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Register a new userDoes weak quasi-o-minimality behave better than weak o-minimality?
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We present a relatively simple description of binary, definable subsets of models of weakly quasi-o-minimal theories. In particular, we closely describe definable linear orders and prove a weak version of the monotonicity theorem. We also prove that weak quasi-o-minimality of a theory with respect to one definable linear order implies weak quasi-o-minimality with respect to any other such order.
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It is shown how tools from the area of Model Theory, specifically from the Theory of o-minimality, can be used to prove that a class of functions is VC-subgraph (in the sense of Dudley, 1987), and therefore satisfies a uniform polynomial metric entropy bound. We give examples where the use of these methods significantly improves the existing metric entropy bounds. The methods proposed here can be applied to finite dimensional parametric families of functions without the need for the parameters to live in a compact set, as is sometimes required in theorems that produce similar entropy bounds (for instance Theorem 19.7 of van der Vaart, 1998).
It is well-known that any finite $Pi^{0}_{1}$-class of $2^{mathbb N}$ has a computable member. Then, how can we understand this in the context of reverse mathematics? In this note, we consider several very weak fragments of KH{o}nigs lemma to answer this qeustion.
In this paper, we study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis (e_k) is said to be subsequentially minimal if for every normalized block basis (x_k) of (e_k), there is a further block (y_k) of (x_k) such that (y_k) is equivalent to a subsequence of (e_k). Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal and connections with Bourgains ell^{1}-index are established. It is also shown that a large class of mixed Tsirelson spaces fails to be subsequentially minimal in a strong sense.
A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group $H_+(X)$ of order-preserving homeomorphisms of a compact linearly ordered connected space $X$. We provide a sufficient condition on $X$ under which the topological group $H_+(X)$ is minimal. This condition is satisfied, for example, by: the unit interval, the ordered square, the extended long line and the circle (endowed with its cyclic order). In fact, these groups are even $a$-minimal, meaning, in this setting, that the compact-open topology on $G$ is the smallest Hausdorff group topology on $G$. One of the key ideas is to verify that for such $X$ the Zariski and the Markov topologies on the group $H_+(X)$ coincide with the compact-open topology. The technique in this article is mainly based on a work of Gartside and Glyn.
We study containment and uniqueness problems concerning matrix convex sets. First, to what extent is a matrix convex set determined by its first level? Our results in this direction quantify the disparity between two product operations, namely the product of the smallest matrix convex sets over $K_i subseteq mathbb{C}^d$, and the smallest matrix convex set over the product of $K_i$. Second, if a matrix convex set is given as the matrix range of an operator tuple $T$, when is $T$ determined uniquely? We provide counterexamples to results in the literature, showing that a compact tuple meeting a minimality condition need not be determined uniquely, even if its matrix range is a particularly friendly set. Finally, our results may be used to improve dilation scales, such as the norm bound on the dilation of (non self-adjoint) contractions to commuting normal operators, both concretely and abstractly.
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