# How to apply

For more information on studying with this Department go to our** Research Opportunities page.**

To complete an application to study, go to the** University's application page. **

The goal of this project is to explore fundamental structural questions concerning sets of permutations that avoid a single pattern. The study of these combinatorial objects was initiated by Donald Knuth in the 1960s when he investigated which permutations could be sorted by being passed through a stack.

#### Project Details

A *permutation class* is a set of permutations closed downwards under the subpermutation (or containment) order. See [1] for a brief introduction or [5] for a full exposition. A *principal* class is one that can be defined as avoiding a single permutation: Av(τ) = { σ : σ avoids τ }. This project will explore fundamental questions concerning the structure of permutations in principal classes. In particular, two important monotonicity conjectures will be addressed:

Inversion Monotonicity Conjecture [4]: For all *k* and every non-increasing permutation τ, the number of τ-avoiding *n*-permutations with *k* inversions increases as *n *increases.

Block Monotonicity Conjecture [3]: For all *n* and every skew-indecomposable permutation τ of length greater than 2, the number of τ-avoiding *n*-permutations with *m* skew blocks decreases as *m *increases.

For both of these, the smallest case for which the conjecture is open is τ = 1324. Resolving the Inversion Monotonicity Conjecture for 1324-avoiders would yield an improved upper bound on the growth rate of the class. Understanding Av(1324) is one of the primary open problems in permutation classes research [2].

[1] David Bevan, Permutation patterns: basic definitions and notation, 2015. https://arxiv.org/abs/1506.06673

[2] Bevan David, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, *European Journal of Combinatorics*, 88:103115, 2020.

[3] Miklós Bóna, Supercritical sequences, and the nonrationality of most principal permutation classes, *European Journal of Combinatorics*, 83:103020, 2020.

[4] Anders Claesson, Vít Jelínek and Einar Steingrímsson, Upper bounds for the Stanley–Wilf limit of 1324 and other layered patterns,* Journal of Combinatorial Theory, Series A*, 119:1680–1691, 2012.

[5] Vincent Vatter. Permutation classes. In Miklós Bóna, *Handbook of enumerative combinatorics*, 2015. Preprint version: https://arxiv.org/abs/1409.5159

#### Eligibility

You should have (or expect to have) a UK Honours Degree (or equivalent) at 2.1 or above in Mathematics or a closely related subject with a high mathematical content.

The goal of this project is the exact enumeration of classes of permutations that can be plotted on monotone curves in grids. At present, an effective procedure is only known for so-called “skinny” classes. The general problem is challenging because permutations have multiple ways of being plotted on the curves.

#### Project Details

The monotone grid class Grid(M) consists of permutations that can be plotted on monotone curves in a grid. This grid is specified by the matrix M, all of whose entries are in {0, 1, –1}, where 1 and –1 denote monotone increasing and decreasing curves in a cell, and 0 denotes an empty cell. This project concerns the enumeration of these classes.

For a general overview of permutation classes, see Vatter’s recent survey [5]. An introduction to monotone grid classes can be found in [3, Chapter 2]. There is also an interactive demonstration, available from http://demonstrations.wolfram.com/PermutationGridClasses.

Associated with a grid class is a graph whose structure influences its enumerative properties. If the graph is acyclic, then the class has a rational generating function [1]. If the graph is unicyclic, it is conjectured that the generating function is algebraic; otherwise, it is believed that the class is D-finite [3, Chapter 4]. The exponential growth rate of any monotone grid class can be determined from its matrix [2, 4], but an effective procedure for exact enumeration is only known for “skinny” grid classes, those defined by a k × 1 vector [3, Chapter 3]; these are acyclic. The general problem is hard because permutations typically have multiple “griddings” (different ways of being plotted on the curves).

The project is likely to focus on classes defined by a k × 2 matrix, seeking procedures for enumerating acyclic and unicyclic classes of this sort, and in the latter case proving algebraicity.

