Tag

Bounds

All articles tagged with #bounds

mathematics2 years ago•9 min saved

"Triangle Shrinkage: The Incredible Shrinking Shape"

Mathematicians Alex Cohen, Cosmin Pohoata, and Dmitrii Zakharov have made a breakthrough in the Heilbronn triangle problem, a mathematical puzzle that has stumped researchers for decades. The problem involves finding the largest possible smallest triangle formed by three points within a square. The trio of mathematicians discovered a new upper bound for the size of the smallest triangle, using techniques from intersecting shapes, number theory, and Fourier analysis. Their findings have revitalized research on the problem and revealed new connections to other areas of mathematics.

via Quanta Magazine|

#bounds #dimension #heilbronn-triangle-problem

mathematics2 years ago•5 min saved

"Revolutionary Breakthrough: Mathematicians Push the Limits of the Coloring Problem"

Mathematicians have made a breakthrough in the coloring problem, which involves determining how much of an infinite plane can be colored while ensuring that no two colored points are exactly one unit of distance apart. After decades of research, a team of mathematicians from the Budapest University of Technology and Economics, the Hungarian Academy of Sciences, and other institutions used machine learning models and optimization techniques to prove that no more than 24.7% of the plane can be colored without unit-distance pairs, breaking the previous upper bound set by Paul Erdős. The researchers are now focusing on determining the chromatic number of the plane, which represents the minimum number of colors needed to completely cover the plane while avoiding unit-distance pairs.

via Quanta Magazine|

#bounds #chromatic-number #coloring-problem