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1.
Raimundas Vidūnas 《Linear algebra and its applications》2007,422(1):39-57
Let V denote a vector space with finite positive dimension, and let (A, A∗) denote a Leonard pair on V. As is known, the linear transformations A, A∗ satisfy the Askey-Wilson relations
2.
Let K denote a field and let V denote a vector space over K with finite positive dimension.We consider a pair of K-linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0?i?d, where V-1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W,A∗W⊆W,W≠0,W≠V.We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide.In this paper we show that the following (i)-(iv) hold provided that K is algebraically closed: (i) Each of has dimension 1.(ii) There exists a nondegenerate symmetric bilinear form 〈,〉 on V such that 〈Au,v〉=〈u,Av〉 and 〈A∗u,v〉=〈u,A∗v〉 for all u,v∈V.(iii) There exists a unique anti-automorphism of End(V) that fixes each of A,A∗.(iv) The pair A,A∗ is determined up to isomorphism by the data , where θi (resp.) is the eigenvalue of A (resp.A∗) on Vi (resp.), and is the split sequence of A,A∗ corresponding to and . 相似文献
3.
In the present paper we deal with the polynomials Ln(α,M,N) (x) orthogonal with respect to the Sobolev inner product
4.
Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A∗ : V → V that satisfy the following two conditions:
- (i)
- There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.
5.
Ding-Gong Yang 《Applied mathematics and computation》2010,215(9):3473-3481
Let Tn(A,B,α) denote the class of functions of the form:
6.
Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfy (i) and (ii) below:
- (i)
- There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.
7.
Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i) and (ii) below:
- (i)
- There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.
8.
9.
Steven D. Taliaferro 《Journal of Differential Equations》2011,250(2):892-928
We study classical nonnegative solutions u(x,t) of the semilinear parabolic inequalities
10.
When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?
Manuel Alfaro Francisco Marcellán M. Luisa Rezola 《Journal of Computational and Applied Mathematics》2010,233(6):1446-1452
Given {Pn}n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e.,
11.
Let Γ be a finitely generated, torsion-free, two-step nilpotent group. Let C*(Γ) denote the universal C*-algebra of Γ. We show that , where for a unital C*-algebra A, sr(A) is the stable rank of A, and where is the space of one-dimensional representations of Γ. In process, we give a stable rank estimate for maximal full algebras of operator fields over metric spaces. 相似文献
12.
13.
We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely and . Proofs and numerical counterexamples are given in situations where the zeros of Rn, and Sn, respectively, interlace (or do not in general) with the zeros of , , k=n or n−1. The results we prove hold for continuous, as well as integral, shifts of the parameter α. 相似文献
14.
Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i) and (ii) below:
- (i)
- There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.
15.
Guy Bouchitté 《Journal of Functional Analysis》2003,204(1):228-267
For a given positive measure μ on , we consider integral functionals of the kind
16.
Let (X,T) be a topological dynamical system and be a sub-additive potential on C(X,R). Let U be an open cover of X. Then for any T-invariant measure μ, let . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle:
17.
Let A=(A1,…,Am) be a sequence of finite subsets from an additive abelian group G. Let Σ?(A) denote the set of all group elements representable as a sum of ? elements from distinct terms of A, and set . Our main theorem is the following lower bound:
18.
Hao Pan 《Journal of Combinatorial Theory, Series A》2009,116(8):1374-1381
Let A1,…,An be finite subsets of a field F, and let
19.
V.V. Fedorchuk 《Topology and its Applications》2010,157(4):716-678
We investigate a dimension function L-dim (L is a class of ANR-compacta). Main results are as follows.Let L be an ANR-compactum.(1) If L*L is not contractible, then for every n?0 there is a cube Im with .(2) If L is simply connected and f:X→Y is an acyclic mapping from a finite-dimensional compact Hausdorff space X onto a finite-dimensional space Y, then .(3) If L is simply connected and L*L is not contractible, then for every n?2 there exists a compact Hausdorff space such that , and for an arbitrary closed set either or . 相似文献
20.
Let denote a field and V denote a nonzero finite-dimensional vector space over . We consider an ordered pair of linear transformations A:V→V and A*:V→V that satisfy (i)–(iii) below.
- 1. [(i)]Each of A,A* is diagonalizable on V.
- 2. [(ii)]There exists an ordering of the eigenspaces of A such thatwhere V-1=0, Vd+1=0.
- 3. [(iii)]There exists an ordering of the eigenspaces of A* such thatwhere , .
Keywords: Leonard pair; Tridiagonal pair; q-Inverting pair; Split decomposition 相似文献