Australian Mathematical Sciences Institute

Abstract Partitioned and mixture models are widely employed in Maximum Likelihood phylogenetic analyses of large genomic datasets. Comparing the fit of the two types of models has been challenging, because standard information-theoretic approaches cannot be applied. Mixture models are increasingly popular for the analysis of amino acid datasets and can lead to different conclusions compared to partitioned models. This raises an important question – which type of model tends to perform better? Susko et al. (2026) recently introduced the marginal Akaike information criterion (mAIC), which allows mixture models and partitioned models to be directly compared for the first time. Here, we use the mAIC and a range of other approaches to compare the fit of mixture and partitioned models across a diverse set of empirical datasets. We show that mixture models are universally favoured on amino acid datasets. This has important implications for interpreting empirical analyses and suggests that continued development of mixture models is an important avenue for future research.

Abstract Parametric factor copula models typically work well in modeling multivariate dependencies due to their flexibility and ability to capture complex dependency structures. However, accurately estimating the linking copulas within these models remains challenging, especially when working with high-dimensional data. This paper proposes a novel approach for estimating linking copulas based on a non-parametric kernel estimator. Unlike conventional parametric methods, our approach utilizes the flexibility of kernel density estimation to capture the underlying dependencies more accurately, particularly in scenarios where the underlying copula structure is complex or unknown. We show that the proposed estimator is consistent under mild conditions and demonstrate its effectiveness through extensive simulation studies. Our findings suggest that the proposed approach offers a promising avenue for modeling multivariate dependencies, particularly in applications requiring robust and efficient estimation of copula-based models.

We introduce a strategy to study irreducible representations of automorphism groups of finite modules over finite rings. We prove that these automorphism groups fit in a hierarchy that facilitates a stratification of their irreducible representations in terms of smaller building blocks using symmetric monoidal categories and invariant theory.

Abstract The pro-isomorphic zeta function of a finitely generated nilpotent group is a Dirichlet generating series that enumerates all finite-index subgroups whose profinite completion is isomorphic to that of the ambient group. We study the pro-isomorphic zeta functions of ℚ-indecomposable D *-groups of even Hirsch length. These groups are building blocks of finitely generated class-two nilpotent groups with rank-two centre, up to commensurability. Due to a classification by Grunewald and Segal, they are parameterised by primary polynomials whose companion matrices define commutator relations for an explicit presentation. For Grunewald–Segal representatives of even Hirsch length of type f ( t ) = t m , we give a complete description of the algebraic automorphism groups of associated Lie lattices. Utilising the automorphism groups, we determine the local pro-isomorphic zeta functions of groups associated to t 2 and t 3 . In both cases, the local zeta functions are uniform in the prime p and satisfy functional equations. The functional equations for these groups, not predicted by the currently available theory, prompt us to formulate a conjecture which prescribes, in particular, information about the symmetry factor appearing in local functional equations for pro-isomorphic zeta functions of nilpotent groups. Our description of the local zeta functions also yields information about the analytic properties of the corresponding global pro-isomorphic zeta functions. Some of our results for the D *-groups associated to t 2 and t 3 generalise to two infinite families of class-two nilpotent groups that result naturally from the initial groups via ‘base extensions’.

Abstract Periodic elements in finite-type Artin–Tits groups are elements some positive power of which is central. We give a dynamical characterisation of periodic elements via their action on the corresponding 2-Calabi–Yau category and on its space of (fusion equivariant) Bridgeland stability conditions. The main theorem is that an element $\beta $ is periodic if and only if $\beta $ has a fixed point in the stability manifold.

Abstract We give an equivalence of categories between: (i) Möbius vertex algebras which are equipped with a choice of generating family of quasiprimary vectors, and (ii) (not-necessarily-unitary) Möbius-covariant Wightman conformal field theories on the unit circle. We do not impose any technical restrictions on the theories considered (such as finite-dimensional conformal weight spaces or simplicity), yielding the most general equivalence between these two axiomatizations of two-dimensional chiral conformal field theory.This provides new opportunities to study non-unitary vertex algebras using the lens of algebraic conformal field theory and operator algebras, which we demonstrate by establishing a non-unitary version of the Reeh-Schlieder theorem.

We investigate the solutions of the relaxed magnetohydrodynamic (MHD) model (RxMHD) of R. Dewar and Z. Qu [J. Plasma Phys. 88, 835880101 (2022)]. This model generalizes Taylor relaxation by including the ideal Ohm's law constraint using an augmented Lagrangian method, providing a pathway to extend the multi-region relaxed MHD (MRxMHD) model. We present the first numerical solution of the RxMHD model by Dewar and Qu, demonstrating that it is mathematically well-defined and computationally feasible for constructing MHD equilibria in slab geometry. We also show that a cross-field flow can exist without enforcing an arbitrary constraint on the angular momentum, as is done in the case of MRxMHD with flow. Our results also demonstrate the self-organization of fully relaxed regions during the optimization, which was an important motivation behind developing this model.