Reorthogonalized block classical Gram–Schmidt

A reorthogonalized block classical Gram–Schmidt algorithm is proposed that factors a full column rank matrix $A$ into $A=QR$ where $Q$ is left orthogonal (has orthonormal columns) and $R$ is upper triangular and nonsingular. This block Gram–Schmidt algorithm can be implemented using matrix–matrix operations making it more efficient on modern architectures than orthogonal factorization algorithms based upon matrix-vector operations and purely vector operations. Gram–Schmidt orthogonal factorizations are important in the stable implementation of Krylov space methods such as GMRES and in approaches to modifying orthogonal factorizations when columns and rows are added or deleted from a matrix. With appropriate assumptions about the diagonal blocks of $R$ , the algorithm, when implemented in floating point arithmetic with machine unit $\varepsilon _M$ , produces $Q$ and $R$ such that $\Vert I- Q ^T\!~ Q \Vert =O(\varepsilon _M)$ and $\Vert A-QR \Vert =O(\varepsilon _M\Vert A \Vert )$ . The first of these bounds has not been shown for a block Gram–Schmidt procedure before. As consequence of these results, we provide a different analysis, with a slightly different assumption, that re-establishes a bound of Giraud et al. (Num Math, 101(1):87–100, 2005) for the CGS2 algorithm.     

从(n 1)/n×(n 1)=(n 1)/n (n 1)谈起

一问题的提出本刊2003年第5期刊载了《运用发现法解题》(以下简称《解题》)一文,文章在谈到“归纳发现法”时,提到这样一个例子: 1.观察下列各式     

The n �� n KdV hierarchy

We introduce two new soliton hierarchies that are generalizations of the KdV hierarchy. Our hierarchies are restrictions of the AKNS n × n hierarchy coming from two unusual splittings of the loop algebra. These splittings come from automorphisms of the loop algebra instead of automorphisms of sl (n, \mathbbC){sl (n, \mathbb{C})} . The flows in the hierarchy include systems of coupled nonlinear Schr?dinger equations. Since they are constructed from a Lie algebra splitting, the general method gives formal inverse scattering, bi-Hamiltonian structures, commuting flows, and B?cklund transformations for these hierarchies.     

Balanced path decomposition of λ K n,n and λ K n,n *

Let P _k denote a path with k edges and λ K _n,n denote the λ-fold complete bipartite graph with both parts of size n. In this paper, we obtain the necessary and sufficient conditions for λ K _n,n to have a balanced P _k -decomposition. We also obtain the directed version of this result.     

数列{n√n!/n}极限的简便求法

给出了数列{^n√n!/n}l的极限的一种简便求法。     

数列{n/n／n!}极限的教学价值

给出数列(n)/(n)/(n!)极限的八种灵活新颖的求解方法,进一步讨论了它的教学价值.     

Schur’s theorem for a block Hadamard product

We define a generalized Kronecker product for block matrices, mention some of its properties, and apply it to the study of a block Hadamard product of positive semidefinite matrices, which was defined by Horn, Mathias, and Nakamura. Under strong commutation assumptions we obtain generalizations of Schur’s theorem and of Oppenheim’s inequality.     

The local rank of an ergodic symmetric power T ⊙n does not exceed n!n −n

The infinity of the rank of ergodic symmetric powers of automorphisms of the Lebesgue space is proved, and sharp upper bounds for their local rank are found.     

On the longest block in Lüroth expansion

In this paper, for a finite subset $A?\{2,3,?\}$, we introduce the notion of longest block function $L_n(x,A)$ for the Lüroth expansion of $x\in [0,1)$ with respect to A and consider the asymptotic behavior of $L_n(x,A)$ as n tends to ∞. We also obtain the Hausdorff dimensions of the level sets and exceptional set arising from the longest block function.     

Directed packings with block size 5 and even ν

Let andk be positive integers. A transitively orderedk-tuple (a ₁,a ₂,...,a _k) is defined to be the set {(a _i, a_j) 1i<jk} consisting ofk(k–1)/2 ordered pairs. A directed packing with parameters ,k and index =1, denoted byDP(k, 1; ), is a pair (X, A) whereX is a -set (of points) andA is a collection of transitively orderedk-tuples ofX (called blocks) such that every ordered pair of distinct points ofX occurs in at most one block ofA. The greatest number of blocks required in aDP(k, 1; ) is called packing number and denoted byDD(k, 1; ). It is shown in this paper that for all even integers , where [x] is the floor ofx.     

The Moore–Penrose inverse of block magic rectangles

As is known, a semi-magic square is an n?×?n matrix having the sum of entries in each row and each column equal to a constant. This note generalizes this notion and introduce a special class of block matrices called block magic rectangles. It is proved that the Moore–Penrose inverse of a block magic rectangle is also a block magic rectangle.     

Minima of decomposable forms of degree n in n variables for n ≥ 3

The following theorem is proved: if for all (X0)one has ¦ F(x) ¦ >0, where F(x) is a decomposable form of degree n of n variables, then, for n 3, F(x) is proportional to an integral form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 142–154, 1990.     

级数∞∑n=1nαsin(nβ)的收敛性

通过推导给出了使级数∞∑n=1nαsin(nβ)收敛的α与β的值.     

A convergence analysis of block accelerated over-relaxation iterative methods for weak block H-matrices to partition π

The aim of this paper is to establish the convergence of the block iteration methods such as the block successively accelerated over-relaxation method (BAOR) and the symmetric block successively accelerated over-relaxation method (BSAOR): Let be a weak block H-matrix to partition π, then for ,

     

Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n

In this paper, we introduce the subfamilies H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.