Quantum physics is primarily concerned with real eigenvalues,stemming from the unitarity of time evolutions.With the introduction of PT symmetry,a widely accepted consensus is that,even if the Hamiltonian of the syste...Quantum physics is primarily concerned with real eigenvalues,stemming from the unitarity of time evolutions.With the introduction of PT symmetry,a widely accepted consensus is that,even if the Hamiltonian of the system is not Hermitian,the eigenvalues can still be purely real under specific symmetry.Hence,great enthusiasm has been devoted to exploring the eigenvalue problem of non-Hermitian systems.In this work,from a distinct perspective,we demonstrate that real eigenvalues can also emerge under the appropriate recursive condition of eigenstates.Consequently,our findings provide another path to extract the real energy spectrum of non-Hermitian systems,which guarantees the conservation of probability and stimulates future experimental observations.展开更多
A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential ...A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential function for the Zakharov–Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, tanh(ax) mapping, and Chebyshev nodes, the Zakharov–Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma–Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.展开更多
Wave propagation in horizontally layered media is a classical problem in seismic-wave theory.In semi-infinite space,a nondispersive Rayleigh wave mode exists,and the eigendisplacement decays exponentially with depth.I...Wave propagation in horizontally layered media is a classical problem in seismic-wave theory.In semi-infinite space,a nondispersive Rayleigh wave mode exists,and the eigendisplacement decays exponentially with depth.In a layered model with increasing layer velocity,the phase velocity of the Rayleigh wave varies between the S-wave velocity of the bottom half-space and that of the classical Rayleigh wave propagated in a supposed half-space formed by the parameters of the top layer.If the phase velocity is the same as the P-or S-wave velocity of the layer,which is called the critical mode or critical phase velocity of surface waves,the general solution of the wave equation is not a homogeneous(expressed by trigonometric functions)or inhomogeneous(expressed by exponential functions)plane wave,but one whose amplitude changes linearly with depth(expressed by a linear function).Theories based on a general solution containing only trigonometric or exponential functions do not apply to the critical mode,owing to the singularity at the critical phase velocity.In this study,based on the classical framework of generalized reflection and transmission coefficients,the propagation of surface waves in horizontally layered media was studied by introducing a solution for the linear function at the critical phase velocity.Therefore,the eigenvalues and eigenfunctions of the critical mode can be calculated by solving a singular problem.The eigendisplacement characteristics associated with the critical phase velocity were investigated for different layered models.In contrast to the normal mode,the eigendisplacement associated with the critical phase velocity exhibits different characteristics.If the phase velocity is equal to the S-wave velocity in the bottom half-space,the eigendisplacement remains constant with increasing depth.展开更多
Letλ=(λ_(1),…,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβ,withβvarying with n.Set■.Suppose that■and 0≤σγ<1.We offer the large deviation for p_(1)+p_(2)/p_(1)■λ_(i)whenγ>0 via the...Letλ=(λ_(1),…,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβ,withβvarying with n.Set■.Suppose that■and 0≤σγ<1.We offer the large deviation for p_(1)+p_(2)/p_(1)■λ_(i)whenγ>0 via the large deviation of the corresponding empirical measure and via a direct estimate,respectively,whenγ=0.展开更多
The Dashuigou tellurium deposit is the world’s only known independent tellurium deposit.By restoring metamorphic rocks’protolith,we seek to understand not only the development and evolution trajectory of the region ...The Dashuigou tellurium deposit is the world’s only known independent tellurium deposit.By restoring metamorphic rocks’protolith,we seek to understand not only the development and evolution trajectory of the region but also the origin of the relevant deposits.While there are many ways to restore metamorphic rocks’protolith,we take the host metamorphic rocks of Dashuigou tellurium deposit and leverage various petrochemical eigenvalues and related diagrams previously proposed to reveal the deposit’s host metamorphic rocks’protolith.The petrochemical eigenvalues include molecular number,Niggli’s value,REE parity ratio,CaO/Al_(2)O_(3)ratio,Fe^(3+) /(Fe^(3+) -+Fe^(2+) )ratio,chondrite-normalized REE value,logarithmic REE value,various REE eigenvalues including scandium,Eu/Sm ratio,total REE amount,light and heavy REEs,δEu,Eu anomaly,Sm/Nd ratio,and silicon isotope δ^(30) SiNBS-29‰,etc.The petrochemical plots include ACMs,100 mg-c-(al+alk),SiO_(2)-(Na_(2)O+K_(2)O),(al+fm)-(c+alk)versus