Cohen–Macaulay binomial edge ideals and accessible graphs

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen–Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen–Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness.

Geometric Constraint-based Reconfiguration and Self-motions of a 4-CRU Parallel Mechanism

This paper aims to investigate the reconfiguration and self-motions of a 4-CRU parallel mechanism based on the mechanism geometric constraints. The targeted application of such mechanism in this research is for 3D-printing buildings of multi-directional nozzle as a new technology for constructing sustainable housing. By using primary decomposition, four geometric constraints are identified and the reconfiguration analysis is carried out in each of these. It reveals that each geometric constraint will have three distinct operation modes, namely Schoenflies mode, reversed Schoenflies mode and an additional mode. The additional mode can be either 4-DOF mode or it degenerates into 3-DOF mode, depending on the type of the geometric constraint. By taking into account the actuation and constraint singularities, the workspace of each operation mode is analysed and geometrically illustrated. It allows us to determine the regions in which the reconfiguration takes place. Furthermore, the moving-platform can still perform at least 1-DOF self-motion. It occurs at two specific actuated leg lengths. Demonstration of reconfiguration process and self-motions are also provided through a mock-up prototype.

Primary decomposition in the smooth concordance group of topologically slice knots

We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the conjectures are true and there are infinitely many primary parts, each of which has infinite rank. This supports the conjectures for topologically slice knots. We also prove analogues for the associated graded groups of the bipolar filtration of topologically slice knots. Among ingredients of the proof, we use amenable $L^2$ -signatures, Ozsváth-Szabó d-invariants and Némethi’s result on Heegaard Floer homology of Seifert 3-manifolds. In an appendix, we present a general formulation of the notion of primary decomposition.

Aqueous-phase fates of α-alkoxyalkyl-hydroperoxides derived from the reactions of Criegee intermediates with alcohols

Criegee intermediates react with alcohols to produce α-alkoxyalkyl-hydroperoxides, R1R2C(–OOH)(–OR′). We found that a primary decomposition product of R1R2C(–OOH)(–OR′) in an acidic aqueous solution was a hemiacetal R1R2C(–OH)(–OR′) species.