In this article, we prove that the symmetric function Fn（x,r）=∑i1＋i2＋……in=r（x1（i1x2^i2……xn^in）1/r is Schur harmonic convex for x ∈ R＋n and r ∈N -=（1, 2, 3,...} As its applications, some analytic inequa...In this article, we prove that the symmetric function Fn（x,r）=∑i1＋i2＋……in=r（x1（i1x2^i2……xn^in）1/r is Schur harmonic convex for x ∈ R＋n and r ∈N -=（1, 2, 3,...} As its applications, some analytic inequalities are established.展开更多
The Schur convexity and concavity of a class of symmetric functions are discussed, and an open problem proposed by Guan in "Some properties of a class of symmetric functions" is answered. As consequences, so...The Schur convexity and concavity of a class of symmetric functions are discussed, and an open problem proposed by Guan in "Some properties of a class of symmetric functions" is answered. As consequences, some inequalities are established by use of the theory of majorization.展开更多
The Schur convexity or concavity problem of the Gini mean values S(a, b; x, y) with respect to (x, y) ∈ (0, ∞) × (0, ∞) for fixed (a, b) ∈ ? × ? is still open. In this paper, we prove that S(a, b; x, y) ...The Schur convexity or concavity problem of the Gini mean values S(a, b; x, y) with respect to (x, y) ∈ (0, ∞) × (0, ∞) for fixed (a, b) ∈ ? × ? is still open. In this paper, we prove that S(a, b; x, y) is Schur convex with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): a ? 0, b ? 0, a + b ? 1}, and Schur concave with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): b ? 0, b ? a, a + b ? 1} ∩ {(a, b): a ? 0, a ? b, a + b ? 1}.展开更多
We prove that the Gini mean values S（a,b; x,y） are Schur harmonic convex with respect to （x,y）∈（0,∞）×（0,∞） if and only if （a, b） ∈{（a, b）：a≥0,a ≥ b,a＋b＋1≥0}∪{（a,b）：b≥0,b≥a,a＋b＋1≥0} ...We prove that the Gini mean values S（a,b; x,y） are Schur harmonic convex with respect to （x,y）∈（0,∞）×（0,∞） if and only if （a, b） ∈{（a, b）：a≥0,a ≥ b,a＋b＋1≥0}∪{（a,b）：b≥0,b≥a,a＋b＋1≥0} and Schur harmonic concave with respect to （x,y） ∈ （0,∞）×（0,∞） if and only if （a,b）∈{（a,b）：a≤0,b≤0,a｜b｜1≤0}.展开更多
Robust stability test algorithms of interval 2\|D polynomials and rank\|one polytope of 2\|D polynomials have been developed. To simplify the robust stability test procedure of interval 2\|D polynomials and rank on po...Robust stability test algorithms of interval 2\|D polynomials and rank\|one polytope of 2\|D polynomials have been developed. To simplify the robust stability test procedure of interval 2\|D polynomials and rank on polytope of 2\|D polynomials, we introduce the definition of perturbation radius of 2\|D uncertain polynomials. Based on the perturbation radius of 2\|D polynomials, we establish sufficient conditions of robust Schur stability for the two kinds of uncertain 2\|D polynomials. Examples are given to illustrate the applicaton of our test theorems.展开更多
Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,su...Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.展开更多
In this paper, a new complete and simplified proof for the Husainov-Nikiforova Theorem is given. Then this theorem is generalized to the case where the coefficients may have different signs as well as nonlinear system...In this paper, a new complete and simplified proof for the Husainov-Nikiforova Theorem is given. Then this theorem is generalized to the case where the coefficients may have different signs as well as nonlinear systems. By these results, the robust stability and the bound for robustness for high-order interval discrete dynamical systems are studied, which can be applied to designing stable discrete control system as well as stabilizing a given unstable control system.展开更多
The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infini...The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.展开更多
The preconditioned iterative solvers for solving Sylvester tensor equations are considered in this paper.By fully exploiting the structure of the tensor equation,we propose a projection method based on the tensor form...The preconditioned iterative solvers for solving Sylvester tensor equations are considered in this paper.By fully exploiting the structure of the tensor equation,we propose a projection method based on the tensor format,which needs less flops and storage than the standard projection method.The structure of the coefficient matrices of the tensor equation is used to design the nearest Kronecker product(NKP) preconditioner,which is easy to construct and is able to accelerate the convergence of the iterative solver.Numerical experiments are presented to show good performance of the approaches.展开更多
