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
We examined the fractional second-order singular Lagrangian systems. We wrote the action principal function and equations of motion as fractional total differential equations. Also, we constructed the set of Hamilton-...We examined the fractional second-order singular Lagrangian systems. We wrote the action principal function and equations of motion as fractional total differential equations. Also, we constructed the set of Hamilton-Jacobi partial differential equations (HJPDEs) within fractional calculus. We formulated the fractional path integral quantization for these systems. A mathematical example is examined with first- and second-class constraints.展开更多
Singular spectrum analysis is widely used in geodetic time series analysis.However,when extracting time-varying periodic signals from a large number of Global Navigation Satellite System(GNSS)time series,the selection...Singular spectrum analysis is widely used in geodetic time series analysis.However,when extracting time-varying periodic signals from a large number of Global Navigation Satellite System(GNSS)time series,the selection of appropriate embedding window size and principal components makes this method cumbersome and inefficient.To improve the efficiency and accuracy of singular spectrum analysis,this paper proposes an adaptive singular spectrum analysis method by combining spectrum analysis with a new trace matrix.The running time and correlation analysis indicate that the proposed method can adaptively set the embedding window size to extract the time-varying periodic signals from GNSS time series,and the extraction efficiency of a single time series is six times that of singular spectrum analysis.The method is also accurate and more suitable for time-varying periodic signal analysis of global GNSS sites.展开更多
Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial pertur...Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial perturbation method tends only to capture synoptic scale initial uncertainty rather than mesoscale uncertainty in global ensemble prediction. To address this issue, a multiscale SV initial perturbation method based on the China Meteorological Administration Global Ensemble Prediction System(CMA-GEPS) is proposed to quantify multiscale initial uncertainty. The multiscale SV initial perturbation approach entails calculating multiscale SVs at different resolutions with multiple linearized physical processes to capture fast-growing perturbations from mesoscale to synoptic scale in target areas and combining these SVs by using a Gaussian sampling method with amplitude coefficients to generate initial perturbations. Following that, the energy norm,energy spectrum, and structure of multiscale SVs and their impact on GEPS are analyzed based on a batch experiment in different seasons. The results show that the multiscale SV initial perturbations can possess more energy and capture more mesoscale uncertainties than the traditional single-SV method. Meanwhile, multiscale SV initial perturbations can reflect the strongest dynamical instability in target areas. Their performances in global ensemble prediction when compared to single-scale SVs are shown to(i) improve the relationship between the ensemble spread and the root-mean-square error and(ii) provide a better probability forecast skill for atmospheric circulation during the late forecast period and for short-to medium-range precipitation. This study provides scientific evidence and application foundations for the design and development of a multiscale SV initial perturbation method for the GEPS.展开更多
For singularly perturbed convection-diffusion problems,supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes,especially in the case of 2D.The difficulties arise from the width of ...For singularly perturbed convection-diffusion problems,supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes,especially in the case of 2D.The difficulties arise from the width of the mesh in the layer adjacent to the transition point,resulting in a suboptimal estimate for convergence.Existing analysis techniques cannot handle these difficulties well.To fill this gap,here a novel interpolation is designed delicately for the smooth part of the solution,bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method.Our theoretical result is uniform in the singular perturbation parameterεand is supported by the numerical experiments.展开更多
