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Branch and Bound Algorithm for Globally Solving Minimax Linear Fractional Programming
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作者 WANG Hui-man SHEN Pei-ping LIANG Yu-xin 《Chinese Quarterly Journal of Mathematics》 2024年第4期388-398,共11页
In this paper,we study the minimax linear fractional programming problem on a non-empty bounded set,called problem(MLFP),and we design a branch and bound algorithm to find a globally optimal solution of(MLFP).Firstly,... In this paper,we study the minimax linear fractional programming problem on a non-empty bounded set,called problem(MLFP),and we design a branch and bound algorithm to find a globally optimal solution of(MLFP).Firstly,we convert the problem(MLFP)to a problem(EP2)that is equivalent to it.Secondly,by applying the convex relaxation technique to problem(EP2),a convex quadratic relaxation problem(CQRP)is obtained.Then,the overall framework of the algorithm is given and its convergence is proved,the worst-case iteration number is also estimated.Finally,experimental data are listed to illustrate the effectiveness of the algorithm. 展开更多
关键词 Minimax linear fractional programming Global optimal solution Branch and bound
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A Dimensional Reduction Approach Based on Essential Constraints in Linear Programming
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作者 Eirini I. Nikolopoulou George S. Androulakis 《American Journal of Operations Research》 2024年第1期1-31,共31页
This paper presents a new dimension reduction strategy for medium and large-scale linear programming problems. The proposed method uses a subset of the original constraints and combines two algorithms: the weighted av... This paper presents a new dimension reduction strategy for medium and large-scale linear programming problems. The proposed method uses a subset of the original constraints and combines two algorithms: the weighted average and the cosine simplex algorithm. The first approach identifies binding constraints by using the weighted average of each constraint, whereas the second algorithm is based on the cosine similarity between the vector of the objective function and the constraints. These two approaches are complementary, and when used together, they locate the essential subset of initial constraints required for solving medium and large-scale linear programming problems. After reducing the dimension of the linear programming problem using the subset of the essential constraints, the solution method can be chosen from any suitable method for linear programming. The proposed approach was applied to a set of well-known benchmarks as well as more than 2000 random medium and large-scale linear programming problems. The results are promising, indicating that the new approach contributes to the reduction of both the size of the problems and the total number of iterations required. A tree-based classification model also confirmed the need for combining the two approaches. A detailed numerical example, the general numerical results, and the statistical analysis for the decision tree procedure are presented. 展开更多
关键词 linear programming Binding Constraints Dimension Reduction Cosine Similarity Decision Analysis Decision Trees
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Glia-to-neuron reprogramming to the rescue?
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作者 Jack W.Hickmott Cindi M.Morshead 《Neural Regeneration Research》 SCIE CAS 2025年第5期1395-1396,共2页
Over the last two decades,the dogma that cell fate is immutable has been increasingly challenged,with important implications for regenerative medicine.The brea kth rough discovery that induced pluripotent stem cells c... Over the last two decades,the dogma that cell fate is immutable has been increasingly challenged,with important implications for regenerative medicine.The brea kth rough discovery that induced pluripotent stem cells could be generated from adult mouse fibroblasts is powerful proof that cell fate can be changed.An exciting extension of the discovery of cell fate impermanence is the direct cellular reprogram ming hypothesis-that terminally differentiated cells can be reprogrammed into other adult cell fates without first passing through a stem cell state. 展开更多
关键词 programming PASSING proof
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Neurocircuit regeneration by extracellular matrix reprogramming
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作者 Shengzhang Su Ian N.Levasseur Kimberly M.Alonge 《Neural Regeneration Research》 SCIE CAS 2025年第8期2300-2301,共2页
