Mathematics, Department of
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The Department of Mathematics has built a strong reputation for innovation and leadership in targeted areas of pure and applied mathematics. Indicators of success are:

- In Research Assessment Exercise 2001 the Department received ranking 5 in both pure and applied mathematics. Within the mathematical field, only a small number received a coveted 5/5*-rating which is characterised by forefront position within the international academic community 5-ranking in each unit.

- External funding: a stream of grants have been obtained in recent years in all main subject areas. In 2006, our Department was the best funded mathematics department per capita from EPSRC.

- Interdisciplinary research, linking mathematics with biology, chemistry, engineering, geology and physics. The Centre for Mathematical Modelling (MMC) coordinates this type of activity.

- Organisation of numerous international conferences and workshops both at Leicester and elsewhere.

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Recent Submissions

A circular order on edge-coloured trees and RNA m-diagrams

We study a circular order on labelled, m-edge-coloured trees with k vertices, and show that the set of such trees with a fixed circular order is in bijection with the set of RNA m-diagrams of degree k, combinatorial objects which can be regarded as RNA secondary structures of a certain kind. We enumerate these sets and show that the set of trees with a fixed circular order can be characterized as an equivalence class for the transitive closure of an operation which, in the case m=3, arises as...

Marsh, Robert J.; Schroll, Sibylle

Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces

In the context of representation theory of finite dimensional algebras, string algebras have been extensively studied and almost all aspects of their representation theory are well-understood. One exception to this is the classification of extensions between indecomposable modules. In this paper we explicitly describe such extensions for a class of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. W...

Canakci, Ilke; Schroll, Sibylle

Trivial Extensions of Gentle Algebras and Brauer Graph Algebras

We show that two well-studied classes of tame algebras coincide: namely, the class of symmetric special biserial algebras coincides with the class of Brauer graph algebras. We then explore the connection between gentle algebras and symmetric special biserial algebras by explicitly determining the trivial extension of a gentle algebra by its minimal injective co-generator. This is a symmetric special biserial algebra and hence a Brauer graph algebra of which we explicitly give the Brauer graph...

Schroll, Sibylle

The geometry of Brauer graph algebras and cluster mutations

In this paper we establish a connection between ribbon graphs and Brauer graphs. As
a result, we show that a compact oriented surface with marked points gives rise to a unique Brauer
graph algebra up to derived equivalence. In the case of a disc with marked points we show that a dual
construction in terms of dual graphs exists. The rotation of a diagonal in an m-angulation gives rise
to a Whitehead move in the dual graph, and we explicitly construct a tilting complex on the related
Braue...

Marsh, Robert J.; Schroll, Sibylle

A circular order on edge-coloured trees and RNA m-diagrams

We study a circular order on labelled, m-edge-coloured trees with k vertices, and show that the set of such trees with a fixed circular order is in bijection with the set of RNA m-diagrams of degree k , combinatorial objects which can be regarded as RNA secondary structures of a certain kind. We enumerate these sets and show that the set of trees with a fixed circular order can be characterized as an equivalence class for the transitive closure of an operation which, in the case m=3, arises ...

Marsh, Robert J.; Schroll, Sibylle

The Ext algebra of a Brauer graph algebra

In this paper we study finite generation of the Ext algebra of a Brauer graph algebra by determining the degrees of the generators. As a consequence we characterize the Brauer graph algebras that are Koszul and those that are K_2.

Green, Edward L.; Schroll, Sibylle; Snashall, Nicole; Taillefer, Rachel

Group actions and coverings of Brauer graph algebras

We develop a theory of group actions and coverings on Brauer graphs that parallels
the theory of group actions and coverings of algebras. In particular, we show that any Brauer
graph can be covered by a tower of coverings of Brauer graphs such that the topmost covering has
multiplicity function identically one, no loops, and no multiple edges. Furthermore, we classify
the coverings of Brauer graph algebras that are again Brauer graph algebras.

Green, E. L.; Schroll, Sibylle; Snashall, Nicole

Gaussian process regression with multiple response variables

Gaussian process regression (GPR) is a Bayesian non-parametric technology that has
gained extensive application in data-based modelling of various systems, including
those of interest to chemometrics. However, most GPR implementations model only a
single response variable, due to the difficulty in the formulation of covariance function
for correlated multiple response variables, which describes not only the correlation
between data points, but also the correlation between responses. In t...

Wang, Bo; Chen, Tau

n-Fold groupoids, n-types and n-track categories

For each n ≥ 1, we introduce two new Segal-type models of ntypes
of topological spaces: weakly globular n-fold groupoids, and a lax version
of these. We show that any n-type can be represented up to homotopy by
such models via an explicit algebraic fundamental n-fold groupoid functor.
We compare these models to Tamsamani’s weak n-groupoids, and extract from
them a model for (k − 1)-connected n-types.

