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The order of accuracy related to the error from source approximation in the method of characteristics for purely absorbing materials in planar geometry 认领 引用
1
作者 Ji-Pu Wang Zhen-Wen Wei +5 位作者 Can Huang William Martin Brendan Kochunas Qi-Cang Shen Si-Juan Chen Martin Pilch 《Nuclear Science and Techniques》 SCIE EI CAS CSCD 2026年第6期257-270,共14页
The method of characteristics(MoC)is a well-established tool for lattice physics calculations,offering advantages such as accurate representation of both lattice geometry and boundary conditions.The flat source(FS)app... The method of characteristics(MoC)is a well-established tool for lattice physics calculations,offering advantages such as accurate representation of both lattice geometry and boundary conditions.The flat source(FS)approximation is the most commonly used approach,whereas the linear source(LS)approximation enhances the accuracy by preserving the higherorder spatial moments of the neutron source.However,determining the order of accuracy(OoA)for spatial discretization in the MoC is challenging,particularly for the LS approximation.This complexity arises because MoC employs two spatial meshes:the fission source region(FSR)mesh and a set of characteristic rays used to integrate the transport equation over the FSR mesh.In this study,we analyzed the spatial order of accuracy of the MoC in planar geometry for both FS and LS approximations in relation to the distributed source.Our theoretical predictions are consistent with the numerical results obtained using the Method of Manufactured Solutions(MMS).The results demonstrate that the FS approximation achieves second-order accuracy,whereas the LS approximation attains fourth-order accuracy. 展开更多
关键词 Method of characteristics Order of accuracy Flat source approximation Linear source approximation Method of manufactured solutions
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Improved probabilistic seismic AVO inversion constrained by instantaneous phase using quadratic PP-reflectivity approximation and IA2RMS-Gibbs algorithm 认领 引用
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作者 Shuang-Shuang Zhou Xing-Yao Yin +1 位作者 Kun Li Ya-Ming Yang 《Petroleum Science》 SCIE EI CAS CSCD 2026年第1期127-142,共16页
Seismic amplitude variation with offset(AVO)inversion is a cornerstone of oil and gas reservoir prediction,enabling the estimation of subsurface elastic parameters and characterization of stratigraphic interfaces.Howe... Seismic amplitude variation with offset(AVO)inversion is a cornerstone of oil and gas reservoir prediction,enabling the estimation of subsurface elastic parameters and characterization of stratigraphic interfaces.However,balancing inversion accuracy and computational efficiency remains a critical challenge.To address this,we propose a novel probabilistic AVO inversion framework integrating three key innovations.First,we derive a high-precision quadratic approximation for compressional(P-wave)reflectivity by retaining first-and second-order terms from the exact Zoeppritz equations through a perturbation strategy.This approach significantly enhances accuracy compared to conventional linear approximations,particularly in reflecting the true amplitude variation at large angles.Subsequently,to improve lateral continuity and stratigraphic resolution,we introduce an instantaneous phase constraint derived via the Hilbert transform.This constraint leverages phase sensitivity to seismic waveform coherence,ensuring geologically consistent interface characterization during stochastic inversion.Furthermore,we develop a hybrid Markov Chain Monte Carlo(MCMC)algorithm combining adaptive Gibbs sampling with the independent doubly adaptive rejection Metropolis sampling(IA2RMS)method.This framework efficiently samples high-dimensional posterior probability density functions(PDFs)of elastic pa rameters:Gibbs sampling gene rates adaptive proposal distributions,while IA2RMS accele rates Markov chain convergence through location-and scale-adjustable proposals.Numerical experiments and field seismic data demonstrate the robustness and feasibility of the proposed probabilistic AVO inversion method. 展开更多
关键词 Quadratic approximation Instantaneous phase constraint Gibbs sampling Independent doubly adaptive rejection metropolis sampling(IA2RMS) Probability density functions(PDFs)
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Improved branched multi-point approximation method and its application in spacecraft topology optimization 认领 引用
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作者 Shuanjun LIU Hai HUANG +1 位作者 Pengcheng WANG Weipeng LI 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2026年第3期431-446,共16页