In the development of linear quadratic regulator(LQR) algorithms, the Riccati equation approach offers two important characteristics——it is recursive and readily meets the existence condition. However, these attribu...In the development of linear quadratic regulator(LQR) algorithms, the Riccati equation approach offers two important characteristics——it is recursive and readily meets the existence condition. However, these attributes are applicable only to transformed singular systems, and the efficiency of the regulator may be undermined if constraints are violated in nonsingular versions. To address this gap, we introduce a direct approach to the LQR problem for linear singular systems, avoiding the need for any transformations and eliminating the need for regularity assumptions. To achieve this goal, we begin by formulating a quadratic cost function to derive the LQR algorithm through a penalized and weighted regression framework and then connect it to a constrained minimization problem using the Bellman's criterion. Then, we employ a dynamic programming strategy in a backward approach within a finite horizon to develop an LQR algorithm for the original system. To accomplish this, we address the stability and convergence analysis under the reachability and observability assumptions of a hypothetical system constructed by the pencil of augmented matrices and connected using the Hamiltonian diagonalization technique.展开更多
Studying the seasonal deformation in GPS time series is important to interpreting geophysical contributors and identifying unmodeled and mismodeled seasonal signals.Traditional seasonal signal extraction used the leas...Studying the seasonal deformation in GPS time series is important to interpreting geophysical contributors and identifying unmodeled and mismodeled seasonal signals.Traditional seasonal signal extraction used the least squares method,which models seasonal deformation as a constant seasonal amplitude and phase.However,the seasonal variations are not constant from year to year,and the seasonal amplitude and phase are time-variable.In order to obtain the time-variable seasonal signal in the GPS station coordinate time series,singular spectrum analysis(SSA)is conducted in this study.We firstly applied the SSA on simulated seasonal signals with different frequencies 1.00 cycle per year(cpy),1.04 cpy and with time-variable amplitude are superimposed.It was found that SSA can successfully obtain the seasonal variations with different frequencies and with time-variable amplitude superimposed.Then,SSA is carried out on the GPS observations in Yunnan Province.The results show that the time-variable amplitude seasonal signals are ubiquitous in Yunnan Province,and the timevariable amplitude change in 2019 in the region is extracted,which is further explained by the soil moisture mass loading and atmospheric pressure loading.After removing the two loading effects,the SSA obtained modulated seasonal signals which contain the obvious seasonal variations at frequency of 1.046 cpy,it is close with the GPS draconitic year,1.040 cpy.Hence,the time-variable amplitude changes in 2019 and the seasonal GPS draconitic year in the region could be discriminated successfully by SSA in Yunnan Province.展开更多
(Multichannel)Singular spectrum analysis is considered as one of the most effective methods for seismic incoherent noise suppression.It utilizes the low-rank feature of seismic signal and regards the noise suppression...(Multichannel)Singular spectrum analysis is considered as one of the most effective methods for seismic incoherent noise suppression.It utilizes the low-rank feature of seismic signal and regards the noise suppression as a low-rank reconstruction problem.However,in some cases the seismic geophones receive some erratic disturbances and the amplitudes are dramatically larger than other receivers.The presence of this kind of noise,called erratic noise,makes singular spectrum analysis(SSA)reconstruction unstable and has undesirable effects on the final results.We robustify the low-rank reconstruction of seismic data by a reweighted damped SSA(RD-SSA)method.It incorporates the damped SSA,an improved version of SSA,into a reweighted framework.The damping operator is used to weaken the artificial disturbance introduced by the low-rank projection of both erratic and random noise.The central idea of the RD-SSA method is to iteratively approximate the observed data with the quadratic norm for the first iteration and the Tukeys bisquare norm for the rest iterations.The RD-SSA method can suppress seismic incoherent noise and keep the reconstruction process robust to the erratic disturbance.The feasibility of RD-SSA is validated via both synthetic and field data examples.展开更多
