Rooting the concept and skills of engineering sense in Egyptian and Arabic research and studies

Geometric sense is one of the types of mathematical sense in its general concept and there is a difference between researchers in Egypt and the Arab countries about the concept of geometric sense and its main and sub-skills and there is also a confusion among researchers between geometric sense measurement tests and achievement tests in geometry. The aim of the research is to accurately define the concept of geometric sense that eliminates differences between researchers, to refute the geometric skills identified by researchers within Egypt and the Arab countries for geometric sense, which confuses geometric sense, geometric thinking, geometric proof and achievement in geometry, to identify key and sub-skills for geometric sense according to the writings of the American Council of Mathematicians NTCM so that researchers can commit to them in future research.

Cluster Structures on Double Bott–Samelson Cells

Let $\mathsf {C}$ be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to $\mathsf {C}$ . We prove that the decorated double Bott–Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras. We explicitly describe the Donaldson–Thomas transformations on double Bott–Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock–Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson–Thomas transformations on a family of double Bott–Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov’s periodicity conjecture in the cases of $\Delta \square \mathrm {A}_r$ . When $\mathsf {C}$ is of type $\mathrm {A}$ , the double Bott–Samelson cells are isomorphic to Shende–Treumann–Zaslow’s moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their $\mathbb {F}_q$ -points we obtain rational functions that are Legendrian link invariants.