The cover illustrates a neat result that Anna B. discovered and explains in this issue’s Anna’s Math Journal. She continued her investigation of paraboloids and discovered that orthogonal projection from a paraboloid coincides with the composition of stereographic projection and a special map M inspired by the optical properties of a paraboloid. For details, check out her column!
We also feature an interview with University of Minnesota assistant professor of mathematics Christine Berkesch Zamaere.
Next, Akamai Technologies computer scientist Kate Jenkins concludes her discussion of algorithms that find the “maximal subsequence” of a sequence. Were you able to figure out an algorithm that determines the maximum subsequence of N numbers using O(N) computations? Kate’s article is just one example of how mathematics applies to problems in industry. In the past decades, so much information has been digitized, including books, pictures, video, weather, architectural plans, music, etc. Where there are numbers, there is the potential for mathematical analysis.
Emily and Jasmine return, this time designing star patterns for different numbers of states. We received positive feedback about their last project where they designed a stained glass window (see Volume 7, Number 4), so we plan to feature them more in the future. The two show how, with a bit of inquisitiveness, there’s mathematics.
We conclude with solutions to this summer’s batch of Summer Fun problem sets. Incidentally, if we had more room, we would have liked to include one more problem in the Summer Fun problem set on permutations. That problem set ended with a result of Zolotarev connecting the signs of certain permutations to the theory of squares modulo p, where p is a prime number. With more room, we’d have outlined Zolotarev’s proof of quadratic reciprocity using permutations. This proof is “just around the corner” from the material in the permutation problem set and Cailan’s Summer Fun problem set on quadratic reciprocity. As a challenge, you could try to reconstruct Zolotarev’s beautiful proof. Here’s a hint: The idea is to take a deck of pq playing cards, where p and q are distinct odd prime numbers. Consider the following 3 arrangements of the cards into a p by q rectangle:
Arrangement 1: Deal the cards out row by row, from left to right.
Arrangement 2: Deal the cards out column by column, from top to bottom.
Arrangement 3: Deal the cards out going along a NW-SE diagonal, with wraparound.
Consider the permutations defined in going from arrangement 1 to 2, from 2 to 3, and from 3 to 1.
We hope you enjoy our latest issue!
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