Related papers: NIP formulas and Baire 1 definability

We answer in the affirmative a conjecture of Berarducci, Peterzil and Pillay \cite{BPP10} for solvable groups, which is an o-minimal version of a particular case of Milnor's isomorphism conjecture \cite{jM83}. We prove that every abstract…

Many computational problems can be modelled as the class of all finite structures $\mathbb A$ that satisfy a fixed first-order sentence $\phi$ hereditarily, i.e., we require that every (induced) substructure of $\mathbb A$ satisfies $\phi$.…

We show that every fsg group externally definable in an NIP structure is definably isomorphic to a group interpretable in it. Our proof relies on honest definitions and a group chunk result reconstructing a hyper-definable group from its…

In this short note we show that if we add predicate for a dense complete indiscernible sequence in a dependent theory then the result is still dependent. This answers a question of Baldwin and Benedikt and implies that every unstable…

We propose a definition of computable manifold by introducing computability as a structure that we impose to a given topological manifold, just in the same way as differentiability or piecewise linearity are defined for smooth and PL…

We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely $n$-complicial sets, which are a proposed model for $(\infty,n)$-categories, based on previous work of Verity and…

A unified approach to the construction of weighing matrices and certain symmetric designs is presented. Assuming the weight $p$ in a weighing matrix $W(n,p)$ is a prime power, it is shown that there is a…

We prove the following two results. 1. If $X$ is a completely regular space such that for every topological space $Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the first Baire class, then every Lindel\"of subspace…

A description is an entity that can be interpreted as true or false of an object, and using feature structures as descriptions accrues several computational benefits. In this paper, I create an explicit interpretation of a typed feature…

We propose the notion of countable decomposability of maps on C*-algebras: a bounded linear map $\varphi : \mathscr{A}\to B(\mathcal{H})$, where $\mathscr{A}$ is a C*-algebra and $\mathcal{H}$ a Hilbert space, will be called countably…

We introduce trace definability, a weak notion of interpretability, and trace equivalence, a weak notion of equivalence for first order structures and theories. In particular we get an interesting weak equivalence notion for $\mathrm{NIP}$…

We study invariant types in NIP theories. Amongst other things: we prove a definable version of the (p,q)-theorem in theories of small or medium directionality; we construct a canonical retraction from the space of M-invariant types to that…

A map $\phi:M_m(\bC)\to M_n(\bC)$ is decomposable if it is of the form $\phi=\phi_1+\phi_2$ where $\phi_1$ is a CP map while $\phi_2$ is a co-CP map. It is known that if $m=n=2$ then every positive map is decomposable. Given an extremal…

Given a field $K$, a rational function $\phi \in K(x)$, and a point $b \in \mathbb{P}^1(K)$, we study the extension $K(\phi^{-\infty}(b))$ generated by the union over $n$ of all solutions to $\phi^n(x) = b$, where $\phi^n$ is the $n$th…

We prove a model theoretic Baire category theorem for $\tilde\tau_{low}^f$-sets in a countable simple theory in which the extension property is first-order and show some of its applications. We also prove a trichotomy for minimal types in…

Let $\alpha:[0,1]\to [0,1]$ be a measurable function. It was proved by P. Marchal \cite{Mar15} that the function $$ \phi^{(\alpha)}(\lambda):=\exp\left[ \int_0^1\frac{\lambda-1}{1+(\lambda-1)x}\,\alpha(x)\,d x \right],\quad \lambda>0 $$ is…

We define constructive truth for arithmetic and for intuitionistic analysis, and investigate its properties. We also prove that the set of constructively true (first order) arithmetical statements is Pi-1-2 and Sigma-1-2 hard, and we…

If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. This in particular applies to…

Let K be an algebraically bounded structure and T be its theory. If T is model complete, then the theory of K endowed with a derivation, denoted by $T^{\delta}$, has a model completion. Additionally, we prove that if the theory T is…

We investigate (2,1):1 structures, which consist of a countable set $A$ together with a function $f: A \to A$ such that for every element $x$ in $A$, $f$ maps either exactly one element or exactly two elements of $A$ to $x$. These…