 64.5%and 71.0%,respectively,while the specificity was 96.8%and 67.7%,respectively.Compared to the CISS alone,the combination of the CISS and COVD-QOL demonstrated lower sensitivity and specificity.The areas under the receiver operating characteristic curve of CISS,COVD-QOL and CISS combined with COVD-QOL were 0.806,0.694 and 0.782,respectively.CONCLUSION:Considering the low sensitivity of the CISS and the low specificity of the COVD-QOL,it is recommended to supplement these questionnaires with other screening tests for the detection of convergence insufficiency. 展开更多
关键词 convergence insufficiency symptom survey College of Optometrists Vision Development Quality of Life Questionnaire convergence insufficiency ASTHENOPIA
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GLOBAL CONVERGENCE OF A CAUTIOUS PROJECTION BFGS ALGORITHM FOR NONCONVEX PROBLEMS WITHOUT GRADIENT LIPSCHITZ CONTINUITY
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作者 Gonglin YUAN Xiong ZHAO Jiajia YU 《Acta Mathematica Scientia》 SCIE CSCD 2024年第5期1735-1746,共12页
A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems.The global convergence of this method as well as a stronger general convergence result can be proven without a gr... A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems.The global convergence of this method as well as a stronger general convergence result can be proven without a gradient Lipschitz continuity assumption,which is more in line with the actual problems than the existing modified BFGS methods and the traditional BFGS method.Under some additional conditions,the method presented has a superlinear convergence rate,which can be regarded as an extension and supplement of BFGS-type methods with the projection technique.Finally,the effectiveness and application prospects of the proposed method are verified by numerical experiments. 展开更多
关键词 cautious BFGS nonconvex problems Lipschitz continuity projection technique global convergence
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Convergence Zone Width Analysis Based on Ray Cluster Theory and Its Application in the Array Depth Optimization of Active Sensors
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作者 HAN Zhibin SONG Jun +3 位作者 PENG Zhaohui MENG Lei YANG Hua SU Bing 《Journal of Ocean University of China》 SCIE CAS CSCD 2024年第6期1460-1468,共9页
The active sensor often uses the convergence zone mode to detect a distant target in the deep ocean.However,convergence zones are regions with limited widths that only appear at some discrete distances.Thus,widening t... The active sensor often uses the convergence zone mode to detect a distant target in the deep ocean.However,convergence zones are regions with limited widths that only appear at some discrete distances.Thus,widening the width by adjusting the transmitting array depth facilitates target observation and detection.Traversal search is an effective method for determining the optimal depth,but the heavy computation burden resulting from the calculation of the transmission losses at all source depths impedes its application.To solve the problem,a fast method based on ray cluster theory is proposed.Due to the coherent sound field structure in the deep ocean,several ray clusters with different departure angles radiate from the source,where ray clusters with small departure angles reverse in the water and form a convergence zone.When the source is set to a depth that only the first ray cluster inverts in water,the maximum width of the convergence zone is obtained.Based on this,an optimal transmitting array depth selection method utilizing the reversion condition of the first ray cluster is formulated.Simulation results show that the active sensor can achieve a large convergence zone width with real-time performance using the proposed method. 展开更多
关键词 convergence zone deep ocean active sensor transmitting array depth ray cluster
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Quantitative analysis and prediction of the sound field convergence zone in mesoscale eddy environment based on data mining methods
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作者 Ming Li Yuhang Liu +1 位作者 Yiyuan Sun Kefeng Liu 《Acta Oceanologica Sinica》 SCIE CAS CSCD 2024年第5期110-120,共11页
The mesoscale eddy(ME)has a significant influence on the convergence effect in deep-sea acoustic propagation.This paper use statistical approaches to express quantitative relationships between the ME conditions and co... The mesoscale eddy(ME)has a significant influence on the convergence effect in deep-sea acoustic propagation.This paper use statistical approaches to express quantitative relationships between the ME conditions and convergence zone(CZ)characteristics.Based on the Gaussian vortex model,we construct various sound propagation scenarios under different eddy conditions,and carry out sound propagation experiments to obtain simulation samples.With a large number of samples,we first adopt the unified regression to set up analytic relationships between eddy conditions and CZ parameters.The sensitivity of eddy indicators to the CZ is quantitatively analyzed.Then,we adopt the machine learning(ML)algorithms to establish prediction models of CZ parameters by exploring the nonlinear relationships between multiple ME indicators and CZ parameters.Through the research,we can express the influence of ME on the CZ quantitatively,and achieve the rapid prediction of CZ parameters in ocean eddies.The prediction accuracy(R)of the CZ distance(mean R:0.9815)is obviously better than that of the CZ width(mean R:0.8728).Among the three ML algorithms,Gradient Boosting Decision Tree has the best prediction ability(root mean square error(RMSE):0.136),followed by Random Forest(RMSE:0.441)and Extreme Learning Machine(RMSE:0.518). 展开更多