The Branched Multi-point Approximation(BMA)method holds immense potential in tackling topology optimization challenges involving discrete variables.This research focuses on enhancing the adaptability,approximation eff... The Branched Multi-point Approximation(BMA)method holds immense potential in tackling topology optimization challenges involving discrete variables.This research focuses on enhancing the adaptability,approximation efficiency and engineering application of the method.Four key advancements are presented below.Firstly,an adjusted BMA function is introduced,which comprehensively incorporates the topologies of known and current points,utilizing topological difference between them as the foundation for function segmentation.This function further mitigates singularities as continuous variables approach zero,and is verified for its approximation capability through randomized numerical simulations.Secondly,the concept of adjacent individuals is proposed to ensure that the topological disparities between the current point and current optimal solution remain within manageable limits.This approach prevents approximation failures while maintaining the diversity of the population generated by genetic algorithm.Thirdly,for optimizing continuous variable in complex problems,an improved separable-variable approximation is formulated to ensure strong duality validity.Finally,an adaptive technique for move limit of the retained continuous variables is established.Benchmark examples demonstrate the effectiveness of the improved method in iterations and optimal results.Optimization systems capable of performing topology optimization for spacecraft such as stiffened satellite and carrier rocket have been rare in prior research. 展开更多
关键词 Adaptive technique Adjacent individual Branched multi-point approximation Spacecraft Topology optimization
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Research on heuristic approximation algorithm of the densest k-subgraph discovery in large-scale dynamic graphs 认领 引用
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作者 HAN Tao TIAN Yuxi +1 位作者 ZHAO Jianwei WANG Senzhang 《Journal of Southeast University(English Edition)》 EI CAS 2026年第1期74-79,共6页
To address the issue that static densest subgraph mining algorithms often exhibit low efficiency when handling large scale dynamic graphs,this paper proposes a heuristic approximation algorithm.The algorithm approxima... To address the issue that static densest subgraph mining algorithms often exhibit low efficiency when handling large scale dynamic graphs,this paper proposes a heuristic approximation algorithm.The algorithm approximates the densest k-subgraphs of the entire graph through four steps:partitioning the large-scale dynamic graph,constructing a partial set of the densest k-subgraphs,heuristically merging the subgraph sets,and finally extracting the densest k-subgraphs.This approach significantly reduces the computational time for large-scale dynamic graphs while simultaneously improving the quality of the resulting subgraphs.This algorithm is applicable to various definitions of“density”and can accommodate diverse requirements on the number of edges.When integrated with existing static densest subgraph detection algorithms,it achieves scalability and computational efficiency.Theoretical analysis demonstrates that the optimal density of the densest k-subgraphs extracted by the proposed algorithm reaches 0.9.To evaluate the performance of the algorithm,experiments were conducted on four billion-scale datasets:Friendster,Orkut,YouTube,and DBLP.The results indicate that the proposed algorithm outperforms static methods in both runtime efficiency and subgraph quality on large-scale dynamic graphs. 展开更多
关键词 the densest k-subgraph features heuristic approximation algorithm optimal density
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HYDROSTATIC APPROXIMATION OF THE NAVIER-STOKES-ALPHA MODEL IN A THIN STRIP 认领 引用
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作者 Léo GLANGETAS Van-Sang NGO El Mehdi SAID 《Acta Mathematica Scientia》 SCIE CSCD 2026年第3期1287-1323,共37页
In this paper we study the hydrostatic approximation of the Navier-Stokes-alpha(NS-α)small initial data in appropriate analytic function spaces.We also justify this approximation by proving the convergence of the Nav... In this paper we study the hydrostatic approximation of the Navier-Stokes-alpha(NS-α)small initial data in appropriate analytic function spaces.We also justify this approximation by proving the convergence of the Navier-Stokes-alpha system towards its hydrostatic limit. 展开更多
关键词 Navier-Stokes-αmodel hydrostatic approximation analyticity
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Approximation Scheme for the Single-Client Capacitated Facility Location Problem With Operational Cost Budget Constraint 认领 引用
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作者 Lu Chen Zheng Chen +1 位作者 Long Wan Wen-Chang Luo 《Journal of the Operations Research Society of China》 EI CSCD 2026年第2期687-699,共13页
