Searcharxiv⌕ Search

arXiv subjects

Mark B. Villarino

Publications and source records attributed to Mark B. Villarino.

At least 19 recordsLinked to original sources
arXiv2026

The Error in Rayleigh's Approximative Period

We obtain rigorous a priori upper and lower bounds to the exact period of the celebrated Rayleigh stretched string differential equation. We use them to show that Rayleigh's approximative period overestimates the true period and that the relative error is, to a first approximation, directly proportional to the initial fractional displacement and inversely proportional to the initial stretch. Thus, for a given length and stretch, one can determine the initial displacement so as to guarantee a prescribed accuracy in Rayleigh's period while for a given displacement one can see why the relative error blows up of the initial stretch is tiny. We have replaced the big-O terms with explicit inequalities and a new elegant formula for the relative error.

math.CA↗
arXiv2025

Huygens and $π$

The Dutch scientist Christiaan Huygens refined Archimedes' celebrated geometrical computation of $π$ to its highest point. Yet the rich content of his beautiful treatise \emph{De circuli magnitudine inventa} (1654) has apparently never been presented in modern form. Here we offer a detailed and contemporary development of several of his most striking results. We also make a historical conjecture concerning Archimedes' trisection figure.

math.HO↗
arXiv2024

Archimedes' Revenge

We offer an instructive solution to the problem of computing the volume of the orthogonal intersection of three hyperboloids.

math.HO↗
arXiv2022

Legendre-Teege Reciprocity

Legendre published the first attempted proof of the law of Quadratic Reciprocity. But, in its final form (1797), it had a gap. Some 125 years later Herman Teege published the first rigorous proof of the unproven hypothesis which formed that gap. Then, 48 years later Kenneth Rogers published a second (but implicit) proof. These proofs lifted Legendre's attempt to the list of complete proofs. No detailed exposition of these proofs appears in the literature. Our paper fills that gap.

math.HO↗
arXiv2022

A Quadratic Harmonic Approximation

We obtain the quadratic term in Euler's asymptotic expansion of the nth harmonic number by a simple modification of Young's elementary determination of the linear term.

math.CA↗
arXiv2020

Legendre's Singular Modulus

We elaborate Legendre's original proof of his discovery of the first singular modulus in the history of mathematics, as well as its appearance in Ramanujan's formula for the arc length of an ellipse with said singular modulus as eccentricity, and its appearance in the three-body choreography of Bernoulli's lemniscate.

math.HO↗
arXiv2017

Algebraic Addition Theorems

We present a self-contained development of the Weierstrass theory of those analytic functions (single-valued or multiform) which admit an algebraic addition theorem. We review the history of the theory and present detailed proofs of the major theorems.

math.CA↗
arXiv2017

Hilbert's Proof of His Irreducibility Theorem

Hilbert's Irreducibility Theorem is a cornerstone that joins areas of analysis and number theory. Both the genesis and genius of its proof involved combining real analysis and combinatorics. We try to expose the motivations that led Hilbert to this synthesis. Hilbert's famous Cube Lemma supplied fuel for the proof but without the analytical foundation and framework it would have been heating empty air. The lemma is said to presage Ramsey Theory but we note differences in motivation.

math.HO↗
arXiv2015

The error in an alternating series

We give a new proof of Johnsonbaugh's refined error estimates of an alternating series based on an idea of R. M. Young. We also give a new proof of the error estimate and convergence of the Euler transform.

math.CA↗
arXiv2015

A Tripos Surd

We analyze the accuracy and the origin of a fourth root approximation that appeared in the 1886 Mathematical Tripos.

math.CA↗
arXiv2014

The AGM Simple Pendulum

We present a self-contained development of Gauss' Arithmetic-Geometric Mean (AGM) and the work of A.E. Ingham who obtained rigorous error bounds for the AGM approximations to the period of a simple pendulum

math.CA↗
arXiv2013

A Cubic Surface of Revolution

We develop a direct and elementary (calculus-free) exposition of the famous cubic surface of revolution x^3+y^3+z^3-3xyz=1.12 pages. We have added a second elementary proof that the surface is of revolution.

math.HO↗
arXiv2012

Rayleigh's Stretched String

We obtain rigorous \emph{a priori} upper and lower bounds to the exact period of the celebrated Rayleigh stretched string differential equation

math.CA↗