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.
- Strong cutting-edge PhD programme.
- 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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The classical notion of the Darboux transformation of isothermic surfaces can be generalised to a transformation for conformal immersions. Since a minimal surface is Willmore, we can use the associated C∗–family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so–called µ–Darboux transforms. We show that a µ–Darboux transform of a minimal surface is not minimal but a Willmore surface in 4–space. More precisely, we show that a µ–Darboux transform of ...
Leschke, K.; Moriya, K.
A quantum mechanics approach is proposed to model non-life insurance risks and to compute the future reserve amounts and the ruin probabilities. The claim data, historical or simulated, are treated as coming from quantum observables and analyzed with traditional machine learning tools. They can then be used to forecast the evolution of the reserves of an insurance company. The following methodology relies on the Dirac matrix formalism and the Feynman path-integral method.
Lefèvre, Claude; Loisel, Stéphane; Tamturk, Muhsin; Utev, Sergey
We provide an axiomatic approach for studying support varieties of objects in a triangulated category via the action of a tensor triangulated category, where the tensor product is not necessarily symmetric. This is illustrated by examples, taken in particular from representation theory of finite dimensional algebras.
Buan, Aslak Bakke; Krause, Henning; Snashall, Nicole; Solberg, Øyvind
Modelling evolution of virulence in host-parasite systems is an actively developing area of research with ever-growing literature. However, most of the existing studies overlook the fact that individuals within an infected population may have a variable infection load, i.e. infected populations are naturally structured with respect to the parasite burden. Empirical data suggests that the mortality and infectiousness of individuals can strongly depend on their infection load; moreover, the sha...
Sandhu, Simran K.; Morozov, Andrew Yu.; Farkas, József Z.
The work concerns the problem of reducing a pre-trained deep neuronal network to a smaller network, with just few layers, whilst retaining the network’s functionality on a given task. In this particular case study, we are focusing on the networks developed for the purposes of face recognition. The proposed approach is motivated by the observation that the aim to deliver the highest accuracy possible in the broadest range of operational conditions, which many deep neural networks models strive...
Gorban, Alexander N.; Mirkes, Evgeny M.; Tyukin, Ivan Y.
We study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and show that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.
Baranov, Alexander; Shlaka, Hasan M.
Let Q be a (not necessarily unital) simple ring or algebra. A nonempty subset S of Q is said to have zero product if S 2 = 0. We classify all maximal zero product subsets of Q by proving that the map R 7→ R∩LeftAnn(R) is a bijection from the set of all proper nonzero annihilator right ideals of Q onto the set of all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.
Baranov, Alexander; Fernández López, Antonio
Shape sensitivity measures the impact of small perturbations of geometric features of a problem on certain quantities of interest. The shape sensitivity of PDE (partial differential equation) constrained output functionals is derived with the help of shape gradients. In electromagnetic scattering problems, the standard procedure of the Lagrangian approach cannot be applied because of solution of the state problem is complex valued. We derive a closed-form formula of the shape gradient of a ge...
Sargheini, Sahar; Paganini, Alberto; Hiptmair, Ralf; Hafner, Christian
We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spaces (RKHSs). First, we study the properties of these basis vector fields and show how to approximate them. Then, we employ this basis to discretise shape Newton methods and investigate the impact of this discretisation on convergence rates.
Paganini, Alberto; Sturm, Kevin
We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of PDE-constraints. This shape optimization method is based on discretized deformation diffeomorphisms and allows for arbitrarily high resolution of shapes with arbitrary smoothness. Numerical experiments show that it allows the solution of PDE-constrained shape op...
Paganini, Alberto; Wechsung, Florian; Farrell, Patrick E.
In this thesis, we are interested in the study of cohomology of differentiable stacks and we want to provide a good notion of equivariant cohomology for differentiable stacks. For this we describe in detail some of the cohomology theories found in the literature and give some relations between them. As we want a notion of equivariant cohomology, we discuss the notion of an action on a stack by a Lie group G and how to define the quotient stack for this action. We find that this quotient stack...
Barbosa Torres, Luis A.
