Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted...Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.展开更多
In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the ba...In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation.Furthermore,using the approach of Deift-Zhou nonlinear steepest descent,the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived.For the reverse space-time nonlocal Hirota equation,since the symmetries of its scattering matrix are different with the local Hirota equation,the v(λ_(i))(i=0,1)would like to be imaginary,which results in theδ_(λi)^(0)contains an increasing t(±Imv(λ_(i)))/2,and then the asymptotic behavior for nonlocal Hirota equation becomes differently.展开更多
The paper provides mathematical analysis of sensitivity of different combination rules in the DS/AHP method when an alternative is added to the set of decision alternatives while solving foresight problems. Different ...The paper provides mathematical analysis of sensitivity of different combination rules in the DS/AHP method when an alternative is added to the set of decision alternatives while solving foresight problems. Different cases of rank reversals are defined and two sets of conditions for these cases using the method DS/AHP are considered. Rank reversals are illustrated when the DS/AHP method is used to solve practical problem of critical technologies of energy conservation and power efficiency evaluation in Ukraine. It is shown that the DS/AHP method is not sensitive to exclusion (or addition) of an irrelevant decision alternative from (or to) the set of decision alternatives.展开更多
Non-standard backward heat conduction problem is ill-posed in the sense that the solution(if it exists) does not depend continuously on the data. In this paper, we propose a regularization strategy-quasi-reversibility...Non-standard backward heat conduction problem is ill-posed in the sense that the solution(if it exists) does not depend continuously on the data. In this paper, we propose a regularization strategy-quasi-reversibility method to analysis the stability of the problem. Meanwhile, we investigate the roles of regularization parameter in this method. Numerical result show that our algorithm is effective and stable.展开更多
In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Che...In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Chen-Lee-Liu equation can be obtained.Based on the spectral problem,the specific matrix Riemann-Hilbert problem is constructed for this nonlocal equation.Through solving this associated Riemann-Hilbert problem,the N-soliton solutions to this nonlocal equation can be obtained in the case of the jump matrix as an identity matrix.展开更多
This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contain...This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibilky method and regularized semigroups. Finally, an example is given.展开更多
TRIZ(俄语缩写)或TIPS(Theory of Inventive Problem Solving)是在理论界和实践中公认的一种创造性解决问题的方法。它通过系统化的方式解决(技术)冲突,从而推动产品或工艺的创新。德累斯顿应用科学大学的研究表明,该方法也可以反向使用...TRIZ(俄语缩写)或TIPS(Theory of Inventive Problem Solving)是在理论界和实践中公认的一种创造性解决问题的方法。它通过系统化的方式解决(技术)冲突,从而推动产品或工艺的创新。德累斯顿应用科学大学的研究表明,该方法也可以反向使用,称之为"TRIZReverse",即TRIZ逆向方法。详细阐释这两种方法,对于人才培养具有重要意义。展开更多
In this paper, on the one hand, we take the conventional quasi-reversibility method to obtain the error estimates of approximate solutions of the Cauchy problems for parabolic equations in a sub-domain of QT with stro...In this paper, on the one hand, we take the conventional quasi-reversibility method to obtain the error estimates of approximate solutions of the Cauchy problems for parabolic equations in a sub-domain of QT with strong restrictions to the measured boundary data. On the other hand, weakening the conditions on the measured data, then combining the duality method in optimization with the quasi-reversibility method, we solve the Cauchy problems for parabolic equations in the presence of noisy data. Using this method, we can get the proper regularization parameter ε that we need in the quasi-reversibility method and obtain the convergence rate of approximate solutions as the noise of amplitude δ tends to zero.展开更多
基金supported by the Key Project of Gansu Provincial National Science Foundation(23JRRA1022)the National Natural Science Foundation of China(12071431)+1 种基金the Fundamental Research Funds for the Central Universities(lzujbky-2021-ey18)the Innovative Groups of Basic Research in Gansu Province(22JR5RA391).
文摘Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.
基金supported by the National Natural Science Foundation of China(No.12175069 and No.12235007)Science and Technology Commission of Shanghai Municipality(No.21JC1402500 and No.22DZ2229014)Natural Science Foundation of Shanghai(No.23ZR1418100)。
文摘In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation.Furthermore,using the approach of Deift-Zhou nonlinear steepest descent,the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived.For the reverse space-time nonlocal Hirota equation,since the symmetries of its scattering matrix are different with the local Hirota equation,the v(λ_(i))(i=0,1)would like to be imaginary,which results in theδ_(λi)^(0)contains an increasing t(±Imv(λ_(i)))/2,and then the asymptotic behavior for nonlocal Hirota equation becomes differently.
文摘The paper provides mathematical analysis of sensitivity of different combination rules in the DS/AHP method when an alternative is added to the set of decision alternatives while solving foresight problems. Different cases of rank reversals are defined and two sets of conditions for these cases using the method DS/AHP are considered. Rank reversals are illustrated when the DS/AHP method is used to solve practical problem of critical technologies of energy conservation and power efficiency evaluation in Ukraine. It is shown that the DS/AHP method is not sensitive to exclusion (or addition) of an irrelevant decision alternative from (or to) the set of decision alternatives.
文摘Non-standard backward heat conduction problem is ill-posed in the sense that the solution(if it exists) does not depend continuously on the data. In this paper, we propose a regularization strategy-quasi-reversibility method to analysis the stability of the problem. Meanwhile, we investigate the roles of regularization parameter in this method. Numerical result show that our algorithm is effective and stable.
基金supported by the National Natural Science Foundation of China under Grant No.11975145。
文摘In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Chen-Lee-Liu equation can be obtained.Based on the spectral problem,the specific matrix Riemann-Hilbert problem is constructed for this nonlocal equation.Through solving this associated Riemann-Hilbert problem,the N-soliton solutions to this nonlocal equation can be obtained in the case of the jump matrix as an identity matrix.
基金This project was supported by TRAPOYT, the Key Project of Chinese Ministry of Education(104126) the NNSF of China(10371046)
文摘This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibilky method and regularized semigroups. Finally, an example is given.
文摘TRIZ(俄语缩写)或TIPS(Theory of Inventive Problem Solving)是在理论界和实践中公认的一种创造性解决问题的方法。它通过系统化的方式解决(技术)冲突,从而推动产品或工艺的创新。德累斯顿应用科学大学的研究表明,该方法也可以反向使用,称之为"TRIZReverse",即TRIZ逆向方法。详细阐释这两种方法,对于人才培养具有重要意义。
基金supported by National Natural Science Foundation of China (Grant No.11226166)Scientific Research Fund of Hu'nan Provincial Education Department (Grant No.11C0052)
文摘In this paper, on the one hand, we take the conventional quasi-reversibility method to obtain the error estimates of approximate solutions of the Cauchy problems for parabolic equations in a sub-domain of QT with strong restrictions to the measured boundary data. On the other hand, weakening the conditions on the measured data, then combining the duality method in optimization with the quasi-reversibility method, we solve the Cauchy problems for parabolic equations in the presence of noisy data. Using this method, we can get the proper regularization parameter ε that we need in the quasi-reversibility method and obtain the convergence rate of approximate solutions as the noise of amplitude δ tends to zero.