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Pengarang	:	Raphael Cerf & Carlo Mariconda
Nama Majalah/Jurnal	:	The American Mathematical Monthly
Volume / Edisi	:	131 (No. 10)
Halaman	:	881-898
Abstrak	:	The basic problem of the calculus of variations consists of finding a function that minimizes an energy, like finding the fastest trajectory between two points for a point mass in a gravity field or finding the best shape of a wing. The existence of a solution may be established in quite abstract spaces, much larger than the space of smooth functions. An important practical problem is that of being able to approach the value of the infimum of the energy. However, numerical methods work with very “concrete” functions and sometimes they are unable to approximate the infimum: this is the surprising Lavrentiev phenomenon. The papers that ensure the nonoccurrence of the phenomenon form a recent saga, and the most general result formulated in the early ’90s was actually fully proved just recently, more than 30 years later. Our aim here is to introduce the reader to the calculus of variations, to illustrate the Lavrentiev phenomenon with the simplest known example, and to give an elementary proof of the nonoccurrence of the phenomenon.

Pengarang	:	Noga Alon, Olivier Bousquet, Kasper Green Larsen, Shay Moran & Shlomo Moran
Nama Majalah/Jurnal	:	The American Mathematical Monthly
Volume / Edisi	:	131 (No. 10)
Halaman	:	866-879
Abstrak	:	We study several variants of a combinatorial game which is based on Cantor’s diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by the known fact that Leopold Kronecker did not appreciate Georg Cantor’s arguments about the infinite, and even referred to him as a “scientific charlatan.” In the game Kronecker maintains a list of m binary vectors, each of length n, and Cantor’s goal is to produce a new binary vector which is different from each of Kronecker’s vectors, or prove that no such vector exists. Cantor does not see Kronecker’s vectors but he is allowed to ask queries of the form What is bit number j of vector number i? What is the minimal number of queries with which Cantor can achieve his goal? How much better can Cantor do if he is allowed to pick his queries adaptively, based on Kronecker’s previous replies? The case when m = n is solved by diagonalization using n (nonadaptive) queries. We study this game more generally, and prove an optimal bound in the adaptive case and nearly tight upper and lower bounds in the nonadaptive case.

Pengarang	:	M. Suresh Kumar & V. Kamalakannan
Nama Majalah/Jurnal	:	The American Mathematical Monthly
Volume / Edisi	:	131 (No. 10)
Halaman	:	855-865
Abstrak	:	A QR code is a two-dimensional bar code that encodes a message into a finite collection of black and white squares which are arranged in a square region with a specific order. Considering a QR code as a compact subset of ?2, we construct an Iterated Function System (IFS) on ?2 whose attractor is the given QR code.

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