Related papers: Beyond Borel-amenability: scales and superamenable…
In this paper we consider some properties of a space B(X) of Borel functions on a set of reals X, with pointwise topology, that are stronger than separability.
In this paper we obtained the formula for the number of irreducible polynomials with degree $n$ over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al.(2003)…
We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets A of an abelian group.
We study quantum mechanical systems with a discrete spectrum. We show that the asymptotic series associated to certain paths of steepest-descent (Lefschetz thimbles) are Borel resummable to the full result. Using a geometrical approach…
Large Language Models (LLMs) have benefited enormously from scaling, yet these gains are bounded by five fundamental limitations: (1) hallucination, (2) context compression, (3) reasoning degradation, (4) retrieval fragility, and (5)…
We survey the recent investigations on (bounded, sequential) approximate amenability/contractibility and pseudo-amenability/contractibility for Banach algebras. We will discuss the core problems concerning these notions and address the…
We answer one of the main questions in generalized descriptive set theory, the Friedman-Hyttinen-Kulikov conjecture on the Borel reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel…
We study projective functions. We prove that projective functions generalise lower and upper-semianalytic ones while being stable by composition and difference. We show that the class of projective functions is closed under sums,…
Reasoning under uncertainty is a fundamental challenge in Artificial Intelligence. As with most of these challenges, there is a harsh dilemma between the expressive power of the language used, and the tractability of the computational…
While several classes of integer linear optimization problems are known to be solvable in polynomial time, far fewer tractability results exist for integer nonlinear optimization. In this work, we narrow this gap by identifying a broad…
We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…
Differential geometry may be generalized to allow infinitesimals to any order. The purpose of the present contribution is to show that the theory so developed expands received geometrical ideas in an interesting way, rich in potential for…
We consider the problem of translating between irreducible closed sets and implicational bases in closure systems. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph…
Recent years have seen rapid progress at the intersection between causality and machine learning. Motivated by scientific applications involving high-dimensional data, in particular in biomedicine, we propose a deep neural architecture for…
Interpretability research on large language models (LLMs) has yielded important insights into model behaviour, yet recurring pitfalls persist: findings that do not generalise, and causal interpretations that outrun the evidence. Our…
Systems of decision rules and decision trees are widely used as a means for knowledge representation, as classifiers, and as algorithms. They are among the most interpretable models for classifying and representing knowledge. The study of…
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…
This paper proves a representation theorem regarding sequences of random elements that take values in a Borel space and are measurable with respect to the sigma algebra generated by an arbitrary union of sigma algebras. This, together with…
We introduce and study several notions of computability-theoretic reducibility between subsets of $\omega$ that are "robust" in the sense that if only partial information is available about the oracle, then partial information can be…
We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…