From N -tensor powers of the Toeplitz algebra, we construct a multi-pullback C*-algebra that is a noncommutative deformation of the complex projective space \mathbb{P}^N\!(\mathbb{C}) . Using Birkhoff’s Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multi-pullback C*-algebra is free. This shows that our deformation preserves the freeness of the lattice of subsets generated by the affine covering of the complex projective space.

The purpose of this paper is to determine the variables of high importance affecting the mathematics achievement of the students of 12 Asia-Pacific countries participating in the Program for International Student Assessment (PISA) 2018. For this purpose, we used random forest (RF), logistic regression (LR) and support vector machine (SVM) models to classify student achievement in mathematics. The variables affecting the student achievement in mathematics were examined by the feature importance method. We observed that the variables with the highest importance for all of the 12 Asia-Pacific countries we considered are the educational status of the parents, having access to educational resources, age, the time allocated to weekly lessons, and the age of starting kindergarten. Then we applied two different clustering analysis by using the variable importance values and socio-economic variables of these countries. We observed that Korea, Japan and Taipei form one group of Asia-Pacific countries, while Thailand, China, Indonesia, and Malaysia form another meaningful group in both clustering analyses. The results we obtained strongly suggest that there is a quantifiable relationship between the educational attainment and socio-economic levels of these 12 Asia-Pacific countries.

We define a Hopf cyclic (co)homology theory in an arbitrary symmetric strict monoidal category. Thus we unify all different types of Hopf cyclic (co)homologies under one single universal theory. We recover Hopf cyclic (co)homology of module algebras, comodule algebras and module coalgebras along with Hopf–Hochschild (co)homology of module algebras, and describe the missing theory for comodule coalgebras.