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
In this article,we established the structure of all eigenvalues and the oscillation property of corresponding eigenfunctions for discrete clamped beam equation △^(4)u(k-2)=λm(k)u(k),k∈[2,N+1]Z,u(0)=△u(0)=0=u(N+2)=...In this article,we established the structure of all eigenvalues and the oscillation property of corresponding eigenfunctions for discrete clamped beam equation △^(4)u(k-2)=λm(k)u(k),k∈[2,N+1]Z,u(0)=△u(0)=0=u(N+2)=△u(N+2)with the weight function m:[2,N+1]Z→(0,∞),[2,N+1]_(Z)={2,3,...,N+1}.As an application,we obtain the global structure of nodal solutions of the corresponding nonlinear problems based on the nonlinearity satisfying suitable growth conditions at zero and infinity.展开更多
We propose an ？~1 regularized method for numerical differentiation using empirical eigenfunctions. Compared with traditional methods for numerical differentiation, the output of our method can be considered directly ...We propose an ？~1 regularized method for numerical differentiation using empirical eigenfunctions. Compared with traditional methods for numerical differentiation, the output of our method can be considered directly as the derivative of the underlying function. Moreover,our method could produce sparse representations with respect to empirical eigenfunctions.Numerical results show that our method is quite effective.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 introduce a decimation scheme of constructing renormalized Hamiltonian flows,which is useful in the study of properties of energy eigenfunctions,such as localization,as well as in approximate calculation of eigenen...We introduce a decimation scheme of constructing renormalized Hamiltonian flows,which is useful in the study of properties of energy eigenfunctions,such as localization,as well as in approximate calculation of eigenenergies.The method is based on a generalized Brillouin-Wigner perturbation theory.Each flow is specific for a given energy and,at each step of the flow,a finite subspace of the Hilbert space is decimated in order to obtain a renormalized Hamiltonian for the next step.Eigenenergies of the original Hamiltonian appear as unstable fixed points of renormalized flows.Numerical illustration of the method is given in the Wigner-band random-matrix model.展开更多
We obtain the energy spectrum and all the corresponding eigenfunctions of N-body Bose and Fermi systems with Quadratic Pair Potentials in one dimension. The original first excited state or energy level is disappeared ...We obtain the energy spectrum and all the corresponding eigenfunctions of N-body Bose and Fermi systems with Quadratic Pair Potentials in one dimension. The original first excited state or energy level is disappeared in one dimension, which results from the operation of symmetry or antisymmetry of identical particles. In two and higher dimensions, we give the energy spectrum and the analytical ground state wave [unctions and the degree of degeneracy. By comparison, we refine A vinash Khare＇s results by making some items in his article precisely.展开更多
In this paper, non-self-adjoint Sturm-Liuville operators in Weyl＇s limit-circle case are studied. We first determine all the non-self-adjoint boundary conditions yielding dissipative operators for each allowed Sturm-...In this paper, non-self-adjoint Sturm-Liuville operators in Weyl＇s limit-circle case are studied. We first determine all the non-self-adjoint boundary conditions yielding dissipative operators for each allowed Sturm-Liouville differential expression. Then, using the characteristic determinant, we prove the completeness of the system of eigenfunctions and associated functions for these dissipative operators.展开更多
Let{e_(j)}be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold(M,g).Let H■M be a submanifold and{ψ_(k)}be an orthonormal basis of Laplace eigenfunctions of H with the induced metric.We ...Let{e_(j)}be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold(M,g).Let H■M be a submanifold and{ψ_(k)}be an orthonormal basis of Laplace eigenfunctions of H with the induced metric.We obtain joint asymptotics for the Fourier coefficients<γHe_(j),ψ_(k)>L^(2)(H)=∫He_(j),ψ_(k)dV_(H)of restrictionsγHe_(j)of e_(j)to H.In particular,we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum{(μ_(k),λ_(j))}^(∞)_(j,k-0)of the(square roots of the)Laplacian△_(M)on M and the Laplacian△_(H)on H in a family of suitably‘thick'regions in R^(2).Thick regions include(1)the truncated coneμ_(k)/λ_(j)∈[a,b]■(0,1)andλ_(j)≤λ,and(2)the slowly thickening strip|μ_(k)-cλ_(j)|≤w(λ)andλ_(j)≤λ,where w(λ)is monotonic and 1■w(λ)≤λ^(1/2).Key tools for obtaining the asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.展开更多