关键词 convergence zone mesoscale eddy statistic analysis quantitative prediction machine learning
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Valence Bands Convergence in p-Type CoSb_(3) through Electronegative Fluorine Filling
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作者 黄写格 李家良 +5 位作者 马浩钦 李昌隆 刘天乐 段波 翟鹏程 李国栋 《Chinese Physics Letters》 SCIE EI CAS CSCD 2024年第7期87-94,共8页
Band convergence is considered to be a strategy with clear benefits for thermoelectric performance,generally favoring the co-optimization of conductivity and Seebeck coefficients,and the conventional means include ele... Band convergence is considered to be a strategy with clear benefits for thermoelectric performance,generally favoring the co-optimization of conductivity and Seebeck coefficients,and the conventional means include elemental filling to regulate the band.However,the influence of the most electronegative fluorine on the CoSb_(3) band remains unclear.We carry out density-functional-theory calculations and show that the valence band maximum gradually shifts downward with the increase of fluorine filling,lastly the valence band maximum converges to the highly degenerated secondary valence bands in fluorine-filled skutterudites. 展开更多
关键词 VALENCE convergence BANDS
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On power series statistical convergence and new uniform integrability of double sequences
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作者 Sevda Y■ld■z Kamil Demirci 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2024年第3期519-532,共14页
In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p... In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings. 展开更多
关键词 power series methods statistical convergence uniform integrability double sequences
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Spatial Differentiation and Convergence Trend of High-quality Development Level of China’s Tourism Economy
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作者 CAO Shanshan HE Zhaoli +2 位作者 NIU Jinlan WANG Songmao ZHAO Lei 《Chinese Geographical Science》 SCIE CSCD 2024年第2期230-249,共20页
This paper aims to interpret the connotation of high-quality development of tourism economy(HQTE)from the perspective of the new development concepts of innovation,coordination,green,openness and sharing,and then to e... This paper aims to interpret the connotation of high-quality development of tourism economy(HQTE)from the perspective of the new development concepts of innovation,coordination,green,openness and sharing,and then to evaluate the spatial differenti-ation of China’s HQTE based on provincial panel data from 2009 to 2018.Specifically,we employ the spatial convergence model to ex-plore the absolute and conditionalβconvergence trends of HQTE in the whole country and the eastern,central and western regions of China.Our empirical results reveal that:1)within the decade,from 2009 to 2018,regions of China with the highest HQTE index is its eastern region followed by the central region and then the western region,but the fastest growing one is the western region of China fol-lowed by the central region and then the eastern region.2)Whether or not the spatial effect is included,there are absolute and condition-alβconvergence in HQTE in the whole country and aforementioned three regions.3)The degree of government attention as well as the level of economic development and location accessibility are the positive driving factors for the convergence of HQTE in the whole country and the three regions.The degree of marketization and human capital have not passed the significance test either in the whole country or in the three regions.The above conclusions could deepen the understanding of the regional imbalance and spatial conver-gence characteristics of HQTE,clarify the primary development objects,and accomplish the goal of China’s HQTE. 展开更多
关键词 new development concepts high-quality development of tourism economy(HQTE) spatial convergence China
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Call for Papers——Feature Topic Vol.22,No.3,2025 Special Issue of China Communications:Convergence of 6G empowered Edge Intelligence and Generative AI:Theories,Algorithms,and Applications
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《China Communications》 SCIE CSCD 2024年第1期F0003-F0003,共1页