A physics-informed neural network(PINN)is a powerful tool for solving differential equations in solid and fluid mechanics.However,it suffers from singularly perturbed boundary-layer problems in which there exist sharp...A physics-informed neural network(PINN)is a powerful tool for solving differential equations in solid and fluid mechanics.However,it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives.In this paper,we introduce Chien's composite expansion method into PINNs,and propose a novel architecture for the PINNs,namely,the Chien-PINN(C-PINN)method.This novel PINN method is validated by singularly perturbed differential equations,and successfully solves the wellknown thin plate bending problems.In particular,no cumbersome matching conditions are needed for the C-PINN method,compared with the previous studies based on matched asymptotic expansions.展开更多
This paper is mainly about the spectral properties of a class of Jacobi operators(H_(c,b)u)(n)=c_(n)u(n+1)+c_(n-1)u(n-1)+b_(n)u(n),.where∣c_(n)−1∣=O(n^(−α))and b_(n)=O(n^(−1)).We will show that,forα≥1,the singula...This paper is mainly about the spectral properties of a class of Jacobi operators(H_(c,b)u)(n)=c_(n)u(n+1)+c_(n-1)u(n-1)+b_(n)u(n),.where∣c_(n)−1∣=O(n^(−α))and b_(n)=O(n^(−1)).We will show that,forα≥1,the singular continuous spectrum of the operator is empty.展开更多
This article studies the adaptive optimal output regulation problem for a class of interconnected singularly perturbed systems(SPSs) with unknown dynamics based on reinforcement learning(RL).Taking into account the sl...This article studies the adaptive optimal output regulation problem for a class of interconnected singularly perturbed systems(SPSs) with unknown dynamics based on reinforcement learning(RL).Taking into account the slow and fast characteristics among system states,the interconnected SPS is decomposed into the slow time-scale dynamics and the fast timescale dynamics through singular perturbation theory.For the fast time-scale dynamics with interconnections,we devise a decentralized optimal control strategy by selecting appropriate weight matrices in the cost function.For the slow time-scale dynamics with unknown system parameters,an off-policy RL algorithm with convergence guarantee is given to learn the optimal control strategy in terms of measurement data.By combining the slow and fast controllers,we establish the composite decentralized adaptive optimal output regulator,and rigorously analyze the stability and optimality of the closed-loop system.The proposed decomposition design not only bypasses the numerical stiffness but also alleviates the high-dimensionality.The efficacy of the proposed methodology is validated by a load-frequency control application of a two-area power system.展开更多
In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be r...In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.展开更多
We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We ob...We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.展开更多
Making exact approximations to solve equations distinguishes applied mathematicians from pure mathematicians, physicists, and engineers. Perturbation problems, both regular and singular, are pervasive in diverse field...Making exact approximations to solve equations distinguishes applied mathematicians from pure mathematicians, physicists, and engineers. Perturbation problems, both regular and singular, are pervasive in diverse fields of applied mathematics and engineering. This research paper provides a comprehensive overview of algebraic methods for solving perturbation problems, featuring a comparative analysis of their strengths and limitations. Serving as a valuable resource for researchers and practitioners, it offers insights and guidance for tackling perturbation problems in various disciplines, facilitating the advancement of applied mathematics and engineering.展开更多
In this paper we develop and study, as the second part of one more general development, the energy transmutation equation for the material singularity, previously obtained through the symmetrisation of a wave packet, ...In this paper we develop and study, as the second part of one more general development, the energy transmutation equation for the material singularity, previously obtained through the symmetrisation of a wave packet, that is, we develop the correlation between the terms of this equation, which accounts for the formation of matter from a previous vibrational state, and the different possible energy species. These energetic species are ascribed, in a simplified form, to the equation E¯ω=E¯k+E¯f, which allows us, through its associated phase factor, to gain an insight into the wave character of the kinetic energy and thus to attain the basis of the matter-wave, and all sorts of related phenomenologies, including that concerning quantum entanglement. The formation of the matter was previously identified as an energetic process, analogous to the kinetic one, in which finally the inertial mass is consolidated as a mass in a different phase, now, in addition, the mass of the material singularity is identified as a volumetric density of waves of toroidal geometry created in the process of singularisation or energy transfer between species, which makes it possible to establish the real relation or correspondence between the corpuscular and photonic energy equation (E=mc2=hν), i.e. to explain through m the intimate sense of the first equivalence, which explains what νis in the second one.展开更多
