Amalgamations of classes of Banach spaces with a monotone basis

It was proved by Argyros and Dodos that, for many classes $C$ of separable Banach spaces which share some property $P$, there exists an isomorphically universal space that satisfies $P$ as well. We introduce a variant of their amalgamation technique which provides an isometrically universal space in the case that $C$ consists of spaces with a monotone Schauder basis. For example, we prove that if $C$ is a set of separable Banach spaces which is analytic with respect to the Effros-Borel structure and every $X \in C$ is reflexive and has a monotone Schauder basis, then there exists a separable reflexive Banach space that is isometrically universal for $C$

Complexity of distances: Theory of generalized analytic equivalence relations

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov-Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that $E_1$ is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben-Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov-Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided.Comment: Based on the feedback we received, we decided to split the original version into two parts. The new version is now the first part of this spli

BIOREFINERIES AND BIOBASED PRODUCTS FROM THE CONSUMER'S POINT OF VIEW

Paper prepared for presentation at the 13th ICABR International Conference on Agricultural Biotechnology: “The emerging bio-economy” Ravello (Italy), 18th to 20th June 2009Industrial biotechnology, Biorefinery, Consumer behaviour, Demand and Price Analysis, M39, R20,

Large separated sets of unit vectors in Banach spaces of continuous functions

The paper concerns the problem whether a nonseparable \C(K) space must contain a set of unit vectors whose cardinality equals to the density of \C(K) such that the distances between every two distinct vectors are always greater than one. We prove that this is the case if the density is at most continuum and we prove that for several classes of \C(K) spaces (of arbitrary density) it is even possible to find such a set which is $2$-equilateral; that is, the distance between every two distinct vectors is exactly 2.Comment: The second version does not contain new results, but it is reorganized in order to distinguish our main contributions from what was essentially know

Optimal quality of exceptional points for the Lebesgue density theorem

In spite of the Lebesgue density theorem, there is a positive $\delta$ such that, for every non-trivial measurable set $S$ of real numbers, there is a point at which both the lower densities of $S$ and of the complement of $S$ are at least $\delta$. The problem of determining the supremum of possible values of this $\delta$ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work.Comment: 45 page

Decidability and Universality in Symbolic Dynamical Systems

Many different definitions of computational universality for various types of dynamical systems have flourished since Turing's work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as undecidability of a model-checking problem. For Turing machines, counter machines and tag systems, our definition coincides with the classical one. It yields, however, a new definition for cellular automata and subshifts. Our definition is robust with respect to initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for undecidability and universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have infinite number of subsystems. We also discuss the thesis according to which computation should occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2: minor orthographic changes v3: section 5.2 (collatz functions) mathematically improved v4: orthographic corrections, one reference added v5:27 pages. Important modifications. The formalism is strengthened: temporal logic replaced by finite automata. New results. Submitte