A stability property for locally uniformly rotund renorming

It is proved that C(K,E) (the space of all continuous functions on a Hausdorff compact space K taking values in a Banach space E) admits an equivalent locally uniformly rotund norm if C(K) and E do so. Moreover, if the equivalent LUR norms on C(K) and E are lower semicontinuous with respect to some weak topologies, the LUR norm on C(K,E) can be chosen to be lower semicontinuous with respect to an appropriate weak topology. As a consequence we prove that if X and Y are two Hausdorff compacta and C(X), C(Y) admit equivalent (pointwise lower semicontinuous) LUR norms, then so does C(X×Y).     

The gap between the Schur group and the subgroup generated by cyclic cyclotomic algebras

Let K be an abelian extension of the rationals. Let S(K) be the Schur group of K and let CC(K) be the subgroup of S(K) generated by classes containing cyclic cyclotomic algebras. We characterize when CC(K) has finite index in S(K) in terms of the relative position of K in the lattice of cyclotomic extensions of the rationals.     

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K → 2^Y a point-to-set mapping such that for any χ ε K, C(χ) is a pointed, closed, and convex cone in Y and int C(χ) ≠ 0. Given a mapping g : K → K and a vector valued bifunction f : K × K → Y, we consider the implicit vector equilibrium problem (IVEP) of finding χ^* ε K such that f g(χ^*), y) -int C(χ) for all y ε K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.     

Fields for which the projective schur subgroup is the whole Brauer group

LetBr(K) denote the Brauer group of a fieldK andPS(K) the projective Schur subgroup. 1. LetK be a finitely generated infinite field. ThenPS(K)=Br(K) if and only ifK is a global field. 2.|LetK be a finitely generated infinite field, and letK((t)) denote the field of formal power series int overK. ThenPS(K((t)))=Br(K((t))) if and only ifK=ℚ.     

Universal entire functions with gap power series

Let be the family of all compact sets in which have connected complement. For K ε M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior.Suppose that {z_n} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of ₀.If Q has density Δ(Q) = 1 then there exists a universal entire function with lacunary power series

1. (1) (z) = ε^∞_{v = 0 v}Z^v, _v = 0 for v Q, which has for all K ε M the following properties simultaneously

2. (2) the sequence {(Z + Z_n)} is dense in A(K)

3. (3) the sequence { (ZZ_n)} is dense in A(K) if 0 K.

Also a converse result is proved: If is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δ_max(Q) = 1.     

A problem in extremal quasiconformal extensions

A constantK ₀ ^(m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK ₀(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established forK ₀ ^(m) (h) =K ₁(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK ₀ ^(m) (h)<K ₁(h), wherek ₁(h) is the maximal dilatation ofh.     

Indecomposables in the composition algebra of the kronecker algebra

Let; K be the Kronecker algebra; M a K-indecomposable; C(K) the composition algebra of K. By using the triangular decomposition of C(K) we prove that [M] ? C(K) if and only if M is either a preprojective or a preinjective.     

The Algebra of Flows in Graphs

We define a contravariant functorKfrom the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graphX, an abelian groupB, and a nonnegative integerj, an element of Hom(K^j(X), B) is a coherent family ofB-valued flows on the set of all graphs obtained by contracting some (j − 1)-set of edges ofX; in particular, Hom(K¹(X), ) is the familiar (real) “cycle-space” ofX. We show thatK ^· (X) is torsion-free and that its Poincaré polynomial is the specializationt^{n − k}T_X(1/t, 1 + t) of the Tutte polynomial ofX(hereXhasnvertices andkcomponents). Functoriality ofK ^· induces a functorial coalgebra structure onK ^· (X); dualizing, for any ringBwe obtain a functorialB-algebra structure on Hom(K ^· (X), B). WhenBis commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincaré polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows inX, and conclude with 10 open problems.     

Diameters of iterated clique graphs of chordal graphs

The clique graph K(G) of a graph is the intersection graph of maximal cliques of G. The iterated clique graph Kⁿ(G) is inductively defined as K(K^n?1(G)) and K¹(G) = K(G). Let the diameter diam(G) be the greatest distance between all pairs of vertices of G. We show that diam(Kⁿ(G)) = diam(G) — n if G is a connected chordal graph and n ≤ diam(G). This generalizes a similar result for time graphs by Bruce Hedman.     

