arXiv subjects

Thomas G. Brooks

Publications and source records attributed to Thomas G. Brooks.

4 recordsLinked to original sources

arXiv2024

Sampling Spiked Wishart Eigenvalues

Efficient schemes for sampling from the eigenvalues of the Wishart distribution have recently been described for both the uncorrelated central case (where the covariance matrix is $\mathbf{I}$) and the spiked Wishart with a single spike (where the covariance matrix differs from $\mathbf{I}$ in a single entry on the diagonal). Here, we generalize these schemes to the spiked Wishart with an arbitrary number of spikes. This approach also applies to the spiked pseudo-Wishart distribution. We describe how to differentiate this procedure for the purposes of stochastic gradient descent, allowing the fitting of the eigenvalue distribution to some target distribution.

stat.CO↗

arXiv2024

Generating Sets of Mathieu Groups

Julius Whiston calculated the maximum size of an irredundant generating set for $S_n$ and $A_n$ by examination of maximal subgroups. Using analogous considerations, we will compute upper bounds to this value for the first two Mathieu groups, $M_{11}$ and $M_{12}$. Computational results gave explicit irredundant generating sets of $M_{11}$ and $M_{12}$ of size 5 and 6, respectively. Together these give the full results that the maximum size of an irredundant generating set for $M_{11}$ is 5 and for $M_{12}$ it is 6.

math.GR↗

arXiv2021

3-Manifolds with Constant Ricci Eigenvalues $(λ, λ, 0)$

We consider complete Riemannian $3$-manifolds whose Ricci tensors have constant eigenvalues $(λ, λ, 0)$. When $π_1$ is finitely generated, we classify the topology of such manifolds by showing that they have a free fundamental group if non-trivial and that every free group is obtained. We give a description up to isometry, when the metric is locally irreducible or when it is analytic.

math.DG↗

arXiv2019

Non-negative Curvature and Conullity of the Curvature Tensor

The conullity of a curvature tensor is the codimension of its kernel. We consider the cases of conullity two in any dimension and conullity three in dimension four. We show that these conditions are compatible with non-negative sectional curvature only if either the manifold is diffeomorphic to $\mathbb{R}^n$ or the universal cover is an isometric product with a Euclidean factor. Moreover, we show that finite volume manifolds with conullity 3 are locally products.

math.DG↗