We investigate the single-client capacitated facility location problem with operational cost budget constraint.Specifically speaking,we are given a single client with demand,a set of potential facilities with capaciti... We investigate the single-client capacitated facility location problem with operational cost budget constraint.Specifically speaking,we are given a single client with demand,a set of potential facilities with capacities,and a positive integer specifying the operational cost budget.For each open facility,one should pay its opening cost and operational cost.For each open facility to service the client,one should pay the service cost.The objective is to open enough facilities serving the client’s demand and satisfying the capacity and operational cost budget constraints while minimizing the sum of opening and service costs.In this paper,based on dynamic programming and sparse techniques,we derive a fully polynomial time approximation scheme(FPTAS)with violating the operational cost budget constraint at most an arbitrarily small factor or claim that there is no feasible solution. 展开更多
关键词 Single client Capacitated facility location Dynamic programming Sparse Approximation scheme
Approximation Algorithms for k-Submodular Maximization Subject to a Knapsack Constraint 认领 引用
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作者 Hao Xiao Qian Liu +1 位作者 Yang Zhou Min Li 《Journal of the Operations Research Society of China》 EI CSCD 2026年第2期397-412,共16页
In this paper,we study the problem of maximizing k-submodular functions subject to a knapsack constraint.For monotone objective functions,we present a 1/2(1−e-2)≈0.432 greedy approximation algorithm,improving the ... In this paper,we study the problem of maximizing k-submodular functions subject to a knapsack constraint.For monotone objective functions,we present a 1/2(1−e-2)≈0.432 greedy approximation algorithm,improving the previous best-known ratio1(1-e-1)≈0.316.We also consider the non-monotone knapsack problem and provide two algorithms.The first is a greedy-type combinatorial algorithm with approximation ratio 1/3(1−e-3)≈0.317,while the second is a multilinear-extension-based algorithm with approximation ratio 1/3-ε,whereε>0. 展开更多
关键词 k-submodular Knapsack Approximation algorithm
An Approximation Theorem and Generic Uniqueness of Weakly Pareto-Nash Equilibrium for Multiobjective Population Games 认领 引用
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作者 Hua-Xin Chen Wen-Sheng Jia 《Journal of the Operations Research Society of China》 EI CSCD 2026年第2期591-602,共12页
We mainly study an approximation theorem and generic uniqueness of weakly Pareto-Nash equilibrium(PNE)for multiobjective population games.First,we define the concept of approximate weakly PNE of multiobjective populat... We mainly study an approximation theorem and generic uniqueness of weakly Pareto-Nash equilibrium(PNE)for multiobjective population games.First,we define the concept of approximate weakly PNE of multiobjective population games.Then,an approximation theorem is proved under very mild conditions.Furthermore,we prove the generic uniqueness of weakly PNE for multiobjective population games by establishing a function space.Finally,we obtain the generic uniqueness of weakly PNE for multiobjective population games in the case of function perturbation.The above results are new and improve the related results of the recent literature. 展开更多
关键词 Multiobjective population game Approximation theorem Weakly Pareto-Nash equilibrium
Fast Ion Gates without the Lamb-Dicke Approximation by Robust Quantum Optimal Control 认领 引用
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作者 Ran Liu Xiaodong Yang +2 位作者 Yiheng Lin Yao Lu Jun Li 《Chinese Physics Letters》 SCIE EI CAS CSCD 2025年第8期75-82,共8页
We present a robust quantum optimal control framework for implementing fast entangling gates on ion-trap quantum processors.The framework leverages tailored laser pulses to drive the multiple vibrational sidebands of ... We present a robust quantum optimal control framework for implementing fast entangling gates on ion-trap quantum processors.The framework leverages tailored laser pulses to drive the multiple vibrational sidebands of the ions to create phonon-mediated entangling gates and,unlike the state of the art,requires neither weakcoupling Lamb-Dicke approximation nor perturbation treatment.With the application of gradient-based optimal control,it enables finding amplitude-and phase-modulated laser control protocols that work without the Lamb-Dicke approximation,promising gate speeds on the order of microseconds comparable to the characteristic trap frequencies.Also,robustness requirements on the temperature of the ions and initial optical phase can be conveniently included to pursue high-quality fast gates against experimental imperfections.Our approach represents a step in speeding up quantum gates to achieve larger quantum circuits for quantum computation and simulation,and thus can find applications in near-future experiments. 展开更多