We study finite dimensional Koszul algebras and their generalisations including d-Koszul algebras and (D, A)-stacked algebras, together with their projective resolutions and Hochschild cohomology. Then we introduce the stretched algebra ~Λ and give a functorial construction of the projective resolution of ~Λ =~r and the projective bimodule resolution of A. Following this, we show that if E(Λ) is finitely generated then so is E(~Λ). We investigate the connection between HH*( Λ) and HH*(~ Λ) an...
Jawad, Ruaa Y.
Nonparametric functional regression is of considerable importance due to its impact on the development of data analysis in a number of elds, least cost and saving time. In this thesis, we focus on nonparametric functional regression and its extensions, and its application to functional data. We rst review nonparametric functional regression, followed by a detailed discussion about model structures, semi-metrics and kernel functions. Secondly, we extend the independent response model to mult...
Omar, Kurdistan M.
We construct quantization of semisimple conjugacy classes of the exceptional group G = G2 along with and by means of their representations on highest weight modules over the quantum group Uq(g). With every point t of a fixed maximal torus we associate a highest weight module Mt over Uq(g) and realize the quantized polynomial algebra of the class of t by linear operators on Mt . Quantizations corresponding to points of the same orbit of the Weyl group are isomorphic.
Baranov, Alexander; Mudrov, Andrey; Ostapenko, Vadim
The most reliable estimates of the population abundance of ground-dwelling arthropods are obtained almost entirely through trap counts. Trap shape can be easily controlled by the researcher, commonly the same trap design is employed in all sites within a given study. Few researchers really try to compare abundances (numbers of collected individuals) between studies because these are heavily influenced by environmental conditions, e.g. temperature, habitat structure and food sources available,...
Ahmed, Danish A.; Petrovskii, Sergei V.
An a posteriori error estimator for the error in the (L 2 (H 1 )+L ∞ (L 2 ))-type norm for an interior penalty discontinuous Galerkin (dG) spatial discretisation and backward Euler temporal discretisation of linear non-stationary convection–diffusion initial/boundary value problems is derived, allowing for anisotropic elements. The proposed error estimator is used to drive an hp-space–time adaptive algorithm wherein directional mesh refinement is employed to give rise to highly anisotropic el...
Cangiani, A.; Georgoulis, E. H.; Giani, S.; Metcalfe, S.
Living neuronal networks in dissociated neuronal cultures are widely known for their ability to generate highly robust spatiotemporal activity patterns in various experimental conditions. Such patterns are often treated as neuronal avalanches that satisfy the power scaling law and thereby exemplify self-organized criticality in living systems. A crucial question is how these patterns can be explained and modeled in a way that is biologically meaningful, mathematically tractable and yet broad ...
Tyukin, IY; Iudin, D; Iudin, F; Tyukina, T; Kazantsev, V... et al.
In this thesis, we study general cocycles of dynamical systems in topological, measurable and smooth (differentiable) settings. Dynamical systems are viewed here as given by actions of a discrete, topological, measurable or Lie group on a set, topological space, measurable space or smooth manifold respectively depending on the given geometrical setting to be considered. We will mostly concentrate on the topological and smooth settings in this thesis, but will comment about the necessary alter...
Al-Bayati, Mudheher A. H.
In this thesis, we introduce a novel consensus-based group decision making (CGDM) model by integrating the notions of Social Network Analysis (SNA), clustering and Social Influence Network (SIN). Four main contributions are presented in order to handle a number of issues in CGDM. In dealing with the issue of the consistency of preferences, we introduce a consistency operator and construct a consistency control module for the purpose of securing the correctness of expert preferences. The propo...
Kamis, Nor H.
We give a criterion for complete reducibility of tensor product V ⊗ Z of two irreducible highest weight modules V and Z over a classical or quantum semi-simple group in terms of a contravariant symmetric bilinear form on V ⊗ Z. This form is the product of the canonical contravariant forms on V and Z. Then V ⊗ Z is completely reducible if and only if the form is non-degenerate when restricted to the sum of all highest weight submodules in V ⊗ Z or equivalently to the span of singular vectors.
Mudrov, Andrey I.