In this paper we prove that if is a cardinal in , then there is an inner model such that has no elementary end extension. In particular if exists, then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than of uncountable cofinality in is Mahlo in every strict inner model of .
A commutative Banach algebra is said to have the property if the following holds: Let be a closed subspace of finite codimension such that, for every , the Gelfand transform has at least distinct zeros in , the maximal ideal space of . Then there exists a subset of of cardinality such that vanishes on , the set of common zeros of . In this paper we show that if is compact and nowhere dense, then , the uniform closure of the space of rational functions with poles off , has the property for all . We also investigate the property for the algebra of real continuous functions on a compact Hausdorff space.
Let be a -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ). Let be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset of and let be arbitrary. Then the iteration sequence defined by , converges strongly to a fixed point of , provided that and have certain properties. If is a Hilbert space, then converges strongly to the unique fixed point of closest to .
Let be a certain Banach space consisting of continuous functions defined on the open unit disk. Let be a univalent function defined on , and assume that denotes the operator of multiplication by . We characterize the structure of the operator such that . We show that for some function in . We also characterize the commutant of under certain conditions.
Let be a compact manifold, and let be a transitive homologically full Anosov flow on . Let be a -cover for , and let be the lift of to . Babillot and Ledrappier exhibited a family of measures on , which are invariant and ergodic with respect to the strong stable foliation of . We provide a new short proof of ergodicity.
Suppose is a complex Hilbert space and is a bounded operator. For each closed set let denote the corresponding spectral manifold. Let denote the set of all points with the property that for any open neighborhood of In this paper we show that if is dominating in some bounded open set, then has a nontrivial invariant subspace. As a corollary, every Hilbert space operator which is a quasiaffine transform of a subdecomposable operator with large spectrum has a nontrivial invariant subspace.
Let and be two Del Pezzo fibrations of degrees , respectively. Assume that and differ by a flop. Then we prove that and give a short list of values of other basic numerical invariants of and .
Let be a polynomial of degree with integer coefficients, any prime, any positive integer and the exponential sum . We establish that if is nonconstant when read , then . Let , let be a zero of the congruence of multiplicity and let be the sum with restricted to values congruent to . We obtain for odd, and . If, in addition, , then we obtain the sharp upper bound .
(1) If all maximal subgroups of any Sylow subgroup of are -supple- mented in , then ;
(2) If all minimal subgroups and all cyclic subgroups with order 4 of are -supplemented in , then .
Let be a locally compact Hausdorff space. We define a quasi-measure in , a quasi-integral on , and a quasi-integral on . We show that all quasi-integrals on are bounded, continuity properties of the quasi-integral on , representation of quasi-integrals on in terms of quasi-measures, and unique extension of quasi-integrals on to .
In this note, we show that when is a torsion group the second center of the group of units of the integral group ring is generated by its torsion subgroup and by the center of . This extends a result of Arora and Passi (1993) from finite groups to torsion groups, and completes the characterization of hypercentral units in when is a torsion group.
Let be a positive matrix-valued measure on a locally compact abelian group such that is the identity matrix. We give a necessary and sufficient condition on for the absence of a bounded non-constant matrix-valued function on satisfying the convolution equation . This extends Choquet and Deny's theorem for real-valued functions on .