共查询到20条相似文献,搜索用时 46 毫秒
1.
《Communications in Nonlinear Science & Numerical Simulation》2011,16(8):3015-3023
Starting from the study of the symmetries of systems of 4 second-order linear ODEs with constant real coefficients, we determine the dimension and generators of the symmetry algebra for systems of n equations described by a diagonal Jordan canonical form. We further prove that some dimensions between the lower and upper bounds cannot be attained in the diagonal case, and classify the Levi factors of the symmetry algebras. 相似文献
2.
On split Lie algebras with symmetric root systems 总被引:1,自引:1,他引:0
Antonio J. Calderón Martín 《Proceedings Mathematical Sciences》2008,118(3):351-356
We develop techniques of connections of roots for split Lie algebras with symmetric root systems. We show that any of such
algebras L is of the form L = + Σ
j
I
j
with
a subspace of the abelian Lie algebra H and any I
j
a well described ideal of L, satisfying [I
j
, I
k
] = 0 if j ≠ k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system
and having all its nonzero roots connected. 相似文献
3.
Clestin Wafo Soh 《Communications in Nonlinear Science & Numerical Simulation》2010,15(2):139-143
We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24. 相似文献
4.
We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J
*, there exists a Lie supercoalgebra (L
c
(J),
L
) whose dual algebra (L
c
(J))* is the Lie KKT-superalgebra for the Jordan superalgebra J
*. It is well known that some Jordan coalgebra J
0 can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L
c
(J
0),L) to be isomorphic to the coalgebra (Loc(L
in
(J)0),
L
0), where L
in
(J) is the adjoint Lie KKT-algebra for the Jordan algebra J. 相似文献
5.
Let Jn be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G -gradings on Jn where G is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n−1, where n is the dimension of the vector space V defining Jn. We prove that in this case the algebra Jn is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form. 相似文献
6.
Alessio Martini 《Journal of Functional Analysis》2011,260(9):2767-2814
The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L1,…,Ln on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a “weighted subcoercive operator” of ter Elst and Robinson (1998) [52]. The joint spectrum of L1,…,Ln in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L1,…,Ln. Connections with the theory of Gelfand pairs are established in the case L1,…,Ln generate the algebra of K-invariant left-invariant differential operators on G for some compact subgroup K of Aut(G). 相似文献
7.
K.C. O’Meara 《Linear algebra and its applications》2006,412(1):39-74
A collection A1, A2, …, Ak of n × n matrices over the complex numbers C has the ASD property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable. Such a collection must perforce be commuting. We show by a direct matrix proof that the ASD property holds for three commuting matrices when one of them is 2-regular (dimension of eigenspaces is at most 2). Corollaries include results of Gerstenhaber and Neubauer-Sethuraman on bounds for the dimension of the algebra generated by A1, A2, …, Ak. Even when the ASD property fails, our techniques can produce a good bound on the dimension of this subalgebra. For example, we establish for commuting matrices A1, …, Ak when one of them is 2-regular. This bound is sharp. One offshoot of our work is the introduction of a new canonical form, the H-form, for matrices over an algebraically closed field. The H-form of a matrix is a sparse “Jordan like” upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular. (The Jordan form itself does not accommodate this.) 相似文献
8.
Célestin Wafo Soh 《Journal of Mathematical Analysis and Applications》2008,345(1):387-395
We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: 2A1, A1⊕A2, 3A1, or A3,3⊕A1. Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra A3,3⊕A1 corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity). We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams [H.P.W. Gottlieb, Isospectral Euler-Bernoulli beam with continuous density and rigidity functions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 413 (1987) 235-250]. 相似文献
9.
Jon Wilkening 《Linear algebra and its applications》2007,427(1):6-25
We present an efficient algorithm for obtaining a canonical system of Jordan chains for an n × n regular analytic matrix function A(λ) that is singular at the origin. For any analytic vector function b(λ), we show that each term in the Laurent expansion of A(λ)−1b(λ) may be obtained from the previous terms by solving an (n + d) × (n+d) linear system, where d is the order of the zero of det A(λ) at λ = 0. The matrix representing this linear system contains A(0) as a principal submatrix, which can be useful if A(0) is sparse. The last several iterations can be eliminated if left Jordan chains are computed in addition to right Jordan chains. The performance of the algorithm in floating point and exact (rational) arithmetic is reported for several test cases. The method is shown to be forward stable in floating point arithmetic. 相似文献
10.
