In this paper we present a generalized perturbative approximate series expansion in terms of non-orthogonal component functions. The expansion is based on a perturbative formulation where, in the non-orthogonal case, ...In this paper we present a generalized perturbative approximate series expansion in terms of non-orthogonal component functions. The expansion is based on a perturbative formulation where, in the non-orthogonal case, the contribution of a given component function, at each point, in the time domain or frequency in the Fourier domain, is assumed to be perturbed by contributions from the other component functions in the set. In the case of orthogonal basis functions, the formulation reduces to the non-perturbative case approximate series expansion. Application of the series expansion is demonstrated in the context of two non-orthogonal component function sets. The technique is applied to a series of non-orthogonalized Bessel functions of the first kind that are used to construct a compound function for which the coefficients are determined utilizing the proposed approach. In a second application, the technique is applied to an example associated with the inverse problem in electrophysiology and is demonstrated through decomposition of a compound evoked potential from a peripheral nerve trunk in terms of contributing evoked potentials from individual nerve fibers of varying diameter. An additional application of the perturbative approximation is illustrated in the context of a trigonometric Fourier series representation of a continuous time signal where the technique is used to compute an approximation of the Fourier series coefficients. From these examples, it will be demonstrated that in the case of non-orthogonal component functions, the technique performs significantly better than the generalized Fourier series which can yield nonsensical results.展开更多
It is difficult to obtain exact solutions of the nonlinear partial differential equations （PDEs） due to their complexity and nonlinearity, especially for non-integrable systems. In this paper, some reasonable approx...It is difficult to obtain exact solutions of the nonlinear partial differential equations （PDEs） due to their complexity and nonlinearity, especially for non-integrable systems. In this paper, some reasonable approximations of real physics are considered, and the invariant expansion is proposed to solve real nonlinear systems. A simple invariant expansion with quite a universal pseudopotential is used for some nonlinear PDEs such as the Korteweg-de Vries （KdV） equation with a fifth-order dispersion term, the perturbed fourth-order KdV equation, the KdV-Burgers equation, and a Boussinesq-type equation.展开更多
One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Anot...One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random parameters.We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems.The first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random inputs.The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty.To enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability measure.We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the distribution.We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.展开更多
In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x)∈L2 continuous in a finite interval (a, b) which is much superi...In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x)∈L2 continuous in a finite interval (a, b) which is much superior to those of approximation by polynomial operators and by expansions of classical orthogonal series.展开更多
A novel method for obtaining the approximate symmetry of a partial differential equation with a small parameter is introduced. By expanding the independent variable and the dependent variable in the small parameter se...A novel method for obtaining the approximate symmetry of a partial differential equation with a small parameter is introduced. By expanding the independent variable and the dependent variable in the small parameter series, we obtain more affluent approximate symmetries. The method is applied to two perturbed nonlinear partial differential equations and new approximate solutions are derived.展开更多
In this work, the magnetic properties of Ising and XY antiferromagnetic thin-films are investigated each as a function of Neel temperature and thickness for layers （n = 2, 3, 4, 5, 6, and bulk （∞） by means of a me...In this work, the magnetic properties of Ising and XY antiferromagnetic thin-films are investigated each as a function of Neel temperature and thickness for layers （n = 2, 3, 4, 5, 6, and bulk （∞） by means of a mean-field and high temperature series expansion （HTSE） combined with Pade approximant calculations. The scaling law of magnetic susceptibility and magnetization is used to determine the critical exponent γ, veff （mean）, ratio of the critical exponents γ/v, and magnetic properties of Ising and XY antiferromagnetic thin-films for different thickness layers n = 2, 3, 4, 5, 6, and bulk （∞）.展开更多
The （3＋1）-dimensional Jimbo-Miwa （JM） equation is solved approximately by using the conformal invariant asymptotic expansion approach presented by Ruan. By solving the new （3＋1）-dimensional integrable models, ...The （3＋1）-dimensional Jimbo-Miwa （JM） equation is solved approximately by using the conformal invariant asymptotic expansion approach presented by Ruan. By solving the new （3＋1）-dimensional integrable models, which are conformal invariant and possess Painlevé property, the approximate solutions are obtained for the JM equation, containing not only one-soliton solutions but also periodic solutions and multi-soliton solutions. Some approximate solutions happen to be exact and some approximate solutions can become exact by choosing relations between the parameters properly.展开更多