Digital watermarking is one of the most powerful tools used in ownership and copyrights protection in digital media. This paper presents a blind digital video watermarking technique based on a combination scheme betwe...Digital watermarking is one of the most powerful tools used in ownership and copyrights protection in digital media. This paper presents a blind digital video watermarking technique based on a combination scheme between the Discrete Wavelet transform in (DWT) and the real Schur Decomposition. The scheme starts with applying twolevel DWT to the video scene followed by Schur decomposition in which the binary watermark bits are embedded in the resultant block upper triangular matrix. The proposed technique shows high efficiency due to the use of Schur decomposition which requires fewer computations compared to other transforms. The imperceptibility of the scheme is also very high due to the use of DWT transform;therefore, no visual distortion is noticed in the watermarked video after embedding. Furthermore, the technique proves to be robust against set of standard attacks like: Gaussian, salt and pepper and rotation and some video attacks such as: frame dropping, cropping and averaging. Both capacity and blindness features are also considered and achieved in this technique.展开更多
Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant ...Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.展开更多
This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solutio...This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solution of test equations u'(t) = a 11 u(t) + a12v(t) + b11 u(t - τ1) + b12v(t-τ2,v'(t) = a21 u(t) + a22 v(t) + b21 u(t -τ1,) + b22 v(t -τ2), t>0,with initial conditionsu(t)=u0(t),v(t) =v0(t), t≤0.where aij, bij∈C, τj >0, i,j = 1,2,, and u0(t), v0(t)are continuous and complex valued. Sufficient conditions for the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2≤θ≤1 and the one-leg θ-method is stable if and only if θ= 1.展开更多
Cluster synchronization in a network of non-identical dynamic systems is studied in this paper, using two-cluster synchronization for detailed analysis and discussion. The results show that the common intercluster cou...Cluster synchronization in a network of non-identical dynamic systems is studied in this paper, using two-cluster synchronization for detailed analysis and discussion. The results show that the common intercluster coupling condition is not always needed for the diffusively coupled network. Several sufficient conditions are obtained by using the Schur unitary triangularization theorem, which extends previous results. Some numerical examples are presented for illustration.展开更多
In this paper,we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems.One type of the preconditioners are based on the nested(or recursive)Schur complement...In this paper,we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems.One type of the preconditioners are based on the nested(or recursive)Schur complement,the other is based on an additive type Schur complement after permuting the original saddle point systems.We analyze different preconditioners incorporating the exact Schur complements.We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements.These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly.Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.展开更多
Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of t...Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space,but not every infinite matrix corresponds to an operator.The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators,which is considered a respectable mathematical accomplishment.In this paper,we prove the compact version of the Schur test.Moreover,we provide the Schur test for the Schatten class S_(2).It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers.We finally provide the Schur test for compact operators from l_(p) into l_(q).展开更多
Using matrix model,Mironov and Morozov recently gave a formula which represents Kontsevich-Witten tau function as a linear expansion of Schur Q-polynomials.In this paper,we will show directly that the Q-polynomial exp...Using matrix model,Mironov and Morozov recently gave a formula which represents Kontsevich-Witten tau function as a linear expansion of Schur Q-polynomials.In this paper,we will show directly that the Q-polynomial expansion in this formula satisfies the Virasoro constraints,and consequently obtains a proof of this formula without using matrix model.We also give a proof for Alexandrov’s conjecture that Kontsevich-Witten tau function is a hypergeometric tau function of theBKPhierarchy after re-scaling.展开更多
We develop and study the generalization of rational Schur algebras to the super setting.Similar to the classical case,this provides a new method for studying rational supermodules of the general linear supergroup GL(m...We develop and study the generalization of rational Schur algebras to the super setting.Similar to the classical case,this provides a new method for studying rational supermodules of the general linear supergroup GL(m|n).Furthermore,we establish a Schur-Weyl duality result for rational Schur superalgebras and conclude that under certain conditions these objects will be semisimple.展开更多
基金supported by NSFC (60850005)NSF of Zhejiang Province(D7080080, Y7080185, Y607128)
文摘In this article, we prove that the symmetric function Fn（x,r）=∑i1＋i2＋……in=r（x1（i1x2^i2……xn^in）1/r is Schur harmonic convex for x ∈ R＋n and r ∈N -=（1, 2, 3,...} As its applications, some analytic inequalities are established.
