On injective or dense-range operators leaving a given chain of subspaces invariant

In this paper we prove the existence of dense-range or one-to-one compact operators on a separable Banach space leaving a given finite chain of subspaces invariant. We use this result to prove that a semigroup of bounded operators is reducible if and only if there exists an appropriate one-to-one compact operator such that the collection of compact operators is reducible.

     

A brief remark on van der Waerden spaces

We demonstrate that Martin's axiom for -centered notions of forcing implies the existence of a van der Waerden space that is not a Hindman space. Our proof is an adaptation of the one given by M. Kojman and S. Shelah that such a space exists if one assumes the continuum hypothesis to be true.

     

Topological Vortices in the Modified Two-Dimensional Gross—Pitaevskii Theory

It is shown that there exists an elementary vortex topological current in the modified two-dimensional Gross—Pitaevskii theory. An explicit expression for the vortex velocity field as a function of the order parameter is derived. The evolution of vortices is studied from the topological properties of the order parameter field. It is found that vortices generate or annihilate at the limit points and encounter, split, or merge at the bifurcation points of the order parameter field .     

De Rham and Infinitesimal Cohomology in Kapranov's Model for Non commutative Algebraic Geometry

The title refers to the nilcommutative or NC-schemes introduced by M. Kapranov in Noncommutative Geometry Based on Commutator Expansions, J. Reine Angew. Math 505 (1998) 73–118. The latter are noncommutative nilpotent thickenings of commutative schemes. We also consider the parallel theory of nil-Poisson or NP-schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for NC- and NP-schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck in Crystals and the de Rham Cohomology of Schemes, Dix exposés sur la cohomologie des schémas, Masson, Paris (1968), pp. 306–358. It turns out that each of these noncommutative variants admits a kind of Hodge decomposition which allows one to express the cohomology groups of a noncommutative scheme Y as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme X (Theorems 6.1, 6.4, 6.7). As a byproduct we obtain new proofs for classical results of Grothendieck (Corollary 6.2) and of Feigin and Tsygan (Corollary 6.8) on the relation between de Rham and infinitesimal cohomology and between the latter and periodic cyclic homology.     

Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

Condensations of projective sets onto compacta

For a coanalytic-complete or -complete subspace of a Polish space we prove that there exists a continuous bijection of onto the Hilbert cube . This extends results of Pytkeev. As an application of our main theorem we give an answer to some questions of Arkhangelskii and Christensen.

Under the assumption of Projective Determinacy we also give some generalizations of these results to higher projective classes.

     

On non-measurability of in its second dual

We show that with the weak topology is not an intersection of Borel sets in its Cech-Stone extension (and hence in any compactification). Assuming (CH), this implies that has no continuous injection onto a Borel set in a compact space, or onto a Lindelöf space. Under (CH), this answers a question of Arhangel'ski.

     

Production systems with interruptions, arbitrary topology and finite buffers

We consider a production system with finite buffers and arbitrary topology where service time is subject to interruptions in one of three ways, viz. machine breakdown, machine vacations or Npolicy. We develop a unified approximation (analytical) methodology to calculate the throughput of the system using queueing networks together with decomposition, isolation and expansion techniques. The methodology is rigorously tested covering a large experimental region. Orthogonal arrays are used to design the experiments in order to keep the number of experiments manageable. The results obtained using the approximation methodology are compared to the simulation results. The ttests carried out to investigate the differences between the two results show that they are statistically insignificant. Finally, we test the methodology by applying it to several arbitrary topology networks. The results show that the performance of the approximation methodology is consistent, robust and produces excellent results in a variety of experimental conditions.     

L-拓扑空间中的相对强F紧性与相对超F紧性

In this paper,two concepts of relative compactness-the relative strong fuzzy compactness and the relative ultra-fuzzy compactness are defined in L-topological spaces for an arbitrary L-set.Properties of relative strong fuzzy sets and relative ultra-fuzzy compact sets are studied in detail and some characteristic theorems are given.Some examples are illustrated.