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.

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 by a succinct matrix method, a non-linear recurrence identity class for terms of the so called Horadam sequence. A particular instance was established (in equivalent form) over half a century ago by A.F. Horadam, which provides a starting point for the discussion and an introduction to our formulation technique.

This paper describes a novel hackathon-style system engineering process and its value as an agile approach to the rapid generation and development of early design concepts of complex engineered products–in this case a future aircraft. Complex product design typically requires a diverse range of stakeholders to arrive at a consensus of key decision criteria and design factors, which requires effective articulation and communication of information across traditional engineering and operational disciplines. The application of the methodology is highlighted by means of a case study inspired by Airbus where stakeholder involvement and internal collaboration among team members were essential to achieve a set of agreed goals. This paper shows that a hackathon grounded on systems engineering approaches and structured around the technical functions within an engineering company has the capability and capacity to communicate a coherent vision and rationale for the conceptual design of a complex engineered product. The hackathon method offers significant benefits to these stakeholders to better manage, prioritize, and decrease excessive complexities in the overall design process. A significant benefit of this agile process is that it can achieve useful results in a very short timeframe (i.e., 80% reduction), where it could take up to a year to accomplish compared with using current/regular internal methods.

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.