Related papers: A Groszek-Laver pair of undistinguishable $E_0$ cl…

Let $K$ be a local field with residue characteristic $p$ and let $L/K$ be a totally ramified extension of degree $p^k$. In this paper we show that if $L/K$ has only two distinct indices of inseparability then there exists a uniformizer…

We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued…

Scott considered the determinant of 1/(y-z)^2, with y,z running over two sets X,Y of size n, and determined its specialisation when Y and Z are the roots of y^n-a and z^n-b. We give the same specialisation for the determinant…

Let p be a prime number which is split in an imaginary quadratic field k. Let \mathfrak{p} be a place of k above p. Let k_\infty be the unique Z_p-extension of k which unramified outside of \mathfrak{p}, and let K_\intfy be a finite…

We construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p \in K$ for all $x\in F$).

We show that if a first-order structure $\mathcal{M}$, with universe $\mathbb{Z}$, is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,<,0)$, then $\mathcal{M}$ must be interdefinable with $(\mathbb{Z},+,0)$ or…

Let $\xi\geq 1$ be a countable ordinal. We study the Borel subsets of the plane that can be made ${\bf\Pi}^0_\xi$ by refining the Polish topology on the real line. These sets are called potentially ${\bf\Pi}^0_\xi$. We give a Hurewicz-like…

The Gruenberg--Kegel graph (or the prime graph) $\Gamma(G)$ of a finite group $G$ is defined as follows. The vertex set of $\Gamma(G)$ is the set of all prime divisors of the order of $G$. Two distinct primes $r$ and $s$ regarded as…

We prove that if $\bI$ is a p.\ o. set in a countable transitive model $\gM$ of $\ZFC$ then $\gM$ can be extended by a generic sequence of reals $\a_\i,$ $\i\in\bI,$ such that $\aleph_1^\gM$ is preserved and every $\a_\i$ is Sacks generic…

Suppose that $\Omega$ is a complex lattice that is closed under complex conjugation and that $I$ is a small real interval, and that $D$ is a disc in $ \mathbb{C}$. Then the restriction $\wp|_D$ is definable in the structure…

We observe \cite[Proposition 4.1]{LaLe} that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra $A_1$ considered as a Poisson version of the…

In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups,…

We describe a finite presentation of $ \mathcal{T}_{g,r} $ for $g \geq 3$. % or $(g,r)=(2,0)$. Here $\mathcal{T}_{g,r} $ is the universal central extension of the mapping class group of the surface of genus $g$ with $r$-boundaries. We also…

It is well known that the classic {\L}o\'s-Tarski preservation theorem fails in the finite: there are first-order definable classes of finite structures closed under extensions which are not definable (in the finite) in the existential…

A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. In the present paper we give an elementary…

We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices $\mathfrak{gl}^M\left(V,V_*\right)$. Here…

This paper concerns the expansion of the real ordered additive group by a predicate for a subset of $[0,1]$ whose base-$r$ representations are recognized by a B\"uchi automaton. In the case that this predicate is closed, a dichotomy is…

We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the second…

Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we give a complete classification of extensions of finite irreducible…

We show that finitely generated irreducible $\mathrm{II}_1$ subfactors are generic in the following sense. Given a separable $\mathrm{II}_1$ factor $M$ and an integer $n\geq 2$, equip the set of $n$-tuples of self-adjoint operators in $M$…