Completions of Leavitt path algebras

We introduce a class of topologies on the Leavitt path algebra $L(\Gamma)$ of a finite directed graph and decompose a graded completion $\widehat{L}(\Gamma)$ as a direct sum of minimal ideals.Comment: 16 pages and 2 figure

Decomposition of Singular Matrices into Idempotents

In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and apply these factorizations for proving our main results. We generalize works due to Laffey (Products of idempotent matrices. Linear Multilinear A. 1983) and Rao (Products of idempotent matrices. Linear Algebra Appl. 2009) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems. We also consider singular matrices over B\'ezout domains as to when such a matrix is a product of idempotent matrices.Comment: 15 page

On self-dual double circulant codes

Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic. Exact counting formulae are derived for them and used to show they contain families of codes with relative distance satisfying a modified Gilbert-Varshamov bound.Comment: 8 page

Spatial Optimization of Urban Cellular Automata Model

Although cellular automata (CA) offer a modelling framework and set of techniques for modelling the dynamic processes of urban growth, determining the optimal value of weights or parameters for elements or factors of urban CA models is challenging. This chapter demonstrates the implementation of a calibration module in a fuzzy cellular urban growth model (FCUGM) for optimizing the weights and parameters of an urban CA model using three types of algorithms: (i) genetic algorithm (GA), (ii) parallel simulated annealing (PSA) and (iii) expert knowledge (EK). It was found that the GA followed by EK produced better and more accurate and consistent results compared with PSA. This suggests that the GA was able to some extent to understand the urban growth process and the underlying relationship between input factors in a way similar to human experts. It also suggests that the two algorithms (GA and EK) have similar agreement about the efficiency of scenarios in terms of modelling urban growth. In contrast, the results of the PSA do not show results corresponding to those of the GA or EK. This suggests that the complexity of the urban process is beyond the algorithm’s capability or could be due to being trapped in local optima. With this satisfactory calibration of the FCUGM for the urban growth of Riyadh city in Saudi Arabia by using CALIB-FCUGM, these calibrated parameters can be passed into the SIM-FCUGM to simulate the spatial patterns of urban growth of Riyadh