Determining the parameters in the i-th order approximation in the cumulant expansion from the requirement that the correction to the zeroth order approximation is zero or the smallest,we show in the example of calcula...Determining the parameters in the i-th order approximation in the cumulant expansion from the requirement that the correction to the zeroth order approximation is zero or the smallest,we show in the example of calculating the Polyakov line<L>of U(1)gauge model at finite temperature with N_(T)=1-to 5-th order that the expansion works well not only in the strong and weak coupling regions,but also in the intermediate coupling region except the very vicinity of the phase transition point.The calculated<L>is in agreement with Monte Carlo simulations.展开更多
Quantitative thickness estimation below tuning thickness is a great challenge in seismic exploration. Most studies focus on the thin-beds whose top and bottom reflection coefficients are of equal magnitude and opposit...Quantitative thickness estimation below tuning thickness is a great challenge in seismic exploration. Most studies focus on the thin-beds whose top and bottom reflection coefficients are of equal magnitude and opposite polarity. There is no systematic research on the other thin-bed types. In this article, all of the thin-beds are classified into four types： thin-beds with equal magnitude and opposite polarity, thin-beds with unequal magnitude and opposite polarity, thin-beds with equal magnitude and identical polarity, and thin-beds with unequal magnitude and identical polarity. By analytical study, an equation describing the general relationship between seismic peak frequency and thin-bed thickness was derived which shows there is a Complex implicit non-linear relationship between them and which is difficult to use in practice. In order to solve this problem, we simplify the relationship by Taylor expansion and discuss the precision of the approximation formulae with different orders for the four types of thin-beds. Compared with the traditional amplitude method for thin-bed thickness calculation, the method we present has a higher precision and isn＇t influenced by the absolute value of top or bottom reflection coefficient, so it is convenient for use in practice.展开更多
The Jaynes-Cummings model （JCM） is studied in the absence of the rotating-wave approximation （RWA） by a coherent-state expansion technique. In comparison with the previous paper in which the coherent-state expansi...The Jaynes-Cummings model （JCM） is studied in the absence of the rotating-wave approximation （RWA） by a coherent-state expansion technique. In comparison with the previous paper in which the coherent-state expansion was performed only to the third order, we carry out in this paper a complete expansion to demonstrate exactly the dynamics of the JCM without the RWA. Our study gives a systematic method to solve the non-RWA problem, which would be useful in various physical systems, e.g., in a system with an ultracold trapped ion experiencing the running waves of lasers.展开更多
As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the researc...As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.展开更多
In this paper,we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padéapproximant.The virial expansion is ...In this paper,we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padéapproximant.The virial expansion is a high-temperature and low-density expansion and in practice,often,only the first several virial coefficients can be obtained.For Bose gases,we determine the BEC phase transition from a truncated virial expansion.For Fermi gases,we recover the low-temperature and high-density result from the virial expansion.展开更多
The random phase approximation is applied to the coupled-cluster expansions of lattice gauge theory (LGT). Using this method, wavefunctions are approximated by linear combination of graphs consisting of only one conne...The random phase approximation is applied to the coupled-cluster expansions of lattice gauge theory (LGT). Using this method, wavefunctions are approximated by linear combination of graphs consisting of only one connected Wilson loop. We study the excited state energy and wavefunction in (2+1)-D SU(3) LGT up to the third order. The glueball mass shows a good scaling behavior.展开更多
This paper presents two new non-iterative approximations of the power flow in a network. Real and reactive power are simultaneously modelled in complex equations. Also, resistances are not set to zero. This is a gener...This paper presents two new non-iterative approximations of the power flow in a network. Real and reactive power are simultaneously modelled in complex equations. Also, resistances are not set to zero. This is a generalization of the DC approximation, where only real power is modelled with zero line resistance. Hence the proposed approximations are more accurate than the DC approximation. The voltage lag over a link in a short, low voltage, network link is ten times as accurate as with the DC approximation. In the Appendix a new mathematical constant is introduced.展开更多
We develop the theory of multivariate saddlepoint approximations. Our treatment differs from the one in Barndorff-Nielsen and Cox （1979, equation （4.7）） in two aspects： 1） our results are satisfied for random ve...We develop the theory of multivariate saddlepoint approximations. Our treatment differs from the one in Barndorff-Nielsen and Cox （1979, equation （4.7）） in two aspects： 1） our results are satisfied for random vectors that are not necessarily sums of independent and identically distributed random vectors, and 2） we consider that the sample is taken from any distribution, not necessarily a member of the exponential family of densities. We also show the relationship with the corresponding multivariate Edgeworth approximations whose general treatment was developed by Durbin in 1980, emphasizing that the basic assumptions that support the validity of both approaches are essentially similar.展开更多
