A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were integers---hence the name, from the acronym BOunded HEight Matrix of Integers (BOHEMI)---but other kinds of entries are also interesting. Some kinds of questions about Bohemian matrices can be answered by numerical computation, but sometimes exact computation is better. In this paper we explore some Bohemian families (symmetric, upper Hessenberg, or Toeplitz) computationally, and answer some open questions posed about the distributions of eigenvalue densities.

Hamiltonian Monte Carlo (HMC) is a powerful Markov chain Monte Carlo (MCMC) algorithm for estimating expectations with respect to continuous un-normalized probability distributions. MCMC estimators typically have higher variance than classical Monte Carlo with i.i.d. samples due to autocorrelations; most MCMC research tries to reduce these autocorrelations. In this work, we explore a complementary approach to variance reduction based on two classical Monte Carlo ’swindles’: first, running an auxiliary coupled chain targeting a tractable approximation to the target distribution, and using the auxiliary samples as control variates; and second, generating anti-correlated ("antithetic") samples by running two chains with flipped randomness. Both ideas have been explored previously in the context of Gibbs samplers and random-walk Metropolis algorithms, but we argue that they are ripe for adaptation to HMC in light of recent coupling results from the HMC theory literature. For many posterior distributions, we find that these swindles generate effective sample sizes orders of magnitude larger than plain HMC, as well as being more efficient than analogous swindles for Metropolis-adjusted Langevin algorithm and random-walk Metropolis.

There is certain diverse class of diagram that is found in a variety of branches of mathematics and which all share this property: there is a common scheme for translating all of these diagrams into useful functional code. These diagrams include Bayesian networks, quantum computer circuits [1], trace diagrams for multilinear algebra [2], Feynman diagrams and even knot diagrams [3]. I will show how a common thread lying behind these diagrams is the presence of a commutative monad and I will show how we can use this fact to translate these diagrams directly into Haskell code making use of do-notation for monads. I will also show a number of examples of such translated code at work and use it to solve problems ranging from Bayesian inference to the topological problem of untangling tangled strings. Along the way I hope to give a little insight into the subjects mentioned above and illustrate how a functional programming language can be a valuable tool in mathematical research and experimentation.

No abstract available.

Subdivision surfaces solve numerous problems related to the geometry of character and animation models. However, unlike on parametrised surfaces there is no natural choice of texture coordinates on subdivision surfaces. Existing algorithms for generating texture coordinates on non-parametrised surfaces often find solutions that are locally acceptable but globally are unsuitable for use by artists wishing to paint textures. In addition, for topological reasons there is not necessarily any choice of assignment of texture coordinates to control points that can satisfactorily be interpolated over the entire surface. We introduce a technique, pelting, for finding both optimal and intuitive texture mapping over almost all of an entire subdivision surface and then show how to combine multiple texture mappings together to produce a seamless result.