关键词 quantum optimal control framework gradient based optimal control quantum computation Lamb Dicke approximation fast ion gates tailored laser pulses entangling gates robust quantum optimal control
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Multi-Interval-Aggregation Failure Point Approximation for Remaining Useful Life Prediction 认领 引用
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作者 Linchuan Fan Xiaolong Chen +1 位作者 Shuo Li Yi Chai 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2025年第3期639-641,共3页
Dear Editor,This letter focuses on the remaining useful life(RUL)prediction task under limited labeled samples.Existing machine-learning-based RUL prediction methods for this task usually pay attention to mining degra... Dear Editor,This letter focuses on the remaining useful life(RUL)prediction task under limited labeled samples.Existing machine-learning-based RUL prediction methods for this task usually pay attention to mining degradation information to improve the prediction accuracy of degradation value or health indicator for the next epoch.However,they ignore the cumulative prediction error caused by iterations before reaching the failure point. 展开更多
关键词 remaining useful life prediction failure point degradation value health indicator multi interval aggregation failure point approximation machine learning based mining degradation information
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Optimal convex approximations of qubit states based on l1-norm of coherence 认领 引用
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作者 Li-Qiang Zhang Yan-Dong Du Chang-Shui Yu 《Chinese Physics B》 SCIE EI CAS CSCD 2025年第8期475-491,共17页
Determining the minimal distance between the target state and the convex combination of given states is a fundamental problem in quantum resource theory,offering critical guidance for experimental implementations.In t... Determining the minimal distance between the target state and the convex combination of given states is a fundamental problem in quantum resource theory,offering critical guidance for experimental implementations.In this paper,we embark on an in-depth exploration of the use of a quantum state prepared by the convex combination of given qubit states to optimally approximate the l1-norm of coherence of the target quantum state,striving to make the prepared state and the target state as similar as possible.Here,we present the analytical solution for the optimal distance for any N given quantum states.We find that the optimal approximation problem for any N>4 quantum states can be transformed into an optimal approximation problem for no more than four quantum states,which not only significantly streamlines the problem but also proves advantageous for laboratories in terms of material conservation.Ultimately,a one-to-one comparison between the analytical and numerical solutions verifies the effectiveness of our approach. 展开更多
关键词 quantum information processing quantum resource theory quantum coherence optimal convex approximations
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An Approximation Method for Singular Trudinger-Moser Inequality 认领 引用
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作者 ZHU Maochun XU Wenyan JIANG Rou 《应用数学》 北大核心 2025年第1期64-70,共7页
In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to... In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball. 展开更多
关键词 Singular Trudinger-Moser inequality Approximation functional Concentration level
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A new oolitic content test method for green sand by repeated approximation 认领 引用 被引量：1
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作者 Yu-yang Qi Jian-xin Zhou +6 位作者 Xin Peng Si-yuan Pan Ya-jun Yin Xiao-yuan Ji Yuan-cai Li Peng-fei Lin Peng Wan 《China Foundry》 SCIE EI CAS CSCD 2025年第6期701-709,共9页
The reuse of green sand in casting production is hindered by the accumulation of oolitic deposits,primarily composed of clay binder with surface degradation,which may adversely affect the the moulding sand performance... The reuse of green sand in casting production is hindered by the accumulation of oolitic deposits,primarily composed of clay binder with surface degradation,which may adversely affect the the moulding sand performance.Currently,there is a lack of standardized methods for quantifying the oolitic content.Accurate measurement of oolitic content is of great significance to the reuse of green sand.Attempts to determine oolitic content using potassium hydroxide(KOH)and phosphoric acid(H3PO4)methods encounter challenges due to their excessive reactions with SiO2 in the sand.In this study,an improved method for measuring the