The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane (pictured), as well as related higher-dimensional objects. They may also be defined as collections of permutations (rearrangements) of the integers that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied in the fields of combinatorics and representation theory. Each of these groups is an infinite extension of a finite symmetric group, and many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. The affine symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection groups. Many of their combinatorial and geometric properties extend to the broader family of affine Coxeter groups.
Read more: https://en.wikipedia.org/wiki/Affine_symmetric_group
_______________________________ Today's selected anniversaries:
1918:
World War I: After his platoon suffered heavy casualties during the Meuse–Argonne offensive in France's Forest of Argonne, American Corporal Alvin York led the 7 remaining men on an attack against a German machine gun nest; 25 German soldiers were killed and 132 captured. https://en.wikipedia.org/wiki/Alvin_York
1932:
The Indian Air Force was founded as an auxiliary air force of the British Royal Air Force. https://en.wikipedia.org/wiki/Indian_Air_Force
2016:
Yemen War: A funeral in Sanaa was hit by two consecutive airstrikes by a Saudi-led coalition, leaving 143–155 civilians dead and more than 525 injured. https://en.wikipedia.org/wiki/Sanaa_funeral_airstrike
2019:
Anti-government protests calling for free and fair elections began in Baku, Azerbaijan. https://en.wikipedia.org/wiki/2019_Baku_protests
_____________________________ Wiktionary's word of the day:
refusenik: 1. (Soviet Union, historical, slang) One of the citizens of the former Soviet Union who was refused permission to emigrate (typically but not exclusively a Jewish citizen denied permission to emigrate to Israel). 2. (by extension, informal) A person who refuses to do something, usually as a protest; for example, one who refuses conscription or vaccination. [...] https://en.wiktionary.org/wiki/refusenik
___________________________ Wikiquote quote of the day:
Political extremism involves two prime ingredients: an excessively simple diagnosis of the world's ills, and a conviction that there are identifiable villains back of it all. Such extremism comes easily to men who have doped themselves with delusions of their own overriding virtue and the rascality of others. --John W. Gardner https://en.wikiquote.org/wiki/John_W._Gardner