基金supported by National Natural Science Foundation of China (Grant Nos.60850005, 10771195)the Natural Science Foundation of Zhejiang Province (Grant Nos. D7080080, Y7080185,Y607128)
文摘The Schur convexity and concavity of a class of symmetric functions are discussed, and an open problem proposed by Guan in "Some properties of a class of symmetric functions" is answered. As consequences, some inequalities are established by use of the theory of majorization.
基金supported by National Natural Science Foundation of China (Grant Nos. 60850005, 10771195)the Natural Science Foundation of Zhejiang Province (Grant Nos. D7080080, Y607128, Y7080185)
文摘The Schur convexity or concavity problem of the Gini mean values S(a, b; x, y) with respect to (x, y) ∈ (0, ∞) × (0, ∞) for fixed (a, b) ∈ ? × ? is still open. In this paper, we prove that S(a, b; x, y) is Schur convex with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): a ? 0, b ? 0, a + b ? 1}, and Schur concave with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): b ? 0, b ? a, a + b ? 1} ∩ {(a, b): a ? 0, a ? b, a + b ? 1}.
基金Supported by the NSFC (11071069)the NSF of Zhejiang Province (D7080080 and Y7080185)the Innovation Team Foundation of the Department of Education of Zhejiang Province (T200924)
文摘We prove that the Gini mean values S（a,b; x,y） are Schur harmonic convex with respect to （x,y）∈（0,∞）×（0,∞） if and only if （a, b） ∈{（a, b）：a≥0,a ≥ b,a＋b＋1≥0}∪{（a,b）：b≥0,b≥a,a＋b＋1≥0} and Schur harmonic concave with respect to （x,y） ∈ （0,∞）×（0,∞） if and only if （a,b）∈{（a,b）：a≤0,b≤0,a｜b｜1≤0}.
基金This w ork is supported by N S F C of P. R. China and D F G of Germ any.
文摘Robust stability test algorithms of interval 2\|D polynomials and rank\|one polytope of 2\|D polynomials have been developed. To simplify the robust stability test procedure of interval 2\|D polynomials and rank on polytope of 2\|D polynomials, we introduce the definition of perturbation radius of 2\|D uncertain polynomials. Based on the perturbation radius of 2\|D polynomials, we establish sufficient conditions of robust Schur stability for the two kinds of uncertain 2\|D polynomials. Examples are given to illustrate the applicaton of our test theorems.
文摘Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.
基金This work was supported by the National Natural Science Foundation of China (GrantNo. 69874016) the National Key Basic Research Special Fund (Grant No. 1998020319) the "95" Climbing Program (Grant No. PD9521907).
文摘In this paper, a new complete and simplified proof for the Husainov-Nikiforova Theorem is given. Then this theorem is generalized to the case where the coefficients may have different signs as well as nonlinear systems. By these results, the robust stability and the bound for robustness for high-order interval discrete dynamical systems are studied, which can be applied to designing stable discrete control system as well as stabilizing a given unstable control system.
文摘The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.
基金supported by National Natural Science Foundation of China (Grant No.10961010)Science and Technology Foundation of Guizhou Province (Grant No. LKS[2009]03)
文摘The preconditioned iterative solvers for solving Sylvester tensor equations are considered in this paper.By fully exploiting the structure of the tensor equation,we propose a projection method based on the tensor format,which needs less flops and storage than the standard projection method.The structure of the coefficient matrices of the tensor equation is used to design the nearest Kronecker product(NKP) preconditioner,which is easy to construct and is able to accelerate the convergence of the iterative solver.Numerical experiments are presented to show good performance of the approaches.
