We study the relations between two groups related to cluster automorphism which are defined by Assem, Schiffler and Shamchenko. establish relationships among (strict) direct those consisting of periodicities labeled seeds exchange matrices, respectively, in language short exact sequences. As an application, we characterize automorphism-finite algebras cases bipartite or finite mutation type. Fi...

We introduce a new cluster character with coefficients for category $\mathcal{C}$ and rather than using Frobenius $2$-Calabi-Yau realization to incorporate into the representation-theoretic model algebra, as done by Fu Keller, we exploit intrinsic properties of $\mathcal{C}$. For this purpose, define an ice quiver associated each tilting object in In Dynkin case $\mathbb{A}_n$, also prove that ...

Clusters are geographically close groups of related companies or institutions related to a certain area which are inherently more efficient than the other companies due to advantages such as being located in one place, networks, external knowledge, variability of human capital, etc. Today, development through clusters plays a pivotal role in the economic and industrial policies of developed...

Sometimes in order to estimate population parameters such as mean and total values, we extract a random sample by cluster sampling method, and after completing sampling, we are interested in using the same sample to estimate the desired parameters in a subset of the population, which is said subpopulation. In this paper, we try to estimate subpopulation parameters in different cases when one-st...