On order-polynomial completeness of lattices

In this note we prove that the cardinality of an infinite order-polynomially complete lattice (if such a lattice exists) must be greater than each , where is defined such that and for . This strengthens the result in [4]. Received November 10, 1995; accepted in final form April 6, 1998.     

On completeness of quadratic systems

A dynamical system is called complete if every solution of it exists for all t∈R. Let K be the dimension of the vector space of quadratic systems. The set of complete quadratic systems is shown to contain a vector subspace of dimension 2K/3. We provide two proofs, one by the Gronwall lemma and the second by compactification that is capable of showing incompleteness as well. Characterization of a vector subspace of complete quadratic systems is provided. The celebrated Lorenz system for all real ranges of its parameters is shown to belong to this subspace. We also provide a sufficient condition for a system to be incomplete.     

On completeness in quasi-metric spaces

A notion of Cauchy sequence in quasi-metric spaces is introduced and used to define a standard completion for a special class of spaces.     

On quantification of weak sequential completeness

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in L-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space X with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.     

On the completeness of normed spaces

Let $(G,+)$ be an abelian group and let $E$ be a normed space. A mapping $f:G\to E$ is called $\epsilon$-quadratic if for a given $\epsilon >0$ it satisfies $6f(x+y)+f(x?y)?2f\left(x\right)?2f\left(y\right)6\le \epsilon$ for all $x,y\in G$. In this work, we show that $E$ is complete if every $\epsilon$-quadratic mapping $f:X\to E$ can be estimated by a quadratic mapping, where $X$ is ${\mathbb{N}}_0$ or a finitely generated free abelian group.     

On equivalence of derived categories

Let A be an Abelian category and B be a thick subcategory of A. Let D ^b(B) denote the derived category of cohomologically bounded chain complexes of objects in A and D _B ^b (A) denote the derived category of cohomologically bounded chain complexes of objects in A with cohomology in B. We give two if and only if conditions for equivalence of D(B) and D _B ^b (A), and we give an example where D ^b (B) and D _B ^b (A) are not equivalent.     

On Bergman completeness and Bergman stability

It is proved that any bounded pseudoconvex domain in is complete w.r.t. the Bergman metric if its boundary can be described locally as the graph of a continuous function in suitable coordinates for . Further arguments are given concerning the stability problems of the Bergman kernel on non-smooth pseudoconvex domains. Received November 30, 1998 / Revised December 21, 1999 / Published online September 5, 2000     

On universal categories of coalgebras

A category ${\mathcal{K}}$ is called universal if for every accessible functor F : Set → Set the category of all F-coalgebras and the category of all F-algebras can be fully embedded into ${\mathcal{K}}$ . We prove that for a functor G preserving intersections, the category Coalg G of all G-coalgebras is universal unless the functor G is linear, that is, of the form GX = X × A + B for some fixed sets A and B. Other types of universality are also investigated.     

On the functional completeness of simple tournaments

The theory of multitraces provides a new proof that any simple tournament with more than two elements is functionally complete.