The brain's extracellular matrix(ECM),which is comprised of protein and glycosaminoglycan(GAG)scaffolds,constitutes 20%-40% of the human brain and is considered one of the largest influencers on brain cell functio... The brain's extracellular matrix(ECM),which is comprised of protein and glycosaminoglycan(GAG)scaffolds,constitutes 20%-40% of the human brain and is considered one of the largest influencers on brain cell functioning(Soles et al.,2023).Synthesized by neural and glial cells,the brain's ECM regulates a myriad of homeostatic cellular processes,including neuronal plasticity and firing(Miyata et al.,2012),cation buffering(Moraws ki et al.,2015),and glia-neuron interactions(Anderson et al.,2016).Considering the diversity of functions,dynamic remodeling of the brain's ECM indicates that this understudied medium is an active participant in both normal physiology and neurological diseases. 展开更多
关键词 MATRIX programming
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High cycle fatigue performance at 650℃and corresponding fracture behaviors of GH4169 joint produced by linear friction welding 被引量:2
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作者 Shitong MA Xiawei YANG +6 位作者 Zhenguo GUO Yu SU Xinyuan HE Ju LI Jun TAO Bo XIAO Wenya LI 《Chinese Journal of Aeronautics》 2025年第1期622-637,共16页
GH4169 joints manufactured by Linear Friction Welding(LFW)are subjected to tensile test and stair-case method to evaluate the High Cycle Fatigue(HCF)performance at 650℃.The yield and ultimate tensile strengths are 58... GH4169 joints manufactured by Linear Friction Welding(LFW)are subjected to tensile test and stair-case method to evaluate the High Cycle Fatigue(HCF)performance at 650℃.The yield and ultimate tensile strengths are 582 MPa and 820 MPa,respectively.The HCF strength of joint reaches 400 MPa,which is slightly lower than that of Base Metal(BM),indicating reliable quality of this type of joint.The microstructure observation results show that all cracks initiate at the inside of specimens and transfer into deeper region with decrease of external stress,and the crack initiation site is related with microhardness of matrix.The Electron Backscattered Diffraction(EBSD)results of the observed regions with different distances to fracture show that plastic deformation plays a key role in HCF,and the Schmid factor of most grains near fracture exceeds 0.4.In addition,the generation of twins plays a vital role in strain concentration release and coordinating plastic deformation among grains. 展开更多
关键词 High cycle fatigue GH4169 superalloy linear friction welding Fracture mechanism Microstructure evolution
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Safe flight corridor constrained sequential convex programming for efficient trajectory generation of fixed-wing UAVs 被引量:2
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作者 Jing SUN Guangtong XU +2 位作者 Zhu WANG Teng LONG Jingliang SUN 《Chinese Journal of Aeronautics》 2025年第1期537-550,共14页
Generating dynamically feasible trajectory for fixed-wing Unmanned Aerial Vehicles(UAVs)in dense obstacle environments remains computationally intractable.This paper proposes a Safe Flight Corridor constrained Sequent... Generating dynamically feasible trajectory for fixed-wing Unmanned Aerial Vehicles(UAVs)in dense obstacle environments remains computationally intractable.This paper proposes a Safe Flight Corridor constrained Sequential Convex Programming(SFC-SCP)to improve the computation efficiency and reliability of trajectory generation.SFC-SCP combines the front-end convex polyhedron SFC construction and back-end SCP-based trajectory optimization.A Sparse A^(*)Search(SAS)driven SFC construction method is designed to efficiently generate polyhedron SFC according to the geometric relation among obstacles and collision-free waypoints.Via transforming the nonconvex obstacle-avoidance constraints to linear inequality constraints,SFC can mitigate infeasibility of trajectory planning and reduce computation complexity.Then,SCP casts the nonlinear trajectory optimization subject to SFC into convex programming subproblems to decrease the problem complexity.In addition,a convex optimizer based on interior point method is customized,where the search direction is calculated via successive elimination to further improve efficiency.Simulation experiments on dense obstacle scenarios show that SFC-SCP can generate dynamically feasible safe trajectory rapidly.Comparative studies with state-of-the-art SCP-based methods demonstrate the efficiency and reliability merits of SFC-SCP.Besides,the customized convex optimizer outperforms off-the-shelf optimizers in terms of computation time. 展开更多
关键词 Fixed-wing unmanned aerial vehicle Efficient trajectory planning Safe flight corridor Sequential convex programming Customized convex optimizer
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Effect of initial microstructure on performance and corrosion behavior of GH4169 superalloy joint produced by linear friction welding 被引量:1
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作者 Xiawei YANG Tingxi MENG +5 位作者 Yu SU Rui XU Zhenguo GUO Yaxin XU Tiejun MA Wenya LI 《Chinese Journal of Aeronautics》 2025年第3期662-675,共14页
