Some generalizations of Lindelöf and paracompact spaces

In this paper notions ofm-Lindelöf, meta-m-Lindelöf, para-m-Lindelöf andm-closure preserving property are defined, wherem is any infinite cardinal. The main results are the following:

1. A topological space ism-Lindelöf if and only if it is meta-m-Lindelöf and it ism-Lindelöf in the sense of complete accumulation point.
2. A regular topological space is paracompact if and only if it is para-m-Lindelöf and it hasm-closure preserving property for somem.

     

Resolvability of Lindelöf spaces

We prove the ω-resolvability of hereditarily finally compact spaces and the resolvability of Lindelöf spaces whose dispersion character is uncountable.     

Some consequences of the Lindelöf conjecture

Suppose that the Lindelöf conjecture is valid in the following quantitative form: $$|\zeta (\frac{1}{2} + it)| \leqslant c_0 |t|^{\varepsilon (|t|)} $$ , where ε(t) is a monotone decreasing function, $\varepsilon (2t) \geqslant \tfrac{1}{2}\varepsilon (t),\varepsilon (t) \geqslant \tfrac{1}{{\sqrt {log t} }}$ . Then it is proved that for |t|≥T₀ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant v\} $ contains at most 20v log |t| zeros of ζ(s) if $\tfrac{1}{2} \geqslant v \geqslant \sqrt {\varepsilon (t)} $ . There exists an absolute constant A such that for |t|≥T₁ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant A\varepsilon ^{\tfrac{1}{3}} (t)\} $ contains at least one zero of ζ(s). Bibliography: 2 titles.     

A result on monotonically Lindelöf generalized ordered spaces

In this paper, we show that a generalized ordered space representable as the union of two closed monotonically Lindelöf subspaces is monotonically Lindelöf, which partially answers a question [7, Question 2] of Levy and Matveev. In addition, we show that the monotone Lindelöf property is hereditary with respect to open Lindelöf subsets in generalized ordered spaces.     

Some remarks on Teichmüller spaces and modular groups

We study the fixed point sets of subgroups of modular groups acting on Teichmiiller spaces.     

Some remarks on localization of Schrödinger means

We study localization and localization almost everywhere of Schrödinger means of functions in Sobolev spaces.     

Some remarks on the theorem of Möbius

H.S.M. Coxeter in his book Introduction to Geometry quotes a theorem of Möbius. In the paper two counterexamples are given. A corrected version of the theorem is stated and proved.     

On countable unions of nonmeager sets in hereditarily Lindelöf spaces

It is well known that any Vitali set on the real line ? does not possess the Baire property. The same is valid for finite unions of Vitali sets. What can be said about infinite unions of Vitali sets? Let S be a Vitali set, S _r be the image of S under the translation of ? by a rational number r and F = {S _r : r is rational}. We prove that for each non-empty proper subfamily F′ of F the union ∪F′ does not possess the Baire property. We say that a subset A of ? possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ? O \ M. Then we characterize those non-empty proper subfamilies F′ of F which unions ∪F′ possess the Vitali property.     

Remarks on Completely Regular Lindelöf Reflection of Locales

We prove, under the axiom of countable choice, that the l-ring structure of C(L) of real-valued continuous functions on a locale L determines the completely regular Lindelöf reflection lL of L up to isomorphism. Mathematics Subject Classifications (2000) 54C30, 46E25.Project supported by NSF of China (Grant No. 10271056 and Grant No. 10331011).     

A note on weakly Lindelöf frames

The notion of σ^?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.     

The Lindelöf Number of Functional Spaces on Monolithic Compacta

Let X be a compactum, τ be an infinite cardinal, and t(X) ≤ τ. In this case, l(C_p(X)) ≤ 2^τ. If X is τ-monolitliic, then l(C_p(X)) ≤ τ⁺. In addition, if X is zero-dimensional and there are no τ⁺-Aronszajn trees, then l(C_p(X)) ≤ τ.     

