Vector Variational Inequalities and the (S)<Subscript>+</Subscript> Condition

Let $${\cal Z}$$ and X be Hausdorff real topological vector spaces and let $${\cal L}_b(X,{\cal Z})$$ be the space of continuous linear mappings from X into $${\cal Z}$$ equipped with the topology of bounded convergence. In this paper, we define the (S)₊ condition for operators from a nonempty subset of X into $${\cal L}_b(X,{\cal Z})$$ and derive some existence results for vector variational inequalities with operators of the class (S)₊. Some applications to vector complementarity problems are given.     

On the cohomology of orbit space of free ℤ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-actions on lens spaces

Let G = ℤ _p , p an odd prime, act freely on a finite-dimensional CW-complex X with mod p cohomology isomorphic to that of a lens space L ^2m−1(p; q ₁, …, q _m ). In this paper, we determine the mod p cohomology ring of the orbit space X/G, when p ² ∤ m.     

关于CSS空间及相关结论

CSS空间是指空间中的紧集都是一致G_δ集的空间.该文的第一部分,主要证明了具有拟G_δ(2)对角线的空间是CSS空间.另外,还证明了如果X是可数个闭的CSS空间的并,则X是CSS空间.CSS空间的可数积空间是CSS空间;第二部分证明了如果空间X可以表示成可数个闭的β空间(或半层空间)的并,则X是β空间(或半层空间).     

Dimensionality Reductions in ℓ<Subscript>2</Subscript> that Preserve Volumes and Distance to Affine Spaces

Let X be a subset of n points of the Euclidean space, and let 0 < ε < 1. A classical result of Johnson and Lindenstrauss [JL] states that there is a projection of X onto a subspace of dimension O(ε^-2 log n) with distortion ≤ 1+ ε. We show a natural extension of the above result to a stronger preservation of the geometry of finite spaces. By a k-fold increase of the number of dimensions used compared with [JL], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show how to embed a subset of size n of the Euclidean space into a O(ε^-2 log n)-dimensional Euclidean space, so that no set of size s ≤ k changes its volume by more than (1 + ε^s-1. Moreover, distances of points from affine hulls of sets of at most k - 1 points in the space do not change by more than a factor of 1 + ε. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL]. Our method can be applied to many problems with high-dimensional nature such as Projective Clustering and Approximated Nearest Affine Neighbour Search. In particular, it shows a first polylogarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [F], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.     

New examples of <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>-saturated Banach spaces

For every p ∈ (1, ∞), an isomorphically polyhedral Banach space E _p is constructed which has an unconditional basis and does not embed isomorphically into a C(K) space for any countable and compact metric space K. Moreover, E _p admits a quotient isomorphic to ℓ _p .     

The Cayley Transform of the Generator of a Bounded C<Subscript>0</Subscript>-Semigroup

Let A be the generator of a uniformly bounded C₀-semigroup on the Banach space X. We present sufficient conditions on the resolvent , under which the Cayley transform is a power-bounded operator, i.e., .     

On the b-Coloring of Cographs and <Emphasis Type="Italic">P</Emphasis><Subscript>4</Subscript>-Sparse Graphs

A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromatic number of a graph G, denoted by χ _b (G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is b-continuous if it admits a b-coloring with t colors, for every . We define a graph G to be b-monotonic if χ _b (H ₁) ≥ χ _b (H ₂) for every induced subgraph H ₁ of G, and every induced subgraph H ₂ of H ₁. In this work, we prove that P ₄-sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic. Besides, we describe a dynamic programming algorithm to compute the b-chromatic number in polynomial time within these graph classes. Flavia Bonomo: Partially supported by ANPCyT PICT-2007-00533 and PICT-2007-00518, and UBACyT Grants X069 and X606 (Argentina). Guillermo Durán: Partially supported by FONDECyT Grant 1080286 and Millennium Science Institute “Complex Engineering Systems” (Chile), and ANPCyT PICT-2007-00518 and UBACyT Grant X069 (Argentina). Javier Marenco: Partially supported by ANPCyT PICT-2007-00518 and UBACyT Grant X069 (Argentina).     

Solvability of a Model Oblique Derivative Problem for the Heat Equation in the Zygmund Space <Emphasis Type="Italic">H</Emphasis><Subscript>1</Subscript>

We consider the oblique derivative problem for the heat equation in a model statement. We introduce a difference matching condition for the initial and boundary functions, under which we establish conditions on the data of the problem sufficient for the solution to belong to the parabolic Zygmund space H₁, which is an analog of the parabolic Hölder space for the case of an integer smoothness exponent. We present an example showing that if the above-mentioned matching condition is not satisfied, then the solution may fail to belong to the space H₁.     

