M-partitions: Optimal partitions of weight for one scale pan

An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M-partition is a log-linear function of m and the M-partitions of m correspond to lattice points in a polytope. We exhibit a recurrence relation for counting the number of M-partitions of m and, for ``half'' of the positive integers, this recurrence relation will have a generating function. The generating function will be, in some sense, the same as the generating function for counting the number of distinct binary partitions for a given integer.Comment: 11 page

A Civic Republican Analysis of Mental Capacity Law

This article draws upon the civic republican tradition to offer new conceptual resources for the normative assessment of mental capacity law. The republican conception of liberty as non-domination is used to identify ways in which such laws generate arbitrary power that can underpin relationships of servility and insecurity. It also shows how non-domination provides a basis for critiquing legal tests of decision-making that rely upon ‘diagnostic’ rather than ‘functional’ criteria. In response, two main civic republican strategies are recommended for securing freedom in the context of the legal regulation of psychological disability: self-authorisation techniques and participatory shaping of power. The result is a series of proposals for the reform of decisional capacity law, including a transition towards purely functional assessment of decisional capacity, surer legal footing for advanced care planning, and greater control over the design and administration of decision-making capacity laws by those with psychological disabilities

Multiples of Pfister forms

The isotropy of multiples of Pfister forms is studied. In particular, an improved lower bound on the values of their first Witt indices is obtained. A number of corollaries of this result are outlined. An investigation of generic Pfister multiples is also undertaken. These results are applied to distinguish between properties preserved by Pfister products.Comment: 14 page

Group and round quadratic forms

We offer some elementary characterisations of group and round quadratic forms. These characterisations are applied to establish new (and recover existing) characterisations of Pfister forms. We establish "going-up" results for group and anisotropic round forms with respect to iterated Laurent series fields, which contrast with the established results with respect to rational function field extensions. For forms of two-power dimension, we determine when there exists a field extension over which the form becomes an anisotropic group form that is not round.Comment: 12 page

Semi-Supervised Radio Signal Identification

Radio emitter recognition in dense multi-user environments is an important tool for optimizing spectrum utilization, identifying and minimizing interference, and enforcing spectrum policy. Radio data is readily available and easy to obtain from an antenna, but labeled and curated data is often scarce making supervised learning strategies difficult and time consuming in practice. We demonstrate that semi-supervised learning techniques can be used to scale learning beyond supervised datasets, allowing for discerning and recalling new radio signals by using sparse signal representations based on both unsupervised and supervised methods for nonlinear feature learning and clustering methods