Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1611.01502

The algebraic structure of quantity calculus

Alvaro P. Raposo
View original source ↗Read source document ↗Export metadata ↓

Abstract

An algebraic structure underlying the quantity calculus is proposed consisting in an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them bound by algebraic restrictions. Subspaces, tensor product and quotient spaces are considered as well as homomorphisms to end with a classification theorem of these structures. The new structure provides an axiomatic foundation for quantity calculus and gives complete justification within its framework of the way that quantity calculus is actually performed. It is hoped that this exposition helps to clarify the role of the interviening concepts of quantity, quantity value, quantity dimension and their relation with a system of units, particularly, the SI.

Explore related subjects

math.GM

Keep this discovery

BibTeXRIS

Alvaro P. Raposo. 2016-10-21. The algebraic structure of quantity calculus. https://arxiv.org/abs/1611.01502

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