Generative artificial intelligence(AI),as an emerging paradigm in content generation,has demonstrated its great potentials in creating high-fidelity data including images,texts,and videos.Nowadays wireless networks an... Generative artificial intelligence(AI),as an emerging paradigm in content generation,has demonstrated its great potentials in creating high-fidelity data including images,texts,and videos.Nowadays wireless networks and applications have been rapidly evolving from achieving“connected things”to embracing“connected intelligence”.Generative AI has been recognized as a fundamentally innovative technology to drive the advancement of intelligent wireless communications and networks. 展开更多
关键词 CONNECTED convergence networks
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Multi-modal knowledge graph inference via media convergence and logic rule
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作者 Feng Lin Dongmei Li +5 位作者 Wenbin Zhang Dongsheng Shi Yuanzhou Jiao Qianzhong Chen Yiying Lin Wentao Zhu 《CAAI Transactions on Intelligence Technology》 SCIE EI 2024年第1期211-221,共11页
Media convergence works by processing information from different modalities and applying them to different domains.It is difficult for the conventional knowledge graph to utilise multi-media features because the intro... Media convergence works by processing information from different modalities and applying them to different domains.It is difficult for the conventional knowledge graph to utilise multi-media features because the introduction of a large amount of information from other modalities reduces the effectiveness of representation learning and makes knowledge graph inference less effective.To address the issue,an inference method based on Media Convergence and Rule-guided Joint Inference model(MCRJI)has been pro-posed.The authors not only converge multi-media features of entities but also introduce logic rules to improve the accuracy and interpretability of link prediction.First,a multi-headed self-attention approach is used to obtain the attention of different media features of entities during semantic synthesis.Second,logic rules of different lengths are mined from knowledge graph to learn new entity representations.Finally,knowledge graph inference is performed based on representing entities that converge multi-media features.Numerous experimental results show that MCRJI outperforms other advanced baselines in using multi-media features and knowledge graph inference,demonstrating that MCRJI provides an excellent approach for knowledge graph inference with converged multi-media features. 展开更多
关键词 logic rule media convergence multi-modal knowledge graph inference representation learning
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Smartphone use patterns and the impact on accommodation and convergence system of the eyes among Malaysian teenagers
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作者 Thilageswary Chellapan Norlaila Mat Daud Sumithira Narayanasamy 《International Journal of Ophthalmology(English edition)》 SCIE CAS 2024年第11期2093-2099,共7页
AIM:To determine the smartphone use patterns and effects of smartphone use on accommodation and convergence system of the eyes among Malaysian teenagers.METHODS:A total of 62 participants aged between 13 and 17y were ... AIM:To determine the smartphone use patterns and effects of smartphone use on accommodation and convergence system of the eyes among Malaysian teenagers.METHODS:A total of 62 participants aged between 13 and 17y were involved.A self-administered questionnaires containing 12 items was used to evaluate the smartphone usage patterns.This was followed by an eye examination,involving a battery of accommodation and convergence assessments before and after the smartphone use.The data analysis comprised descriptive statistics,paired t-test,and correlation coefficients.RESULTS:The use of smartphones is at a high level and at an optimal distance daily,with more than 6h a day watching video films,games,and completing school projects.Majority of the participants not reported eye strain factors and eye prescription changes with the use of digital devices.The use of a smartphone continuously for 30min was found to significantly decrease amplitude of accommodation,accommodative facility,and positive relative accommodation(P<0.001).Meanwhile,the lag of accommodation parameters and negative relative accommodation increased with the use of smartphones significantly(P<0.001).The near point of convergence(NPC)and distance and near negative fusional vergence decreased significantly(P<0.001).The NPC parameter was found to have a weak negative association with the frequency of smartphone use(R=-0.276,P<0.05).CONCLUSION:Frequent and continuous use of smartphones have increased visual stress and resulted in weakness of accommodation and vergence functions.Therefore,frequent break is mandatory when using a smartphone and appropriate visual hygiene,the 20-20-20 rule(every 20min,view something 20 feet away for 20s)are required during smartphone use to maintain visual function. 展开更多
关键词 smartphone use patterns accommodation and convergence visual function visual stress
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Some convergence theorems of fuzzy concave integral on fuzzyσ-algebra