The pre-weld heat treatment was carried out to obtain different initial microstructures of the GH4169 superalloy,and then Linear Friction Welding(LFW)was performed.The effect of the pre-weld heat treatment on the micr... The pre-weld heat treatment was carried out to obtain different initial microstructures of the GH4169 superalloy,and then Linear Friction Welding(LFW)was performed.The effect of the pre-weld heat treatment on the microstructure evolution and mechanical properties of the joint was analyzed,and the joint electrochemical corrosion behavior as well as the hot corrosion behavior was studied.The results show that the joint hardness of Base Metal(BM)increases after pre-weld heat treatment,and the strengthening phasesγ′andγ″further precipitate.However,the precipitation phases dissolve significantly in the Weld Zone(WZ)due to the thermal process of LFW.The corrosion resistance in BM is reduced after the pre-weld heat treatment,while it is similar in WZ with a slight decrease.The surface morphology of the BM and WZ can be generally divided into a loose and porous matrix and a scattered oxide particle layer after hot corrosion.The joint cross section exhibits a Cr-depleted zone with the diffusion of Cr to form an oxide film.The corrosion product mainly consists of Fe_(2)O_(3)/Fe_(3)O_(4) as the outer layer and Cr_(2)O_(3) as the inner layer. 展开更多
关键词 linear friction weldingi GH4169 superalloy Heat treatment Mechanical properties Corrosion behavior
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In situ stress inversion using nonlinear stress boundaries achieved by the bubbling method 被引量:1
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作者 Xige Liu Chenchun Huang +3 位作者 Wancheng Zhu Joung Oh Chengguo Zhang Guangyao Si 《Journal of Rock Mechanics and Geotechnical Engineering》 2025年第3期1510-1527,共18页
Due to the heterogeneity of rock masses and the variability of in situ stress,the traditional linear inversion method is insufficiently accurate to achieve high accuracy of the in situ stress field.To address this cha... Due to the heterogeneity of rock masses and the variability of in situ stress,the traditional linear inversion method is insufficiently accurate to achieve high accuracy of the in situ stress field.To address this challenge,nonlinear stress boundaries for a numerical model are determined through regression analysis of a series of nonlinear coefficient matrices,which are derived from the bubbling method.Considering the randomness and flexibility of the bubbling method,a parametric study is conducted to determine recommended ranges for these parameters,including the standard deviation(σb)of bubble radii,the non-uniform coefficient matrix number(λ)for nonlinear stress boundaries,and the number(m)and positions of in situ stress measurement points.A model case study provides a reference for the selection of these parameters.Additionally,when the nonlinear in situ stress inversion method is employed,stress distortion inevitably occurs near model boundaries,aligning with the Saint Venant's principle.Two strategies are proposed accordingly:employing a systematic reduction of nonlinear coefficients to achieve high inversion accuracy while minimizing significant stress distortion,and excluding regions with severe stress distortion near the model edges while utilizing the central part of the model for subsequent simulations.These two strategies have been successfully implemented in the nonlinear in situ stress inversion of the Xincheng Gold Mine and have achieved higher inversion accuracy than the linear method.Specifically,the linear and nonlinear inversion methods yield root mean square errors(RMSE)of 4.15 and 3.2,and inversion relative errors(δAve)of 22.08%and 17.55%,respectively.Therefore,the nonlinear inversion method outperforms the traditional multiple linear regression method,even in the presence of a systematic reduction in the nonlinear stress boundaries. 展开更多
关键词 In situ stress field Inversion method The bubbling method Nonlinear stress boundary Multiple linear regression method
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Evaluation of Integer Programming Solvers to Improve the Efficiency of Individual Work Planning
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作者 Keiichi Takahashi 《Open Journal of Modelling and Simulation》 2025年第1期51-64,共14页
This study proposes a novel approach to optimizing individual work schedules for book digitization using mixed-integer programming (MIP). By leveraging the power of MIP solvers, we aimed to minimize the overall digiti... This study proposes a novel approach to optimizing individual work schedules for book digitization using mixed-integer programming (MIP). By leveraging the power of MIP solvers, we aimed to minimize the overall digitization time while considering various constraints and process dependencies. The book digitization process involves three key steps: cutting, scanning, and binding. Each step has specific requirements and limitations such as the number of pages that can be processed simultaneously and potential bottlenecks. To address these complexities, we formulate the problem as a one-machine job shop scheduling problem with additional constraints to capture