Related papers: On Approximate Classification of Theories

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable…

We consider heterogeneous coupling problems on an abstract level, establishing fundamental principles of domain decomposition agnostic to the solvers of the local subproblems. Introducing a coupling framework reminiscent of FETI methods,…

In the context of continuous first-order logic, special attention is often given to theories that are somehow continuous in an 'essential' way. A common feature of such theories is that they do not interpret any infinite discrete…

Modern data science applications demand solving large-scale optimization problems. The prevalent approaches are first-order methods, valued for their scalability. These methods are implemented to tackle highly irregular problems where…

The classes stable, simple and NSOP$_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one…

Part I: We would like to generalize imaginary elements, weight of ${\rm ortp}(a,M,N),{\mathbf P}$-weight, ${\mathbf P}$-simple types, etc. from [Sh:c, Ch.III,V,\S4] to the context of good frames. This requires allowing the vocabulary to…

This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed.…

For given non-consistent initial conditions, we study the stability of a class of generalised linear systems of difference equations with constant coefficients and taking into account that the leading coefficient can be a singular matrix.…

This survey is intended as an invitation to the theory of stable $\infty$-categories, addressed primarily to mathematicians working in the representation theory of algebras and related subjects.

Rough set theory is a new mathematical approach to imperfect knowledge. The notion of rough sets is generalized by using an arbitrary binary relation on attribute values in information systems, instead of the trivial equality relation. The…

We consider the fundamental question of learnability of a hypotheses class in the supervised learning setting and in the general learning setting introduced by Vladimir Vapnik. We survey classic results characterizing learnability in term…

We propose a construction of a stable category for any pretorsion theory in a lextensive category. We prove the universal property of the stable category, that extends previous results obtained for the stable category of internal preorders…

Ranking entities such as algorithms, devices, methods, or models based on their performances, while accounting for application-specific preferences, is a challenge. To address this challenge, we establish the foundations of a universal…

We study resource allocation in two-sided markets from a fundamental perspective and introduce a general modeling and algorithmic framework to effectively incorporate the complex and multidimensional aspects of fairness. Our main technical…

We develop a general theory of local stability up to belonging to an ideal (e.g. having measure zero). From a model-theoretic perspective, we prove a stationarity principle for almost stable formulas in this sense, and build a topological…

This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling -- a central task in statistical…

We present a soundness theorem for a dependent type theory with context constants with respect to an indexed category of (finite, abstract) simplical complexes. The point of interest for computer science is that this category can be seen to…

We will investigate proof-theoretic and linguistic aspects of first-order linear logic. We will show that adding partial order constraints in such a way that each sequent defines a unique linear order on the antecedent formulas of a sequent…

Consider the stable matching problem on two sets. We introduce the concept of a preference cycle and show how its natural presence in stable matchings proves a series of classical results in an elementary way.

This paper is about the surprising interaction of a foundational result from model theory, about stability of theories, with algorithmic stability in learning. First, in response to gaps in existing learning models, we introduce a new…