In this paper,we review computational approaches to optimization problems of inhomogeneous rods and plates.We consider both the optimization of eigenvalues and the localization of eigenfunctions.These problems are mot...In this paper,we review computational approaches to optimization problems of inhomogeneous rods and plates.We consider both the optimization of eigenvalues and the localization of eigenfunctions.These problems are motivated by physical problems including the determination of the extremum of the fundamental vibration frequency and the localization of the vibration displacement.We demonstrate how an iterative rearrangement approach and a gradient descent approach with projection can successfully solve these optimization problems under different boundary conditions with different densities given.展开更多
In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Ope...In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Operator (LBO). These functions are isometry invariant, so they are independent of the object’s representation including parameterization, spatial position and orientation. Several works have shown that these eigenfunctions provide topological and geometrical information of the surfaces of interest [1] [2]. We propose to make use of that information for the construction of a set of skeletons, associated to each eigenfunction, which can be used as a fingerprint for the surface of interest. The main goal is to develop a classification system based on these skeletons, instead of the surfaces, for the analysis of medical images, for instance.展开更多
We consider the following eigenvalue problem: [GRAPHICS] Where f(x, t) is a continuous function with critical growth. We prove the existence of nontrivial solutions.
The structure of a Hamiltonian matrix for a quantum chaotic system, the nuclear octupole deformation model, has been discussed in detail. The distribution of the eigenfunctions of this system expanded by the eigenstat...The structure of a Hamiltonian matrix for a quantum chaotic system, the nuclear octupole deformation model, has been discussed in detail. The distribution of the eigenfunctions of this system expanded by the eigenstates of a quantum integrable system is studied with the help of generalized Brillouin?Wigner perturbation theory. The results show that a significant randomness in this distribution can be observed when its classical counterpart is under the strong chaotic condition. The averaged shape of the eigenfunctions fits with the Gaussian distribution only when the effects of the symmetry have been removed.展开更多
The fully developed slip flow in an annular sector duct is solved by expansions of eigenfunctions in the radial direction and boundary collocation on the straight sides. The method is efficient and accurate. The flow ...The fully developed slip flow in an annular sector duct is solved by expansions of eigenfunctions in the radial direction and boundary collocation on the straight sides. The method is efficient and accurate. The flow field for slip flow differs much from that of no-slip flow. The Poiseuille number increases with increased inner radius, opening angle, and decreases with slip.展开更多
Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two-dimensional and three-dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurati...Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two-dimensional and three-dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, Wentzel-Kramers-Brillouin (WKB) methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms of Green’s functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary differential equations for which the eigenvalues and the eigenfunctions can be approximated. In the long-wave limit approximation, solutions are expressed in terms of generalized hypergeometric functions. Our procedure works regardless of the values of the Reynolds number.展开更多
We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce ...We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the z-complex plane is divided into four analytic regions of D_(j) : j = 1, 2, 3, 4. Since the second column of Jost eigenfunctions is analytic in D_(j), but in the upper-half or lowerhalf plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in Dj. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries;these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this N-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the N-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.展开更多
We aim to find the eigenvalues and eigenfunctions of the barrier potential case for Strum-Liouville operator on the finite interval [0,π] when λ > 0. Generally, the eigenvalue problem for the Sturm-Liouville oper...We aim to find the eigenvalues and eigenfunctions of the barrier potential case for Strum-Liouville operator on the finite interval [0,π] when λ > 0. Generally, the eigenvalue problem for the Sturm-Liouville operator is often solved by using integral equations, which are sometimes complex to solve, and difficulties may arise in computing the boundary values. Considering the said complexity, we have successfully developed a technique to give the asymptotic formulae of the eigenvalue and the eigenfunction for Sturm-Liouville operator with barrier potential. The results are of significant interest in the field of quantum mechanics and atomic systems to observe discrete energy levels.展开更多
With respect to flows in a two-dimensional sudden expansion and contraction channel having a pair of cavities, numerical simulation was performed by imposing inlet/outlet boundary conditions giving a velocity distribu...With respect to flows in a two-dimensional sudden expansion and contraction channel having a pair of cavities, numerical simulation was performed by imposing inlet/outlet boundary conditions giving a velocity distribution to the inlet. Periodic flows have been reproduced, which have a discrete spectrum about frequency. A fundamental wave occupies most part of the disturbance components, but higher harmonic waves are also included. The disturbance is excited by Kelvin-Helmholtz instability in a cavity section, where only the fundamental wave is generated. A wavenumber is regulated by a channel length under a periodic boundary condition, but there is no restriction in a main flow direction under the inlet/outlet boundary conditions, and therefore, some wavenumbers can occur. Therefore, an arbitrary frequency component of disturbance is a synthesized wave composed of various wave numbers. There are two kinds of components constituting this synthesized wave: a maximum of a velocity distribution is near a wall and in the center of the channel, which are called as wall mode and central mode in linear stability analysis of the plane Poiseuille flow. The synthesized wave composed of some modes shows a tendency to lower wavenumbers at the center of the channel.展开更多
基金supported by the DMS-1853701supported in part by the DMS-2208373.