The eigenfunctions in a stability problem of boundary-layer flow over a viscoelastic compliant wall were studied. Two categories of modes, TSI and CIFI, exist in the eigenvalue solutions. The eigenfunctions of flow-ba...The eigenfunctions in a stability problem of boundary-layer flow over a viscoelastic compliant wall were studied. Two categories of modes, TSI and CIFI, exist in the eigenvalue solutions. The eigenfunctions of flow-based TSI were investigated together with those in the flow over rigid wall, whereas the eigenfunctions of wall-based CIFI were compared with the wall functions in an individual wall without fluid constraint. The physical characteristics of the eigenmodes were discussed based on their eigenfunctions.展开更多
This study investigates the effects of radiation force due to the rotational pitch motion of a wave energy device,which comprises a coaxial bottom-mounted cylindrical caisson in a two-layer fluid,along with a submerge...This study investigates the effects of radiation force due to the rotational pitch motion of a wave energy device,which comprises a coaxial bottom-mounted cylindrical caisson in a two-layer fluid,along with a submerged cylindrical buoy.The system is modeled as a two-layer fluid with infinite horizontal extent and finite depth.The radiation problem is analyzed in the context of linear water waves.The fluid domain is divided into outer and inner zones,and mathematical solutions for the pitch radiating potential are derived for the corresponding boundary valve problem in these zones using the separation of variables approach.Using the matching eigenfunction expansion method,the unknown coefficients in the analytical expression of the radiation potentials are evaluated.The resulting radiation potential is then used to compute the added mass and damping coefficients.Several numerical results for the added mass and damping coefficients are investigated for numerous parameters,particularly the effects of the cylinder radius,the draft of the submerged cylinder,and the density proportion between the two fluid layers across different frequency ranges.The major findings are presented and discussed.展开更多
For a generalized radiation problem,an infinitely long submerged horizontal cylinder is forced to vibrate periodically in the transverse direction,with a described elastic harmonic motion along its longitudinal direct...For a generalized radiation problem,an infinitely long submerged horizontal cylinder is forced to vibrate periodically in the transverse direction,with a described elastic harmonic motion along its longitudinal direction.A critical frequency corresponds to the described wave number of elastic vibration,and the generalized hydrodynamic coefficients abruptly change in the vicinity of critical frequency.In this work,a numerical examination is carried out to study the characteristics of wave profiles and wave propagation in the vicinity of the critical frequency.Results show that below the critical frequency,the real parts of complex wave profiles have large values in the vicinity of the cylinder and decay to zero with the increasing distance from the cylinder.Meanwhile,the imaginary parts of complex wave profiles are all zero,which explains why the generalized radiation damping is zero when the vibration is less than the critical frequency.At far distances,no radiation wave is observed.When the vibration exceeds the critical frequency,the real and imaginary parts of the wave profiles oscillate harmonically and keep steady amplitudes.In addition,the generated radiation wave propagates obliquely outward.The influence of the cylinder’s submergence depth on the wave profile is also studied,and the results indicate that the amplitude of the wave profile decreases as the submergence depth of the cylinder increases.The 3D wave profiles are graphically presented to show the wave propagation characteristics in the vicinity of the critical frequency for this generalized radiation problem.This study provides a good reference for the interaction between fluid and slender elastic structures.展开更多
In this paper we shall consider the nonresonance Dirichlet boundary value problemwhere λ>0 is a parameter, p>0 is a constant. Intervals of A are determined to ensure the existence of a nonnegative solution of t...In this paper we shall consider the nonresonance Dirichlet boundary value problemwhere λ>0 is a parameter, p>0 is a constant. Intervals of A are determined to ensure the existence of a nonnegative solution of the boundary value problem. For λ=1, we shall also offer criteria for the existence of eigenfunctions. The main results include and improve on those of [2,4,6,8].展开更多
Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N. In this article, we characterize the He-valued joint eigenfunctions of L and T havin...Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N. In this article, we characterize the He-valued joint eigenfunctions of L and T having eigenvalues from the quaternionic Heisenberg fan.展开更多
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
