Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p?1. This paper is concerned with the perturbed optimization problem of finding z₀∈Z such that ‖x−z₀p^‖+J(z₀)=infz∈Z{‖x−zp^‖+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z₀, where Z₀ is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z₀.     

On certain spaces of vector measures of bounded variation

If (Σ,X) is a measurable space and X a Banach space we investigate the X-inheritance of copies of ?_∞ in certain subspaces Δ(Σ,X) of bvca(Σ,X), the Banach space of all X-valued countable additive measures of bounded variation equipped with the variation norm. Among the consequences of our main theorem we get a theorem of J. Mendoza on the X-inheritance of copies of ?_∞ in the Bochner space L₁(μ,X) and other of the author on the X-inheritance of copies of ?_∞ in bvca(Σ,X).     

On operator valued sequences of multipliers and R-boundedness

In recent papers (cf. [J.L. Arregui, O. Blasco, (p,q)-Summing sequences, J. Math. Anal. Appl. 274 (2002) 812-827; J.L. Arregui, O. Blasco, (p,q)-Summing sequences of operators, Quaest. Math. 26 (2003) 441-452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166-186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629-644]) the concept of (p,q)-summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of (p,q)-summing multipliers. The present paper is a continuation of this study, whereby multiplier spaces for some classical Banach spaces are considered. The scope of this research is also broadened, by studying other classes of summing multipliers. Let E(X) and F(Y) be two Banach spaces whose elements are sequences of vectors in X and Y, respectively, and which contain the spaces c₀₀(X) and c₀₀(Y) of all X-valued and Y-valued sequences which are eventually zero, respectively. Generally spoken, a sequence of bounded linear operators (un)⊂L(X,Y) is called a multiplier sequence from E(X) to F(Y) if the linear operator from c₀₀(X) into c₀₀(Y) which maps (xi)∈c₀₀(X) onto (unxn)∈c₀₀(Y) is bounded with respect to the norms on E(X) and F(Y), respectively. Several cases where E(X) and F(Y) are different (classical) spaces of sequences, including, for instance, the spaces Rad(X) of almost unconditionally summable sequences in X, are considered. Several examples, properties and relations among spaces of summing multipliers are discussed. Important concepts like R-bounded, semi-R-bounded and weak-R-bounded from recent papers are also considered in this context.     

Lushness, numerical index one and duality

We construct an example of a Banach space which is not lush, but whose dual space is lush. This example shows that lushness is not equivalent to numerical index one. A characterization of lushness for some quotient spaces of L₁(μ) spaces and new results on C-rich subspaces of (scalar- or vector-valued) C(K) spaces are also presented.     

Least and most colored bases

Consider a matroid M=(E,B), where B denotes the family of bases of M, and assign a color c(e) to every element e∈E (the same color can go to more than one element). The palette of a subset F of E, denoted by c(F), is the image of F under c. Assume also that colors have prices (in the form of a function π(?), where ? is the label of a color), and define the chromatic price as: π(F)=∑_?∈c(F)π(?). We consider the following problem: find a base B∈B such that π(B) is minimum. We show that the greedy algorithm delivers a lnr(M)-approximation of the unknown optimal value, where r(M) is the rank of matroid M. By means of a reduction from SETCOVER, we prove that the lnr(M) ratio cannot be further improved, even in the special case of partition matroids, unless . The results apply to the special case where M is a graphic matroid and where the prices π(?) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the ln(n-1)+1 ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function π(F), where F is a common independent set on matroids M₁,…,M_k and, more generally, to independent systems characterized by the k-for-1 property.     

Ball remotality in Banach spaces

In this paper, we continue our study of ball remotality of subspaces of Banach spaces. In particular, we study the problem in classical sequence spaces c₀, c, ?₁ and ?_∞; and also ball remotality of a Banach space in its bidual.     

Upper triangular operator matrices with the single-valued extension property

If A=(Aij)_1?i,j?n∈B(X) is an upper triangular Banach space operator such that AiiAij=AijAjj for all 1?i?j?n, then A has SVEP or satisfies (Dunford's) condition (C) or (Bishop's) property (β) or (the decomposition) property (δ) if and only if Aii, 1?i?n, has the corresponding property.     

A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

On extension of isometries and approximate isometries between unit spheres

In this paper, we will study the isometric extension problem for L¹-spaces and prove that every surjective isometry from the unit sphere of L¹(μ) onto that of a Banach space E can be extended to a linear surjective isometry from L¹(μ) onto E. Moreover, we introduce the approximate isometric extension problem and show that, if E and F are Banach spaces and E satisfies the property (m) (special cases are L^∞(Γ), C₀(Ω) and L^∞(μ)), then every bijective ?-isometry between the unit spheres of E and F can be extended to a bijective 5?-isometry between their closed unit balls. At last, we will give an example to show that the surjectivity assumption cannot be omitted. Using this, we solve the non-surjective isometric extension problem in the negative.     