the unique characteristics of book digitization. We conducted a series of experiments to evaluate the performance of our proposed approach. By comparing the optimized schedules with the baseline approach, we demonstrated significant reductions in the overall processing time. In addition, we analyzed the impact of different weighting schemes on the optimization results, highlighting the importance of identifying and prioritizing critical processes. Our findings suggest that MIP-based optimization can be a valuable tool for improving the efficiency of individual work schedules, even in seemingly simple tasks, such as book digitization. By carefully considering specific constraints and objectives, we can save time and leverage resources by carefully considering specific constraints and objectives. 展开更多
关键词 Mixed Integer programming Scheduling Optimization Work Planning Book Scanning Gantt Chart
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Research on Human-Computer Collaboration Paradigm in AIGC-Empowered High-Level Language Programming Courses
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作者 Hongyuan Wang Baokai Zu +2 位作者 Yafang Li Wanting Zhu Hongli Chen 《Journal of Contemporary Educational Research》 2025年第5期285-289,共5页
With the rapid development of artificial intelligence technology,AIGC(Artificial Intelligence-Generated Content)has triggered profound changes in the field of high-level language programming courses.This paper deeply ... With the rapid development of artificial intelligence technology,AIGC(Artificial Intelligence-Generated Content)has triggered profound changes in the field of high-level language programming courses.This paper deeply explored the application principles,advantages,and limitations of AIGC in intelligent code generation,analyzed the new mode of human-computer collaboration in high-level language programming courses driven by AIGC,discussed the impact of human-computer collaboration on programming efficiency and code quality through practical case studies,and looks forward to future development trends.This research aims to provide theoretical and practical guidance for high-level language programming courses and promote innovative development of high-level language programming courses under the human-computer collaboration paradigm. 展开更多
关键词 Human-computer collaboration AIGC High-level language programming Intelligence programming Efficiency improvement
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A 28 nm 576K RRAM-based computing-in-memory macro featuring hybrid programming with area efficiency of 2.82 TOPS/mm^(2)
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作者 Siqi Liu Songtao Wei +7 位作者 Peng Yao Dong Wu Lu Jie Sining Pan Jianshi Tang Bin Gao He Qian Huaqiang Wu 《Journal of Semiconductors》 2025年第6期112-119,共8页
Computing-in-memory(CIM)has been a promising candidate for artificial-intelligent applications thanks to the absence of data transfer between computation and storage blocks.Resistive random access memory(RRAM)based CI... Computing-in-memory(CIM)has been a promising candidate for artificial-intelligent applications thanks to the absence of data transfer between computation and storage blocks.Resistive random access memory(RRAM)based CIM has the advantage of high computing density,non-volatility as well as high energy efficiency.However,previous CIM research has predominantly focused on realizing high energy efficiency and high area efficiency for inference,while little attention has been devoted to addressing the challenges of on-chip programming speed,power consumption,and accuracy.In this paper,a fabri-cated 28 nm 576K RRAM-based CIM macro featuring optimized on-chip programming schemes is proposed to address the issues mentioned above.Different strategies of mapping weights to RRAM arrays are compared,and a novel direct-current ADC design is designed for both programming and inference stages.Utilizing the optimized hybrid programming scheme,4.67×programming speed,0.15×power saving and 4.31×compact weight distribution are realized.Besides,this macro achieves a normalized area efficiency of 2.82 TOPS/mm2 and a normalized energy efficiency of 35.6 TOPS/W. 展开更多
关键词 computing-in-memory on-chip programming scheme hybrid programming resistive random access memory matrix-vector-multiplication acceleration
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Dark-gap solitons with mixed nonlinear and linear lattices
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作者 Xue-Fei Zhang Xiao-Yang Wang +1 位作者 Hui-Lian Wei Tian-Fu Xu 《Chinese Physics B》 2025年第8期493-499,共7页