Si,FeO+Fe_(2)O^(3+) TiO)-Al_(2)O_(3)-MgO,c-mg,Al_(2)O_(3)-(Na_(2)O+K_(2)O),chondrite-normalized REE model,La/Yb-REE,and Sm/Nd ratio,etc.On the basis of these comprehensive analyses,the following conclusions are drawn,starting from the many mantle-derived types of basalt developed in the study area of different geological ages,combined with the previously published research results on the deposit s fluid inclusions and sulfur and lead isotopes.The deposit is formed by mantle degassing in the form of a mantle plume in the late Yanshanian orogeny.The degassed fluids are rich in nano-sc ale substances including Fe,Te,S,As,Bi,Au,Se,H_(2),CO_(2),N_(2),H_(2)O,and CH_(4),which are enriched by nano-effect,and then rise to a certain part of the crust in the form of mantle plume along the lithospheric fault to form the deposit.The ultimate power for tellurium mineralization was from H_(2)flow with high energy,which was produced through radiation from the melted iron of the Earth’s outer core.The H,flow results in the Earth’s degassing,as well as the mantle and crust’s uplift.展开更多
The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation...The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.展开更多
In this article, an attempt based on Spin Topological Space, STS, to give areasonable detailed account of the cause of photonic fermionization phenomena of light photon is made. STS is an unconventional spin space in ...In this article, an attempt based on Spin Topological Space, STS, to give areasonable detailed account of the cause of photonic fermionization phenomena of light photon is made. STS is an unconventional spin space in quantum mechanics, which can be used to account for where the unconventional half-integer spin eigenvalues phenomenon of light photon comes from. We suggest to dectect the possible existence of photonic one-third-spinization phenomenon of light photon, by using three beams of light photon in interference experiment.展开更多
In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered in...In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of innnit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.展开更多
Let ? be a bounded open domain in Rnwith smooth boundary ??,X =(X_1,X_2,···,X_m) be a system of real smooth vector fields defined on ? and the boundary ?? is non-characteristic for X. If X satisfies the...Let ? be a bounded open domain in Rnwith smooth boundary ??,X =(X_1,X_2,···,X_m) be a system of real smooth vector fields defined on ? and the boundary ?? is non-characteristic for X. If X satisfies the H¨ormander's condition,then the vector field is finitely degenerate and the sum of square operator △X =Σ_(j=1)~mX_j^2 is a finitely degenerate elliptic operator. In this paper,we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △X on ?.展开更多
A new method for estimation the bounds of eigenvalues is presented.In order to show that themethod propose is as effective as Qiu’s,an undamping spring-mass system with 5 nodes and 5 degrees offreedom is given.To ill...A new method for estimation the bounds of eigenvalues is presented.In order to show that themethod propose is as effective as Qiu’s,an undamping spring-mass system with 5 nodes and 5 degrees offreedom is given.To illustrate that the present method can be applied to structures which cannot be treatedby non-negative decompsition,a pland frame with 202 nodes and 357 beam elements is given.The resultsshow that the present method is effective for estimating the bounds of eigenvalues and is more common thanQiu’s.展开更多
This paper presents a method for estimating the upper and lower bounds of eigenvalues ofstructures with uncertainties.The uncertain parameters are described by the convex model.A numerical ex-ample of the frame struct...This paper presents a method for estimating the upper and lower bounds of eigenvalues ofstructures with uncertainties.The uncertain parameters are described by the convex model.A numerical ex-ample of the frame structure is given to illustrate the efficiency of the method.展开更多
The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of 3D Fredholm-Stieltjes integral equations, i.e. 3D Fredholm equations with respect to suitable Stieltjes-type measures...The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of 3D Fredholm-Stieltjes integral equations, i.e. 3D Fredholm equations with respect to suitable Stieltjes-type measures. Some applications are shown, relevant to the problem of computing the eigenvalues of a body charged by a finite number of masses concentrated on points, curves or surfaces lying in.展开更多
From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determinedby the parameters of th...From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determinedby the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a by-product in the computation of the conjugate gradient ifa computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on A^T A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.展开更多
