Related papers: Beginning of stability theory for Polish Spaces

We answer some questions from a paper of Krupi\'nski by giving suitable examples of small Polish structures. First, we present a class of small Polish group structures without generic elements. Next, we construct a first example of a small…

In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard $\kappa$-Borel spaces for $\kappa$ an uncountable (regular) cardinal satisfying $\kappa^{<\kappa}…

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical…

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from…

The Halpern-L\"auchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles…

We deal with stability theory for ``reasonable'' non-elementary classes without any remanents of compactness (like: above Hanf number or definable by L_{omega_1, omega}).

The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant…

We introduce a notion of $\mu$-structures which are certain locally compact group actions and prove some counterparts of results on Polish structures(introduced by Krupinski in \cite{Kru5}). Using the Haar measure of locally compact groups,…

For every countable structure $M$ we construct an $\aleph_0$-stable countable structure $N$ such that $Aut(M)$ and $Aut(N)$ are topologically isomorphic. This shows that it is impossible to detect any form of stability of a countable…

This paper deals with stability of a certain class of fractional order linear and nonlinear systems. The stability is investigated in the time domain and the frequency domain. The general stability conditions and several illustrative…

We indicate a way of distinguishing between structures, for which, two structures are said to be separable.Being separable implies being non-isomorphic. We show that for any first order theory $T$ in a countable language, if it has an…

We study stability patterns in the high dimensional rational homology of unordered configuration spaces of manifolds. Our results follow from a general approach to stability phenomena in the homology of Lie algebras, which may be of…

This article establishes several necessary and sufficient criteria on asymptotic stability and mean ergodicity in various types of topologies for Feller processes taking values in Polish spaces. In particular, asymptotic stability and mean…

We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, resp. pseudonorms, on the countable infinite-dimensional rational vector space. We provide an exhaustive…

We prove the Decomposability Conjecture for functions of Baire class $2$ on a Polish space to a separable metrizable space. This partially answer an important open problem in descriptive set theory.

This paper considers "definable cardinalities" arising from Polish group actions. The first part of the paper answers a question of Becker-Kechris by showing that under suitable determinacy assumptions in ZF+DC, every action by a Polish…

We give, for some Borel sets of a product of two Polish spaces, including the Borel sets with countable sections, a Hurewicz-like characterization of those which cannot become a transfinite difference of open sets by changing the two Polish…

We analyse logic actions of Polish groups which arise in continuous logic. We extend the generalised model theory of H.Becker to the case of Polish G-spaces when G is an arbitrary Polish group.

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.

The first results, both positive and negative, recently obtained in the area of constructing stationary spinning solitons in flat Minkowski space in 3+1 dimensions are discussed.