Blanc, David; Paoli, Simona

Minimum Distance Estimation of Milky Way Model Parameters and Related Inference

We propose a method to estimate the location of the Sun in the disk of the Milky Way using a
method based on the Hellinger distance and construct confidence sets on our estimate of the unknown
location using a bootstrap based method. Assuming the Galactic disk to be two-dimensional, the
sought solar location then reduces to the radial distance separating the Sun from the Galactic center
and the angular separation of the Galactic center to Sun line, from a pre-fixed line on the disk. On
a...

Inverse Bayesian Estimation of Gravitational Mass Density in Galaxies from Missing Kinematic Data

In this paper, we focus on a type of inverse problem in which the data are expressed as an unknown function of
the sought and unknown model function (or its discretised representation as a model parameter vector). In particular,
we deal with situations in which training data are not available. Then we cannot model the unknown
functional relationship between data and the unknown model function (or parameter vector) with a Gaussian
Process of appropriate dimensionality. A Bayesian method ba...

Chakrabarty, Dalia; Saha, P.

Bayesian Density Estimation via Multiple Sequential Inversions of 2-D Images with Application in Electron Microscopy

We present a new Bayesian methodology to learn the unknown material density of
a given sample by inverting its two-dimensional images that are taken with a Scanning Electron
Microscope. An image results from a sequence of projections of the convolution of the density
function with the unknown microscopy correction function that we also learn from the data;
thus learning of the unknowns demands multiple inversions. We invoke a novel design of experiment,
involving imaging at multiple valu...

Chakrabarty, Dalia; Rigat, F.; Gabrielyan, N.; Beanland, R.; Paul, S.

Bayesian Learning of Material Density Function by Multiple Sequential Inversions of 2-D Images in Electron Microscopy

We discuss a novel inverse problem in which the data is generated by the sequential contractive projections
of the convolution of two unknown functions, both of which we aim to learn. The method is illustrated
using an application that relates to the multiple inversions of image data recorded with a Scanning Electron
Microscope, with the aim of learning the density of a given material sample and the microscopy correction
function. Given the severe logistical difficulties in this applicati...

Chakrabarty, Dalia; Paul, S.

Simple Locally Finite Lie Algebras of Diagonal Type

We discuss various characterizations of simple locally finite Lie algebras of diagonal type over an algebraically closed field of characteristic zero.

Baranov, Alexander

On time scale invariance of random walks in confined space.

Animal movement is often modelled on an individual level using simulated random walks. In such applications it is preferable that the properties of these random walks remain consistent when the choice of time is changed (time scale invariance). While this property is well understood in unbounded space, it has not been studied in detail for random walks in a confined domain. In this work we undertake an investigation of time scale invariance of the drift and diffusion rates of Brownian random ...

Bearup, Daniel; Petrovskii, Sergei

Modelling biological invasions : population cycles, waves and time delays

Biological invasions are rapidly gaining importance due to the ever-increasing number
of introduced species. Alongside the plenitude of empirical data on invasive
species there exists an equally broad range of mathematical models that might be
of use in understanding biological invasions.
This thesis aims to address several issues related to modelling invasive species
and provide insight into their dynamics. Part I (Chapter 2) documents a case
study of the gypsy moth, Lymantria dispar, ...

Jankovic, Masha

Special functions and generalized functions

In 1950, Laurent Schwartz marked a convenient starting point for the theory of generalized functions as a subject in its own right. He developed and unified much of the earlier work by Hadamard, Bochner, Sobolev and others. Since then an enormous literature dealing with both theory and applications has grown up, and the subject has undergone extensive further development. The original Schwartz treatment defined a distribution as a linear continuous functional on a space of test functions.;Thi...

Al-Sirehy, Fatma.

Ancient Egyptian astronomy : timekeeping and cosmography in the New Kingdom

The first part of this study analyses and discusses astronomical timekeeping methods used in the New Kingdom. Diagonal star clocks are examined first, looking at classification of sources, decan lists, and the updating of the tables over time. The date list in the Osireion at Abydos is discussed, and issues concerning its place in the history of astronomical timekeeping are raised. The final stellar timekeeping method, the Ramesside star clock, is then examined. The conventional interpretatio...

Symons, Sarah.

Data structures and implementation of an adaptive hp finite element method

For a fully adaptive hp finite element programme to be implemented it is necessary to implement n-irregular meshes efficiently. This requires a sufficiently flexible data structure to be implemented. Because such flexibility is required, the traditional array based approach cannot be used because of its limited applicability. In this thesis this traditional approach has been replaced by an object orientated design and implementation. This leads to an implementation that can be extended ea...

Senior, Bill.

On finite groups of p-local rank one and a conjecture of Robinson

We use the classification of finite simple groups to verify a conjecture of Robinson for finite groups G where G/Op(G) has trivial intersection Sylow p-subgroups. Groups of this type are said to have p-local rank one, and it is hoped that this invariant will eventually form the basis for inductive arguments, providing reductions for the conjecture, or even a proof using the results presented here as a base. A positive outcome for Robinson's conjecture would imply Alperin's weight conjecture...