An inequality for the matrix pressure function and applications

We prove an a priori lower bound for the pressure, or $p$-norm joint spectral radius, of a measure on the set of $d \times d$ real matrices which parallels a result of J. Bochi for the joint spectral radius. We apply this lower bound to give new proofs of the continuity of the affinity dimension of a self-affine set and of the continuity of the singular-value pressure for invertible matrices, both of which had been previously established by D.-J. Feng and P. Shmerkin using multiplicative ergodic theory and the subadditive variational principle. Unlike the previous proof, our lower bound yields algorithms to rigorously compute the pressure, singular value pressure and affinity dimension of a finite set of matrices to within an a priori prescribed accuracy in finitely many computational steps. We additionally deduce a related inequality for the singular value pressure for measures on the set of $2 \times 2$ real matrices, give a precise characterisation of the discontinuities of the singular value pressure function for two-dimensional matrices, and prove a general theorem relating the zero-temperature limit of the matrix pressure to the joint spectral radius.Comment: To appear in Advances in Mathematic

Leeds Met Library Facebook application

At the Leeds Met staff development festival in 2008 a library graduate trainee, Anna Hepworth, took part in a ‘Dragons’ Den’ event which saw staff propose new, innovative ideas to a panel of senior managers. Anna’s suggestion was to develop a Facebook application (or ‘app’) for the library and it was one of the competition winners. Anna’s initial proposal was to create a Leeds Met library catalogue application, but after discussions with members of the library’s ‘technologies for learning’ team it was decided to take the application a stage further, creating a mash-up using data from the library management system (Sirsi-Dynix Symphony). The Facebook application would send a library catalogue search box to a Facebook profile, but would also add value by delivering customised user data, including library record details such as number of issues, reservations and overdues. There would also be links to the library website and online self-service functions from the application