Related papers: Canonical Cauchy sequences for real numbers
A practical version of the polynomial canonical formalism is developed for normal mesoscopic systems consisting of N independent electrons. Drastic simplification of calculations is attained by means of proper ordering excited states of the…
Reordering the terms of a series is a useful mathematical device, and much is known about when it can be done without affecting the convergence or the sum of the series. For example, if a series of real numbers absolutely converges, we can…
We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally in regular 2n-gons). Each infinite trajectory gives a cutting sequence corresponding to the sequence of sides…
The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the…
We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the…
We prescribe a choice of 18 variables in all that casts the equations of the fully nonlinear characteristic formulation of general relativity in first--order quasi-linear canonical form. At the analytical level, a formulation of this type…
There are two incompatible Coq libraries that have a theory of the real numbers; the Coq standard library gives an axiomatic treatment of classical real numbers, while the CoRN library from Nijmegen defines constructively valid real…
The canonical dimension is an invariant attached to admissible representations of p-adic reductive groups, which has only received significant attention in the case of mod-p representations. In the case of complex representations, the…
We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the…
The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's…
Formal languages are sets of strings of symbols described by a set of rules specific to them. In this note, we discuss a certain class of formal languages, called regular languages, and put forward some elementary results. The properties of…
In this work, we illustrate and explore the use of Taylor series as solutions of differential equations. For a large a number of classes of differential equations in the literature, there are plenty of sources where the well known Taylor…
We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…
We give a canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over $\mathbb C$ by taking ultraproducts of absolute integral closures in positive characteristic. This yields a new tight closure…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it…
We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special…
This paper enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: {\sf All $x$ are $y$} and {\sf Some $x$ are…
In this paper fields of quantum reference frames based on gauge transformations of rational string states are described in a way that, hopefully, makes them more understandable than their description in an earlier paper. The approach taken…
The metrical theory of the product of consecutive partial quotients is associated with the uniform Diophantine approximation, specifically to the improvements to Dirichlet's theorem. Achieving some variant forms of metrical theory in…