oolitic content of green sand with repeated approximations was proposed.This method judges the chemical activity of the sample surface through the change of its mass to accurately obtain the mass of the reaction oolitic deposits.The test result of the used sand samples from the foundry shows that the oolitic deposits are completely removed after reacting with KOH solution three times at 300℃ for 20 min.SEM and EDS also show that after three times of reactions,the surface of green sand becomes smooth and the content of Al-containing oolitic deposits is very low.This indicates that the method can accurately control the extent of the reaction.Implementation of this method at Huangshi Dongbei Casting Co.,Ltd.has yielded consistent and reliable test results,effectively mirroring variations in green sand oolitic content on the production line.This new method is expected to be widely adopted to improve the efficiency and quality of reused green sand in casting operations. 展开更多
关键词 green sand oolitic content repeated approximation reuse of green sand
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Statistical analysis on the validity of the cold plasma approximation for chorus waves based on Van Allen Probe observations and their effects on radiation belt electrons 认领 引用 被引量：1
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作者 AoJun Ren Jiang Yu +2 位作者 Jing Wang ZuZheng Chen XiaoMan Liu 《Earth and Planetary Physics》 EI CSCD 2025年第1期159-170,共12页
Theoretical analysis has demonstrated that the dispersion relation of chorus waves plays an essential role in the resonant interaction and energy transformation between the waves and magnetospheric electrons.Previous ... Theoretical analysis has demonstrated that the dispersion relation of chorus waves plays an essential role in the resonant interaction and energy transformation between the waves and magnetospheric electrons.Previous quantitative analyses often simplified the chorus dispersion relation by using the cold plasma assumption.However,the applicability of the cold plasma assumption is doubtful,especially during geomagnetic disturbances.We here present a systematic statistical analysis on the validity of the cold plasma dispersion relation of chorus waves based on observations from the Van Allen Probes over the period from 2012 to 2018.The statistical results show that the observed magnetic field intensities deviate substantially from those calculated from the cold plasma dispersion relation and that they become more pronounced with an increase in geomagnetic activity or a decrease in background plasma density.The region with large deviations is mainly concentrated in the nightside and expands in both the radial and azimuthal directions as the geomagnetic activity increases or the background plasma density decreases.In addition,the bounce-averaged electron scattering rates are computed by using the observed and cold plasma dispersion relation of chorus waves.Compared with usage of the cold plasma dispersion relation,usage of the observed dispersion relation considerably lowers the minimum resonant energy of electrons and lowers the scattering rates of electrons above tens of kiloelectronvolts but enhances those below.Furthermore,these differences are more pronounced with the enhancement of geomagnetic activity or the decrease in background plasma density. 展开更多
关键词 chorus waves cold plasma approximation dispersion relations scattering rates
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Adaptive polynomial approximation-based virtual coupled cooperative control for high-speed trains 认领 引用
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作者 Kai-Xiang Wang Ming-Yue Ren +1 位作者 Qian-Ling Wang Xue Lin 《Chinese Physics B》 SCIE EI CAS CSCD 2025年第10期589-596,共8页
Virtual coupling is a novel technology that enables trains to run closely together without physical connections through communication and automation systems.The paper addresses an adaptive polynomial approximation alg... Virtual coupling is a novel technology that enables trains to run closely together without physical connections through communication and automation systems.The paper addresses an adaptive polynomial approximation algorithm for the cooperative control of high-speed trains(HSTs)under virtual coupling.It aims to solve the cooperative tracking control problem of HST formation operations under various scenarios,including known and unknown parameters.To enable the HST formation system to achieve cooperative operation while ensuring an appropriate spacing distance,the tracking errors of displacement and speed throughout the entire operation converge to zero.The proposed control strategy focuses on adopting polynomial approximation to handle unknown parameters,which are estimated via adaptive laws.Additionally,the unknown parameters of the HSTs are estimated online through adaptive laws.Experimental results verify the effectiveness of this method. 展开更多