Global warming has led to major melting of ice in the polar Arctic,making it possible to open Arctic shipping lanes.In this case,the large number of ice sheets are extremely dangerous for ship navigation,so in this pa...Global warming has led to major melting of ice in the polar Arctic,making it possible to open Arctic shipping lanes.In this case,the large number of ice sheets are extremely dangerous for ship navigation,so in this paper,a body floating on water confined between two finite ice sheets is investigated.The linearized potential flow theory is adopted,and water is considered an incompressible ideal fluid with a finite depth of the fluid domain.The ice sheets are treated as elastic plates,and the problem is solved by matching eigenfunction expansion.The fluid domain is divided into subregions on the basis of the water surface conditions,and the velocity potential of the subdomains is expanded via the separated variable method.By utilizing the continuity of pressure and velocity at the interfaces of two neighboring regions,a system of linear equations is established to obtain the unknown coefficients in the expansion,which in turn leads to analytical solutions for different motion modes in different regions.The effects of different structural drafts,and different lengths of ice sheets on both sides,etc.,on the hydrodynamic characteristics of floats are analyzed.The amplitude of motion of the float is explored,as is the wave elevation between the ice sheets and the float.展开更多
The wave interaction with stratified porous structure combined with a surface-piercing porous block in a stepped seabed is analysed based on the small amplitude wave theory.The study is performed to analyse the effect...The wave interaction with stratified porous structure combined with a surface-piercing porous block in a stepped seabed is analysed based on the small amplitude wave theory.The study is performed to analyse the effectiveness of partial porous structure in increasing the wave attenuation in the nearshore regions consisting of stratified porous structures of different configurations using the eigenfunction expansion method and orthogonal mode-coupling relation.The hydrodynamic characteristics such as wave reflection coefficient,transmission coefficient,dissipation coefficient,wave force impact and surface elevation are investigated due to the presence of both horizontally and vertically stratified porous structures.The effect of varying porosity,structural width,angle of incidence,wavelength and length between the porous block and stratified structure is examined.The numerical results are validated with the results available in the literature.The present study illustrates that the presence of the stratified structure decreases wave transmission and efficient wave attenuation can also be easily achieved.The wave force acting on stratified structure can be decreased if the structure is combined with wider surface-piercing porous blocks.Further,the presence of stratified porous structure combined with porous block helps in creating a tranquil zone in the leeside of the structure.The combination of vertical and horizontal stratified porous structure with surface-piercing porous block is intended to be an effective solution for the protection of coastal facilities.展开更多
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.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1190146411801453)the Young Teachers’Scientific Research Capability Upgrading Project of Northwest Normal University(Grant No.NWNULKQN2020-20).
文摘In this article,we established the structure of all eigenvalues and the oscillation property of corresponding eigenfunctions for discrete clamped beam equation △^(4)u(k-2)=λm(k)u(k),k∈[2,N+1]Z,u(0)=△u(0)=0=u(N+2)=△u(N+2)with the weight function m:[2,N+1]Z→(0,∞),[2,N+1]_(Z)={2,3,...,N+1}.As an application,we obtain the global structure of nodal solutions of the corresponding nonlinear problems based on the nonlinearity satisfying suitable growth conditions at zero and infinity.
基金Supported by the National Nature Science Foundation of China(Grant Nos.11301052,11301045,11271060,11601064,11671068)the Fundamental Research Funds for the Central Universities(Grant No.DUT16LK33)the Fundamental Research of Civil Aircraft(Grant No.MJ-F-2012-04)
文摘We propose an ？~1 regularized method for numerical differentiation using empirical eigenfunctions. Compared with traditional methods for numerical differentiation, the output of our method can be considered directly as the derivative of the underlying function. Moreover,our method could produce sparse representations with respect to empirical eigenfunctions.Numerical results show that our method is quite effective.
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 under Grant Nos.11275179,11535011,and 11775210
文摘We introduce a decimation scheme of constructing renormalized Hamiltonian flows,which is useful in the study of properties of energy eigenfunctions,such as localization,as well as in approximate calculation of eigenenergies.The method is based on a generalized Brillouin-Wigner perturbation theory.Each flow is specific for a given energy and,at each step of the flow,a finite subspace of the Hilbert space is decimated in order to obtain a renormalized Hamiltonian for the next step.Eigenenergies of the original Hamiltonian appear as unstable fixed points of renormalized flows.Numerical illustration of the method is given in the Wigner-band random-matrix model.