文摘In this paper we present a generalized perturbative approximate series expansion in terms of non-orthogonal component functions. The expansion is based on a perturbative formulation where, in the non-orthogonal case, the contribution of a given component function, at each point, in the time domain or frequency in the Fourier domain, is assumed to be perturbed by contributions from the other component functions in the set. In the case of orthogonal basis functions, the formulation reduces to the non-perturbative case approximate series expansion. Application of the series expansion is demonstrated in the context of two non-orthogonal component function sets. The technique is applied to a series of non-orthogonalized Bessel functions of the first kind that are used to construct a compound function for which the coefficients are determined utilizing the proposed approach. In a second application, the technique is applied to an example associated with the inverse problem in electrophysiology and is demonstrated through decomposition of a compound evoked potential from a peripheral nerve trunk in terms of contributing evoked potentials from individual nerve fibers of varying diameter. An additional application of the perturbative approximation is illustrated in the context of a trigonometric Fourier series representation of a continuous time signal where the technique is used to compute an approximation of the Fourier series coefficients. From these examples, it will be demonstrated that in the case of non-orthogonal component functions, the technique performs significantly better than the generalized Fourier series which can yield nonsensical results.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175092)Scientific Research Fund of Zhejiang Provincial Education Department(Grant No.Y201017148)K.C.Wong Magna Fund in Ningbo University
文摘It is difficult to obtain exact solutions of the nonlinear partial differential equations （PDEs） due to their complexity and nonlinearity, especially for non-integrable systems. In this paper, some reasonable approximations of real physics are considered, and the invariant expansion is proposed to solve real nonlinear systems. A simple invariant expansion with quite a universal pseudopotential is used for some nonlinear PDEs such as the Korteweg-de Vries （KdV） equation with a fifth-order dispersion term, the perturbed fourth-order KdV equation, the KdV-Burgers equation, and a Boussinesq-type equation.
基金supported by the NSF of China(No.11671265)partially supported by NSF DMS-1848508+4 种基金partially supported by the NSF of China(under grant numbers 11688101,11571351,and 11731006)science challenge project(No.TZ2018001)the youth innovation promotion association(CAS)supported by the National Science Foundation under Grant No.DMS-1439786the Simons Foundation Grant No.50736。
文摘One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random parameters.We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems.The first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random inputs.The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty.To enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability measure.We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the distribution.We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.
基金This work is supported by the Natural Science Foundation of Zhejiang,PR China.
文摘In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x)∈L2 continuous in a finite interval (a, b) which is much superior to those of approximation by polynomial operators and by expansions of classical orthogonal series.
文摘A novel method for obtaining the approximate symmetry of a partial differential equation with a small parameter is introduced. By expanding the independent variable and the dependent variable in the small parameter series, we obtain more affluent approximate symmetries. The method is applied to two perturbed nonlinear partial differential equations and new approximate solutions are derived.
文摘In this work, the magnetic properties of Ising and XY antiferromagnetic thin-films are investigated each as a function of Neel temperature and thickness for layers （n = 2, 3, 4, 5, 6, and bulk （∞） by means of a mean-field and high temperature series expansion （HTSE） combined with Pade approximant calculations. The scaling law of magnetic susceptibility and magnetization is used to determine the critical exponent γ, veff （mean）, ratio of the critical exponents γ/v, and magnetic properties of Ising and XY antiferromagnetic thin-films for different thickness layers n = 2, 3, 4, 5, 6, and bulk （∞）.
基金The project supported by the Natural Science Foundation of Zhejiang Province of China under Grant No. Y604036 and State Key Laboratory of 0il/Gas Reservoir Geology and Exploitation ＂PLN0402＂ The authors would like to thank Prof. Sen-Yue Lou for his help and discussion.
文摘The （3＋1）-dimensional Jimbo-Miwa （JM） equation is solved approximately by using the conformal invariant asymptotic expansion approach presented by Ruan. By solving the new （3＋1）-dimensional integrable models, which are conformal invariant and possess Painlevé property, the approximate solutions are obtained for the JM equation, containing not only one-soliton solutions but also periodic solutions and multi-soliton solutions. Some approximate solutions happen to be exact and some approximate solutions can become exact by choosing relations between the parameters properly.
