Mini-Batch k-Means versus k-Means to Cluster English Tafseer Text: View of Al-Baqarah Chapter

Al-Quran is the primary text of Muslims’ religion and practise. Millions of Muslims around the world use al-Quran as their reference guide, and so knowledge can be obtained from it by Muslims and Islamic scholars in general. Al-Quran has been reinterpreted to various languages in the world, for example, English and has been written by several translators. Each translator has ideas, comments and statements to translate the verses from which he has obtained (Tafseer). Therefore, this paper tries to cluster the translation of the Tafseer using text clustering. Text clustering is the text mining method that needs to be clustered in the same section of related documents. The study adapted (mini-batch k-means and k-means) algorithms of clustering techniques to explain and to define the link between keywords known as features or concepts for Al-Baqarah chapter of 286 verses. For this dataset, data preprocessing and extraction of features using Term Frequency-Inverse Document Frequency (TF-IDF) and Principal Component Analysis (PCA) applied. Results showed that two/three-dimensional clustering plotting assigning seven cluster categories (k = 7) for the Tafseer. The implementation time of the mini-batch k-means algorithm (0.05485s) outperformed the time of the k-means algorithm (0.23334s). Finally, the features ‘god’, ‘people’, and ‘believe’ was the most frequent features.

Tropical friezes and the index in higher homological algebra

Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander–Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points. Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers. This paper will define higher dimensional tropical friezes. One of the motivations is the higher dimensional cluster categories of Oppermann and Thomas, which encode (d + 1)-dimensional structures for an integer d ⩾ 1. They are (d + 2)-angulated categories, which belong to the subject of higher homological algebra. We will define higher dimensional tropical friezes as maps from higher cluster categories (and more general (d + 2)-angulated categories) to the integers. Following Palu, we will define a notion of (d + 2)-angulated index, establish some of its properties, and use it to construct higher dimensional tropical friezes.

CLUSTER CATEGORIES FROM GRASSMANNIANS AND ROOT COMBINATORICS

The category of Cohen–Macaulay modules of an algebra $B_{k,n}$ is used in Jensen et al. (A categorification of Grassmannian cluster algebras, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212) to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander–Reiten sequences and study the Auslander–Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen–Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac–Moody algebra in the tame cases.