A Short Note on the Lindelöf Hypothesis

We obtain a quantitative estimate for the measure of the set of large values of the zeta-function on the critical line which is related to the Lindelöf hypothesis and give its application to extreme values (-results). We also investigate the problem of small values.     

A generalization of the Lindelöf theorem

We present a generalization of the Lindelöf theorem on conditions that should be imposed on the coefficients of the Taylor series of an entire transcendental function ? in order that the relation $ln M_f (r) - \tau r^\rho , r \to \infty , M_f (r) = \max \left\{ {\left| {f(r)} \right|:|z| = r} \right\}$ , be satisfied.     

Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations

Let(X,p,μ)d,θ be a space of homogeneous type,(?) ∈(0,θ],|s|＜(?) andmax{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞.The author introduces the new Triebel-Lizorkin spaces (?)_∞q~s(X) and establishes the framecharacterizations of these spaces by first establishing a Plancherel-P(?)lya-type inequalityrelated to the norm of the spaces (?)_∞q~s(X).The frame characterizations of the Besovspace (?)_pq~s(X) with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p≤∞ and 0＜q≤∞and the Triebel-Lizorkin space (?)_pq~s(X)with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p＜∞ and max{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞ are also presented.Moreover,the au-thor introduces the new TriebeI-Lizorkin spaces b(?)_∞q~s(X) and H(?)_∞q~s(X) associated to agiven para-accretive function b.The relation between the space b(?)_∞q~s(X) and the spaceH(?)_∞q~s(X) is also presented.The author further proves that if s＝0 and q＝2,thenH(?)_∞q~s(X)＝(?)_∞q~s(X),which also gives a new characterization of the space BMO(X),since (?)_∞q~s(X)＝BMO(X).     

Some remarks on compact constant mean curvature hypersurfaces in a halfspace of ℍ n+1

We consider embedded compact hypersurfacesM in a halfspace of hyperbolic space with boundaryM in the boundary geodesic hyperplaneP of the halfspace and with non-zero constant mean curvature. We prove the following. Let {M _n } be a sequence of such hypersurfaces withM _n contained in a disk of radiusr _n centered at a point P such thatr _n 0 and that eachM _n is a large. H-hypersurface,H > 1. Then there exists a subsequence of {M _n } converging to the sphere of mean curvatureH tangent toP at. In the case of smallH-hypersurfaces orH 1, if we add a condition on the curvature of the boundary, there exists a subsequence of {M _n } which are graphs. The convergence is smooth on compact subset of ³ .     

Remarks on discretely absolutely star-Lindelöf spaces

In this paper, we prove the following statements(1) There exists a Hausdorff Lindelöf space X such that the Alexandroff duplicate A(X) of X is not discretely absolutely star-Lindelöf.(2) If X is a regular Lindelöf space, then A(X) is discretely absolutely star-Lindelöf.(3) If X is a normal discretely star-Lindelöf space with e(X) < ω ₁, then A(X) is discretely absolutely star-Lindelöf.     

Some remarks on indestructibility and Hamkins’ lottery preparation

In this paper, we first prove several general theorems about strongness, supercompactness, and indestructibility, along the way giving some new applications of Hamkins lottery preparation forcing to indestructibility. We then show that it is consistent, relative to the existence of cardinals < so that is supercompact and is inaccessible, for the least strongly compact cardinal to be the least strong cardinal and to have its strongness, but not its strong compactness, indestructible under -strategically closed forcing. Mathematics Subject Classification (2000):03E35, 03E55     

Ernst Lindelöf in memoriam

Ohne Zusammenfassung     

Some G-M-type Banach spaces and K-groups of operator algebras on them

By providing several new varieties of G-M-type Banach spaces according todecomposable and compoundable properties, this paper discusses the operator structuresof these spaces and the K-theory of the algebra of the operators on these G-M-type Ba-nach spaces through calculation of the K-groups of the operator ideals contained in theclass of Riesz operators.