A Generalized Logarithm for Exponential-Linear Equations
An introduction to the glog function. That is pronounced "gee-log," with a soft "g" and not the way you may have thought. Glog is the inverse of ex/x. Otherwise expressed, y = glog(x) if and only if ey = xy. The glog function has some properties and some uses. The properties are not as nice as those of the logarithm, nor are the uses as widespread, but they are nevertheless interesting.
The Case of the Missing Lottery Number
W. D. Kaigh
When the state of Arizona got into the numbers game in 1998, it drew thirty-two consecutive winning three-digit numbers in which the digit 9 did not appear. Such things did not occur when organized crime ran the rackets. This paper explains why someone should have noticed.
Surface Approximation and Interpolation via Matrix SVD
Clifford Long and Andrew Long
If forced to, most readers of the Journal could probably smooth out a curve: moving averages, least squares, splines, peeking in a numerical analysis book - a number of methods suggest themselves. But how to smooth out a surface? There you might come up empty and never think of using the singular value decomposition of a matrix.
Punxsutawney's Phenomenal Phorecaster
Michael A. Aaron, Brewster B. Boyd, Jr., Melanie J. Curtis, and Paul M. Sommers
Punxsutawney Phil is a semi-legendary groundhog who receives wide media attention every February 2 as an indicator of whether spring will be early or late. How does he (or she) do? The paper contains surprising results. For example, Phil's accuracy increases by 33% when the President is a Democrat.
How you round numbers off matters. For example, three different reasonable methods of rounding give three different values for a variance: 2.88, 0.51, and -21. Be careful - negative variances can get you into trouble.
Speeding Up a Numerical Algorithm
The algorithm is one for calculating the value of an integral, and it may be speeded up by replacing it with another that requires twice the number of computations. The paradox is only apparent, and the moral is that thought at the beginning of a problem may save time in the end.
Against the Odds
A short story.
Tension in Generalized Geometric Sequences
In a geometric sequence, each term is the same multiple of the preceding term. In a generalized geometric sequence the multiple varies. If the multiple approaches 1, what does the sequence do? It turns out that it can do just about anything it likes.
Fallacies, Flaws, and Flimflam
FFFs #167-173, including a proof that if A is idempotent, so is A + I.
Cable-laying and Intuition
Yael Roitberg and Joseph Roitberg
The problem about laying a cable from A to B, partly under a river and partly on shore or one of its variants (swimming and walking, etc.) appears in every calculus book. When is it best to go perpendicularly across the river? You may be surprised by the answer.
Taylor's Formula via Determinants
Taylor's formula is a direct consequence of Rolle's theorem if you are clever enough to see it.
The Cantor Set Contains 1/4? Really?
Michael Green and sarah-marie belcastro
Did you know that 1/4 is in the Cantor set? Be honest, now. After being told that it is, can you show it? Probably, but how about without using ternary expansions?
When do Approximating Polynomials Cross Graphs of Approximated Functions?
Approximate the square root curve at (4, 2) with a parabola that has the same tangent line and the same curvature as the square root at (4, 2) and the approximation crosses the curve at (4, 2). Intuition fails again!
Barrow's Fundamental Theorem
Isaac Barrow had, before Newton and Leibniz, the geometric content of the Fundamental Theorem of Calculus
Problems and Solutions