We study the existence and stability of dark-gap solitons in linear lattice and nonlinear lattices.The results indicate that the combination of linear and nonlinear lattices gives dark-gap solitons unique properties.T... We study the existence and stability of dark-gap solitons in linear lattice and nonlinear lattices.The results indicate that the combination of linear and nonlinear lattices gives dark-gap solitons unique properties.The linear lattice can stabilize dark-gap solitons,while the nonlinear lattice reduces the stability of dark-gap solitons.On the basis of numerical analysis,we investigate the effects of lattice depth,chemical potential,nonlinear lattice amplitude,and nonlinear lattice period on the soliton in mixed lattices with the same and different periods.The stability of dark-gap soliton is studied carefully by means of real-time evolution and linear stability analysis.Dark-gap solitons can exist stably in the band gap,but the solitons formed by the mixed lattices are slightly different when the period is the same or different. 展开更多
关键词 dark-gap solitons nonlinear and linear lattices linear stability analysis
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Adapting High-Level Language Programming(C Language)Education in the Era of Large Language Models
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作者 Baokai Zu Hongyuan Wang +1 位作者 Hongli Chen Yafang Li 《Journal of Contemporary Educational Research》 2025年第5期264-269,共6页
With the widespread application of large language models(LLMs)in natural language processing and code generation,traditional High-Level Language Programming courses are facing unprecedented challenges and opportunitie... With the widespread application of large language models(LLMs)in natural language processing and code generation,traditional High-Level Language Programming courses are facing unprecedented challenges and opportunities.As a core programming language for computer science majors,C language remains irreplaceable due to its foundational nature and engineering adaptability.This paper,based on the rapid development of large model technologies,proposes a systematic reform design for C language teaching,focusing on teaching objectives,content structure,teaching methods,and evaluation systems.The article suggests a teaching framework centered on“human-computer collaborative programming,”integrating prompt training,AI-assisted debugging,and code generation analysis,aiming to enhance students’problem modeling ability,programming expression skills,and AI collaboration literacy. 展开更多
关键词 Large language models(LLMs) High-level language programming C language Human-computer collaborative programming
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Linear-Quadratic-Gaussian控制实现腔频率锁定仿真研究
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作者 宋旭宁 翟泽辉 《量子光学学报》 北大核心 2025年第3期100-107,共8页
本文考虑了Linear-Quadratic-Gaussian(LQG)控制理论在量子光学中腔锁定问题中的应用。锁定光学腔是指将输入激光频率与腔的谐振频率匹配的过程,将腔的频率稳定在激光频率上,以便于在腔内最大限度地积累能量,使得透射光场最强。激光频... 本文考虑了Linear-Quadratic-Gaussian(LQG)控制理论在量子光学中腔锁定问题中的应用。锁定光学腔是指将输入激光频率与腔的谐振频率匹配的过程,将腔的频率稳定在激光频率上,以便于在腔内最大限度地积累能量,使得透射光场最强。激光频率和腔频率之间的误差由“失谐”?表征。本文描述了建立腔体系统的模型,设计了LQG控制器。该控制器已在仿真实验中实现并成功测试。 展开更多
关键词 频率锁定 线性二次高斯控制 最优控制
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Reprogramming induced neurons from olfactory ensheathing glial cells: A feasible approach for spinal cord injury repair
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作者 Javier Sierra María Portela-Lomba +1 位作者 Diana Simón M.Teresa Moreno-Flores 《Neural Regeneration Research》 2026年第1期296-297,共2页
Every year, around the world, between 250,000 and 500,000 people suffer a spinal cord injury(SCI). SCI is a devastating medical condition that arises from trauma or disease-induced damage to the spinal cord, disruptin... Every year, around the world, between 250,000 and 500,000 people suffer a spinal cord injury(SCI). SCI is a devastating medical condition that arises from trauma or disease-induced damage to the spinal cord, disrupting the neural connections that allow communication between the brain and the rest of the body, which results in varying degrees of motor and sensory impairment. Disconnection in the spinal tracts is an irreversible condition owing to the poor capacity for spontaneous axonal regeneration in the affected neurons. 展开更多
关键词 INJURY feasible programming
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From flying dreams to programming pioneer
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作者 孙启禄 《疯狂英语(新读写)》 2025年第7期24-27,76,共5页
More than seventy years before airplanes were invented,a twelve⁃year⁃old girl named Ada Lovelace dreamed of flying.She studied birds and experimented with materials to make wings,even writing a guide called Flyology.B... More than seventy years before airplanes were invented,a twelve⁃year⁃old girl named Ada Lovelace dreamed of flying.She studied birds and experimented with materials to make wings,even writing a guide called Flyology.But her curiosity didnt stop there. 展开更多
关键词 GUIDE flyology ada lovelace programming pioneer FLYING
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AI-Enabled Computer Programming Course Teaching Reform
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作者 Yao Dai 《Journal of Electronic Research and Application》 2025年第3期90-96,共7页
