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After introducing the different boundary geometries of rank one symmetric spaces, we state and prove Fried's theorem in the general setting of all those geometries: a closed manifold with a similarity structure is either complete or the…

In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: of type (I), which hold at a neighborhood of infinity, and of type (II),…

We generalize Iskovskih's theorem about surfaces without irregularity and bigenus from the smooth case to regular surfaces over arbitrary fields, with special focus on the case of imperfect fields. This includes surfaces that are…

We introduce a variant of the $k$-nearest neighbor classifier in which $k$ is chosen adaptively for each query, rather than supplied as a parameter. The choice of $k$ depends on properties of each neighborhood, and therefore may…

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…

We work with semi-algebraic functions on arbitrary real closed fields. We generalize the notion of critical values and prove a Sard type theorem in our framework.

Functions with uniform level sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used, e.g., in multicriteria optimization, decision theory, mathematical…

We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a…

Let $U$ be a smooth and connected curve over an algebraically closed field of positive characteristic, with smooth compactification $X$. We generalize classical Geometric Class Field theory to provide a classification of fppf $G$-torsors…

We prove a triangulation theorem for semi-algebraic sets over a p-adically closed field, quite similar to its real counterpart. We derive from it several applications like the existence of flexible retractions and splitting for…

A complete first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition.…

We introduce a method to estimate the size of the domain of definition of the solutions of a meromorphic vector field on a neighborhood of its pole divisor. The corresponding techniques are, in a certain sense, quantitative versions of some…

Motivated by policy gradient methods in the context of reinforcement learning, we identify a large deviation rate function for the iterates generated by stochastic gradient descent for possibly non-convex objectives satisfying a…

We study mappings that satisfy the inverse modulus inequality of Poletsky type in a fixed domain. It is shown that, under some additional restrictions, the image of a ball under such mappings contains a fixed ball uniformly over the class.…

This paper studies "pro-excision" for the K-theory of one-dimensional (usually semi-local) rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite…

We study abelian varieties defined over function fields of curves in positive characteristic $p$, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the $L$-function of such an abelian variety…

In the famous paper of Deligne and Mumford, they proved that a proper hyperbolic curve over a discrete valuation field has stable reduction if and only if the Jacobian variety of the curve has stable reduction in the case where the residue…

To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this…

The paper proves the intermediate value theorem for polynomials and power series over a valued field with divisible valuation group and infinite residue field. Some further results on the behaviour of the valuation are obtained using…

We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally…