矩阵的WZ分解及其误差分析

在[1]中,对矩阵A进行WZ分解,其中矩阵W=(w_(i,j))和Z=(z_(i,j))形状如下: 1, i=j, w_(i,j)=任意,min(i,n-i+1)相似文献   

The maximum number of disjoint pairs in a family of subsets

Let ? be a family of 2 ⁿ⁺¹ subsets of a 2n-element set. Then the number of disjoint pairs in ? is bounded by (1+o(1))2²ⁿ . This proves an old conjecture of Erdös. Let ? be a family of 2^1/(k+1)+δ)n subsets of ann-element set. Then the number of containments in ? is bounded by (1-1/k+o(1))( ₂ ^|?| ). This verifies a conjecture of Daykin and Erdös. A similar Erdös-Stone type result is proved for the maximum number of disjoint pairs in a family of subsets.     

On the minimum number of disjoint pairs in a family of finite sets

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2ⁿ. Then there is a family $\text{F}$, |$\text{F}$| = m, such that F ∈ $\text{F}$, G ? X, | G | > | F | implies G ∈ $\text{F}$ and $\text{F}$ minimizes the number of pairs (F₁, F₂), F₁, F₂ ∈ $\text{F}$ F₁ ∩ F₂ = ? over all families consisting of m subsets of X.     

A note on groups and permutation-partition pairs

ANNALI DELL'UNIVERSITA' DI FERRARA - A link between permutation-partition pairs and finite presentations of groups, leading to a graph-theoretical representation of groups, is presented. A...     

A classification of sharp tridiagonal pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of 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^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+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. The pair A,A^∗ is called sharp whenever . It is known that if F is algebraically closed then A,A^∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.     

Existence of best proximity pairs and equilibrium pairs

In this paper, using the fixed point theorem for Kakutani factorizable multifunctions, we shall prove new existence theorems of best proximity pairs and equilibrium pairs for free abstract economies, which include the previous fixed point theorems and equilibrium existence theorems.     

A new class of Gelfand pairs

The well-known result for rank 1 real symmetric spacesX=G/H, stating the multiplicity-free decomposition of the natural action ofG onL ²(X), is extended to the classical rank 1p-adic symmetric spaces. The absence of ap-adic analogue of the Laplace-Beltrami operator causes additional complications and necessitates a global study of the invariant distributions on the quotient space.     

A trace formula for isometric pairs

It is well known that for every isometry , This fact for the shift operator is a basis for many important developments in operator theory and topology. In this paper we prove an analogous formula for a pair of isometries , namely

where is the complete anti-symmetric sum and is the Fredholm index of the pair . The major tool is what we call the fringe operator. Two examples are considered.

     

A reciprocity relation for WP-Bailey pairs

We derive a new general transformation for WP-Bailey pairs by considering a certain limiting case of a WP-Bailey chain previously found by the authors, and examine several consequences of this new transformation. These consequences include new summation formulae involving WP-Bailey pairs. Other consequences include new proofs of some classical identities due to Jacobi, Ramanujan and others, and indeed extend these identities to identities involving particular specializations of arbitrary WP-Bailey pairs.     

A note on commuting pairs of matrices

We give a short proof of the Motzkin-Taussky result that the variety of commuting pairs of matrices is irreducible. An easy consequence of this is that any two generated commutative subalgebra of n×n matrices has dimension at most n. We also answer an old question of Gerstenhaber by showing that the set of commuting triples of n×n matrices is not irreducible for n≥32.     

A note on commuting pairs of matrices

We give a short proof of the Motzkin-Taussky result that the variety of commuting pairs of matrices is irreducible. An easy consequence of this is that any two generated commutative subalgebra of n×n matrices has dimension at most n. We also answer an old question of Gerstenhaber by showing that the set of commuting triples of n×n matrices is not irreducible for n≥32.     

A topological invariant for pairs of maps

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝⁿ, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘ f. For this latter set we obtain a generalization of Sharkovsky’s theorem.     

Some new transformations for Bailey pairs and WP-Bailey pairs

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.     

Best proximity pairs and equilibrium pairs for Kakutani multimaps

New existence theorems of best proximity pairs for a finite family of Kakutani multimaps are proved in a normed space setting. As applications, new existence theorems of equilibrium pairs for free n$n$-person and 1-person games are obtained. Our results generalize and improve several well-known results.     

A characterization of going-down-ring pairs of commutative rings

SoientA ⊆B des anneaux (commutaifs et unitaires). On dit que (A,B) est une paire d’anneaux de going-down siD est un anneau de going-down pour tout anneauD tel queA ⊆D ⊆B. On preuve que (A,B) est une paire d’anneaux de going-down si et seulement siA[b ₁,b ₂] est un anneau de going-down pour toutb ₁,b ₂ εB.