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

Detailed balance in micro- and macrokinetics and micro-distinguishability of macro-processes

We develop a general framework for the discussion of detailed balance and analyse its microscopic background. We find that there should be two additions to the well-known T- or PT-invariance of the microscopic laws of motion:
1. Equilibrium should not spontaneously break the relevant T- or PT-symmetry.
2. The macroscopic processes should be microscopically distinguishable to guarantee persistence of detailed balance in the model reduction from micro- to macrokinetics.
We briefly discuss ...

Gorban, A. N.

Multilevel sparse grid kernels collocation with radial basis functions for elliptic and parabolic problems

Radial basis functions (RBFs) are well-known for the ease implementation as
they are the mesh-free method [31, 37, 71, 72]. In this thesis, we modify the
multilevel sparse grid kernel interpolation (MuSIK) algorithm proposed in [48]
for use in Kansa’s collocation method (referred to as MuSIK-C) to solve elliptic
and parabolic problems. The curse of dimensionality is a significant challenge
in high dimension approximation. A full grid collocation method requires O(Nd)
nodal points to con...

Zhao, Yangzhang

Dynamic Cooperative Investment

In this thesis we develop dynamic cooperative investment schemes in discrete and
continuous time. Instead of investing individually, several agents may invest joint
capital into a commonly agreed trading strategy, and then split the uncertain
outcome of the investment according to the pre-agreed scheme, based on their
individual risk-reward preferences. As a result of cooperation, each investor is able
to get a share, which cannot be replicated with the available market instruments,
and...

Almualim, Anwar Hassan Ali

Gaussian Process and Functional Data Methods for Mortality Modelling

Modelling the demographic mortality trends is of great importance due to its considerable impact on welfare policy, resource allocation and government planning. In this thesis, we propose to use various statistical methods, including Gaussian process (GP), principal curve, multilevel functional principal component analysis (MFPCA) for forecasting and clustering of human mortality data. This thesis is actually composed of three main topics regarding mortality modelling. In the first topic, we ...

Wu, Ruhao

Discontinuous Galerkin Methods on Polytopic Meshes

This thesis is concerned with the analysis and implementation of the hp-version
interior penalty discontinuous Galerkin finite element method (DGFEM) on computational
meshes consisting of general polygonal/polyhedral (polytopic) elements.
Two model problems are considered: general advection-diffusion-reaction boundary
value problems and time dependent parabolic problems. New hp-version a
priori error bounds are derived based on a specific choice of the interior penalty
parameter which a...

Dong, Zhaonan

Efficient Option Pricing under Levy Processes, with CVA and FVA

We generalize the Piterbarg [1] model to include (1) bilateral default risk as in Burgard and Kjaer [2], and (2) jumps in the dynamics of the underlying asset using general classes of Lévy processes of exponential type. We develop an efficient explicit-implicit scheme for European options and barrier options taking CVA-FVA into account. We highlight the importance of this work in the context of trading, pricing and management a derivative portfolio given the trajectory of regulations.

Shek, C. K.; Law, J.; Levendorskiĭ, Sergei

Quantifying non-Newtonian effects in rotating boundary-layer flows

The stability of the boundary-layer on a rotating disk is considered for fluids that adhere to a non-Newtonian governing viscosity relationship. For fluids with shear-rate dependent viscosity the base flow is no longer an exact solution of the Navier–Stokes equations, however, in the limit of large Reynolds number the flow inside the three-dimensional boundary-layer can be determined via a similarity solution. The convective instabilities associated with flows of this nature are described bot...

Griffiths, P. T.; Garrett, S. J.; Stephen, S. O.; Hussain, Z.

Representations of Quantum Conjugacy Classes of Non-Exceptional Quantum Groups

Let G be a complex non-exceptional simple algebraic group and g its Lie algebra. With every point x of the maximal torus T ʗ G we associate a highest weight module Mx over the Drinfeld-Jimbo quantum group Uq(g) and an equivariant quantization of the conjugacy class of x by operators in End(Mx). These equivariant quantizations are isomorphic for x lying on the same orbit of the Weyl group, and Mx support different exact representations of the same quantum conjugacy class.
This recovers all qu...

Ashton, Thomas Stephen

Comparison of the effects of surface roughness and confinement on rotor–stator cavity flow

Results of a computational study are discussed which investigate roughness-induced and geometry-induced (confinement) effects on the steady-state velocity components in 3-D boundary-layer flow over the rotor disc in a rotor–stator flow configuration. It is found that, for the rotor–stator flow investigated, the roughness-induced effects are very similar to geometry-induced effects, both in nature and magnitude. The overall aim was to compare these two types of effects with corresponding rough...

