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

Time Dependent Diffusion as a Mean Field Counterpart of Levy Type Random Walk

Insect trapping is commonly used in various pest insect monitoring programs as well as in many ecological field studies. An individual is said to be trapped if it falls within a well defined capturing zone, which it cannot escape. The accumulation of trapped individuals over time forms trap counts or alternatively, the flux of the population density into the trap. In this paper, we study the movement of insects whose dynamics are governed by time dependent diffusion and Lévy walks. We demonst...

Ahmed, D. A.; Petrovskii, S.

Mathematical Modelling of Spatiotemporal Dynamics of Oxygen in a Plankton System

Oxygen production due to phytoplankton photosynthesis is a crucial phenomenon underlying the dynamics of marine ecosystems. However, most of the existing literature focus on other aspects of the plankton community functioning, thus leaving the issue of the coupled oxygen-plankton dynamics understudied. In this paper, we consider a generic model of the oxygen-phytoplankton-zooplankton dynamics to make an insight into the basic properties of the plankton-oxygen interactions. The model is analyz...

Sekerci, Y.; Petrovskii, S.

Statistical mechanics of animal movement: Animals's decision-making can result in superdiffusive spread

Peculiarities of individual animal movement and dispersal have been a major focus of recent research as they are thought to hold the key to the understanding of many phenomena in spatial ecology. Superdiffusive spread and long-distance dispersal have been observed in different species but the underlying biological mechanisms often remain obscure. In particular, the effect of relevant animal behavior has been largely unaddressed. In this paper, we show that a superdiffusive spread can arise na...

Tilles, Paulo F. C.; Petrovskii, Sergei V.

The Role of Host and Microbial Factors in the Pathogenesis of Pneumococcal Bacteraemia Arising from a Single Bacterial Cell Bottleneck

The pathogenesis of bacteraemia after challenge with one million pneumococci of three isogenic variants was investigated. Sequential analyses of blood samples indicated that most episodes of bacteraemia were monoclonal events providing compelling evidence for a single bacterial cell bottleneck at the origin of invasive disease. With respect to host determinants, results identified novel properties of splenic macrophages and a role for neutrophils in early clearance of pneumococci. Concerning ...

Gerlini, A.; Colomba, L.; Furi, L.; Braccini, T.; Manso, A. S.... et al.

Feeding on Multiple Sources: Towards a Universal Parameterization of the Functional Response of a Generalist Predator Allowing for Switching

Understanding of complex trophic interactions in ecosystems requires correct descriptions of the rate at which predators consume a variety of different prey species. Field and laboratory data on multispecies communities are rarely sufficient and usually cannot provide an unambiguous test for the theory. As a result, the conventional way of constructing a multi-prey functional response is speculative, and often based on assumptions that are difficult to verify. Predator responses allowing for ...

Morozov, Andrew; Petrovskii, Sergei

Revisiting the Role of Individual Variability in Population Persistence and Stability

Populations often exhibit a pronounced degree of individual variability and this can be important when constructing ecological models. In this paper, we revisit the role of inter-individual variability in population persistence and stability under predation pressure. As a case study, we consider interactions between a structured population of zooplankton grazers and their predators. Unlike previous structured population models, which only consider variability of individuals according to the a...

Morozov, Andrew; Pasternak, A. F.; Arashkevich, E. G.

General H-theorem and Entropies that Violate the Second Law

H-theorem states that the entropy production is nonnegative and, therefore, the entropy of a closed system should monotonically change in time. In information processing, the entropy production is positive for random transformation of signals (the information processing lemma). Originally, the H-theorem and the information processing lemma were proved for the classical Boltzmann-Gibbs-Shannon entropy and for the correspondent divergence (the relative entropy). Many new entropies and divergenc...

Gorban, Alexander N.

On Multiple Convolutions and Time Scales

The properties of the multiple Laplace transform and convolutions on a time scale are studied. Further, some related results are also obtained by utilizing the double Laplace transform. We also provide an example in order to illustrate the main result.

Eltayeb, Hassan; Kılıçman, Adem; Fisher, Brian

Adaptive radial basis functions for option pricing

In this thesis, we have developed meshless adaptive radial basis functions (RBFs)
method for the pricing of financial contracts by solving the Black-Scholes partial
differential equation (PDE). In the 1-D problem, we priced the financial contracts
of a European call option, Greeks (Delta, Gamma and Vega), an American put
option and a barrier up and out call option with this method. In the BENCHOP
project with Challenge Parameter Set (Parameter Set 2) [97], we have shown
that our adaptiv...

Li, Juxi

Inequalities and eigenvalues of Sturm-Liouville problems near a singular boundary

We study the behavior of eigenvalues of Sturm-Liouville problems (SLP) when an endpoint of the underlying interval approaches a singularity.

Marletta, Marco; Everitt, W. N.; Zettl, A.

Model Reductions in Biochemical Reaction Networks

Many complex kinetic models in the field of biochemical reactions
contain a large number of species and reactions. These models often require a
huge array of computational tools to analyse. Techniques of model reduction,
which arise in various theoretical and practical applications in systems biology,
represent key critical elements (variables and parameters) and substructures of
the original system. This thesis aims to study methods of model reduction for
biochemical reaction networks....

Khoshnaw, Sarbaz Hamza Abdullah

Approximation with Random Bases: Pro et Contra

In this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a Hilbert space of functions. We consider and analyze published procedures, both randomized and deterministic, for selecting elements from these families that have been shown to ensure the rate of convergence in $L_2$ norm of order $O(1/N)$, where $N$ is the number of elements. We show that both strategies are successful providing that addit...

Gorban, Alexander N.; Tyukin, Ivan Yu.; Prokhorov, D. V.; Sofeikov, Konstantin I.

Derivative pricing in lévy driven models

We consider an important class of derivative contracts written on multiple assets
which are traded on a wide range of financial markets. More specifically, we are
interested in developing novel methods for pricing financial derivatives using approximation
theoretic methods which are not well-known to the financial engineering
community. The problem of pricing of such contracts splits into two parts.
First, we need to approximate the respective density function which depends on the
adapt...

Kushpel, Alexander

Leaders do not look back, or do they?

We study the effect of adding to a directed chain of interconnected systems a
directed feedback from the last element in the chain to the first. The problem is closely related
to the fundamental question of how a change in network topology may influence the behavior of
coupled systems. We begin the analysis by investigating a simple linear system. The matrix that
specifies the system dynamics is the transpose of the network Laplacian matrix, which codes
the connectivity of the network. O...

Gorban, A. N.; Jarman, N.; Steur, E.; van Leeuwen, C.; Tyukin, I.

The center of a convex set and capital allocation

A capital allocation scheme for a company that has a random total profit Y and uses a coherent risk measure ρ has been suggested. The scheme returns a unique real number Λρ*(X,Y), which determines the capital that should be allocated to company’s subsidiary with random profit X. The resulting capital allocation is linear and diversifying as defined by Kalkbrener (2005). The problem is reduced to selecting the “center” of a non-empty convex weakly compact subset of a Banach space, and the solu...

Grechuk, Bogdan

Computational diagnosis of canine lymphoma

One out of four dogs will develop cancer in their lifetime and 20% of those will be lymphoma cases. PetScreen developed a lymphoma blood test using serum samples collected from several veterinary practices. The samples were fractionated and analysed by mass spectrometry. Two protein peaks, with the highest diagnostic power, were selected and further identified as acute phase proteins, C-Reactive Protein and Haptoglobin. Data mining methods were then applied to the collected data for the devel...

Mirkes, E. M.; Alexandrakis, I.; Slater, K.; Tuli, R.; Gorban, A. N.

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