文摘In this paper,we review computational approaches to optimization problems of inhomogeneous rods and plates.We consider both the optimization of eigenvalues and the localization of eigenfunctions.These problems are motivated by physical problems including the determination of the extremum of the fundamental vibration frequency and the localization of the vibration displacement.We demonstrate how an iterative rearrangement approach and a gradient descent approach with projection can successfully solve these optimization problems under different boundary conditions with different densities given.
文摘In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Operator (LBO). These functions are isometry invariant, so they are independent of the object’s representation including parameterization, spatial position and orientation. Several works have shown that these eigenfunctions provide topological and geometrical information of the surfaces of interest [1] [2]. We propose to make use of that information for the construction of a set of skeletons, associated to each eigenfunction, which can be used as a fingerprint for the surface of interest. The main goal is to develop a classification system based on these skeletons, instead of the surfaces, for the analysis of medical images, for instance.
文摘We consider the following eigenvalue problem: [GRAPHICS] Where f(x, t) is a continuous function with critical growth. We prove the existence of nontrivial solutions.
文摘The structure of a Hamiltonian matrix for a quantum chaotic system, the nuclear octupole deformation model, has been discussed in detail. The distribution of the eigenfunctions of this system expanded by the eigenstates of a quantum integrable system is studied with the help of generalized Brillouin?Wigner perturbation theory. The results show that a significant randomness in this distribution can be observed when its classical counterpart is under the strong chaotic condition. The averaged shape of the eigenfunctions fits with the Gaussian distribution only when the effects of the symmetry have been removed.
文摘The fully developed slip flow in an annular sector duct is solved by expansions of eigenfunctions in the radial direction and boundary collocation on the straight sides. The method is efficient and accurate. The flow field for slip flow differs much from that of no-slip flow. The Poiseuille number increases with increased inner radius, opening angle, and decreases with slip.
文摘Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two-dimensional and three-dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, Wentzel-Kramers-Brillouin (WKB) methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms of Green’s functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary differential equations for which the eigenvalues and the eigenfunctions can be approximated. In the long-wave limit approximation, solutions are expressed in terms of generalized hypergeometric functions. Our procedure works regardless of the values of the Reynolds number.
基金supported by the National Natural Science Foundation of China(12175069 and 12235007)the Science and Technology Commission of Shanghai Municipality (21JC1402500 and 22DZ2229014)。
文摘We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the z-complex plane is divided into four analytic regions of D_(j) : j = 1, 2, 3, 4. Since the second column of Jost eigenfunctions is analytic in D_(j), but in the upper-half or lowerhalf plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in Dj. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries;these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this N-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the N-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.
文摘We aim to find the eigenvalues and eigenfunctions of the barrier potential case for Strum-Liouville operator on the finite interval [0,π] when λ > 0. Generally, the eigenvalue problem for the Sturm-Liouville operator is often solved by using integral equations, which are sometimes complex to solve, and difficulties may arise in computing the boundary values. Considering the said complexity, we have successfully developed a technique to give the asymptotic formulae of the eigenvalue and the eigenfunction for Sturm-Liouville operator with barrier potential. The results are of significant interest in the field of quantum mechanics and atomic systems to observe discrete energy levels.
文摘With respect to flows in a two-dimensional sudden expansion and contraction channel having a pair of cavities, numerical simulation was performed by imposing inlet/outlet boundary conditions giving a velocity distribution to the inlet. Periodic flows have been reproduced, which have a discrete spectrum about frequency. A fundamental wave occupies most part of the disturbance components, but higher harmonic waves are also included. The disturbance is excited by Kelvin-Helmholtz instability in a cavity section, where only the fundamental wave is generated. A wavenumber is regulated by a channel length under a periodic boundary condition, but there is no restriction in a main flow direction under the inlet/outlet boundary conditions, and therefore, some wavenumbers can occur. Therefore, an arbitrary frequency component of disturbance is a synthesized wave composed of various wave numbers. There are two kinds of components constituting this synthesized wave: a maximum of a velocity distribution is near a wall and in the center of the channel, which are called as wall mode and central mode in linear stability analysis of the plane Poiseuille flow. The synthesized wave composed of some modes shows a tendency to lower wavenumbers at the center of the channel.