Finiteness conditions and PD<Subscript><Emphasis Type="Italic">r</Emphasis></Subscript>-group covers of PD<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-complexes

We show that an infinite cyclic covering space M′ of a PD _n -complex M is a PD _n-1-complex if and only if χ(M) = 0, M′ is homotopy equivalent to a complex with finite [(n−1)/2]-skeleton and π₁(M′) is finitely presentable. This is best possible in terms of minimal finiteness assumptions on the covering space. We give also a corresponding result for covering spaces M _ν with covering group a PD _r -group under a slightly stricter finiteness condition.      

Bornological and ultrabornological C(X;E) spaces

Denote by C_s(X;E) the space of the continuous functions defined on the completely regular and Hausdorff space X, with values in the locally convex topological vector space E, when it is endowed with the simple or point-wise convergence topology. We give here some conditions on X and on E under which the space C_s(X;E) is bornological or ultrabornological and characterize in some cases the corresponding associated spaces. We give also a few results concerning the case of the compact connvergence topology.     

SOME PROPERTIES OF STRONGLY MONOTONICALLY T2 SPACES

This paper investigates Buck＇s question about which class of spaces is strongly monotonically T2,and if other properties are combined with strongly monotonically T2,which class of spaces could be got. Based on having a cushioned pair-base space and compact strongly monotonically T2 space,some results （Theorems 1--3） are obtained.     

On generalized <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> and <Emphasis Type="Italic">R</Emphasis><Subscript>1</Subscript> spaces

Characterizations of some properties of generalized R ₀ and R ₁ topological spaces by using closure operator defined on a generalized topological space will be given. It is also shown that many results done in this area in some previous papers can be considered as special cases of our results.     

Perturbations of Bi-continuous Semigroups with Applications to Transition Semigroups on C<Subscript>b</Subscript>(H)

We prove an unbounded perturbation theorem for bi-continuous semigroups on the space of bounded, continuous functions on the Hilbert space H. This is applied to the Ornstein-Uhlenbeck semigroup, thus providing a purely functional analytic approach to the existence of transition semigroups on C_b(H) with bounded non-linear drift.     

Uniqueness of unconditional bases in nonlocally convex c<Subscript>0</Subscript>-products

We show that the c ₀-product (X ⊕ X ⊕ ... ⊕ X ⊕ ...)₀ of a natural quasi-Banach space X with strongly absolute unconditional basis has a unique unconditional basis up to permutation. Our results apply to a wide range of cases, including most of the c ₀-products of the nonlocally convex classical quasi-Banach spaces.     

Cone characterization of Grothendieck spaces and Banach spaces containing <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>

In this article we study the embeddability of cones in a Banach space X. First we prove that c ₀ is embeddable in X if and only if its positive cone c₀⁺{c_0^+} is embeddable in X and we study some properties of Banach spaces containing c ₀ in the light of this result. So, unlike with the positive cone of ℓ ₁ which is embeddable in any non-reflexive space, c₀⁺{c_0^+} has the same behavior as the whole space c ₀. In the second part of this article we give a characterization of Grothendieck spaces X according to the geometry of cones of X*. By these results we give a partial positive answer to a problem of J.H. Qiu concerning the geometry of cones.     

Embedding c<Subscript>0</Subscript> in bvca(Σ, <Emphasis Type="Italic">X</Emphasis>)

If (Ω,Σ) is a measurable space and X a Banach space, we provide sufficient conditions on Σ and X in order to guarantee that bvca(Σ, X) the Banach space of all X-valued countably additive measures of bounded variation equipped with the variation norm, contains a copy of c₀ if and only if X does. This work was supported by the project MTM2005-01182 of the Spanish Ministry of Education and Science, co-financed by the European Community (Feder projects).     

Regularity of the moduli space of instanton bundles MI<Subscript>P3</Subscript>(5)

We prove that the moduli space of mathematical instanton bundles on P³ with c₂ = 5 is smooth.     

Banach spaces not containing<Emphasis Type="Italic">l</Emphasis><Subscript>1</Subscript>

We show that ifE is a complex Banach space which contains no subspace isomorphic tol ₁, then each infinite dimensional subspace ofE′ contains a normalized sequence which converges to zero for the weak star topology. Dedicated to the memory of Klaus Floret (1941–2002).     

<Emphasis Type="Italic">w</Emphasis>*-Basic Sequences and Reflexivity of Banach Spaces

We observe that a separable Banach space X is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if ℒ(X, Y) is not reflexive for reflexive X and Y then ℒ(X ₁, Y) is is not reflexive for some X ₁ ⊂ X, X ₁ having a basis. This work was supported by the grants No. 201/03/0041 and No. 201/04/0090 of the Grant Agency of the Czech Republic and by the grant No. A1019801 of the Academy of Sciences of the Czech Republic.