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作者 SUN Rong 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2024年第3期438-447,共10页
In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some ... In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved. 展开更多
关键词 convergence theorems fuzzy concave integral fuzzyσ-algebra
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Convergence of Hyperbolic Neural Networks Under Riemannian Stochastic Gradient Descent
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作者 Wes Whiting Bao Wang Jack Xin 《Communications on Applied Mathematics and Computation》 EI 2024年第2期1175-1188,共14页
We prove,under mild conditions,the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model,both in batch gradient descent and stochastic gradient descent.We also discuss a ... We prove,under mild conditions,the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model,both in batch gradient descent and stochastic gradient descent.We also discuss a Riemannian version of the Adam algorithm.We show numerical simulations of these algorithms on various benchmarks. 展开更多
关键词 Hyperbolic neural network Riemannian gradient descent Riemannian Adam(RAdam) Training convergence
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Distribution dynamics,regional differences,and convergence of sustainable development of cities and communities in China
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作者 Guang Yang Mingle Li Chaofeng Shao 《Chinese Journal of Population,Resources and Environment》 2024年第4期443-454,共12页
China is one of the most populated and rapidly urbanizing countries worldwide and was among the earliest countries to integrate sustainable development into urban construction.To achieve high-quality development and i... China is one of the most populated and rapidly urbanizing countries worldwide and was among the earliest countries to integrate sustainable development into urban construction.To achieve high-quality development and implement the objectives of“Transforming Our World:The 2030 Agenda for Sustainable Development”,it is crucial to measure and analyze the current level of sustainable development of cities in China.Following the principles of relevance,scientific rigor,universality,reliability,and timeliness,this study constructs an assessment index system for sustainable development,covering seven themes corresponding to the UN Sustainable Development Goal 11.Through detailed calculations,we obtained sustainable development indices for 139 Chinese cities from 2016 to 2022 and analyzed them in three dimensions:distribution dynamics,regional differences,and convergence.The key findings are as follows.First,the level of sustainable development showed improvement,with the average score of included cities increasing by 11.88%from 2016 to 2022.Second,the level of sustainable development was relatively balanced,maintaining low Gini coefficients between 0.05 and 0.06.Third,a weak overallσconvergence feature existed,with increased differentiation in 2021.From a regional perspective,aσconvergence feature was observed in the northeastern but not in the western region.Fourth,both overall absoluteβ-convergence and conditionalβeffects were significant.Regional absoluteβ-convergence and conditionalβ-convergence were also significant.This study contributes to the literature by providing evidence of China’s urban sustainable development,offering policy insights for deepening the implementation of development goals in the future,and providing experiential reference for other developing countries to achieve sustainable development. 展开更多
关键词 Urban sustainability Sustainable Development Goal 11 Dynamic evolution of distribution Regional differences Spatial convergence
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Exploring the Dynamics of Industrial Convergence Among RCEP Member Countries: An Empirical Study
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作者 Zheng Yuanfang 《Contemporary Social Sciences》 2024年第5期26-50,共25页
The evolving dynamics of industrial convergence among the member countries of the Regional Comprehensive Economic Partnership(RCEP)framework have emerged as a significant subject that merits in-depth consideration and... The evolving dynamics of industrial convergence among the member countries of the Regional Comprehensive Economic Partnership(RCEP)framework have emerged as a significant subject that merits in-depth consideration and analysis.This study initially employs multi-regional input-output(MRIO)data and the social network analysis(SNA)method to delineate the levels and variation trends of this industrial convergence across the RCEP member countries.It then delves into the positive effects of this convergence phenomenon on the trade and investment fields of the member countries.The research findings indicate:(a)In 2006 and 2015,before the implementation of the RCEP,the RCEP member countries displayed a relatively close industrial convergence.The convergence levels exhibited a general upward trend on both the supply and the demand sides,but there were significant disparities in the levels of industrial convergence among the member countries.Furthermore,while the convergence in the three economic sectors showed an increasing trend,the development was uneven across the board.