The Theory of General Singularity is presented, unifying quantum field theory, general relativity, and the standard model. This theory posits phonons as fundamental excitations in a quantum vacuum, modeled as a Bose-E...The Theory of General Singularity is presented, unifying quantum field theory, general relativity, and the standard model. This theory posits phonons as fundamental excitations in a quantum vacuum, modeled as a Bose-Einstein condensate. Through key equations, the role of phonons as intermediaries between matter, energy, and spacetime geometry is demonstrated. The theory expands Einsteins field equations to differentiate between visible and dark matter, and revises the standard model by incorporating phonons. It addresses dark matter, dark energy, gravity, and phase transitions, while making testable predictions. The theory proposes that singularities, the essence of particles and black holes, are quantum entities ubiquitous in nature, constituting the very essence of elementary particles, seen as micro black holes or quantum fractal structures of spacetime. As the theory is refined with increasing mathematical rigor, it builds upon the foundation of initial physical intuition, connecting the spacetime continuum of general relativity with the hydrodynamics of the quantum vacuum. Inspired by the insights of Tesla and Majorana, who believed that physical intuition justifies the infringement of mathematical rigor in the early stages of theory development, this work aims to advance the understanding of the fundamental laws of the universe and the perception of reality.展开更多
As traced by the Big Bang theory,the starting point of the universe,is called the“Singularity”in Da Ci Hai,an unabridged,comprehensive dictionary.According to cosmological reasoning,the singularity has an infinite d...As traced by the Big Bang theory,the starting point of the universe,is called the“Singularity”in Da Ci Hai,an unabridged,comprehensive dictionary.According to cosmological reasoning,the singularity has an infinite density of matter,an infinite curvature of space and time,and it is invisible and infinite.These characteristics are analogous to the human imagination at the level of innovation.For the innovation of cosmetic raw materials,there is also the possibility of infinite evolution.For example,in recent years,the scientific research in cosmetic industry the for promoting upgrade in raw materials is quite proactive.From the raw material enterprises,down to the brand company,the investment in raw material innovation is also strengthened at a visible rate.展开更多
This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utiliz...This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.展开更多
The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent ...The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.展开更多
文摘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.
文摘We examined the fractional second-order singular Lagrangian systems. We wrote the action principal function and equations of motion as fractional total differential equations. Also, we constructed the set of Hamilton-Jacobi partial differential equations (HJPDEs) within fractional calculus. We formulated the fractional path integral quantization for these systems. A mathematical example is examined with first- and second-class constraints.
基金supported by the National Natural Science Foundation of China(Grants:42204006,42274053,42030105,and 41504031)the Open Research Fund Program of the Key Laboratory of Geospace Environment and Geodesy,Ministry of Education,China(Grants:20-01-03 and 21-01-04)。
文摘Singular spectrum analysis is widely used in geodetic time series analysis.However,when extracting time-varying periodic signals from a large number of Global Navigation Satellite System(GNSS)time series,the selection of appropriate embedding window size and principal components makes this method cumbersome and inefficient.To improve the efficiency and accuracy of singular spectrum analysis,this paper proposes an adaptive singular spectrum analysis method by combining spectrum analysis with a new trace matrix.The running time and correlation analysis indicate that the proposed method can adaptively set the embedding window size to extract the time-varying periodic signals from GNSS time series,and the extraction efficiency of a single time series is six times that of singular spectrum analysis.The method is also accurate and more suitable for time-varying periodic signal analysis of global GNSS sites.
