Related papers: Comparing angles in Euclid's Elements
In this paper, the notion of a uniformly distributed systems of elements on the variety of metabelian Lie algebras is introduced. This notion is analogous to one of a measure preserving systems of elements on group varieties. As the main…
Mixed metaphors have been neglected in recent metaphor research. This paper suggests that such neglect is short-sighted. Though mixing is a more complex phenomenon than straight metaphors, the same kinds of reasoning and knowledge…
A brief personal perspective is given of issues, questions, formulations, methods, some answers and selected extensions posed by the spin glass problem, showing how considerations of an apparently insignificant and practically unimportant…
To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of trees have been the subject of much…
A large-scale smoothed-out model of the universe ignores small-scale inhomogeneities, but the averaged effects of those inhomogeneities may alter both observational and dynamical relations at the larger scale. This article discusses these…
We give precise details to support that observations of gravitational lensing at scales of individual, groups and clusters of galaxies can be understood in terms of non-Newtonian gravitational interactions with a relativistic structure…
When compared with pure mathematicians, applied ones have a clear preference for proofs that go beyond a chain of reasonings and do exhibit the fact to be proved. Here we exhibit the bijection between the 60 icosahedron rotations of the…
Recently, we have endowed various categories of groups with topologies. The purpose of this paper is to introduce on these categories others topologies which are statistically more suitable to study well-known problems in groups theory. We…
Extracting cosmological information from the Euclid galaxy survey will require modelling numerous systematic effects during the inference process. This implies varying a large number of nuisance parameters, which have to be marginalised…
In a real expert system, one may have unreliable, unconfident, conflicting estimates of the value for a particular parameter. It is important for decision making that the information present in this aggregate somehow find its way into use.…
As an example of empirical metamathematics, we present a detailed study of the dependency structure of the 465 theorems in Euclid's Elements, finding empirical signatures of concepts such as the power of a theorem. We apply similar methods…
We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++…
Translated from the Latin original, "Observationes generales circa series, quarum termini secundum sinus vel cosinus angulorum multiplorum progrediuntur" (1777). E655 in the Enestrom index. Euler looks at the binomial expansion $(1+x)^n$…
We propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems…
The article introduces the concept of uniformity, which is formulated as a scheme of axioms. The connection of this concept with ordered sets is studied. The effectiveness of using axiom schemes as a convenient and short way of replacing…
Two of the pillars of combinatorics are the notion of choosing an arbitrary subset of a set with $n$ elements (which can be done in $2^n$ ways), and the notion of choosing a $k$-element subset of a set with $n$ elements (which can be done…
It is shown that Euler's theorem for graphs can be generalized for 2-complexes. Two notions that generalize cycle and Eulerian tour are introduced (``circlet'' and ``Eulerian cover''), and we show that for a strongly-connected, pure…
The cosmological principle posits that the universe does not exhibit any specific preference for position or direction. However, it remains unclear whether the universe has a distinct preference for parity: whether certain properties are…
Reading a visualization is like reading a paragraph. Each sentence is a comparison: the mean of these is higher than those; this difference is smaller than that. What determines which comparisons are made first? The viewer's goals and…
We address the issue of angular measure, which is a contested issue for the International System of Units (SI). We provide a mathematically rigorous and axiomatic presentation of angular measure that leads to the traditional way of…