Related papers: Higher Order Tsirelson Spaces and their Modified V…

We study the family of isomorphisms and strictly singular operators in mixed Tsirelson spaces and their modified versions setting. We show sequential minimality of modified mixed Tsirelson spaces $T_M[(\mc{S}_n,\theta_n)]$ satisfying some…

We extend existing results that characterize isometries on the Tsirelson-type spaces $T\big[\frac{1}{n}, \mathcal{S}_1\big]$ ($n\in \mathbb{N}, n\geq 2$) to the class $T[\theta, \mathcal{S}_{\alpha}]$ ($\theta \in \big(0, \frac{1}{2}\big]$,…

We prove that any normalized block sequence in a Schreier space $X_\xi$, of arbitrary order $\xi<\omega_1$, admits a subsequence equivalent to a subsequence of the canonical basis of some Schreier space. The analogous result is proved for…

We study the modified and boundedly modified mixed Tsirelson spaces $T_M[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ and $T_{M(s)}[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ respectively, defined by a subsequence $({\cal F}_{k_n})$ of the…

We give a complete classification of mixed Tsirelson spaces T[(F\_i, theta\_i)\_{i=1}^r ] for finitely many pairs of given compact and hereditary families F\_i of finite sets of integers and 0<theta\_i<1 in terms of the Cantor-Bendixson…

The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present…

Any regular mixed Tsirelson space $T(\theta_n,S_n)_{\N}$ for which $\frac{\theta_n}{\theta^n} \to 0$, where $\theta=\lim_n \theta_n^{1/n}$, is shown to be arbitrarily distortable. Certain asymptotic $\ell_1$ constants for those and other…

A coarse space $X$, endowed with a linear order compatible with the coarse structure of $X$, is called linearly ordered. We prove that every linearly ordered coarse space $X$ is locally convex and the asymptotic dimension of $X$ is either…

Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of $c_{0}$ is a complete analytic set with respect to the Effros Borel structure…

We extend the existing results on surjective isometries of unit spheres in the Tsirelson space $T\left[\frac{1}{2}, S_1\right]$ to the class $T[\theta,S_{\alpha}]$ for any integer $\theta^{-1} \geq 2$ and $1 \leqslant \alpha < \omega_1$,…

The problem of characterizing normed ordered spaces which admit a representation in the algebraic, order and norm sense as a subspace of $C(X)$, the space of all continuous functions on a compact Hausdorff space is a classical problem that…

Tsirelson's space $T$ is known to be distortable but it is open as to whether or not $T$ is arbitrarily distortable. For $n\in {\Bbb N}$ the norm $\|\cdot\|_n$ of the Tsirelson space $T(S_n,2^{-n})$ is equivalent to the standard norm on…

It is shown that the isomorphism relation between continuous t-norms is Borel bireducible with the relation of order isomorphism between linear orders on the set of natural numbers, and therefore, it is a Borel complete equivalence…

We show that a topometric space $X$ is topometrically isomorphic to a type space of some continuous first-order theory if and only if $X$ is compact and has an open metric (i.e., satisfies that $\{p : d(p,U) < \varepsilon\}$ is open for…

For each countable ordinal $\alpha$ let $\mathcal{S}_{\alpha}$ be the Schreier set of order $\alpha$ and $X_{\mathcal{S}_\alpha}$ be the corresponding Schreier space of order $\alpha$. In this paper we prove several new properties of these…

This paper studies tree-automatic ordinals (or equivalently, well-founded linearly ordered sets) together with the ordinal addition operation +. Informally, these are ordinals such that their elements are coded by finite trees for which the…

We show that the irreducible components of any moduli space of semistable representations of a special biserial algebra are always isomorphic to products of projective spaces of various dimensions. This is done by showing that irreducible…

We find all possible isomorphisms and 3-birational maps (i.e., birational maps which induce an isomorphism between open subsets whose respective complements have codimension at least 3) between moduli spaces of parabolic vector bundles with…

We show that the Lascar group $\operatorname{Gal}_L(T)$ of a first order theory $T$ is naturally isomorphic to the fundamental group $\pi_1(|\operatorname{Mod}(T)|)$ of the classifying space of the category of models of $T$ and elementary…

We present a method for constructing bounded strictly singular non-compact operators on mixed Tsirelson spaces defined either by the families (A_n) or (S_n) of a certain class, as well as on spaces built on them, including hereditarily…