基金supported by the Joint Funds of the Chinese National Natural Science Foundation (NSFC)(Grant No.U2242213)the National Key Research and Development (R&D)Program of the Ministry of Science and Technology of China(Grant No. 2021YFC3000902)the National Science Foundation for Young Scholars (Grant No. 42205166)。
文摘Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial perturbation method tends only to capture synoptic scale initial uncertainty rather than mesoscale uncertainty in global ensemble prediction. To address this issue, a multiscale SV initial perturbation method based on the China Meteorological Administration Global Ensemble Prediction System(CMA-GEPS) is proposed to quantify multiscale initial uncertainty. The multiscale SV initial perturbation approach entails calculating multiscale SVs at different resolutions with multiple linearized physical processes to capture fast-growing perturbations from mesoscale to synoptic scale in target areas and combining these SVs by using a Gaussian sampling method with amplitude coefficients to generate initial perturbations. Following that, the energy norm,energy spectrum, and structure of multiscale SVs and their impact on GEPS are analyzed based on a batch experiment in different seasons. The results show that the multiscale SV initial perturbations can possess more energy and capture more mesoscale uncertainties than the traditional single-SV method. Meanwhile, multiscale SV initial perturbations can reflect the strongest dynamical instability in target areas. Their performances in global ensemble prediction when compared to single-scale SVs are shown to(i) improve the relationship between the ensemble spread and the root-mean-square error and(ii) provide a better probability forecast skill for atmospheric circulation during the late forecast period and for short-to medium-range precipitation. This study provides scientific evidence and application foundations for the design and development of a multiscale SV initial perturbation method for the GEPS.
基金supported by National Natural Science Foundation of China(11771257)the Shandong Provincial Natural Science Foundation of China(ZR2023YQ002,ZR2023MA007,ZR2021MA004)。
文摘For singularly perturbed convection-diffusion problems,supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes,especially in the case of 2D.The difficulties arise from the width of the mesh in the layer adjacent to the transition point,resulting in a suboptimal estimate for convergence.Existing analysis techniques cannot handle these difficulties well.To fill this gap,here a novel interpolation is designed delicately for the smooth part of the solution,bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method.Our theoretical result is uniform in the singular perturbation parameterεand is supported by the numerical experiments.
基金supported by the European Union’s Horizon Europe research and innovation programme (101120657)project ENFIELD (European Lighthouse to Manifest Trustworthy and Green AI), the Estonian Research Council (PRG658, PRG1463)the Estonian Centre of Excellence in Energy Efficiency, ENER (TK230) funded by the Estonian Ministry of Education and Research。
文摘In the development of linear quadratic regulator(LQR) algorithms, the Riccati equation approach offers two important characteristics——it is recursive and readily meets the existence condition. However, these attributes are applicable only to transformed singular systems, and the efficiency of the regulator may be undermined if constraints are violated in nonsingular versions. To address this gap, we introduce a direct approach to the LQR problem for linear singular systems, avoiding the need for any transformations and eliminating the need for regularity assumptions. To achieve this goal, we begin by formulating a quadratic cost function to derive the LQR algorithm through a penalized and weighted regression framework and then connect it to a constrained minimization problem using the Bellman's criterion. Then, we employ a dynamic programming strategy in a backward approach within a finite horizon to develop an LQR algorithm for the original system. To accomplish this, we address the stability and convergence analysis under the reachability and observability assumptions of a hypothetical system constructed by the pencil of augmented matrices and connected using the Hamiltonian diagonalization technique.
基金funded by National Natural Science Foundation of China(Grant No.11803065)Natural Science Foundation of Shanghai(Grant No.22ZR1472800)。
文摘Studying the seasonal deformation in GPS time series is important to interpreting geophysical contributors and identifying unmodeled and mismodeled seasonal signals.Traditional seasonal signal extraction used the least squares method,which models seasonal deformation as a constant seasonal amplitude and phase.However,the seasonal variations are not constant from year to year,and the seasonal amplitude and phase are time-variable.In order to obtain the time-variable seasonal signal in the GPS station coordinate time series,singular spectrum analysis(SSA)is conducted in this study.We firstly applied the SSA on simulated seasonal signals with different frequencies 1.00 cycle per year(cpy),1.04 cpy and with time-variable amplitude are superimposed.It was found that SSA can successfully obtain the seasonal variations with different frequencies and with time-variable amplitude superimposed.Then,SSA is carried out on the GPS observations in Yunnan Province.The results show that the time-variable amplitude seasonal signals are ubiquitous in Yunnan Province,and the timevariable amplitude change in 2019 in the region is extracted,which is further explained by the soil moisture mass loading and atmospheric pressure loading.After removing the two loading effects,the SSA obtained modulated seasonal signals which contain the obvious seasonal variations at frequency of 1.046 cpy,it is close with the GPS draconitic year,1.040 cpy.Hence,the time-variable amplitude changes in 2019 and the seasonal GPS draconitic year in the region could be discriminated successfully by SSA in Yunnan Province.