Multicoloring and Mycielski construction

The generalized Mycielskians of graphs (also known as cones over graphs) are the natural generalization of the Mycielskians of graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer p?0, one can transform G into a new graph μ_p(G), the p-Mycielskian of G. In this paper, we study the kth chromatic numbers χ_k of Mycielskians and generalized Mycielskians of graphs. We show that χ_k(G)+1?χ_k(μ(G))?χ_k(G)+k, where both upper and lower bounds are attainable. We then investigate the kth chromatic number of Mycielskians of cycles and determine the kth chromatic number of p-Mycielskian of a complete graph K_n for any integers k?1, p?0 and n?2. Finally, we prove that if a graph G is a/b-colorable then the p-Mycielskian of G, μ_p(G), is (at+b^p+1)/bt-colorable, where . And thus obtain graphs G with m(G) grows exponentially with the order of G, where m(G) is the minimal denominator of a a/b-coloring of G with χ_f(G)=a/b.     

On time-relaxed broadcasting networks

Given a communication network (modelled as a graph), a message is transmitted to at one vertex transmits to all other vertices in such a way that each message transmission takes one time unit and each vertex participates in at most one transmission to its neighbor per time step. We call this process broadcasting. For t≥0, let B_t(n) be the number of edges in the sparsest possible graph on n vertices in which broadcasting can be accomplished in ⌈log₂n⌉+t steps regardless of the originator. Shastri [A. Shastri, Time-relaxed broadcasting in communication networks, Discrete Applied mathematics 83 (1998) 263-278] conjectured that B₁(22)=24 and B₂(n)=n+1 for 25≤n≤29. In this paper, we show that B₁(22)=24, B₂(n)=n for 25≤n≤28 and 37≤n≤44, B₂(n)≤n+1 for 45≤n≤49, B₂(n)≤n+4 for 50≤n≤56, and B₃(n)=n for 55≤n≤64.     

Positive Quaternion Kähler Manifolds with fourth Betti number equal to one

Positive Quaternion Kähler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. In this article we are mainly concerned with Positive Quaternion Kähler Manifolds M satisfying b₄(M)=1. Generalising a result of Galicki and Salamon we prove that M4n in this case is homothetic to a quaternionic projective space if 2≠n?6.     

An extremal problem on non-full colorable graphs

For a given graph G of order n, a k-L(2,1)-labelling is defined as a function f:V(G)→{0,1,2,…k} such that |f(u)-f(v)|?2 when d_G(u,v)=1 and |f(u)-f(v)|?1 when d_G(u,v)=2. The L(2,1)-labelling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2,1)-labelling. The hole index ρ(G) of G is the minimum number of integers not used in a λ(G)-L(2,1)-labelling of G. We say G is full-colorable if ρ(G)=0; otherwise, it will be called non-full colorable. In this paper, we consider the graphs with λ(G)=2m and ρ(G)=m, where m is a positive integer. Our main work generalized a result by Fishburn and Roberts [No-hole L(2,1)-colorings, Discrete Appl. Math. 130 (2003) 513-519].     

Noncommutative maximal ergodic theorems for positive contractions

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ. Let T be a positive linear contraction on M such that τ○T?τ and such that the numerical range of T as an operator on L₂(M) is contained in a Stoltz region with vertex 1. We show that Junge and Xu's noncommutative Stein maximal ergodic inequality holds for the powers of T on Lp(M), 1<p?∞. We apply this result to obtain the noncommutative analogue of a recent result of Cohen concerning the iterates of the product of a finite number of conditional expectations.     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p, let . In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that for every prime p, where ?=max{k₍₂₎,(k−1)₍₂₎}, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k=2^? or k=2^?+1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k-hypergraphs which have prime order p in the case where k=2^? or k=2^?+1 and , and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order , up to isomorphism.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

Weighted norm inequalities of Bochner-Riesz means

Let w be a Muckenhoupt weight and be the weighted Hardy spaces. We use the atomic decomposition of and their molecular characters to show that the Bochner-Riesz means are bounded on for 0<p?1 and δ>max{n/p−(n+1)/2,[n/p]rw⁻¹(rw−1)−(n+1)/2}, where rw is the critical index of w for the reverse Hölder condition. We also prove the boundedness of the maximal Bochner-Riesz means for 0<p?1 and δ>n/p−(n+1)/2.     

Colouring games on outerplanar graphs and trees

Let f be a graph function which assigns to each graph H a non-negative integer f(H)≤|V(H)|. The f-game chromatic number of a graph G is defined through a two-person game. Let X be a set of colours. Two players, Alice and Bob, take turns colouring the vertices of G with colours from X. A partial colouring c of G is legal (with respect to graph function f) if for any subgraph H of G, the sum of the number of colours used in H and the number of uncoloured vertices of H is at least f(H). Both Alice and Bob must colour legally (i.e., the partial colouring produced needs to be legal). The game ends if either all the vertices are coloured or there are uncoloured vertices with no legal colour. In the former case, Alice wins the game. In the latter case, Bob wins the game. The f-game chromatic number of G, χ_g(f,G), is the least number of colours that the colour set X needs to contain so that Alice has a winning strategy. Let be the graph function defined as , for any n≥3 and otherwise. Then is called the acyclic game chromatic number of G. In this paper, we prove that any outerplanar graph G has acyclic game chromatic number at most 7. For any integer k, let ?_k be the graph function defined as ?_k(K₂)=2 and ?_k(P_k)=3 (P_k is the path on k vertices) and ?_k(H)=0 otherwise. This paper proves that if k≥8 then for any tree T, χ_g(?_k,T)≤9. On the other hand, if k≤6, then for any integer n, there is a tree T such that χ_g(?_k,T)≥n.