In this paper, the fluid flow differential equation based on the homogenous reservoirs model is first reviewed. Then a theorem about the formal similarity of solutions in the Laplace space with outer boundary conditio...In this paper, the fluid flow differential equation based on the homogenous reservoirs model is first reviewed. Then a theorem about the formal similarity of solutions in the Laplace space with outer boundary conditions and inner boundary condition is presented and proved. Lastly, a corollary of our theorem is given particularly on inner boundary. The obtained results are very helpful for understanding inherent laws of relevant engineering science and designing practical analysis software.展开更多
Let P(∆)be a polynomial of the Laplace operator∆=n∑j=1∂^(2)∂x^(2)_(j) on R^(n).We prove the existence of a bounded right inverse of the differential operator P(∆)in the weighted Hilbert space with the Gaussian measur...Let P(∆)be a polynomial of the Laplace operator∆=n∑j=1∂^(2)∂x^(2)_(j) on R^(n).We prove the existence of a bounded right inverse of the differential operator P(∆)in the weighted Hilbert space with the Gaussian measure,i.e.,L^(2)(R^(n),e^(−|x|^(2))).展开更多
Mathematical models for phenomena in the physical sciences are typically parameter-dependent, and the estimation of parameters that optimally model the trends suggested by experimental observation depends on how model...Mathematical models for phenomena in the physical sciences are typically parameter-dependent, and the estimation of parameters that optimally model the trends suggested by experimental observation depends on how model-observation discrepancies are quantified. Commonly used parameter estimation techniques based on least-squares minimization of the model-observation discrepancies assume that the discrepancies are quantified with the L<sup>2</sup>-norm applied to a discrepancy function. While techniques based on such an assumption work well for many applications, other applications are better suited for least-squared minimization approaches that are based on other norm or inner-product induced topologies. Motivated by an application in the material sciences, the new alternative least-squares approach is defined and an insightful analytical comparison with a baseline least-squares approach is provided.展开更多
Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>（Ω）={f∈L<sup>1</sup>（Ω）: there is an F∈ B<sub>1</sub><sup>0.1</sup>（R<s...Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>（Ω）={f∈L<sup>1</sup>（Ω）: there is an F∈ B<sub>1</sub><sup>0.1</sup>（R<sup>n</sup>） such that F|Ω=f| and B<sub>1,z</sub><sup>0.1</sup>（Ω）={f∈B<sub>1</sub><sup>0.1</sup>（R<sup>n</sup>）: f=0 on R<sup>n</sup>\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R<sub>+</sub><sup>n</sup>.展开更多
We analyze the behaviour of TE, TM electromagnetic fields in a toroidal space through Maxwell and wave equations. Their solutions are discussed in a space endowed with a refractive index making separable the wave equa...We analyze the behaviour of TE, TM electromagnetic fields in a toroidal space through Maxwell and wave equations. Their solutions are discussed in a space endowed with a refractive index making separable the wave equations.展开更多
The solutions of the Laplace equation in n-dimensional space are studied. The angular eigenfunctions have the form of associated Jacob/polynomials. The radial solution of the Helmholtz equation is derived.
文摘In this paper, the fluid flow differential equation based on the homogenous reservoirs model is first reviewed. Then a theorem about the formal similarity of solutions in the Laplace space with outer boundary conditions and inner boundary condition is presented and proved. Lastly, a corollary of our theorem is given particularly on inner boundary. The obtained results are very helpful for understanding inherent laws of relevant engineering science and designing practical analysis software.
文摘Let P(∆)be a polynomial of the Laplace operator∆=n∑j=1∂^(2)∂x^(2)_(j) on R^(n).We prove the existence of a bounded right inverse of the differential operator P(∆)in the weighted Hilbert space with the Gaussian measure,i.e.,L^(2)(R^(n),e^(−|x|^(2))).
文摘Mathematical models for phenomena in the physical sciences are typically parameter-dependent, and the estimation of parameters that optimally model the trends suggested by experimental observation depends on how model-observation discrepancies are quantified. Commonly used parameter estimation techniques based on least-squares minimization of the model-observation discrepancies assume that the discrepancies are quantified with the L<sup>2</sup>-norm applied to a discrepancy function. While techniques based on such an assumption work well for many applications, other applications are better suited for least-squared minimization approaches that are based on other norm or inner-product induced topologies. Motivated by an application in the material sciences, the new alternative least-squares approach is defined and an insightful analytical comparison with a baseline least-squares approach is provided.
基金This research was partially supported by the SEDFthe NNSF of China.
文摘Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>（Ω）={f∈L<sup>1</sup>（Ω）: there is an F∈ B<sub>1</sub><sup>0.1</sup>（R<sup>n</sup>） such that F|Ω=f| and B<sub>1,z</sub><sup>0.1</sup>（Ω）={f∈B<sub>1</sub><sup>0.1</sup>（R<sup>n</sup>）: f=0 on R<sup>n</sup>\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R<sub>+</sub><sup>n</sup>.
文摘We analyze the behaviour of TE, TM electromagnetic fields in a toroidal space through Maxwell and wave equations. Their solutions are discussed in a space endowed with a refractive index making separable the wave equations.
基金Supported by the Nationa1 Natural Science Foundation of China under Grant No.10874018"the Fundamental Research Funds for the Central Universities"
文摘The solutions of the Laplace equation in n-dimensional space are studied. The angular eigenfunctions have the form of associated Jacob/polynomials. The radial solution of the Helmholtz equation is derived.