基金supported by the National Natural Science Foundation of China under grant no.42374133the Beijing Nova Program under grant no.2022056+1 种基金the Fundamental Research Funds for the Central Universities under grant no.2462020YXZZ006the Young Elite Scientists Sponsorship Program by CAST(YESS)under grant no.2018QNRC001。
文摘(Multichannel)Singular spectrum analysis is considered as one of the most effective methods for seismic incoherent noise suppression.It utilizes the low-rank feature of seismic signal and regards the noise suppression as a low-rank reconstruction problem.However,in some cases the seismic geophones receive some erratic disturbances and the amplitudes are dramatically larger than other receivers.The presence of this kind of noise,called erratic noise,makes singular spectrum analysis(SSA)reconstruction unstable and has undesirable effects on the final results.We robustify the low-rank reconstruction of seismic data by a reweighted damped SSA(RD-SSA)method.It incorporates the damped SSA,an improved version of SSA,into a reweighted framework.The damping operator is used to weaken the artificial disturbance introduced by the low-rank projection of both erratic and random noise.The central idea of the RD-SSA method is to iteratively approximate the observed data with the quadratic norm for the first iteration and the Tukeys bisquare norm for the rest iterations.The RD-SSA method can suppress seismic incoherent noise and keep the reconstruction process robust to the erratic disturbance.The feasibility of RD-SSA is validated via both synthetic and field data examples.
基金Project supported by the National Natural Science Foundation of China Basic Science Center Program for“Multiscale Problems in Nonlinear Mechanics”(No.11988102)the National Natural Science Foundation of China(No.12202451)。
文摘A physics-informed neural network(PINN)is a powerful tool for solving differential equations in solid and fluid mechanics.However,it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives.In this paper,we introduce Chien's composite expansion method into PINNs,and propose a novel architecture for the PINNs,namely,the Chien-PINN(C-PINN)method.This novel PINN method is validated by singularly perturbed differential equations,and successfully solves the wellknown thin plate bending problems.In particular,no cumbersome matching conditions are needed for the C-PINN method,compared with the previous studies based on matched asymptotic expansions.
文摘This paper is mainly about the spectral properties of a class of Jacobi operators(H_(c,b)u)(n)=c_(n)u(n+1)+c_(n-1)u(n-1)+b_(n)u(n),.where∣c_(n)−1∣=O(n^(−α))and b_(n)=O(n^(−1)).We will show that,forα≥1,the singular continuous spectrum of the operator is empty.
基金supported by the National Natural Science Foundation of China (62073327,62273350)the Natural Science Foundation of Jiangsu Province (BK20221112)。
文摘This article studies the adaptive optimal output regulation problem for a class of interconnected singularly perturbed systems(SPSs) with unknown dynamics based on reinforcement learning(RL).Taking into account the slow and fast characteristics among system states,the interconnected SPS is decomposed into the slow time-scale dynamics and the fast timescale dynamics through singular perturbation theory.For the fast time-scale dynamics with interconnections,we devise a decentralized optimal control strategy by selecting appropriate weight matrices in the cost function.For the slow time-scale dynamics with unknown system parameters,an off-policy RL algorithm with convergence guarantee is given to learn the optimal control strategy in terms of measurement data.By combining the slow and fast controllers,we establish the composite decentralized adaptive optimal output regulator,and rigorously analyze the stability and optimality of the closed-loop system.The proposed decomposition design not only bypasses the numerical stiffness but also alleviates the high-dimensionality.The efficacy of the proposed methodology is validated by a load-frequency control application of a two-area power system.