基金Supported by the National Natural Science Foundation of China under Grant No.10975125
文摘We obtain the energy spectrum and all the corresponding eigenfunctions of N-body Bose and Fermi systems with Quadratic Pair Potentials in one dimension. The original first excited state or energy level is disappeared in one dimension, which results from the operation of symmetry or antisymmetry of identical particles. In two and higher dimensions, we give the energy spectrum and the analytical ground state wave [unctions and the degree of degeneracy. By comparison, we refine A vinash Khare＇s results by making some items in his article precisely.
基金The author is partially supported by the Nature Science Foundation of Guangdong(5012285)the"Thousand,Hundred,Ten"Science Foundation of Guangdong(Q02052)the Nature Science Foundation of Education Bureau of Guangdong(Z02075)
文摘In this paper, non-self-adjoint Sturm-Liuville operators in Weyl＇s limit-circle case are studied. We first determine all the non-self-adjoint boundary conditions yielding dissipative operators for each allowed Sturm-Liouville differential expression. Then, using the characteristic determinant, we prove the completeness of the system of eigenfunctions and associated functions for these dissipative operators.
基金supported by National Science Foundation of USA (Grant Nos.DMS-1810747 and DMS-1502632)supported by National Natural Science Foundation of China (Grant No.12171424)。
文摘Let{e_(j)}be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold(M,g).Let H■M be a submanifold and{ψ_(k)}be an orthonormal basis of Laplace eigenfunctions of H with the induced metric.We obtain joint asymptotics for the Fourier coefficients<γHe_(j),ψ_(k)>L^(2)(H)=∫He_(j),ψ_(k)dV_(H)of restrictionsγHe_(j)of e_(j)to H.In particular,we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum{(μ_(k),λ_(j))}^(∞)_(j,k-0)of the(square roots of the)Laplacian△_(M)on M and the Laplacian△_(H)on H in a family of suitably‘thick'regions in R^(2).Thick regions include(1)the truncated coneμ_(k)/λ_(j)∈[a,b]■(0,1)andλ_(j)≤λ,and(2)the slowly thickening strip|μ_(k)-cλ_(j)|≤w(λ)andλ_(j)≤λ,where w(λ)is monotonic and 1■w(λ)≤λ^(1/2).Key tools for obtaining the asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.
文摘The eigenfunctions in a stability problem of boundary-layer flow over a viscoelastic compliant wall were studied. Two categories of modes, TSI and CIFI, exist in the eigenvalue solutions. The eigenfunctions of flow-based TSI were investigated together with those in the flow over rigid wall, whereas the eigenfunctions of wall-based CIFI were compared with the wall functions in an individual wall without fluid constraint. The physical characteristics of the eigenmodes were discussed based on their eigenfunctions.
基金supported by MHRD as researcher C.K.Neog received the MHRD Institute GATE scholarship from Govt.of India.
文摘This study investigates the effects of radiation force due to the rotational pitch motion of a wave energy device,which comprises a coaxial bottom-mounted cylindrical caisson in a two-layer fluid,along with a submerged cylindrical buoy.The system is modeled as a two-layer fluid with infinite horizontal extent and finite depth.The radiation problem is analyzed in the context of linear water waves.The fluid domain is divided into outer and inner zones,and mathematical solutions for the pitch radiating potential are derived for the corresponding boundary valve problem in these zones using the separation of variables approach.Using the matching eigenfunction expansion method,the unknown coefficients in the analytical expression of the radiation potentials are evaluated.The resulting radiation potential is then used to compute the added mass and damping coefficients.Several numerical results for the added mass and damping coefficients are investigated for numerous parameters,particularly the effects of the cylinder radius,the draft of the submerged cylinder,and the density proportion between the two fluid layers across different frequency ranges.The major findings are presented and discussed.
基金supported by the National Key Research and Development Program of China(2022YFB2602800)National Natural Science Foundation of China(No.52271261).
