Using a Gradient Vector to Find Multiple Periodic Oscillations in Suspension Bridge Models
L.D. Humphreys and P.J. McKenna
This paper describes how the method of steepest descent can be used to find periodic solutions of differential equations. Applications to two suspension bridge models are discussed, and the method is used to find non-obvious large-amplitude solutions.
Error Estimates For Numerical Integration Rules
Peter R. Mercer
The starting point for this discussion of error estimates is the fact that integrals that arisein Fourier series have properties that can be used to get improved bounds. This idea is extended to more general situations.
Discovering Roots: Ancient, Medieval, and Serendipitous
A new algorithm for computing roots of any order is presented, along with the historical roots of its development.
Finding the Sums of Harmonic Series of Even Order
An Apothem Apparently Appears
Pat Cade and Russell A. Gordon
The standard calculus problem of determining the minimum combined area of an equilateral triangle and a square formed by cutting a piece of wire of given length is the starting point of this note. It is observed that the minimum occurs when their inscribed circles have the same radius. This is first extended to any pair of regular polygons, and then to more than two.
Irrational Roots of Integers
Ayshhyah Khazad and Allen J. Schwenk
The authors give a unified proof of the fact that the rth root of a natural number that is not the rth power of an integer is irrational.
A Quotient Rule Integration by Parts Formula
This note gives the quotient-rule analogue to the usual integration by parts formula.
Estimating Definite Integrals
Some new techniques are given for estimating integrals using comparisons.
A Painless Approach to Least Squares
Eric S. Key
The basic idea in this approach is that the least squares line for a set of points goes through the average point.
A Quick Proof that the Least Squares
Eric S. Key
The Cauchy-Schwarz inequality is the key ingredient to the proof of the result given here.
A Recursive Formula for Moments of a Binomial Distribution
Árpá Bényi and Saverio M. Manago
This note gives a method for finding higher moments for a binomial distribution than the standard first and second.