(b)Since the implementation of the RCEP,the trade ties among the member countries within the region have strengthened significantly,and the interplay between the countries’industrial and supply chains has been characterized by high-quality collaboration and demonstrated remarkable resilience.In addition,the convergence in the investment fields of the RCEP member countries and their respective industries has unleashed a wave of positive synergies.These findings offer valuable insights that can serve as a robust foundation for formulating effective policies to advance the growth and prosperity of the RCEP region. 展开更多
关键词 RCEP international industrial convergence input-output table social network analysis method
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Selective Technology Importation and the Convergence in the Late Qing Dynasty: Evidence from Wired Telegraph
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作者 Jiao Yinyi Cai Meng 《China Economist》 2024年第5期117-136,共20页
Using the adoption and expansion of wired telegraph in China during the late 19^(th) century,this paper investigates the effect of cost reductions for knowledge exchange on China’s industrial growth before the outbre... Using the adoption and expansion of wired telegraph in China during the late 19^(th) century,this paper investigates the effect of cost reductions for knowledge exchange on China’s industrial growth before the outbreak of the War of Resistance Against Japanese Aggression(1931-1945),thus testing Baldwin’s theory of“the Great Convergence”in which developing countries are empowered by information and communication technologies.Based on panel data of 1858-1937,we found that wired telegraph access had a significantly positive effect,as well as a long-term growth effect,on the entry of industrial enterprises.Our mechanism analysis indicates that wired telegraph access accelerated early-stage industrialization in localities by encouraging market integration,human capital accumulation,and auxiliary commercial organizations.Only a few countries firmly asserted their telegraph sovereignty and set up their own workforce educational system during telegraph adoption.This explains why the Great Convergence arising from technology importation only occurred in a small number of countries.Our findings contribute to understanding the source of China’s modern industrial progress,as well as why global inequities remain. 展开更多
关键词 wired telegraph entry of industrial enterprises the Great convergence human capital
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On the Local Convergence and Dynamics of New Iterative Method with Sixth Order Convergence
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作者 Lyu Borui Chu Xue Wang Haijun 《数学理论与应用》 2024年第3期50-66,共17页
In this paper,we construct a new sixth order iterative method for solving nonlinear equations.The local convergence and order of convergence of the new iterative method is demonstrated.In order to check the validity o... In this paper,we construct a new sixth order iterative method for solving nonlinear equations.The local convergence and order of convergence of the new iterative method is demonstrated.In order to check the validity of the new iterative method,we employ several chemical engineering applications and academic test problems.Numerical results show the good numerical performance of the new iterative method.Moreover,the dynamical study of the new method also supports the theoretical results. 展开更多
关键词 Nonlinear equation Sixth order method Local convergence Basin of attraction
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Convergence over AI Governance Amid Differences
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作者 THORSTEN JELINEK 《China Today》 2024年第4期58-61,共4页
As the EU passes the first AI law,the U.S.grapples with bipartisanship,and China proactively advances the issue-based approach and advocates the Global AI Governance Initiative,will AI become“a force for good,ensurin... As the EU passes the first AI law,the U.S.grapples with bipartisanship,and China proactively advances the issue-based approach and advocates the Global AI Governance Initiative,will AI become“a force for good,ensuring safety,and promoting fairness”? 展开更多
关键词 FAIRNESS convergence LAW
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Convergence Phenomenon with Fourier Series of tg(x2)and Alike
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作者 Alfred Wünsche 《Advances in Pure Mathematics》 2024年第7期556-595,共40页
The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generali... The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. . 展开更多
关键词 Gibbs Phenomenon Generalized Functions Weak convergence Chebyshev Polynomials of First and Second Kind Even and Odd Generating Functions for Chebyshev Polynomials POLYLOGARITHMS Completeness Relations
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