We construct one multi-sender authentication code by algebraic combination method from eigenvalues and eigenvectors of the matrix over nite elds. Some parameters and the probabilities of three kinds of successful atta...We construct one multi-sender authentication code by algebraic combination method from eigenvalues and eigenvectors of the matrix over nite elds. Some parameters and the probabilities of three kinds of successful attack of this code are also computed. For multi-sender authentication code,it allows a group of senders to construct an authenticated message for a receiver such that the receiver can verify authenticity of the received message.展开更多
In this article,we study different molecular structures such as Polythiophene network,PLY(n)for n=1,2,and 3,Orthosilicate(Nesosilicate)SiO4,Pyrosilicates(Sorosilicates)Si2O7,Chain silicates(Pyroxenes)(SiO3)n,and Cycli...In this article,we study different molecular structures such as Polythiophene network,PLY(n)for n=1,2,and 3,Orthosilicate(Nesosilicate)SiO4,Pyrosilicates(Sorosilicates)Si2O7,Chain silicates(Pyroxenes)(SiO3)n,and Cyclic silicates(Ring Silicates)Si3O9 for their cardinalities,chromatic numbers,graph variations,eigenvalues obtained from the adjacency matrices which are square matrices in order and their corresponding characteristics polynomials.We convert the general structures of these chemical networks in to mathematical graphical structures.We transform the molecular structures of these chemical networks which are mentioned above,into a simple and undirected planar graph and sketch them with various techniques of mathematics.The matrices obtained from these simple undirected graphs are symmetric.We also label the molecular structures by assigning different colors.Their graphs have also been studied for analysis.For a connected graph,the eigenvalue that shows its peak point(largest value)obtained from the adjacency matrix has multiplicity 1.Therefore,the gap between the largest and its smallest eigenvalues is interlinked with some form of“connectivity measurement of the structural graph”.We also note that the chemical structures,Orthosilicate(Nesosilicate)SiO4,Pyrosilicates(Sorosilicates)Si2O7,Chain silicates(Pyroxenes)(SiO3)n,and Cyclic silicates(Ring Silicates)Si3O9 generally have two same eigenvalues.Adjacency matrices have great importance in the field of computer science.展开更多
A sign pattern(matrix)is a matrix whose entries are the symbols+,-and 0.Foran n×n sign pattern matrix A,the sign pattern class of A,denoted by Q(A),is the set ofall n×n real matrices whose entries have signs...A sign pattern(matrix)is a matrix whose entries are the symbols+,-and 0.Foran n×n sign pattern matrix A,the sign pattern class of A,denoted by Q(A),is the set ofall n×n real matrices whose entries have signs indicated by the corresponding entries of A.We say that a sign pattern matrix A requires a matrix property P if every real matrix in Q(A)has the property P.A matrix with all distinct eigenvalues has many nice展开更多
In this article, we compute the enclosures eigenvalues (upper and lower bounds) using the quadratic method. The Schrodinger operator (A) (harmonic and anharmonic oscillator model) has used as an example. We study a ne...In this article, we compute the enclosures eigenvalues (upper and lower bounds) using the quadratic method. The Schrodinger operator (A) (harmonic and anharmonic oscillator model) has used as an example. We study a new technique to get more accurate bounds. We compare our results with Boulton and Strauss method.展开更多
A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is ...A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is modified to run in parallel on multi-core machines. A basic characteristic of this code (eventually pointing to its parallelization) is that it can proceed with: 1) partitioning the given region into an appropriate number of subregions;2) counting eigenvalues in each subregion;and 3) computing (already counted) eigenvalues in each subregion. Consequently, theoretically speaking, the whole code in itself parallelizes ideally. We carry out several numerical experiments with random complex tridiagonal matrices, and random complex polynomials as well, in order to study the behaviour of the parallel code, especially the degree of declination from theoretical expectations.展开更多
For a bounded linear operator A on a Hilbert space H, let M(A) be the smallest possible constant in the inequality . Here, p is a point on the smooth portion of the boundary of the numerical range of A. is the radius ...For a bounded linear operator A on a Hilbert space H, let M(A) be the smallest possible constant in the inequality . Here, p is a point on the smooth portion of the boundary of the numerical range of A. is the radius of curvature of at this point and ?is the distance from p to the spectrum of A. In this paper, we compute the M(A) for composition operators on Hardy space H2.展开更多
In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an ...In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an NIEP whether is solvable.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant No.62071248)the Natural Science Foundation of Nanjing University of Posts and Telecommunications(Grant No.NY223109)China Postdoctoral Science Foundation(Grant No.2022M721693).