文摘Digital watermarking is one of the most powerful tools used in ownership and copyrights protection in digital media. This paper presents a blind digital video watermarking technique based on a combination scheme between the Discrete Wavelet transform in (DWT) and the real Schur Decomposition. The scheme starts with applying twolevel DWT to the video scene followed by Schur decomposition in which the binary watermark bits are embedded in the resultant block upper triangular matrix. The proposed technique shows high efficiency due to the use of Schur decomposition which requires fewer computations compared to other transforms. The imperceptibility of the scheme is also very high due to the use of DWT transform;therefore, no visual distortion is noticed in the watermarked video after embedding. Furthermore, the technique proves to be robust against set of standard attacks like: Gaussian, salt and pepper and rotation and some video attacks such as: frame dropping, cropping and averaging. Both capacity and blindness features are also considered and achieved in this technique.
文摘Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.
文摘This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solution of test equations u'(t) = a 11 u(t) + a12v(t) + b11 u(t - τ1) + b12v(t-τ2,v'(t) = a21 u(t) + a22 v(t) + b21 u(t -τ1,) + b22 v(t -τ2), t>0,with initial conditionsu(t)=u0(t),v(t) =v0(t), t≤0.where aij, bij∈C, τj >0, i,j = 1,2,, and u0(t), v0(t)are continuous and complex valued. Sufficient conditions for the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2≤θ≤1 and the one-leg θ-method is stable if and only if θ= 1.
基金Project supported by the "13115" Program, China (Grant No. 2008ZDKG-37)the National Natural Science Foundation of China (Grant Nos. 61072139, 61072106, 60804021, and 61001202)the Fundamental Research Funds for the Central Universities of China (Grant Nos. Y10000902036, JY10000902039, JY10000970001, and JY10000902001)
文摘Cluster synchronization in a network of non-identical dynamic systems is studied in this paper, using two-cluster synchronization for detailed analysis and discussion. The results show that the common intercluster coupling condition is not always needed for the diffusively coupled network. Several sufficient conditions are obtained by using the Schur unitary triangularization theorem, which extends previous results. Some numerical examples are presented for illustration.
基金the NIH-RCMI(Grant No.347U54MD013376)the affliated project award from the Center for Equitable Artificial Intelligence and Machine Learning Systems at Morgan State University(Project ID 02232301)+3 种基金the National Science Foundation awards(Grant No.1831950).The work of G.Ju is supported in part by the National Key R&D Program of China(Grant No.2017YFB1001604)the National Natural Science Foundation of China(Grant No.11971221)the Shenzhen Sci-Tech Fund(Grant Nos.RCJC20200714114556020,JCYJ20170818153840322,JCYJ20190809150413261)the Guangdong Provincial Key Laboratory of Computational Science and Material Design(Grant No.2019B030301001).
文摘In this paper,we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems.One type of the preconditioners are based on the nested(or recursive)Schur complement,the other is based on an additive type Schur complement after permuting the original saddle point systems.We analyze different preconditioners incorporating the exact Schur complements.We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements.These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly.Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.
文摘Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space,but not every infinite matrix corresponds to an operator.The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators,which is considered a respectable mathematical accomplishment.In this paper,we prove the compact version of the Schur test.Moreover,we provide the Schur test for the Schatten class S_(2).It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers.We finally provide the Schur test for compact operators from l_(p) into l_(q).
基金partially supported by NSFC grants 11890662 and 11890660.
文摘Using matrix model,Mironov and Morozov recently gave a formula which represents Kontsevich-Witten tau function as a linear expansion of Schur Q-polynomials.In this paper,we will show directly that the Q-polynomial expansion in this formula satisfies the Virasoro constraints,and consequently obtains a proof of this formula without using matrix model.We also give a proof for Alexandrov’s conjecture that Kontsevich-Witten tau function is a hypergeometric tau function of theBKPhierarchy after re-scaling.
文摘We develop and study the generalization of rational Schur algebras to the super setting.Similar to the classical case,this provides a new method for studying rational supermodules of the general linear supergroup GL(m|n).Furthermore,we establish a Schur-Weyl duality result for rational Schur superalgebras and conclude that under certain conditions these objects will be semisimple.
