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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

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.

Mudrov, Andrey

Evolution of adaptation mechanisms: Adaptation energy, stress, and oscillating death

In 1938, Selye proposed the notion of adaptation energy and published ‘Experimental evidence supporting the conception of adaptation energy.’ Adaptation of an animal to different factors appears as the spending of one resource. Adaptation energy is a hypothetical extensive quantity spent for adaptation. This term causes much debate when one takes it literally, as a physical quantity, i.e. a sort of energy. The controversial points of view impede the systematic use of the notion of adaptation ...

Gorban, Alexander N.; Tyukina, Tatiana A.; Smirnova, E. V.; Pokidysheva, L. I.

The Ext algebra and a new generalisation of D-Koszul algebras

We generalise Koszul and D-Koszul algebras by introducing a class of graded
algebras called (D, A)-stacked algebras. We give a characterisation of (D, A)-stacked
algebras and show that their Ext algebra is finitely generated as an algebra in degrees
0, 1, 2 and 3. In the monomial case, we give an explicit description of the Ext algebra
by quiver and relations, and show that the ideal of relations has a quadratic Gr¨obner
basis; this enables us to give a regrading of the Ext algebra under...

Leader, Joanne; Snashall, Nicole

Mechanism of chain collapse of strongly charged polyelectrolytes

We perform extensive molecular dynamics simulations of a charged polymer in a good solvent in the regime where the chain is collapsed. We analyze the dependence of the gyration radius Rg on the reduced Bjerrum length ℓB and find two different regimes. In the first one, called a weak electrostatic regime, Rg∼ℓ−1/2B, which is consistent only with the predictions of the counterion-fluctuation theory. In the second one, called a strong electrostatic regime, we find Rg∼ℓ−1/5B. To explain the novel...

Tom, A. M.; Vemparala, S.; Rajesh, R.; Brilliantov, Nikolai V.

On the stability of the BEK family of rotating boundary-layer flows for power-law fluids

We consider the convective instability of the BEK family of rotating boundary-layer flows for shear-thinning power-law fluids. The Bödewadt, Ekman and von Kármán flows are particular cases within this family. A linear stability analysis is conducted using a Chebyshev polynomial method in order to investigate the effect of shear-thinning fluids on the convective type I (inviscid crossflow) and type II (viscous streamline curvature) modes of instability. The results reveal that an increase in s...

Abdulameer, M. A.; Griffiths, P. T.; Alveroğlu, B.; Garrett, Stephen J.

Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

We investigate the effects of compressibility and wall cooling on the stationary, viscous (Type II) instability mode within the 3D boundary layer over rotating cones with half-angle greater than 40°. The stationary mode is characterised by zero shear stress at the wall and a triple-deck solution is presented in the isothermal case. Asymptotic solutions are obtained which describe the structure of the wavenumber and the orientation of this mode as a function of local Mach number. It is found t...

Towers, P. D.; Hussain, Z.; Griffiths, P. T.; Garrett, S. J.

Equivariant Hochschild Cohomology

In this thesis our goal is to develop the equivariant version of Hochschild cohomology. In the equivariant world there is given a group G which acts on objects. First naive object which can be considered is a G-algebra, that is, an associative algebra A on which G acts via algebra automorphisms. In our work we consider two more general situations. In the first case we develop a cohomology theory for oriented algebras and in the second case we develop a cohomology theory for Green functors.

Koam, Ali Nasser Ali

Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. The case of fast reactions is also included. More specifically, a model problem consisting of a system of semilinear parabolic advection–diffusion–reaction partial different...

Cangiani, Andrea; Georgoulis, Emmanuil H.; Jensen, M.

Conforming and nonconforming virtual element methods for elliptic problems

We present in a unified framework new conforming and nonconforming Virtual Element Methods (VEM) for general second order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and non-symmetric parts and conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal $H^1$- and $L^2$-error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of...

Cangiani, Andrea; Manzini, G.; Sutton, Oliver J.

Optimal Bounds for the Variance of Self-Intersection Local Times

For a Zd-valued random walk (Sn)n N0, let l(n,x) be its local time at the site x Zd. For α N, define the α-fold self-intersection local time as Ln(α) xl(n,x)α. Also let LnSRW(α) be the corresponding quantities for the simple random walk in Zd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely d-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, var(Ln(α))=O(v...

Deligiannidis, G.; Utev, Sergey

Multiserial and special multiserial algebras and their representations

In this paper we study multiserial and special multiserial algebras. These algebras are a natural generalization of biserial and special biserial algebras to algebras of wild representation type. We define a module to be multiserial if its radical is the sum of uniserial modules whose pairwise intersection is either 0 or a simple module. We show that all finitely generated modules over a special multiserial algebra are multiserial. In particular, this implies that, in analogy to special biser...

Green, E. L.; Schroll, Sibylle

Revisiting Brownian motion as a description of animal movement: a comparison to experimental movement data

Summary:
1. Characterization of patterns of animal movement is a major challenge in ecology with applications to conservation, biological invasions and pest monitoring. Brownian random walks, and diffusive flux as their mean field counterpart, provide one framework in which to consider this problem. However, it remains subject to debate and controversy. This study presents a test of the diffusion framework using movement data obtained from controlled experiments.
2. Walking beetles (Tenebri...

Bearup, Daniel; Benefer, Carly M.; Petrovskii, Sergei V.; Blackshaw, Rod P.