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**1 - 2**of**2**### TILING WITH COMMUTATIVE RINGS

"... Consider the collection R of squares obtained from the chessboard by removing two opposite corners: Can it be covered with the vertical and horizontal dominoes so that every square is covered by exactly one domino? In other words, can R be tiled by vertical and horizontal dominoes? The coloring give ..."

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Consider the collection R of squares obtained from the chessboard by removing two opposite corners: Can it be covered with the vertical and horizontal dominoes so that every square is covered by exactly one domino? In other words, can R be tiled by vertical and horizontal dominoes? The coloring gives the answer to this well-known problem away. The region R has 32 black squares and 30 white squares. Since each domino covers exactly one black and one white square, no tiling is possible. The aim of this article is to explain a way to tackle tiling problems using a little commutative algebra. More precisely, we will explain how to obtain coloring arguments, similar to the above chessboard coloring, in a systematic way. I will assume that the reader is familiar with linear algebra and have seen rings and ideals before. 2. Tiles, regions, and tiling problems Let N = {0, 1, 2,...} denote the natural numbers. A tile or region is a finite subset of N 2 considered as a collection of boxes in the first quadrant 1.

### On the Positive Mass, Penrose, and ZAS Inequalities in General Dimension

, 2010

"... After a detailed introduction including new examples, we give an exposition focusing on the Riemannian cases of the positive mass, Penrose, and ZAS inequalities of general relativity, in general dimension. ..."

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After a detailed introduction including new examples, we give an exposition focusing on the Riemannian cases of the positive mass, Penrose, and ZAS inequalities of general relativity, in general dimension.