文摘For a generalized radiation problem,an infinitely long submerged horizontal cylinder is forced to vibrate periodically in the transverse direction,with a described elastic harmonic motion along its longitudinal direction.A critical frequency corresponds to the described wave number of elastic vibration,and the generalized hydrodynamic coefficients abruptly change in the vicinity of critical frequency.In this work,a numerical examination is carried out to study the characteristics of wave profiles and wave propagation in the vicinity of the critical frequency.Results show that below the critical frequency,the real parts of complex wave profiles have large values in the vicinity of the cylinder and decay to zero with the increasing distance from the cylinder.Meanwhile,the imaginary parts of complex wave profiles are all zero,which explains why the generalized radiation damping is zero when the vibration is less than the critical frequency.At far distances,no radiation wave is observed.When the vibration exceeds the critical frequency,the real and imaginary parts of the wave profiles oscillate harmonically and keep steady amplitudes.In addition,the generated radiation wave propagates obliquely outward.The influence of the cylinder’s submergence depth on the wave profile is also studied,and the results indicate that the amplitude of the wave profile decreases as the submergence depth of the cylinder increases.The 3D wave profiles are graphically presented to show the wave propagation characteristics in the vicinity of the critical frequency for this generalized radiation problem.This study provides a good reference for the interaction between fluid and slender elastic structures.
基金the National Natural Science Foundation of China (No.19871048)Natural Science Foundation of Shandong Province of China (No.Z2000A02, Y2001A03).
文摘In this paper we shall consider the nonresonance Dirichlet boundary value problemwhere λ>0 is a parameter, p>0 is a constant. Intervals of A are determined to ensure the existence of a nonnegative solution of the boundary value problem. For λ=1, we shall also offer criteria for the existence of eigenfunctions. The main results include and improve on those of [2,4,6,8].
基金supported by National Natural Science Foundation of China(Grant Nos.10871003 and 10990012)
文摘Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N. In this article, we characterize the He-valued joint eigenfunctions of L and T having eigenvalues from the quaternionic Heisenberg fan.
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Global warming has led to major melting of ice in the polar Arctic,making it possible to open Arctic shipping lanes.In this case,the large number of ice sheets are extremely dangerous for ship navigation,so in this paper,a body floating on water confined between two finite ice sheets is investigated.The linearized potential flow theory is adopted,and water is considered an incompressible ideal fluid with a finite depth of the fluid domain.The ice sheets are treated as elastic plates,and the problem is solved by matching eigenfunction expansion.The fluid domain is divided into subregions on the basis of the water surface conditions,and the velocity potential of the subdomains is expanded via the separated variable method.By utilizing the continuity of pressure and velocity at the interfaces of two neighboring regions,a system of linear equations is established to obtain the unknown coefficients in the expansion,which in turn leads to analytical solutions for different motion modes in different regions.The effects of different structural drafts,and different lengths of ice sheets on both sides,etc.,on the hydrodynamic characteristics of floats are analyzed.The amplitude of motion of the float is explored,as is the wave elevation between the ice sheets and the float.
基金Science and Engineering Research Board(SERB),Department of Science&Technology(DST),Government of India for supporting financially under the research grant No.CRG/2018/004184Ministry of Ports,Shipping and Waterways,Government of India through the research grant No.DW/01013(13)/2/2021.
文摘The wave interaction with stratified porous structure combined with a surface-piercing porous block in a stepped seabed is analysed based on the small amplitude wave theory.The study is performed to analyse the effectiveness of partial porous structure in increasing the wave attenuation in the nearshore regions consisting of stratified porous structures of different configurations using the eigenfunction expansion method and orthogonal mode-coupling relation.The hydrodynamic characteristics such as wave reflection coefficient,transmission coefficient,dissipation coefficient,wave force impact and surface elevation are investigated due to the presence of both horizontally and vertically stratified porous structures.The effect of varying porosity,structural width,angle of incidence,wavelength and length between the porous block and stratified structure is examined.The numerical results are validated with the results available in the literature.The present study illustrates that the presence of the stratified structure decreases wave transmission and efficient wave attenuation can also be easily achieved.The wave force acting on stratified structure can be decreased if the structure is combined with wider surface-piercing porous blocks.Further,the presence of stratified porous structure combined with porous block helps in creating a tranquil zone in the leeside of the structure.The combination of vertical and horizontal stratified porous structure with surface-piercing porous block is intended to be an effective solution for the protection of coastal facilities.