文摘Quantum physics is primarily concerned with real eigenvalues,stemming from the unitarity of time evolutions.With the introduction of PT symmetry,a widely accepted consensus is that,even if the Hamiltonian of the system is not Hermitian,the eigenvalues can still be purely real under specific symmetry.Hence,great enthusiasm has been devoted to exploring the eigenvalue problem of non-Hermitian systems.In this work,from a distinct perspective,we demonstrate that real eigenvalues can also emerge under the appropriate recursive condition of eigenstates.Consequently,our findings provide another path to extract the real energy spectrum of non-Hermitian systems,which guarantees the conservation of probability and stimulates future experimental observations.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.52171251,U2106225,and 52231011)Dalian Science and Technology Innovation Fund (Grant No.2022JJ12GX036)。
文摘A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential function for the Zakharov–Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, tanh(ax) mapping, and Chebyshev nodes, the Zakharov–Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma–Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.
基金supported by the National Natural Science Foundation of China(No.U1839209).
文摘Wave propagation in horizontally layered media is a classical problem in seismic-wave theory.In semi-infinite space,a nondispersive Rayleigh wave mode exists,and the eigendisplacement decays exponentially with depth.In a layered model with increasing layer velocity,the phase velocity of the Rayleigh wave varies between the S-wave velocity of the bottom half-space and that of the classical Rayleigh wave propagated in a supposed half-space formed by the parameters of the top layer.If the phase velocity is the same as the P-or S-wave velocity of the layer,which is called the critical mode or critical phase velocity of surface waves,the general solution of the wave equation is not a homogeneous(expressed by trigonometric functions)or inhomogeneous(expressed by exponential functions)plane wave,but one whose amplitude changes linearly with depth(expressed by a linear function).Theories based on a general solution containing only trigonometric or exponential functions do not apply to the critical mode,owing to the singularity at the critical phase velocity.In this study,based on the classical framework of generalized reflection and transmission coefficients,the propagation of surface waves in horizontally layered media was studied by introducing a solution for the linear function at the critical phase velocity.Therefore,the eigenvalues and eigenfunctions of the critical mode can be calculated by solving a singular problem.The eigendisplacement characteristics associated with the critical phase velocity were investigated for different layered models.In contrast to the normal mode,the eigendisplacement associated with the critical phase velocity exhibits different characteristics.If the phase velocity is equal to the S-wave velocity in the bottom half-space,the eigendisplacement remains constant with increasing depth.
基金supported by the NSFC (12171038,11871008)985 Projects.
文摘Letλ=(λ_(1),…,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβ,withβvarying with n.Set■.Suppose that■and 0≤σγ<1.We offer the large deviation for p_(1)+p_(2)/p_(1)■λ_(i)whenγ>0 via the large deviation of the corresponding empirical measure and via a direct estimate,respectively,whenγ=0.
