On the number of permutatively inequivalent basic sequences in a Banach space

Let X be a Banach space with a Schauder basis (en)_n∈N. The relation E₀ is Borel reducible to permutative equivalence between normalized block-sequences of (en)_n∈N or X is c₀ or ?p saturated for some 1?p<+∞. If (en)_n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c₀ or ?p, 1<p<+∞, or the relation E₀ is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X^∗. If (en)_n∈N is unconditional, then either X is isomorphic to ?₂, or X contains ω² subspaces or ω² quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases.     

The projection constant of finite-dimensional spaces whose unconditional basis constant is 1

The relations of the projection constant λ(E) and the isomorphic distance d(E, 1 _n ^∞ ) of finite-dimensional spacesE whose unconditional basis constant is 1 are investigated. It turns out that both are proportional to the norm of a certain vector inE.     

Minimal and maximal unconditional bases with respect to framings

This paper studies framings in Banach spaces, a concept raised by Casazza, Han and Larson, which is a natural generalization of traditional frames in Hilbert spaces and unconditional bases in Banach spaces. The minimal unconditional bases and the maximal unconditional bases with respect to framings are introduced. Our main result states that, if （xi, fi） is a framing of a Banach space X, and （eimin） and （eimax） are the minimal unconditional basis and the maximal unconditional basis with respect to （xi, fi）, respectively, then for any unconditional basis （ei） associated with （xi, fi）, there are A,B 〉 0 such that A｜｜i=1∑∞aieimin｜｜≤｜｜i=1∑∞aiei｜｜≤B｜｜i=1∑∞aieimax｜｜ for all （ai） ∈ c00.
It means that for any framing, the corresponding associated unconditional bases have common upper and lower bounds.     

On the uniqueness of symmetric bases in finite dimensional Banach spaces

Suppose{e _i} _i=1 ⁿ and{f _i} _i=1 ⁿ are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l _n ² )≧n' for somer>0. Then there is a constantC _r=C_r(C)>0 such that for alla _i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions.     

On separable Lindenstrauss spaces

Isometric embeddings from l_∞ⁿin l_∞^{n + 1} can be described by a_i,n, i ? n, with $\text{∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{¦ a}{}_{\mathrm{i,n}}\text{¦ ? 1}$, such that e_i,n = e_{i,n + 1} + a_i,ne_{n + 1,n + 1}; i = 1,…, n; holds, where e_i,nand e_{i,n + 1} are the elements of the canonical unit vector bases of l_∞ⁿand l_∞^{n + 1}, respectively (negative signs may occur). We study the connections between a triangular substochastic matrix A, whose nth column consists of the elements a_i,n, i = 1,…, n, and the Banach space a_i,n, E_n ? E_{n + 1}, E_n ? l_∞ⁿ, where A determines the embeddings of the E_n. The class of these Banach spaces is the class of all separable Lindenstrauss spaces. Sufficient and necessary conditions are stated for a matrix A to represent c₀and c. Furthermore, we characterize the class of all extreme triangular substochastic matrices which represents C(K), where K is the Cantor set. We investigate how the special biface structure of the dual unit ball of X is reflected in the elements of a matrix A representing the separable Lindenstrauss space X. This is applicable to Gurarij spaces; we give a new proof for the maximality property of Gurarij spaces and show that they are isomorphic to A(S) where S is a Choquet simplex with dense extreme points.     

Riesz product spaces and representation theory

Let {E _i:i∈I} be a family of Archimedean Riesz spaces. The Riesz product space is denoted by ∏ _i∈I E_i. The main result in this paper is the following conclusion: There exists a completely regular Hausdorff spaceX such that ∏ _i∈I E_i is Riesz isomorphic toC(X) if and only if for everyi∈I there exists a completely regular Hausdorff spaceX _i such thatE _i is Riesz isomorphic toC(X _i). Supported by the National Natural Science Foundation of China     

A NOTE ON GRAPHS WHOSE DESTRUCTIONS YIELD COMPLETE SUBGRAPHS

Abstract

A connected graph G of order p =|V| and sise q =| E | is said to be (a_i, b_i)-destructible (with respect to E_i and V_i say) if a_i,b_i are integral factors of p and an a_i-set of edges E_i exists whose removal from G results in exactly b_i components isomorphic to K_i i.e. whose removal from G isolates the vertices in a b_i-set V_i. The operation of removing E_i and V_i from G results in either Ø or a subgraph H of G and is called an (a_i , b_i)-destruction of G. In this paper we show that the only graphs whose every (a_i,b_i)- destruction results in a complete subgraph are K (1,2) and K₄—e, where e ε K₄.     

Vertex pancyclicity in quasi claw-free graphs

This paper generalizes the concept of locally connected graphs. A graph G is triangularly connected if for every pair of edges e₁,e₂∈E(G), G has a sequence of 3-cycles C₁,C₂,…,C_l such that e₁∈C₁,e₂∈C_l and E(C_i)∩E(C_i+1)≠∅ for 1?i?l-1. In this paper, we show that every triangularly connected quasi claw-free graph on at least three vertices is vertex pancyclic. Therefore, the conjecture proposed by Ainouche is solved.     

