Related papers: The real numbers - a survey of constructions
Rewriting is a formalism widely used in computer science and mathematical logic. The classical formalism has been extended, in the context of functional languages, with an order over the rules and, in the context of rewrite based languages,…
We prove astonishing identities generated by compositions of positive integers. In passing, we obtain two new identities for Stirling numbers of the first kind. In the two last sections we clarify an algebraic sense of these identities and…
As suggested by the title, this paper is a survey of recent results and questions on the collection of computably enumerable sets under inclusion. This is not a broad survey but one focused on the author's and a few others' current…
A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by…
The proposed system of integer functions is logically fully independent from the traditional mathematical analysis of the real functions, but there is a well-defined mutual correspondence between the two disciplines. The system of integer…
Recent work has formulated the task for computational construction grammar as producing a constructicon given a corpus of usage. Previous work has evaluated these unsupervised grammars using both internal metrics (for example, Minimum…
Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and…
Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to…
We show how to represent an interval of real numbers in an abstract numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers using the Dyck language. We also show…
We examine the representation of numbers as the sum of two squares in $\mathbb{Z}_n$ for a general positive integer $n$. Using this information we make some comments about the density of positive integers which can be represented as the sum…
Large sets of equiangular lines are constructed from sets of mutually unbiased bases, over both the complex and the real numbers.
Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first…
We consider the problem of counting the number of answers to a first-order formula on a finite structure. We present and study an extension of first-order logic in which algorithms for this counting problem can be naturally and conveniently…
We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.
This is a survey article devoted to the study of real structures on complex algebraic varieties endowed with a reductive group action.
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.
In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
The crossing number of a graph $G$ is the least number of crossings over all possible drawings of $G$. We present a structural characterization of graphs with crossing number one.
We address the problem of counting the number of strings in a collection where a given pattern appears, which has applications in information retrieval and data mining. Existing solutions are in a theoretical stage. We implement these…