基金supported by the National Natural Science Foundation of China (No.12172154)the 111 Project (No.B14044)+1 种基金the Natural Science Foundation of Gansu Province (No.23JRRA1035)the Natural Science Foundation of Anhui University of Finance and Economics (No.ACKYC20043).
文摘In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.
基金supported by Shandong Provincial NSF(ZR2022MA020).
文摘We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.
文摘Making exact approximations to solve equations distinguishes applied mathematicians from pure mathematicians, physicists, and engineers. Perturbation problems, both regular and singular, are pervasive in diverse fields of applied mathematics and engineering. This research paper provides a comprehensive overview of algebraic methods for solving perturbation problems, featuring a comparative analysis of their strengths and limitations. Serving as a valuable resource for researchers and practitioners, it offers insights and guidance for tackling perturbation problems in various disciplines, facilitating the advancement of applied mathematics and engineering.
文摘In this paper we develop and study, as the second part of one more general development, the energy transmutation equation for the material singularity, previously obtained through the symmetrisation of a wave packet, that is, we develop the correlation between the terms of this equation, which accounts for the formation of matter from a previous vibrational state, and the different possible energy species. These energetic species are ascribed, in a simplified form, to the equation E¯ω=E¯k+E¯f, which allows us, through its associated phase factor, to gain an insight into the wave character of the kinetic energy and thus to attain the basis of the matter-wave, and all sorts of related phenomenologies, including that concerning quantum entanglement. The formation of the matter was previously identified as an energetic process, analogous to the kinetic one, in which finally the inertial mass is consolidated as a mass in a different phase, now, in addition, the mass of the material singularity is identified as a volumetric density of waves of toroidal geometry created in the process of singularisation or energy transfer between species, which makes it possible to establish the real relation or correspondence between the corpuscular and photonic energy equation (E=mc2=hν), i.e. to explain through m the intimate sense of the first equivalence, which explains what νis in the second one.
文摘The Theory of General Singularity is presented, unifying quantum field theory, general relativity, and the standard model. This theory posits phonons as fundamental excitations in a quantum vacuum, modeled as a Bose-Einstein condensate. Through key equations, the role of phonons as intermediaries between matter, energy, and spacetime geometry is demonstrated. The theory expands Einsteins field equations to differentiate between visible and dark matter, and revises the standard model by incorporating phonons. It addresses dark matter, dark energy, gravity, and phase transitions, while making testable predictions. The theory proposes that singularities, the essence of particles and black holes, are quantum entities ubiquitous in nature, constituting the very essence of elementary particles, seen as micro black holes or quantum fractal structures of spacetime. As the theory is refined with increasing mathematical rigor, it builds upon the foundation of initial physical intuition, connecting the spacetime continuum of general relativity with the hydrodynamics of the quantum vacuum. Inspired by the insights of Tesla and Majorana, who believed that physical intuition justifies the infringement of mathematical rigor in the early stages of theory development, this work aims to advance the understanding of the fundamental laws of the universe and the perception of reality.
文摘As traced by the Big Bang theory,the starting point of the universe,is called the“Singularity”in Da Ci Hai,an unabridged,comprehensive dictionary.According to cosmological reasoning,the singularity has an infinite density of matter,an infinite curvature of space and time,and it is invisible and infinite.These characteristics are analogous to the human imagination at the level of innovation.For the innovation of cosmetic raw materials,there is also the possibility of infinite evolution.For example,in recent years,the scientific research in cosmetic industry the for promoting upgrade in raw materials is quite proactive.From the raw material enterprises,down to the brand company,the investment in raw material innovation is also strengthened at a visible rate.
文摘This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.
文摘The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.
