On exact triangle inequalities in (q1, q2)-quasimetric spaces

For arbitrary (q1, q2) -quasimetric space, it is proved that there exists a function f, such that f -triangle inequality is more exact than any (q1, q2) -triangle inequality. It is shown that this function f is the least one in the set of all concave continuous functions g for which g -triangle inequality hold.

A CONTACT INVARIANT IN SUTURED MONOPOLE HOMOLOGY

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured monopole Floer homology theory ($SHM$). Our invariant can be viewed as a generalization of Kronheimer and Mrowka’s contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, and Matić’s contact invariant in sutured Heegaard Floer homology ($SFH$). In the process of defining our invariant, we construct maps on $SHM$ associated to contact handle attachments, analogous to those defined by Honda, Kazez, and Matić in $SFH$. We use these maps to establish a bypass exact triangle in $SHM$ analogous to Honda’s in $SFH$. This paper also provides the topological basis for the construction of similar gluing maps in sutured instanton Floer homology, which are used in Baldwin and Sivek [Selecta Math. (N.S.), 22(2) (2016), 939–978] to define a contact invariant in the instanton Floer setting.

Triangle Mesh Based In-Process Workpiece Update for General Milling Processes

A new methodology for modeling and updating the in-process workpiece geometry in milling is presented in this paper. The methodology is developed for general milling processes, in which the cutter can be any shape and follow any tool path trajectory even with self-intersections. And the in-process workpiece is updated with retained sharp features. The associated procedure starts by modeling both the cutter and the workpiece as closed manifold triangle meshes. The mesh model of the cutter swept volume is then generated from repeatedly sampled mesh vertices of the cutter along its trajectory using the ball-pivoting algorithm. The workpiece is updated by a subtraction Boolean operation between the workpiece and the cutter swept volume. An octree space partitioning algorithm is adopted in order to efficiently obtain the exact triangle-to-triangle intersection points. As the last step, a filling operation is performed around the intersection points to establish the closed manifold updated workpiece geometry. Several case studies have been performed to demonstrate the effectiveness of the proposed methodology.