Mathematical aesthetic, having a variety of forms, is commonly experienced by mathematicians as a genuine reality and is available at every level of study. This short essay in hopefully moving beyond standardised hermeneutic critique attests to its authenticity through the words of some mathematical proponents, and reminds us that it should be promoted to children and students as a phenomenon that is as accessible as it is incontestable.

Cloud workloads are increasingly heterogeneous such that a single Cloud job may encompass one to several tasks, and tasks belonging to the same job may behave distinctively during their actual execution. This inherent task heterogeneity imposes increased complexities in achieving an energy efficient management of the Cloud jobs. The phenomenon of a few proportions of tasks characterising increased resource intensity within a given job usually lead the providers to over-provision all the encompassed tasks, resulting in majority of the tasks incurring an increased proportions of resource idleness. To this end, this paper proposes a novel analytics framework which integrates a resource estimation module to estimate the resource requirements of tasks a priori, a straggler classification module to classify tasks based on their resource intensity, and a resource optimisation module to optimise the level of resource provision depending on the task nature and various runtime factors. Performance evaluations conducted both theoretically and through practical experiments prove that the proposed methodology performs better than the compared statistical resource estimation methods and existing models of straggler mitigation, and further demonstrate the effectiveness of the proposed methodology in achieving energy conservation by postulating appropriate level of resource provisioning for task execution.

The so called geometric mean sequence recurrence, with additional scaling variable, produces a sequence for which the general term has a known closed form. Two types of derivative sequence—comprising products of such sequence terms—are examined. In particular, the general term closed forms formulated are shown to depend strongly on a mix of three existing sequences, from which sequence growth rates are deduced and other results given.

We state and prove the invariance, with respect to matrix power, of both the diagonals and anti-diagonals ratio of a special case 2×2 matrix. The proof methodology is new, contrasting with those deployed previously in establishing anti-diagonals matrix invariants.