Related papers: The real numbers - a survey of constructions

This is a survey of results in the enumeration of lattice paths.

We survey some basic mathematical structures, which arguably are more primitive than the structures taught at school. These structures are orders, with or without composition, and (symmetric) monoidal categories. We list several `real life'…

We give a procedure for counting the number of different proofs of a formula in various sorts of propositional logic. This number is either an integer (that may be 0 if the formula is not provable) or infinite.

This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction…

We present a uniform description of sets of $m$ linear forms in $n$ variables over the field of rational numbers whose computation requires $m(n - 1)$ additions.

This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions. The von Neumann construction of the ordinal numbers includes a…

We introduce and study non-uniform expansions of real numbers, given by two non-integer bases.

We present a constructive description of minimal reductions with a given reduction number. This description has interesting consequences on the minimal reduction number, the big reduction number, and the core of an ideal. In particular, it…

Based on the structure of Fibonacci sequence, we give a new proof for the irrationality exponents of the Fibonacci real numbers. Moreover, we obtain all the irrationality exponents of the real numbers corresponding to the differences of…

Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications. While it is hopeless to expect much in general, we know a surprising amount about these…

We describe a formal correctness proof of RANKING, an online algorithm for online bipartite matching. An outcome of our formalisation is that it shows that there is a gap in all combinatorial proofs of the algorithm. Filling that gap…

We introduce the notion of almost realizability, an arithmetic generalization of realizability for integer sequences, which is the property of counting periodic points for some map. We characterize the intersection between the set of…

This work presents a formalization of analogy on numbers that relies on generalized means. It is motivated by recent advances in artificial intelligence and applications of machine learning, where the notion of analogy is used to infer…

Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.

Based on continued fractions with subtractions, we identify the set of real numbers with the set of infinite integer sequences with all terms but the first one greater or equal to two. Each such sequence produces in a canonical way a unique…

The main purpose of this paper is to prove that the positive real numbers can be decomposed into finitely many disjoint pieces which are also closed under addition and multiplication. As a byproduct of the argument we determine all the…

We study convex subsets of buildings, discuss some structural features and derive several characterizations of buildings.

Reconstructing a complete object from its parts is a fundamental problem in many scientific domains. The purpose of this article is to provide a systematic survey on this topic. The reassembly problem requires understanding the attributes…

Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…

Due to the increasing use of machine learning in practice it becomes more and more important to be able to explain the prediction and behavior of machine learning models. An instance of explanations are counterfactual explanations which…