基金supported by Orient Resources Ltd.College of Earth Sciences,Jilin University。
文摘The Dashuigou tellurium deposit is the world’s only known independent tellurium deposit.By restoring metamorphic rocks’protolith,we seek to understand not only the development and evolution trajectory of the region but also the origin of the relevant deposits.While there are many ways to restore metamorphic rocks’protolith,we take the host metamorphic rocks of Dashuigou tellurium deposit and leverage various petrochemical eigenvalues and related diagrams previously proposed to reveal the deposit’s host metamorphic rocks’protolith.The petrochemical eigenvalues include molecular number,Niggli’s value,REE parity ratio,CaO/Al_(2)O_(3)ratio,Fe^(3+) /(Fe^(3+) -+Fe^(2+) )ratio,chondrite-normalized REE value,logarithmic REE value,various REE eigenvalues including scandium,Eu/Sm ratio,total REE amount,light and heavy REEs,δEu,Eu anomaly,Sm/Nd ratio,and silicon isotope δ^(30) SiNBS-29‰,etc.The petrochemical plots include ACMs,100 mg-c-(al+alk),SiO_(2)-(Na_(2)O+K_(2)O),(al+fm)-(c+alk)versus Si,FeO+Fe_(2)O^(3+) TiO)-Al_(2)O_(3)-MgO,c-mg,Al_(2)O_(3)-(Na_(2)O+K_(2)O),chondrite-normalized REE model,La/Yb-REE,and Sm/Nd ratio,etc.On the basis of these comprehensive analyses,the following conclusions are drawn,starting from the many mantle-derived types of basalt developed in the study area of different geological ages,combined with the previously published research results on the deposit s fluid inclusions and sulfur and lead isotopes.The deposit is formed by mantle degassing in the form of a mantle plume in the late Yanshanian orogeny.The degassed fluids are rich in nano-sc ale substances including Fe,Te,S,As,Bi,Au,Se,H_(2),CO_(2),N_(2),H_(2)O,and CH_(4),which are enriched by nano-effect,and then rise to a certain part of the crust in the form of mantle plume along the lithospheric fault to form the deposit.The ultimate power for tellurium mineralization was from H_(2)flow with high energy,which was produced through radiation from the melted iron of the Earth’s outer core.The H,flow results in the Earth’s degassing,as well as the mantle and crust’s uplift.
基金Supported by the National Nature Science Foundation of China(12101356,12101357,12071254,11771253)the National Science Foundation of Shandong Province(ZR2021QA065,ZR2020QA009,ZR2021MA047)the China Postdoctoral Science Foundation(2019M662313)。
文摘The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.
文摘In this article, an attempt based on Spin Topological Space, STS, to give areasonable detailed account of the cause of photonic fermionization phenomena of light photon is made. STS is an unconventional spin space in quantum mechanics, which can be used to account for where the unconventional half-integer spin eigenvalues phenomenon of light photon comes from. We suggest to dectect the possible existence of photonic one-third-spinization phenomenon of light photon, by using three beams of light photon in interference experiment.
文摘In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of innnit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.
基金partially supported by the NSFC(11631011,11626251)
文摘Let ? be a bounded open domain in Rnwith smooth boundary ??,X =(X_1,X_2,···,X_m) be a system of real smooth vector fields defined on ? and the boundary ?? is non-characteristic for X. If X satisfies the H¨ormander's condition,then the vector field is finitely degenerate and the sum of square operator △X =Σ_(j=1)~mX_j^2 is a finitely degenerate elliptic operator. In this paper,we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △X on ?.
基金the National Natural Science Foundation (No.19872028)the Mechanical Technology Development Foundation of China
文摘A new method for estimation the bounds of eigenvalues is presented.In order to show that themethod propose is as effective as Qiu’s,an undamping spring-mass system with 5 nodes and 5 degrees offreedom is given.To illustrate that the present method can be applied to structures which cannot be treatedby non-negative decompsition,a pland frame with 202 nodes and 357 beam elements is given.The resultsshow that the present method is effective for estimating the bounds of eigenvalues and is more common thanQiu’s.
