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

Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory

This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and Möbius inversion, where algebraic identities are realised by taking homotopy cardinality of equivalences of ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in [Figure presented]. Just as the Segal condition expresses composition, the new...

Gálvez-Carrillo, I; Kock, J; Tonks, A

Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness

This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition. In this paper, we introduce various technical conditions on decomposition spaces. Th...

Gálvez-Carrillo, I; Kock, J; Tonks, A

Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees

We prove a Faà di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids.

Gálvez-Carrillo, I; Kock, J; Tonks, A

Dependence Structures and Risk Aggregation Using Copulas

Insurance and reinsurance companies have to calculate solvency capital requirements in order to ensure that they can meet their future obligations to policyholders and beneficiaries. The solvency capital requirement is a risk management tool essential when extreme catastrophic events happen, resulting in high number of possibly interdependent claims. In this thesis, we study the problem of aggregating the risks coming from several insurance lines of business and analyse the effect of reinsura...

Ismail, Isaudin

Simple factor dressing and the López–Ros deformation of minimal surfaces in Euclidean 3-space

The aim of this paper is to investigate a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. We study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that...

Leschke, K.; Moriya, K.

High-Dimensional Bayesian Non-parametric Learning of System Parameters in Different Data Scenarios

The pursuit of the correlation structure of a high-dimensional random construct, underlines my doctoral studies. This thesis reports on the development of methodologies that help undertake learning of functional relationships between variables, given high-dimensional discontinuous data that exhibit non-stationary correlation structure, with such methods tying in with methods needed to undertake such difficult correlation learning–and its possible intuitive graphical representations as network...

Wang, Kangrui

Mathematical Modelling of Population Dynamics in Complex and Fragmented Environments

Understanding the effect of the global environmental change on the dynamics of ecosystems and populations is a major challenge in contemporary science, and mathematical modelling is widely recognised as an efficient research tool to address it. In particular, habitat fragmentation has become a key concern in ecology over the past 20 years as it is thought to increase the threat to biodiversity that causes species extinction worldwide. Mathematical modelling helps to understand the effect of c...

Alharbi, Weam G.

Representations of Crossed Squares and Cat2-Groups

The concept of crossed modules was introduced by J.H.C. Whitehead in the
late 1940s and then Loday [27] reformulated it as cat1-groups. Crossed modules
and cat1-groups are two-dimensional generalisations of a group. Loday showed in
[9] that crossed modules can be understood also as 2-groups. In much the same
way, a higher dimensional analogue of crossed modules, the concept of crossed
squares was introduced by Loday and Guin-Valery [27] and then Arvasi [2] linked
it to the concept of hi...

Al-asady, Jinan

Gaussian process regression method for forecasting of mortality rates

Gaussian process regression (GPR) has long been shown to be a powerful and effective Bayesian nonparametric approach, and has been applied to a wide range of fields. In this paper we present a new application of Gaussian process regression methods for the modelling and forecasting of human mortality rates. The age-specific mortality rates are treated as time series and are modelled by four conventional Gaussian process regression models. Furthermore, to improve the forecasting accuracy we pro...

Wang, Bo; Wu, Ruhao

On the exponent of exponential convergence of p-version FEM spaces

We study the exponent of the exponential rate of convergence in terms of the number of degrees of freedom for various non-standard p-version finite element spaces
employing reduced cardinality basis. More specifically, we show that serendipity
finite element methods and discontinuous Galerkin finite element methods with total
degree Pp basis have a faster exponential convergence with respect to the number of degrees of freedom than their counterparts employing the tensor product Qp
basis ...

Dong, Zhaonan

A comparative thermodynamic analysis of air handling units at variable reference temperature

Ventilation and air conditioning systems are emerging as the major energy consumers in low energy buildings. The objective of this paper is to present new methodology for assessment of Air Handling Units (AHUs) taking into account the variations of reference temperature. The methodology using the concept of coenthalpy, developed for heat exchangers and published by the authors previously has been used. Four AHUs that comprise energy transfer devices, such as: Water-to-Air Heater (WAH), Heat R...

