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

Delay driven spatiotemporal chaos in single species population dynamics models

Questions surrounding the prevalence of complex population dynamics form one of the central themes in ecology. Limit cycles and spatiotemporal chaos are examples that have been widely recognised theoretically, although their importance and applicability to natural populations remains debatable. The ecological processes underlying such dynamics are thought to be numerous, though there seems to be consent as to delayed density dependence being one of the main driving forces. Indeed, time delay ...

How animals move along? Exactly solvable model of superdiffusive spread resulting from animal's decision making

Patterns of individual animal movement have been a focus of considerable attention recently. Of particular interest is a question how different macroscopic properties of animal dispersal result from the stochastic processes occurring on the microscale of the individual behavior. In this paper, we perform a comprehensive analytical study of a model where the animal changes the movement velocity as a result of its behavioral response to environmental stochasticity. The stochasticity is assumed ...

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

Towards roughness-based drag reduction in cross-flow dominated flows

Recent theoretical results are presented from our ongoing study investigating the distinct convective instability properties of the boundary-layer flow over rough rotating disks. In this study, radial anisotropic surface roughness (concentric grooves) is modelled using the partial-slip approach of Miklavčič & Wang (2004) and the surface-geometry approach of Yoon et. Al (2007). An energy analysis reveals that for both instability modes, the main contributors to the energy balance are the ener...

Garrett, Sephen J.; Cooper, A. J.; Ozkan, M.; Thomas, P. J.

An energy analysis of convective instabilities of the Bödewadt and Ekman boundary layers over rough surfaces

An energy balance equation for the three-dimensional Bödewadt and Ekman layers of the so called “BEK family" of rotating boundary-layer flows is derived. A Chebyshev discretisation method is used to solve the equations and investigate the effect of surface roughness on the physical mechanisms of transition. All roughness types lead to a stabilization of the Type I (cross-flow) instability mode for both flows, with the exception of azimuthally-anisotropic roughness (radial grooves) within the ...

Alveroglu, B.; Segalini, A.; Garrett, Stephen J.

On a fixed point theorem of Greguš

We consider two selfmaps T and I of a closed convex subset C of a Banach space X which are weakly commuting in X, i.e.
‖TIx−ITx‖≤‖Ix−Tx‖ for any x in X,
and satisfy the inequality
‖Tx−Ty‖≤a‖Ix−Iy‖+(1−a)max{‖Tx−Ix‖,‖Ty−Iy‖}
for all x, y in C, where 0<a<1. It is proved that if I is linear and non-expansive in C and such that IC contains TC, then T and I have a unique common fixed point in C.

Fisher, Brian; Sessa, S.

On common fixed points of weakly commuting mappings and set-valued mappings

Our main theorem establishes the uniqueness of the common fixed point of two set-valued mappings and of two single-valued mappings defined on a complete metric space, under a contractive condition and a weak commutativity concept. This improves a theorem of the second author.

Sessa, S.; Fisher, B.

On a fixed point theorem of Pathak

It is shown that the continuity of the mapping in Pathak's fixed point theorem for normed spaces is not necessary.

Fisher, Brian

Common fixed point theorems for compatible mappings

By using a compatibility condition due to Jungck we establish some common fixed point theorems for four mappings on complete and compact metric spaces These results also generalize a theorem of Sharma and Sahu.

Taş, K.; Telci, M.; Fisher, Brian

Coincidence theorems for nonlinear hybrid contractions

In this paper, we give some common fixed point theorems for single-valued mappings and multi-valued mappings satisfying a rational inequality. Our theorems generalize some results of B. Fisher, M. L. Diviccaro et al. and V. Popa.

Cho, Y. J.; Fisher, B.; Genga, G. S.

Related fixed point theorems on two complete and compact metric spaces

A new related fixed point theorem on two complete metric spaces is obtained. A generalization is given for two compact metric spaces.

Namdeo, R. K.; Tiwari, N. K.; Fisher, B.; Taş, K.

A note on commutativity of nonassociative rings

A theorem on commutativity of nonassociate ring is given.

Khan, M. S. S.

A Generalization of Prešić Type Mappings in Metric-Like Spaces

We generalize the result of Prešić in metric-like spaces by proving some common fixed point theorems for Prešić type mappings in metric-like spaces. An example is given which shows that the generalization is proper.

