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

Adaptive discontinuous Galerkin methods for nonlinear parabolic problems

This work is devoted to the study of a posteriori error estimation and adaptivity
in parabolic problems with a particular focus on spatial discontinuous Galerkin
(dG) discretisations.
We begin by deriving an a posteriori error estimator for a linear non-stationary
convection-diffusion problem that is discretised with a backward Euler dG method.
An adaptive algorithm is then proposed to utilise the error estimator. The
effectiveness of both the error estimator and the proposed algorithm ...

Metcalfe, Stephen Arthur

Is it possible to predict long-term success with k-NN? Case study of four market indices (FTSE100, DAX, HANGSENG, NASDAQ)

This case study tests the possibility of prediction for 'success' (or 'winner') components of four stock & shares market indices in a time period of three years from 02-Jul-2009 to 29-Jun-2012.We compare their performance ain two time frames: initial frame three months at the beginning (02/06/2009-30/09/2009) and the final three month frame (02/04/2012-29/06/2012).To label the components, average price ratio between two time frames in descending order is computed. The average price ratio is d...

Shi, Y.; Gorban, A. N.; Yang, T. Y.

Multiscale principal component analysis

Principal component analysis (PCA) is an important tool in exploring data. The conventional approach to PCA leads to a solution which favours the structures with large variances. This is sensitive to outliers and could obfuscate interesting underlying structures. One of the equivalent definitions of PCA is that it seeks the subspaces that maximize the sum of squared pairwise distances between data projections. This definition opens up more flexibility in the analysis of principal components w...

Akinduko, A. A.; Gorban, Alexander N.

Multiscale approach to pest insect monitoring: Random walks, pattern formation, synchronization, and networks

Pest insects pose a significant threat to food production worldwide resulting in annual losses worth hundreds of billions of dollars. Pest control attempts to prevent pest outbreaks that could otherwise destroy a sward. It is good practice in integrated pest management to recommend control actions (usually pesticides application) only when the pest density exceeds a certain threshold. Accurate estimation of pest population density in ecosystems, especially in agro-ecosystems, is therefore ver...

Petrovskii, Sergei; Petrovskaya, N.; Bearup, Daniel

Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration.

Monitoring of pest insects is an important part of the integrated pest management. It aims to provide information about pest insect abundance at a given location. This includes data collection, usually using traps, and their subsequent analysis and/or interpretation. However, interpretation of trap count (number of insects caught over a fixed time) remains a challenging problem. First, an increase in either the population density or insects activity can result in a similar increase in the num...

Bearup, D.; Petrovskaya, N.; Petrovskii, Sergei

Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect

One of the main challenges in ecology is to determine the cause of population fluctuations. Both theoretical and empirical studies suggest that delayed density dependence instigates cyclic behavior in many populations; however, underlying mechanisms through which this occurs are often difficult to determine and may vary within species. In this paper, we consider single species population dynamics affected by the Allee effect coupled with discrete time delay. We use two different mathematical ...

Jankovic, Masha; Petrovskii, Sergei

On the composition of the distributions x-s+ lnmx+ and xμ+

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {Fn(f)}, where Fn(x) = F(x)*δn(x) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x-s + lnm x+ and xμ + is proved to exist and be equal to μmx-sμ + lnm x+ for μ > 0 and s,m = 1, 2,....

Fisher, Brian

New Langevin and Gradient Thermostats for Rigid Body Dynamics

We introduce two new thermostats, one of Langevin type and one of gradient (Brownian) type, for rigid body dynamics. We formulate rotation using the quaternion representation of angular coordinates; both thermostats preserve the unit length of quaternions. The Langevin thermostat also ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have constructed three geometric numerical inte...

Davidchack, Ruslan L.; Ouldridge, T. E.; Tretyakov, M. V.

Segal-type algebraic models of n-types

For each n ≥ 1, we introduce two new Segal-type models of n-types 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, D.; Paoli, Simona

The weakly globular double category of fractions of a category

This paper introduces the construction of a weakly globular double category of fractions for a category and studies its universal properties. It shows that this double category is locally small and considers a couple of concrete examples.

Paoli, Simona; Pronk, D.

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...