Matroids with nine elements

We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this cataogue. Our computation confirms and extends the results obtained in the 1960s by Blackburn, Crapo and Higgs. The matroids and associated data are stored in an online database, and we give three short examples of the use of this database.Comment: 22 page

Maximum size binary matroids with no AG(3,2)-minor are graphic

We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank

Paying for Care Costs in Later Life Using the Value in People’s Homes

With the number of U.K. citizens aged 75+ doubling to 10 million by 2040, and with 1.3 million people already receiving social care services in England alone, social care funding is a key public policy challenge. The government has launched a set of reforms designed to get social care funding onto a sustainable footing by establishing a new level for what individuals and the state will pay. The reforms are designed to encourage individuals to explore how best to use their available wealth and assets to meet care costs through a mixed system of local authority and private sector care-funding options. One option is to use the value in the home to bridge the cost between out-of-pocket costs and care home fees. In this article, we consider two new financial arrangements designed to meet the needs of people in different financial circumstances based on releasing equity from the home. These are an equity-backed insurance product and an “equity bank” that lets a person draw down an income from their hom

The NorthStar Ambulatory Assessment in Duchenne muscular dystrophy: considerations for the design of clinical trials.

With the emergence of experimental therapies for Duchenne muscular dystrophy (DMD), it is fundamental to understand the natural history of this disorder to properly design clinical trials. The aims of this study were to assess the effects produced on motor function by different DMD genotypes and early initiation of glucocorticoids