The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems

We show that an effective version of Siegel’s Theorem on finiteness of integer solutions and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the κ ≥ 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. This dichotomy holds even when restricted to planar graphs. A special case of this result is that counting edge κ-colorings is #P-hard over planar 3-regular graphs for κ ≥ 3. In fact, we prove that counting edge κ-colorings is #P-hard over planar r-regular graphs for all κ ≥ r ≥ 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for Holant(f) depends on the fact that a specific polynomial p(x, y) has an explicitly listed finite set of integer solutions, and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials.

An approach to protein structure using information geometry

In the light of recent structural developments in DNA structural diversity crystallographic studies and the Protein Data Bank*, this note is intended to draw attention to an interesting feature of the ordering of amino acids along protein chains. They all exhibited clustering compared to a random distribution, so there is a stable long range ordering that is unexpected. To date we have no clear explanation of why this should be the case. * https://doi.org/10.1016/j.jbc.2021.100553

Topics in Information Geometry

We introduce first some of the background ideas on information theory and its role in studying analytic models for stochastic processes and the geometrization of families of measure functions. This is then used to present the geometry of important examples of the Riemannian manifolds that arise. Next, we obtain the proof of two theorems that characterise the metric neighbourhoods of the two distinguished fundamental states: randomness and independence. These methods have had applications in modelling cryptographic attacks, cosmological void distributions, porous media, clustering of: galaxies, communications, and amino acids along protein chains in genomes

A model for Gaussian perturbations of graphene

Graphene consists nominally of a regular planar hexagonal carbon lattice monolayer. However, its structure experiences perturbations in the presence of external influences, whether from substrate properties, thermal or electromagnetic fields, or ambient fluid movement. Here we give an information geometric model to represent the state space of perturbations as a Riemannian pseudosphere with scalar curvature close to -1/2. This would allow the representation of a trajectory of states under a given ambient or process change, so opening the possibility for geometrically formulated dynamical models to link structural perturbations to the physics

A short review on Landsberg spaces

This short review is concerned with real finite-dimensional Finsler manifolds (M,F) with Finsler structures F:TM-->[0,infty) that satisfy the Landsberg conditions. In particular this includes the case of Berwald manifolds since their Chern connections on the pullback of TM are fibre-independent. The aim is to provide an annotated collection of references to geometric results that seem important in the study of Landsberg spaces and to suggest some areas for further work in this context

Alan Turing’s Manchester by Jonathon Swinton

This is a delightful book, enjoyable for anyone with an interest in science or Manchester or its university. The title precisely captures the subject matter, namely what Turing would have experienced through his window from abstract academia on his arrival in 1948. It describes the scientific–academic–social environment in Manchester during the 1940s and 1950s, when the University of Manchester was drawing talent, which included Turing, away from the Oxford–London–Cambridge triangle

Information distance estimation between mixtures of multivariate Gaussians

There are efficient software programs for extracting from image sequences certain mixtures of distributions, such as multivariate Gaussians, to represent the important features needed for accurate document retrieval from databases. This note describes a method to use information geometric methods to measure distances between distributions in mixtures of multivariate Gaussians. There is no general analytic solution for the information geodesic distance between two k-variate Gaussians, but for many purposes the absolute information distance is not essential and comparative values suffice for proximity testing. For two mixtures of multivariate Gaussians we must resort to approximations to incorporate the weightings. In practice, the relation between a reasonable approximation and a true geodesic distance is likely to be monotonic, which is adequate for many applications. Here we compare several choices for the incorporation of weightings in distance estimation and provide illustrative results from simulations of differently weighted mixtures of multivariate Gaussians

Information distance estimation between mixtures of multivariate Gaussians

There are efficient software programs for extracting from large data sets and image sequences certain mixtures of probability distributions, such as multivariate Gaussians, to represent the important features and their mutual correlations needed for accurate document retrieval from databases. This note describes a method to use information geometric methods for distance measures between distributions in mixtures of arbitrary multivariate Gaussians. There is no general analytic solution for the information geodesic distance between two k-variate Gaussians, but for many purposes the absolute information distance may not be essential and comparative values suffice for proximity testing and document retrieval. Also, for two mixtures of different multivariate Gaussians we must resort to approximations to incorporate the weightings. In practice, the relation between a reasonable approximation and a true geodesic distance is likely to be monotonic, which is adequate for many applications. Here we consider some choices for the incorporation of weightings in distance estimation and provide illustrative results from simulations of differently weighted mixtures of multivariate Gaussians

Information geometry for control of some stochastic processes

A basic requirement in control systems is a metric that measures discrepancies between actual and desired states. For statistically influenced systems information geometric methods provide natural Riemannian metrics on smooth spaces of states; such manifolds arise in minimum-phase linear systems and multi-input systems with known stochastic noise. Commonly recurring practical situations are `nearly' Poisson or `nearly' Uniform with a complementarity in the geometry of these two; another involves multivariate Gaussians and their mixtures. Similarly we encounter `nearly' independent Poisson, and `nearly' independent Gaussian processes. For such cases we have information geometric results and examples. Some of these methods are applicable to control systems for statistically influenced processes, such as monitoring essential features in continuous production of threads, films, foils and fibre networks, and batch processing of stochastic textures