Özkan, M.; Thomas, P. J.; Cooper, A. J.; Garrett, Stephen John

On the diagonal subalgebra of an Ext algebra

Let R be a Koszul algebra over a field k and M be a linear R-module. We study a graded subalgebra ΔM of the Ext-algebra ExtR⁎(M,M) called the diagonal subalgebra and its properties. Applications to the Hochschild cohomology ring of R and to periodicity of linear modules are given. Viewing R as a linear module over its enveloping algebra, we also show that ΔR is isomorphic to the graded center of the Koszul dual of R. When R is selfinjective and not necessarily graded, we study connections bet...

Green, E. L.; Snashall, Nicole Jane; Solberg, O.; Zacharia, D.

Special multiserial algebras are quotients of symmetric special multiserial algebras

In this paper we give a new definition of symmetric special multiserial algebras in terms of defining cycles. As a consequence, we show that every special multiserial algebra is a quotient of a symmetric special multiserial algebra.

Green, E. L.; Schroll, Sibylle

Hourglass stabilization and the virtual element method

In this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of inline image-continuous VEMs on polygonal and polyhedral meshes are ...

Cangiani, A.; Manzini, G.; Russo, A.; Sukumar, N.

On the stability of continuous-discontinuous Galerkin methods for advection-diffusion-reaction problems

We consider a finite element method which couples the continuous Galerkin method away from internal and boundary layers with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. The stability properties of the coupled method are illustrated with a numerical exper...

Cangiani, Andrea; Chapman, J.; Georgoulis, Emmanuil; Jensen, M.

Étale homotopy types of moduli stacks of polarised abelian schemes

We determine the Artin–Mazur étale homotopy types of moduli stacks of polarised abelian schemes using transcendental methods and derive some arithmetic properties of the étale fundamental groups of these moduli stacks. Finally we analyse the Torelli morphism between the moduli stacks of algebraic curves and principally polarised abelian schemes from an étale homotopy point of view.

Frediani, P.; Neumann, Frank

Geometry of moduli stacks of (k, l)-stable vector bundles over algebraic curves

We study the geometry of the moduli stack of vector bundles of fixed rank and degree over an algebraic curve by introducing a filtration made of open substacks build from (k,l)-stable vector bundles. The concept of (k,l)-stability was introduced by Narasimhan and Ramanan to study the geometry of the coarse moduli space of stable bundles. We will exhibit the stacky picture and analyse the geometric and cohomological properties of the moduli stacks of (k,l)-stable vector bundles. For particular...

Mata-Gutiérrez, O.; Neumann, Frank

A New Bayesian Test to test for the Intractability-Countering Hypothesis

We present a new test of hypothesis in which we seek the probability of the null conditioned on the data, where the null is a simplification undertaken to counter the intractability of the more complex model, that the simpler null model is nested within. With the more complex model rendered intractable, the null model uses a simplifying assumption that capacitates the learning of an unknown parameter vector given the data. Bayes factors are shown to be known only up to a ratio of unknown data...

Chakrabarty, Dalia

Pattern, process, scale, and model's sensitivity: Comment on "Phase separation driven by density-dependent movement: A novel mechanism for ecological patterns" by Quan-Xing Liu et al.

Spatial distribution of ecological populations is rarely homogeneous. Typically, the population density exhibits considerable variability of space, in an extreme yet not uncommon case creating a “patchy” pattern where areas of high population density alternate with areas where the population density is much lower or close to zero [1]. This phenomenon, often generically referred to as ecological patterning or ecological pattern formation, has long been a focus of interest in ecology and a numb...

Petrovskii, Sergei

Quantifying uncertainty in partially specified biological models: How can optimal control theory help us?

Mathematical models in biology are highly simplified representations of a complex underlying reality and there is always a high degree of uncertainty with regards to model function specification. This uncertainty becomes critical for models in which the use of different functions fitting the same dataset can yield substantially different predictions-a property known as structural sensitivity. Thus, even if the model is purely deterministic, then the uncertainty in the model functions carries ...

Adamson, M. W.; Morozov, A. Y.; Kuzenkov, O. A.

Piece-wise quadratic approximations of arbitrary error functions for fast and robust machine learning

Most of machine learning approaches have stemmed from the application of minimizing the mean squared distance principle, based on the computationally efficient quadratic optimization methods. However, when faced with high-dimensional and noisy data, the quadratic error functionals demonstrated many weaknesses including high sensitivity to contaminating factors and dimensionality curse. Therefore, a lot of recent applications in machine learning exploited properties of non-quadratic error func...

Gorban, A. N.; Mirkes, E. M.; Zinovyev, A.

R-matrix and inverse Shapovalov form

We construct the inverse Shapovalov form of a simple complex quantum group from its universal R-matrix based on a generalized Nagel-Moshinsky approach to lowering operators. We establish a connection between this algorithm and the ABRR equation for dynamical twist.