Shukla, S.; Fisher, Brian

Phacoemulsification Surgery in Eyes with Neovascular Age-Related Macular Degeneration

Purpose. To evaluate the visual outcomes and effect of phacoemulsification surgery on the progression of neovascular age-related macular degeneration (AMD). Methods. Retrospective, noncomparative, and interventional case series. Thirty eyes from 29 subjects with neovascular AMD treated with intravitreal antivascular endothelial growth factor (VEGF) injections who underwent phacoemulsification and had a postsurgery follow-up of 6 months were included. LogMAR best corrected visual acuity (BCVA)...

Grixti, A.; Papavasileiou, E.; Cortis, Dominic; Kumar, B. V.; Prasad, S.

Stability modes in vortex structure formation: Canonical examples for rotating components

Three rather different physical cases have been studied. All represent very practical geometries for which the modal behavior of vortex structures is not completely understood. The work on these problems is ongoing with the objective of obtaining physical confirmation, enhanced understanding and predictive capability for the vortex structures encountered in rotating machines.

Gostelow, J. Paul; Garrett, Stephen J.; Rona, Aldo; Adebayo, David S.

Cohomology of tiling spaces: beyond primitive substitutions

This thesis explores the combinatorial and topological properties of tiling spaces
associated to 1-dimensional symbolic systems of aperiodic type and their associated
algebraic invariants. We develop a framework for studying systems which are more
general than primitive substitutions, naturally partitioned into two instances: in the
first instance we allow systems associated to sequences of substitutions of primitive
type from a finite family of substitutions (called mixed substitutions)...

Rust, Daniel George

Tiling spaces, codimension one attractors and shape

We establish a close relationship between, on the one hand, expanding, codimension one attractors of diffeomorphisms on closed manifolds (examples of so-called strange attractors), and, on the other, spaces which arise in the study of aperiodic tilings. We show that every such orientable attractor is homeomorphic to a tiling space of either a substitution or a projection tiling, depending on its dimension. We also demonstrate that such an attractor is shape equivalent to a (d+1)-dimensional t...

Clark, Alexander; Hunton, J.

From Ambiguity Aversion to a Generalized Expected Utility. Modeling Preferences in a Quantum Probabilistic Framework

Ambiguity and ambiguity aversion have been widely studied in decision theory and economics both at a theoretical and an experimental level. After Ellsberg's seminal studies challenging subjective expected utility theory (SEUT), several (mainly normative) approaches have been put forward to reproduce ambiguity aversion and Ellsberg-type preferences. However, Machina and other authors have pointed out some fundamental difficulties of these generalizations of SEUT to cope with some variants of E...

Aerts, D.; Sozzo, Sandro

A classification of the point spectrum of constant length substitution tiling spaces and general fixed point theorems for tilings

We examine the point spectrum of the various tiling spaces that result from
different choices of tile lengths for substitution of constant length on a two or a three letter
alphabet. In some cases we establish insensitivity to changes in length. In a wide range
of cases, we establish that the typical choice of length leads to trivial point spectrum.
We also consider a problem related to tilings of the integers and their connection to fixed
point theorems. Using this connection, we prove ...

Abuzaid, Dina Asaad

The centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free stream

In this study, a new centrifugal instability mode, which dominates within the boundary-layer flow over a slender rotating cone in still fluid, is used for the first time to model the problem within an enforced oncoming axial flow. The resulting problem necessitates an updated similarity solution to represent the basic flow more accurately than previous studies in the literature. The new mean flow field is subsequently perturbed, leading to disturbance equations that are solved via numerical a...

Hussain, Z.; Garrett, Stephen J.; Stephen, S. O.; Griffiths, Paul Travis

The neutral curve for stationary disturbances in rotating disk flow for power-law fluids

This paper is concerned with the convective instabilities associated with the boundary-layer flow due to a rotating disk. Shear-thinning fluids that adhere to the power-law relationship are considered. The neutral curves are computed using a sixth-order system of linear stability equations which include the effects of streamline curvature, Coriolis force and the non-Newtonian viscosity model. Akin to previous Newtonian studies it is found that the neutral curves have two critical values, thes...

Griffiths, P. T.; Garrett, Stephen John; Stephen, S. O.