In a recent paper [Linear Algebra Appl., 461:92--122, 2014], Tao et al. proved an analog of Thompson's triangle inequality for simple Euclidean Jordan algebra by using case-by-case analysis. this short note, we provide direct proof that is valid on any algebras.

We are interested in the natural question about which three cevians form a triangle independent of the triangle where they are constructed. This apparently simple question is equivalent to solving a three-parameter family of inequalities which we call Ceva’s triangle inequalities. The parametrization of the solution set is rather surprising and requires a combination of techniques from algebrai...

Associated to any complex simple Lie algebra is a non-reductive complex Lie algebra which we call the intermediate Lie algebra. We propose that these algebras can be included in both the magic square and the magic triangle to give an additional row and column. The extra row and column in the magic square correspond to the sextonions. This is a six-dimensional subalgebra of the split octonions w...

Symmetries of trigonometric integrable two dimensional statistical face models are considered. The corresponding symmetry operators on the Hilbert space of states of the quantum version of these models define a weak *-Hopf algebra isomorphic to the Ocneanu double triangle algebra. R. TRINCHERO 1

Symmetries of trigonometric integrable two dimensional statistical face models are considered. The corresponding symmetry operators on the Hilbert space of states of the quantum version of these models define a weak *-Hopf algebra isomorphic to the Ocneanu double triangle algebra(DTA). R. TRINCHERO 1

The generalized winding number function measures insideness for arbitrary oriented triangle meshes. Exploiting this, I similarly generalize binary boolean operations to act on such meshes. The resulting operations for union, intersection, difference, etc. avoid volumetric discretization or pre-processing. 1 Booleans & Classic Winding Numbers If A ⊂ R and B ⊂ R are open subregions of space, then...

We investigate the problem of cutting a triangle ABC into N congruent triangles (the “tiles”), which may or may not be similar to ABC. We wish to characterize the numbers N for which some triangle ABC can be tiled by N tiles, or more generally to characterize the triples (N,T ) such that ABC can be N-tiled using tile T . In the first part of the paper we exhibit certain families of tilings whic...