The Mathematics of Go To Telescopes
This article presents the mathematics involved in finding and tracking celestial objects with an electronically controlled telescope. The essential idea in solving this problem is to choose several different coordinate systems that simplify the various motions of the earth and other celestial objects. These coordinate systems are then related by rotations and translations of the coordinate axes, which are accomplished using matrix operations studied in elementary linear algebra.
Not Just Hats Anymore: Binomial Inversion and the Problem of Multiple Coincidences
The well-known "hats" problem, in which a number of people enter a restaurant and check their hats, and then receive them back at random, is often used to illustrate the concept of derangements, that is, permutations with no fixed points. In this paper, the problem is extended to multiple items of clothing, and a general solution to the problem of multiple coincidences is obtained using the technique of binomial inversion. This solution also motivates a general formula for the expected number of people who return home with all of their own clothes.
Caps and Robbers: What Can You Expect?
Laura A. Zager and George C. Verghese
The "matching" hats problem is a classic exercise in probability: if n people throw their hats in a box, and then each person randomly draws one out again, what is the expected number of people who draw their own hat? This paper presents several extensions to this problem, with solutions that involve interesting tricks with iterated expectations, the law of total probability, and binomial and hypergeometric distributions.
Newton's Method and the Wada Property: A Graphical Approach
Michael Frame and Nial Neger
Imagine trying to paint a picture with three colors — say red, blue, and yellow — with a blue region between any red and yellow regions, a red region between any blue and yellow regions, and a yellow region between any red and blue regions, down to infinitely fine details. Regions arranged in this way satisfy what is called the Wada property. At first thought, it may not be so clear that any regions at all satisfy the Wada property. Borrowing a simple geometrical construction from a first course on dynamical systems, we show that the Wada property arises naturally in Newton's method as introduced in first-semester calculus.
Some Half-Row Sums from Pascal's Triangle via Laplace Transforms
Thomas P. Dence
This article presents some identities on the sum of the entries in the first half of a row in Pascal's triangle. The results were discovered while the author was working on a problem involving Laplace transforms, which are used in proving of the identities.
A Geometric View of Complex Trigonometric Functions
Given that the sine and cosine functions of a real variable can be interpreted as the coordinates of points on the unit circle, the author of this article asks whether there is something similar for complex variables, and shows that indeed there is.
Fallacies, Flaws, and Flimflam
Ed Barbeau, editor
Ricardo Alfaro and Steven Althoen, editors
Finding Curves with Computable Arc Length
This article derives an algorithm for finding a curve whose arc length can be evaluated exactly by using the arc length formula taught in elementary calculus.
Arch Length and Pythagorean Triples
This note shows that the existence of infinitely many Pythagorean triples with two consecutive members leads to the fact that infinitely many curves in a family commonly encountered in calculus classes have rational length.
On the Convergence of Some Modified p-Series
In this note, we consider the series obtained from the p-series by removing all those terms in which n has a digit in some specified set S. Somewhat surprisingly, the convergence of these series depends only on the size of S, not on its individual elements. Some interesting bounds on the sums of those series that converge are also obtained.
A New and Improved Method for Finding the Center of Gravity of a Quadrilateral
The standard method for finding the center of gravity of a quadrilateral is to use its diagonals to split it into two pairs of triangles and then find the intersection of the lines connecting the centroids of the pairs of triangles that share a common diagonal. In this note we present a much simpler approach, one that involves finding the centroid of just one triangle instead of four.