Banach spaces of continuous functions with few operators

We present two constructions of infinite, separable, compact Hausdorff spaces K for which the Banach space C(K) of all continuous real-valued functions with the supremum norm has remarkable properties. In the first construction K is zero-dimensional and C(K) is non-isomorphic to any of its proper subspaces nor any of its proper quotients. In particular, it is an example of a C(K) space where the hyperplanes, one co-dimensional subspaces of C(K), are not isomorphic to C(K). In the second construction K is connected and C(K) is indecomposable which implies that it is not isomorphic to any C(K) for K zero-dimensional. All these properties follow from the fact that there are few operators on our C(K)s. If we assume the continuum hypothesis the spaces have few operators in the sense that every linear bounded operator T : C (K) C (K) is of the form gI+S where gC(K) and S is weakly compact or equivalently (in C(K) spaces) strictly singular.While conducting research leading to the results presented in this paper, the author was partially supported by a fellowship Produtividade em Pesquisa from National Research Council of Brazil (Conselho Nacional de Pesquisa, Processo Número 300369/01-8). The final stage of the research was realized at the Fields Institute in Toronto where the author was supported by the State of São Paulo Research Assistance Foundation (Fundação de Amparoá Pesquisa do Estado de São Paulo), Processo Número 02/03677-7 and by the Fields Institute.Revised version: 29 January 2004     

Homological properties of Fourier algebras on homogeneous spaces

Let K be a compact subgroup of a locally compact group G. Completely complemented ideals in A(G/K) are characterised. Biprojectivity and biflatness for the Fourier algebra A(G/K) are studied. A(G/K) is operator biprojective precisely when K is open and if this happens, then G does not contain the free group on two generators as a closed subgroup.     

An abstract analogon of the de Branges-Rovnyak functional model

Given a Hilbert spaceK generated by two of its subspaces,K=K ₁K ₂, we investigate the representation ofK inK ₁K ₂ which assigns to eachk K the pairPk=P(K ₁)k P(K ₂)k consisting of its orthogonal projections ontoK ₁ andK ₂.A part of this work was done during the visit of the second author at the University of Sevilla, Spain in September-October 1989. The support of Junta Andalusia is gratefully aknowledged.     

On holomorphic curves in a complex Grassmann manifold <Emphasis Type="Italic">G</Emphasis>(2, <Emphasis Type="Italic">n</Emphasis>)

Let s : S ² → G(2, n) be a linearly full totally unramified non-degenerate holomorphic curve in a complex Grassmann manifold G(2, n), and let K(s) be its Gaussian curvature. It is proved that K(s) = \frac4n-2{K(s) = \frac{4}{n-2}} if K(s) satisfies K(s) 3 \frac4n-2{K(s) \geq \frac{4}{n-2}} or K(s) ￡ \frac4n-2 {K(s) \leq \frac{4}{n-2} } everywhere on S ². In particular, K(s) = \frac4n-2{K(s) = \frac{4}{n-2}} if K(s) is constant.     

Morava K-Theories of p-Groups with Cyclic Maximal Subgroups and Other Related Groups

Let P be a non-Abelian finite p-group, p odd, with cyclic maximal subgroups, and let K(n)^*(–) denote the nth Morava K-theory at p. In this paper we determine the algebras K(n)^*(BP) and K(n)^*(BG) for all groups G with Sylow p-subgroups isomorphic to P, giving further evidence for the fact that Morava K-theory as an invariant of finite groups, is finer than ordinary modp cohomology. Mathematics Subject Classifications (2000): 55N20, 55N22.     