N. Yu. Makarenko 《Siberian Mathematical Journal》2007,48(1):95-111
We prove that if a (?/n?)-graded Lie algebra L = ? i=0 n?1 L i has d nontrivial components L i and the null component L 0 has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (?/n?)-graded Lie rings L = ? i=0 n?1 with few nontrivial components L i if the null component L 0 has finite order m. These results generalize Kreknin’s theorem on the solvability of the (?/n?)-graded Lie rings L = ? i=0 n?1 L i with trivial component L 0 and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components L i . The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2]. 相似文献
11.
A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows
that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras.
Presented by Don Passman. 相似文献
12.
The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L0 that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L0 has finite codimension in L and there are Lie subalgebras L0=L0⊂L1⊂?⊂Lp=L such that Li+1=Li+[Li,Li+1] for all i; (2) L0 is a Lie ideal of L and dim(L0)=∞. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension. 相似文献
13.
We carry out the Lie group classification of the generalized Lane–Emden equation xu″+nu′+xH(u)=0, which has many applications in mathematical physics and astrophysics. We show that the equation admits a three-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra, which in this case is trivial, has seven possible extensions. Three new cases arise for which the Lie point symmetry algebra is non-trivial. Comparison is then made of these cases with the Noether symmetry cases as well as the partial Noether operators. 相似文献
14.
We show that every injective Jordan semi-triple map on the algebra Mn(F) of all n × n matrices with entries in a field F (i.e. a map Φ:Mn(F)→Mn(F) satisfying
Φ(ABA)=Φ(A)Φ(B)Φ(A) 相似文献
15.
This paper proves that the maximum order-index of n × n matrices over an arbitrary commutative incline equals (n − 1)2 + 1. This is an answer to an open problem “Compute the maximum order-index of a member of Mn(L)”, proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984, where Mn(L) is the set of all n × n matrices over an incline L. 相似文献
16.
Jordan isomorphisms of upper triangular matrix rings 总被引:1,自引:0,他引:1
Let R be a 2-torsionfree ring with identity 1 and let Tn(R), n ? 2, be the ring of all upper triangular n × n matrices over R. We describe additive Jordan isomorphisms of Tn(R) onto an arbitrary ring and generalize several results on this line. 相似文献
17.
Yung-Yih Lur 《Linear algebra and its applications》2006,418(1):336-346
Let Ψ be a bounded set of n × n nonnegative matrices in max algebra. In this paper we propose the notions of the max algebra version of the generalized spectral radius μ(Ψ) of Ψ, and the max algebra version of the joint spectral radius η(Ψ) of Ψ. The max algebra version of the generalized spectral radius theorem μ(Ψ) = η(Ψ) is established. We propose the relationship between the generalized spectral radius ρ(Ψ) of Ψ (in the sense of Daubechies and Lagarias) and its max algebra version μ(Ψ). Moreover, a generalization of Elsner and van den Driessche’s lemma is presented as well. 相似文献
18.
Anatoliy P. Petravchuk 《代数通讯》2013,41(6):2311-2316
Let L be a finite dimensional Lie algebra over a field F. It is well known that the solvable radical S(L) of the algebra L is a characteristic ideal of L if char F = 0, and there are counterexamples to this statement in case char F = p > 0. We prove that the sum S(L) of all solvable ideals of a Lie algebra L (not necessarily finite dimensional) is a characteristic ideal of L in the following cases: 1) char F = 0; 2) S(L) is solvable and its derived length is less than log2 p. 相似文献
19.
Yu. Yu. Kochetkov 《Functional Analysis and Its Applications》2006,40(3):228-233
For the nilpotent infinite-dimensional Lie algebra L 3, we compute the second cohomology group H 2(L 3, L 3) with coefficients in the adjoint module. Nontrivial cocycles are found in closed form, and Massey powers are computed for them. 相似文献
20.
Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations
In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of shifted Chebyshev polynomials themselves. We develop also a direct solution technique for solving the linear multi-order fractional differential equations (FDEs) with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials TL,n(x) with x ∈ (0, L), L > 0 and n is the polynomial degree. We presented a shifted Chebyshev collocation method with shifted Chebyshev–Gauss points used as collocation nodes for solving nonlinear multi-order fractional initial value problems. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results. 相似文献