Complete classification of six-dimensional iso-edge domains

We enumerate the 55083357 iso-edge subdivisions of six-dimensional translational lattices. We report on the use of the method of canonical forms that allows us to apply hashing techniques used in modern databases.

The general equation of δ direct methods and the novel SMAR algorithm residuals using the absolute value of ρ and the zero conversion of negative ripples

The general equation of the δ direct methods is established and used to define one of the two residuals of the SMAR phasing algorithm. These residuals utilize the absolute value of ρ and the zero conversion of slightly negative Fourier ripples (≥50% of the cell volume) and lead to overdetermination for atomic resolution diffraction data. Due to its architecture, the SMAR algorithm is particularly well suited for Deep Learning. Alternatively, when solved for ρ, the general equation provides a simple derivation of the already known δ_M tangent formula, the core of the δ_M recycling algorithm.

Periodic graphs with coincident edges: folding-ladder and related graphs

We explore a special class of periodic graphs, ladder graphs, whose edges can coincide when embedded vertices are moved. Many of these exhibit additional non-crystallographic graph symmetries.