On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

On space coverings by unbounded convex sets

Let (C_i) be a sequence of closed convex subsets of Euclidean n-space Eⁿ. This paper is concerned with the problem of finding necessary and sufficient conditions that the sets C_i can be rearranged (by the application of rigid motions or translations) so as to cover all or almost all Eⁿ. Particular attention is paid to the problems that arise if the sets C_i are permitted to be unbounded. It is shown that under certain conditions this covering problem can be reduced to the already thoroughly investigated case of compact sets with bounded diameter set{d(C_i)}, and it is also proved that there are two additional covering possibilities if such a reduction is not possible.     

A characterization of Riesz spaces which are Riesz isomorphic to C(X) for some completely regular space X

Let E be an Archimedean Riesz space possessing a weak unit e and let Ω be the collection of all Riesz homomorphisms ø from E onto ℝ such that ø(e)=1. The Gelfand mapping G :x→x^ on E is defined by x^(ø) = ø(x) for all ø∈Ω. We endow Ω with the topology induced by E (i.e., the weakest topology such that each x^ is continuous on Ω). The principal ideal in E generated by e is denoted by I_d(e). The main theorem in this paper says that the following statements (A) and (B) are equivalent.

• (A)There exists a completely regular space X such that E is Riesz isomorphic to the space C(X) of all real continuous functions on X.
• (B)The following conditions for the Riesz space E hold: (1) E is Archimedean and has a weak unit e; (2) Ω separates the points of E; (3) E is uniformly complete; (4) G(I_d(e)) is norm dense in the space C_b(Ω) of all real bounded continuous functions on Ω; (5) E is 2-universally complete with carrier space Ω.

Some other conditions are mentioned and an example is given to show that condition (5) is necessary for (B) ⇒(A).     

Riesz spaces of real continuous functions

An Archimedean Riesz space E is isomorphic to C(X) for some completely regular Hausdorff space X if and only if there exists a weak order unit e > 0 for which E is e-uniformly complete, e-semisimple, e-separating and 2-universally e-complete.     

Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces

LetX be a Banach space with an unconditional finite-dimensional Schauder decomposition (E _n). We consider the general problem of characterizing conditions under which one can construct an unconditional basis forX by forming an unconditional basis for eachE _n. For example, we show that if sup _n dimE _n<∞ andX has Gordon-Lewis local unconditional structure thenX has an unconditional basis of this type. We also give an example of a non-Hilbertian spaceX with the property that wheneverY is a closed subspace ofX with a UFDD (E _n) such that sup _n dimE _n<∞ thenY has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jaegermann cannot be improved. Both authors were supported by NSF Grant DMS-9201357.     

Hypercyclic subspaces for Fréchet space operators

A continuous linear operator is hypercyclic if there is an x∈X such that the orbit {Tnx} is dense, and such a vector x is said to be hypercyclic for T. Recent progress show that it is possible to characterize Banach space operators that have a hypercyclic subspace, i.e., an infinite dimensional closed subspace H⊆X of, except for zero, hypercyclic vectors. The following is known to hold: A Banach space operator T has a hypercyclic subspace if there is a sequence (ni) and an infinite dimensional closed subspace E⊆X such that T is hereditarily hypercyclic for (ni) and Tni→0 pointwise on E. In this note we extend this result to the setting of Fréchet spaces that admit a continuous norm, and study some applications for important function spaces. As an application we also prove that any infinite dimensional separable Fréchet space with a continuous norm admits an operator with a hypercyclic subspace.     

Complementary bases of a matroid

Let e₁, e′₁, e₂, e′₂, …, e_n, e′_n be the elements of matroid M. Suppose that {e₁, e₂, …;, e_n} is a base of M and that every circuit of M contains at least m + 1 elements. We prove that there exist at least 2^m bases, called complementary bases, of M with the property that only one of each complementary pair e_j, e′_j is contained in any base.We also prove an analogous result for the case where E is partitioned into E₁, E₂, …, E_n and the initial base contains |E_j| ? 1 elements from partition E_j.     

On the uniqueness of the tetrahedral association schemes

Let V be the set of (₃⁴) 3- sets in {1 … n}. Say p, q ∈ V are ith associates, (p, q) ∈ A_i, if 3 = i + |p ∩ q|. An association scheme is tetrahedral if it is isomorphic to the scheme {A₀, A₁, A₂, A₃} and a graph is tetrahedral if it is isomorphic to A₁. Aigner [1] and Bose and Laskar [2] have shown that the tetrahedral graphs are characterized by their characteristic equations, provided n < 9 or n > 16. The present paper extends methods of Hoffman [7] to show that the tetrahedral association schemes are characterized by their structural constants, provided n > 10.     

A RECOGNITION PROBLEM IN CONVERTING LINEAR PROGRAMMING TO NETWORK FLOW MODELS

The main goal of this paper is to study the following combinatorial problem.given a finite set E={e1,e2,…em} and a subset familly σ={S1,S2,…,Sn}of E,does there exist a tree T with the edge set E such that each induced subgraph T[Si] of Si is precisely a path (1≤i≤k)?     

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set {1,2,…,n}, the local direction of each edge (ij) is from i to j if i<j. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence λ=e₁¹e₂²… of a tree on the vertex set {1,2,…,n} is a partition of n−1. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prüfer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a q-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.