Martinaitis, Vytautas; Streckiene, Giedre; Bagdanavicius, Audrius; Bielskus, Juozas

Gaussian process methods for nonparametric functional regression with mixed predictors

Gaussian process methods are proposed for nonparametric functional regression for both scalar and functional responses with mixed multidimensional functional and scalar predictors. The proposed models allow the response variables to depend on the entire trajectories of the functional predictors. They inherit the desirable properties of Gaussian process regression, and can naturally accommodate both scalar and functional variables as the predictors, as well as easy to obtain and express uncert...

Wang, Bo; Xu, Aiping

Explicit Representations Of Periodic Solutions Of Nonlinearly Parameterized Ordinary Differential Equations And Their Applications To Inverse Problems

Developing mathematical models involves joining theory and experimental or observational data. The models often depend on parameters which are not always known or measured. A major task in this process is therefore to determine parameters fitting empirical observations. In this work we consider the fundamental challenge of inferring parameters of systems of ordinary differential equations (ODEs) from the values of their solutions and/or their continuous mappings. To achieve this aim we develo...

Al-Ameri, Jehan M. K.

Quantum mechanics approach to option pricing

Options are financial derivatives on an underlying security. The Schrodinger and
Heisenberg approach to the quantum mechanics together with the Dirac matrix
approaches are applied to derive the Black-Scholes formula and the quantum Cox-
Rubinstein formula.
The quantum mechanics approach to option pricing is based on the interpretation
of the option price as the Schrodinger wave function of a certain quantum mechanics
model determined by Hamiltonian H. We apply this approach to continuou...

Hao, Wenyan

Homotopy of planar Lie group equivariant presheaves

We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.

Balchin, Scott

Crossed Simplicial Group Categorical Nerves

We extend the notion of the nerve of a category for a small class of crossed simplicial groups, explicitly describing them using generators and relations. We do this by first considering a generalised bar construction of a group before looking at twisted versions of some of these nerves. As an application we show how we can use the twisted nerves to give equivariant versions of certain derived stacks.

Balchin, Scott

Optimization of the non-axisymmetric stator casing of a 1.5 stage axial turbine

The interaction of secondary flows with the main passage flow in turbomachines results in entropy generation and in aerodynamic loss. This loss source is most relevant to low aspect ratio blades. One approach for reducing this flow energy loss is by endwall contouring. However, limited work has been reported on using non-axisymmetric endwalls at the stator casing and on its interaction with the tip leakage flow. In this paper, a non-axisymmetric endwall design method for the stator casing is ...

Kadhim, Hakim T.; Rona, Aldo; Gostelow, J. Paul; Leschke, Katrin

Mobility cost and degenerated diffusion in kinesis models

A new critical effect is predicted in population dispersal. It is based on the fact that a trade-off between the advantages of mobility and the cost of mobility breaks with a significant deterioration in living conditions. The recently developed model of purposeful kinesis (Gorban & Çabukoǧlu, Ecological Complexity 33, 2018) is based on the “let well enough alone” idea: mobility decreases for high reproduction coefficient and, therefore, animals stay longer in good conditions and leave quicke...

Gorban, Alexander N.; Çabukoǧlu, Nurdan

Knowledge Transfer Between Artificial Intelligence Systems

We consider the fundamental question: how a legacy “student” Artificial Intelligent (AI) system could learn from a legacy “teacher” AI system or a human expert without re-training and, most importantly, without requiring significant computational resources. Here “learning” is broadly understood as an ability of one system to mimic responses of the other to an incoming stimulation and vice-versa. We call such learning an Artificial Intelligence knowledge transfer. We show that if internal vari...

Tyukin, Ivan Y.; Gorban, Alexander N.; Sofeykov, Konstantin I.; Romanenko, Ilya

A multilevel sparse kernel-based stochastic collocation finite element method for elliptic problems with random coefficients

A new stochastic collocation finite element method is proposed for the numerical solution of elliptic boundary value problems (BVP) with random coefficients, assuming that the randomness is well-approximated by a finite number of random variables with given probability distributions. The proposed method consists of a finite element approximation in physical space, along with a stochastic collocation quadrature approach utilizing the recent Multilevel Sparse Kernel-Based Interpolation (MuSIK) ...