On fractional K‐factors of random graphs

Let K be a graph on r vertices and let G = (V,E) be another graph on ∣V ∣ = n vertices. Denote the set of all copies of K in G by 𝒦. A non‐negative real‐valued function f : 𝒦→ ℝ₊ is called a fractional K‐factor if ∑ _{K:v∈K∈𝒦}f(K) ≤ 1 for every v ∈ V and ∑ _K∈𝒦f(K) = n/r. For a non‐empty graph K let d(K) = e(K)/v(K) and d⁽¹⁾(K) = e(K)/(v(K) ‐ 1). We say that K is strictly K₁‐balanced if for every proper subgraph K^′ ⊊ K, d⁽¹⁾(K^′) < d⁽¹⁾(K). We say that K is imbalanced if it has a subgraph K^′ such that d(K^′) > d(K). Considering a random graph process on n vertices, we show that if K is strictly K₁‐balanced, then with probability tending to 1 as n →∞, at the first moment τ₀ when every vertex is covered by a copy of K, the graph has a fractional K‐factor. This result is the best possible. As a consequence, if K is K₁‐balanced, we derive the threshold probability function for a random graph to have a fractional K‐factor. On the other hand, we show that if K is an imbalanced graph, then for asymptotically almost every graph process there is a gap between τ₀ and the appearance of a fractional K‐factor. We also introduce and apply a criteria for perfect fractional matchings in hypergraphs in terms of expansion properties. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Grothendieck groups of unbounded complexes of finitely generated modules

Let Λ be a left Artinian ring, D⁺(mod Λ) (resp., D⁻(mod Λ), D(mod Λ)) the derived category of bounded below complexes (resp., bounded above complexes, unbounded complexes) of finitely generated left Λ-modules. We show that the Grothendieck groups K₀(D⁺(mod Λ)), K₀(D⁻(mod Λ)) and K₀(D(mod Λ)) are trivial. Received: 7 April 2005     

Tame Galois p-extensions of p-henselian fields

A valued field K is called p-henselian, if its valuation extends uniquely to the maximal Galois p-extension K(p) of K. In this paper we determine the structure of Gal(K(p)/K) in case the residue characteristic is different from p. We do not assume that K contains a p-th root of unity. Received: 3 February 1999     

Constructive Matrix and Orderable Groups

We study into the relationship between constructivizations of an associative commutative ring K with unity and constructivizations of matrix groups GL _n(K) (general), SL _n(K) (special), and UT _n(K) (unitriangular) over K. It is proved that for n 3, a corresponding group is constructible iff so is K. We also look at constructivizations of ordered groups. It turns out that a torsion-free constructible Abelian group is orderly constructible. It is stated that the unitriangular matrix group UT _n(K) over an orderly constructible commutative associative ring K is itself orderly constructible. A similar statement holds also for finitely generated nilpotent groups, and countable free nilpotent groups.     

Measure of central symmetry for fields of convex figures and three-dimensional convex bodies

Let g₂³:E₂( \mathbbR³ ) ? G₂( \mathbbR³ ) \gamma_2^3:{E_2}\left( {{\mathbb{R}^3}} \right) \to {G_2}\left( {{\mathbb{R}^3}} \right) be the tautological vector bundle over the Grassmann manifold of 2-planes in \mathbbR³ {\mathbb{R}^3} , where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of \mathbbR³ {\mathbb{R}^3} . A field of convex figures is given in γ₂³ if a convex figure is distinguished in each fiber so that the figure continuously depends on the fiber. It is proved that each field of convex figures in γ₂³ contains a figure K containing a centrally symmetric convex figure of area ( 4 + 16?2 ) \left( {4 + 16\sqrt {2} } \right) S(K)/31 > 0.858 S(K) (S(K) denotes the area of K), and a figure K′ that is contained in a centrally symmetric convex figure of area ( 12?2 - 8 ) \left( {12\sqrt {2} - 8} \right) S(K′)/7 < 1.282 S(K′). It is also proved that each three-dimensional convex body K is contained in a centrally symmetric convex cylinder of volume ( 36?2 - 24 ) \left( {36\sqrt {2} - 24} \right) V(K)/7 < 3.845 V(K). (Here, V(K) denotes the volume of K.) Bibliography: 5 titles.     

Fourier algebras on locally compact hypergroups

In the present paper we introduce a new definition for the Fourier space A (K) of a locally compact Hausdorff hypergroup K and prove that it is a Banach subspace of B (K). This definition coincides with that of Amini and Medghalchi in the case where K is a tensor hypergroup, and also with that of Vrem which is given only for compact hypergroups. We prove that A_p (K)* = PM_q (K), where q is the exponent conjugate to p, in particular A (K)* = VN (K). Also we show that for Pontryagin hypergroups, A (K) = L²(K) * L²(K) = F (L¹( )), where F stands for the Fourier transform on . Furthermore there is an equivalent norm on A (K) which makes A (K) into a Banach algebra isomorphic with L¹( ). (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)