关键词 high-speed trains(HSTs) cooperative operation adaptive polynomial approximation virtual coupling
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On approximation of Bernstein-Stancu operators in movable interval 认领 引用
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作者 WAGN Feng-feng YU Dan-sheng 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2025年第1期85-101,共17页
In the present paper,we obtain the converse results of approximation of a newly introduced genuine Bernstein-Durrmeyer operators in movable interval.We also get the moments properties of an auxiliary operator which ha... In the present paper,we obtain the converse results of approximation of a newly introduced genuine Bernstein-Durrmeyer operators in movable interval.We also get the moments properties of an auxiliary operator which has its own independent values.The moments of the auxiliary operators play important roles in establishing the main result(Theorem 4). 展开更多
关键词 Bernstein-Stancu operators movable interval approximation rates Voronovskaja's asymptotic estimate
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Exceptional Set for Diophantine Approximation with Primes of Two Squares and Three Biquadrates 认领 引用
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作者 LIU Minmin ZHANG Min LI Jinjiang 《数学进展》 CSCD 北大核心 2025年第1期18-36,共19页
Suppose thatλ1,λ2,λ3,λ4,λ5are nonzero real numbers,not all of the same sign,andλ1/λ2is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for ... Suppose thatλ1,λ2,λ3,λ4,λ5are nonzero real numbers,not all of the same sign,andλ1/λ2is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ1p1~2+λ2p2~2+λ3p3~4+λ4p4~4+λ_5p_5~4-v|<v-δhas no solution in prime variables p1,p2,p3,p4,p5does not exceed O(N29/32+2δ+ε). 展开更多
关键词 Diophantine approximation exponential sum Davenport-Heilbronn method prime variable
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Smoluchowski-Kramers Approximation for Stochastic Differential Equations under Discretization 认领 引用
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作者 Li Ge 《应用概率统计》 CSCD 北大核心 2025年第4期622-635,共14页
This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–M... This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–Maruyama discretization and derive its convergence rate.In particular,the solution of the discretized system converges to the solution of the first-order limit equation in the mean-square sense,and this convergence is independent of the order in which the mass parameterμand the step size h tend to zero. 展开更多
关键词 stochastic differential equations Smoluchowski-Kramers approximation driftimplicit Euler-Maruyama scheme convergence rate
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Wave equation tomographic velocity inversion method based on the Born/Rytov approximation 认领 引用 被引量：9
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作者 张凯 尹正 +1 位作者 李振春 陈永芮 《Applied Geophysics》 SCIE CSCD 2013年第3期314-322,358,359,共9页
This paper discusses Born/Rytov approximation tomographic velocity inversion methods constrained by the Fresnel zone. Calculations of the sensitivity kernel function and traveltime residuals are critical in tomographi... This paper discusses Born/Rytov approximation tomographic velocity inversion methods constrained by the Fresnel zone. Calculations of the sensitivity kernel function and traveltime residuals are critical in tomographic velocity inversion. Based on the Bom/Rytov approximation of the frequency-domain wave equation, we derive the traveltime sensitivity kemels of the wave equation on the band-limited wave field and simultaneously obtain the traveltime residuals based on the Rytov approximation. In contrast to single-ray tomography, the modified velocity inversion method improves the inversion stability. Tests of the near- surface velocity model and field data prove that the proposed method has higher accuracy and Computational efficiency than ray theory tomography and full waveform inversion methods. 展开更多
关键词 Tomographic inversion Fresnel zone sensitivity kernels Bom approximation Rytov approximation
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APPROXIMATION BY COMPLEX SZSZ-DURRMEYER OPERATORS IN COMPACT DISKS 认领 引用 被引量：7
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作者 Sorin G.GAL Vijay GUPTA 《Acta Mathematica Scientia》 SCIE CSCD 2014年第4期1157-1165,共9页
In the present paper, we deal with the complex Szasz-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on comp... In the present paper, we deal with the complex Szasz-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. Also, the exact order of approximation is found. 展开更多
关键词 complex Szasz-Durrmeyer operators Voronovksaja type result exact order of approximation in compact disks simultaneous approximation
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