Jacob Garber | Searcharxiv

arXiv2020

Beth Definability in the Logic KR

The Beth Definability Property holds for an algebraizable logic if and only if every epimorphism in the corresponding category of algebras is surjective. Using this technique, Urquhart in 1999 showed that the Beth Definability Property fails for a wide class of relevant logics, including T, E, and R. However, the counterexample for those logics does not extend to the algebraic counterpart of the super relevant logic KR, the so-called Boolean monoids. Following a suggestion of Urquhart, we use modular lattices constructed by Freese to show that epimorphisms need not be surjective in a wide class of relation algebras. This class includes the Boolean monoids, and thus the Beth Definability Property fails for KR.

math.LO↗

arXiv2018

One Hundred and Twelve Point Three Degree Theorem

It has been known since Fagnano in 1775 that an acute triangle always has a periodic billiard path, namely the orthic triangle. It is currently unknown whether every obtuse triangle has a periodic path. In 2006, Schwartz showed that every obtuse triangle with obtuse angle at most 100 degrees has a periodic path. The aim of this paper is to show that every obtuse triangle with obtuse angle at most 112.3 degrees has a periodic path using a computer assisted proof.

math.DS↗