基金the National Natural Science Foundation of China(No.19872028)
文摘This paper presents a method for estimating the upper and lower bounds of eigenvalues ofstructures with uncertainties.The uncertain parameters are described by the convex model.A numerical ex-ample of the frame structure is given to illustrate the efficiency of the method.
文摘The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of 3D Fredholm-Stieltjes integral equations, i.e. 3D Fredholm equations with respect to suitable Stieltjes-type measures. Some applications are shown, relevant to the problem of computing the eigenvalues of a body charged by a finite number of masses concentrated on points, curves or surfaces lying in.
文摘From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determinedby the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a by-product in the computation of the conjugate gradient ifa computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on A^T A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.
基金Sponsored by the National Natural Science Foundation of China(Grant No.61179026)the Fundamental Research of the Central Universities of China Civil Aviation University of Science Special(Grant No.3122016L005)
文摘We construct one multi-sender authentication code by algebraic combination method from eigenvalues and eigenvectors of the matrix over nite elds. Some parameters and the probabilities of three kinds of successful attack of this code are also computed. For multi-sender authentication code,it allows a group of senders to construct an authenticated message for a receiver such that the receiver can verify authenticity of the received message.
文摘In this article,we study different molecular structures such as Polythiophene network,PLY(n)for n=1,2,and 3,Orthosilicate(Nesosilicate)SiO4,Pyrosilicates(Sorosilicates)Si2O7,Chain silicates(Pyroxenes)(SiO3)n,and Cyclic silicates(Ring Silicates)Si3O9 for their cardinalities,chromatic numbers,graph variations,eigenvalues obtained from the adjacency matrices which are square matrices in order and their corresponding characteristics polynomials.We convert the general structures of these chemical networks in to mathematical graphical structures.We transform the molecular structures of these chemical networks which are mentioned above,into a simple and undirected planar graph and sketch them with various techniques of mathematics.The matrices obtained from these simple undirected graphs are symmetric.We also label the molecular structures by assigning different colors.Their graphs have also been studied for analysis.For a connected graph,the eigenvalue that shows its peak point(largest value)obtained from the adjacency matrix has multiplicity 1.Therefore,the gap between the largest and its smallest eigenvalues is interlinked with some form of“connectivity measurement of the structural graph”.We also note that the chemical structures,Orthosilicate(Nesosilicate)SiO4,Pyrosilicates(Sorosilicates)Si2O7,Chain silicates(Pyroxenes)(SiO3)n,and Cyclic silicates(Ring Silicates)Si3O9 generally have two same eigenvalues.Adjacency matrices have great importance in the field of computer science.
文摘A sign pattern(matrix)is a matrix whose entries are the symbols+,-and 0.Foran n×n sign pattern matrix A,the sign pattern class of A,denoted by Q(A),is the set ofall n×n real matrices whose entries have signs indicated by the corresponding entries of A.We say that a sign pattern matrix A requires a matrix property P if every real matrix in Q(A)has the property P.A matrix with all distinct eigenvalues has many nice
文摘In this article, we compute the enclosures eigenvalues (upper and lower bounds) using the quadratic method. The Schrodinger operator (A) (harmonic and anharmonic oscillator model) has used as an example. We study a new technique to get more accurate bounds. We compare our results with Boulton and Strauss method.
文摘A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is modified to run in parallel on multi-core machines. A basic characteristic of this code (eventually pointing to its parallelization) is that it can proceed with: 1) partitioning the given region into an appropriate number of subregions;2) counting eigenvalues in each subregion;and 3) computing (already counted) eigenvalues in each subregion. Consequently, theoretically speaking, the whole code in itself parallelizes ideally. We carry out several numerical experiments with random complex tridiagonal matrices, and random complex polynomials as well, in order to study the behaviour of the parallel code, especially the degree of declination from theoretical expectations.
文摘For a bounded linear operator A on a Hilbert space H, let M(A) be the smallest possible constant in the inequality . Here, p is a point on the smooth portion of the boundary of the numerical range of A. is the radius of curvature of at this point and ?is the distance from p to the spectrum of A. In this paper, we compute the M(A) for composition operators on Hardy space H2.
文摘In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an NIEP whether is solvable.