基金Supported in part by the National Natural Science Foundation of China。
文摘Determining the parameters in the i-th order approximation in the cumulant expansion from the requirement that the correction to the zeroth order approximation is zero or the smallest,we show in the example of calculating the Polyakov line<L>of U(1)gauge model at finite temperature with N_(T)=1-to 5-th order that the expansion works well not only in the strong and weak coupling regions,but also in the intermediate coupling region except the very vicinity of the phase transition point.The calculated<L>is in agreement with Monte Carlo simulations.
基金supported by National Key S&T Special Projects of Marine Carbonate 2008ZX05000-004CNPC Projects 2008E-0610-10
文摘Quantitative thickness estimation below tuning thickness is a great challenge in seismic exploration. Most studies focus on the thin-beds whose top and bottom reflection coefficients are of equal magnitude and opposite polarity. There is no systematic research on the other thin-bed types. In this article, all of the thin-beds are classified into four types： thin-beds with equal magnitude and opposite polarity, thin-beds with unequal magnitude and opposite polarity, thin-beds with equal magnitude and identical polarity, and thin-beds with unequal magnitude and identical polarity. By analytical study, an equation describing the general relationship between seismic peak frequency and thin-bed thickness was derived which shows there is a Complex implicit non-linear relationship between them and which is difficult to use in practice. In order to solve this problem, we simplify the relationship by Taylor expansion and discuss the precision of the approximation formulae with different orders for the four types of thin-beds. Compared with the traditional amplitude method for thin-bed thickness calculation, the method we present has a higher precision and isn＇t influenced by the absolute value of top or bottom reflection coefficient, so it is convenient for use in practice.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10474118 and 10274093, the National Fundamental Research Program of China under Grant No. 2005CB724502, and the Foundation from Educational Department of Sichuan Province of China under Grant No. 2004C017
文摘The Jaynes-Cummings model （JCM） is studied in the absence of the rotating-wave approximation （RWA） by a coherent-state expansion technique. In comparison with the previous paper in which the coherent-state expansion was performed only to the third order, we carry out in this paper a complete expansion to demonstrate exactly the dynamics of the JCM without the RWA. Our study gives a systematic method to solve the non-RWA problem, which would be useful in various physical systems, e.g., in a system with an ultracold trapped ion experiencing the running waves of lasers.
基金Supported by the NSF of China under the Grant 10471010partially by the NSERC Canada under Grant G121211001
文摘As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.
基金supported in part by The Fundamental Research Funds for the Central Universities under Grant No.2020JKF306Special Funds for theoretical physics Research Program of the NSFC under Grant No.11947124,and NSFC under Grant Nos.11575125 and 11675119。
文摘In this paper,we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padéapproximant.The virial expansion is a high-temperature and low-density expansion and in practice,often,only the first several virial coefficients can be obtained.For Bose gases,we determine the BEC phase transition from a truncated virial expansion.For Fermi gases,we recover the low-temperature and high-density result from the virial expansion.
文摘The random phase approximation is applied to the coupled-cluster expansions of lattice gauge theory (LGT). Using this method, wavefunctions are approximated by linear combination of graphs consisting of only one connected Wilson loop. We study the excited state energy and wavefunction in (2+1)-D SU(3) LGT up to the third order. The glueball mass shows a good scaling behavior.
文摘This paper presents two new non-iterative approximations of the power flow in a network. Real and reactive power are simultaneously modelled in complex equations. Also, resistances are not set to zero. This is a generalization of the DC approximation, where only real power is modelled with zero line resistance. Hence the proposed approximations are more accurate than the DC approximation. The voltage lag over a link in a short, low voltage, network link is ten times as accurate as with the DC approximation. In the Appendix a new mathematical constant is introduced.
文摘We develop the theory of multivariate saddlepoint approximations. Our treatment differs from the one in Barndorff-Nielsen and Cox （1979, equation （4.7）） in two aspects： 1） our results are satisfied for random vectors that are not necessarily sums of independent and identically distributed random vectors, and 2） we consider that the sample is taken from any distribution, not necessarily a member of the exponential family of densities. We also show the relationship with the corresponding multivariate Edgeworth approximations whose general treatment was developed by Durbin in 1980, emphasizing that the basic assumptions that support the validity of both approaches are essentially similar.