With the rapid development of modern science and technology,the era of artificial intelligence has quietly come.Against the background of the new era,students’learning needs,learning resource acquisition methods,teac... With the rapid development of modern science and technology,the era of artificial intelligence has quietly come.Against the background of the new era,students’learning needs,learning resource acquisition methods,teachers’teaching concepts,teaching tools,and so on have changed significantly.How to carry out teaching reform based on this change has become one of the important issues facing educators,and the same is true for the teaching of computer programming courses.This paper focuses on the teaching reform of AI-enabled computer programming courses,analyzes its basic problems,and puts forward corresponding reform countermeasures to provide a useful reference for front-line teachers. 展开更多
关键词 AI Computer programming
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A modified Bis-OPNN phosphorus ligand for Rh-catalyzed linear-selective hydroformylation of alkenes
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作者 Luyun Zhang Ding Liu +2 位作者 Huri Piao Zhenhua Jia Fen-Er Chen 《Chinese Chemical Letters》 2025年第7期391-396,共6页
The hydroformylation of olefins,known as the"oxo reaction",involves the use of syngas(CO/H_(2))to produce aldehyde with an additional carbon atom.However,side reactions such as the isomerization or hydrogena... The hydroformylation of olefins,known as the"oxo reaction",involves the use of syngas(CO/H_(2))to produce aldehyde with an additional carbon atom.However,side reactions such as the isomerization or hydrogenation of olefins often result in unexpected products and other by-products.Recent efforts in developing efficient ligands represent the most effective approach to addressing these challenges.In this study,we described a Bis-OPNN phosphorus ligand facilitated Rh-catalyzed hydroformylation with a high degree of linear selectivity across various olefins.Under mild conditions,a broad range of olefins were efficiently converted into linear aldehydes with high yields and excellent regioselectivity.The protocol also showed impressive functional group tolerance and was successfully applied to modify drugs and natural products,including the total synthesis of(±)-crispine A.Preliminary mechanistic studies revealed that this Bis-OPNN phosphorus ligand anchoring the rhodium catalyst is crucial for controlling the linear selectivity. 展开更多
关键词 Bis-OPNN Phosphorus ligand HYDROFORMYLATION linear selectivity ALKENES
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A neural network based on novel equivalent model for linear complementarity problems
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作者 KE Yi-fen XIE Ya-jun +1 位作者 ZHANG Huai MA Chang-feng 《Applied Mathematics(A Journal of Chinese Universities)》 2025年第2期311-326,共16页
A family of neural networks is proposed to solve linear complementarity problems(LCP).The neural networks are constructed from the novel equivalent model of LCP,which is reformulated by utilizing the modulus and smoot... A family of neural networks is proposed to solve linear complementarity problems(LCP).The neural networks are constructed from the novel equivalent model of LCP,which is reformulated by utilizing the modulus and smoothing technologies.Some important properties of the proposed novel equivalent model are summarized.In addition,the stability properties of the proposed steepest descent-based neural networks for LCP are analyzed.In order to illustrate the theoretical results,we provide some numerical simulations and compare the proposed neural networks with existing neural networks based on the NCP-functions.Numerical results indicate that the performance of the proposed neural networks is effective and robust. 展开更多
关键词 linear complementarity problem neural network MODULUS STABILITY
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Output Consensus of Heterogeneous Linear MASs via Adaptive Event-Triggered Feedback Combination Control
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作者 Shuo Yuan Chengpu Yu Jian Sun 《IEEE/CAA Journal of Automatica Sinica》 2025年第1期285-287,共3页
Dear Editor,This letter studies output consensus problem of heterogeneous linear multiagent systems over directed graphs. A novel adaptive dynamic event-triggered controller is presented based only on the feedback com... Dear Editor,This letter studies output consensus problem of heterogeneous linear multiagent systems over directed graphs. A novel adaptive dynamic event-triggered controller is presented based only on the feedback combination of the agent's own state and neighbors' output,which can achieve exponential output consensus through intermittent communication. The controller is obtained by solving two linear matrix equations, and Zeno behavior is excluded. 展开更多
关键词 intermittent communication feedback combination heterogeneous linear multiagent systems exponential output consensus directed graphs output consensus problem output consensus solving two linear matrix equations
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