[1] M. Albert, M.Atkinson, M. Bouvel, N. Ruškuc, V. Vatter. Geometric grid classes of permutations. Trans. Amer. Math. Soc., 2013.

[2] M. Albert, V. Vatter. An elementary proof of Bevan’s theorem on the growth of grid classes of permutations. Proc. Edinburgh Math. Soc., 2019.

[3] D. Bevan. On the growth of permutation classes. PhD thesis, 2015. https://arxiv.org/pdf/1506.06688

[4] D. Bevan. Growth rates of permutation grid classes, tours on graphs, and the spectral radius. Trans. Amer. Math. Soc., 2015.

[5] V. Vatter. Permutation classes. In M. Bóna, Handbook of enumerative combinatorics, 2015. Preprint version: https://arxiv.org/abs/1409.5159

#### Eligibility

You should have (or expect to have) a UK Honours Degree (or equivalent) at 2.1 or above in Mathematics or a closely related subject with a high mathematical content.

Mathematics and statistics are at the heart of all scientific and engineering disciplines. Strathclyde has a strong international reputation in the use of mathematical modelling and analysis to solve many problems relevant to industry, business and wider society. The department's research interests are in Applied Analysis, **Continuum Mechanics & Industrial Mathematics** (CMIM), Numerical Analysis & Scientific Computing, Population Modelling & Epidemiology, and Stochastic Analysis.

A large number of faculty members in the CMIM group are interested in problems related to fluid mechanics. The study of droplet electrohydrodynamics is a classic fluid mechanical problem, pioneered by G. I. Taylor and has continued to intrigue scientists to the present day. The proposed research project will further the field of droplet EHD by studying the dynamics of a pair of interacting droplets in an electric field. Some articles relevant to the project can be found in the references below. The student will be jointly supervised by Dr Debasish Das and Prof Stephen Wilson. Visit the webpages below for more details.

https://sites.google.com/view/debasishdas

https://www.strath.ac.uk/staff/wilsonstephenprof/

**Qualifications:** A bachelors degree (upper second class honours or higher) in Mathematics or a related subject like Mechanical, Chemical Engineering or Physics is required. Applicant should have taken at least one undergraduate course on Fluid Mechanics (any advanced Fluid Mechanics course is highly desirable). Some experience in basic coding in any programming language is a plus.

**Studentship details: **The studentship covers home fees and stipend (UKRI minimum level - currently £15,609 for 21/22). All candidates are eligible, but international candidates would need to pay the fees difference between Home and Overseas rates. The duration of PhD is 36 months (extendable depending on availability of extra funds). Apply by **March 14th, 2021** for full consideration.

**Application:** Visit the webpage below to apply and make sure to mention the title of the project and name of supervisors listed above.

https://www.strath.ac.uk/studywithus/postgraduateresearch/yourapplicationoffer/

1) D. Das and D. Saintillan, "Electrohydrodynamics of viscous drops in strong electric fields: Numerical simulations", Journal of Fluid Mechanics 829, 127-152 (2017).

2) D. Das and D. Saintillan, "A nonlinear small-deformation theory for transient droplet electrohydrodynamics", Journal of Fluid Mechanics 810, 225-253 (2017).

3) A. Bricard, J.-B. Caussin, D. Das, C. Savoie, V. Chikkadi, K. Shitara, O. Chepizhko, F. Peruani, D. Saintillan, D. Bartolo, "Emergent vortices in populations of colloidal rollers", Nature Communications 6, 7470 (2015).

4) L. T. Corson, C. Tsakonas, B. R. Duffy, N. J. Mottram, I. C. Sage, C. V. Brown, & S. K. Wilson, "Deformation of a nearly hemispherical conducting drop due to an electric field: Theory and experiment". Physics of Fluids, 26, 122106 (2014).

5) D. Das and D. Saintillan, "Electrohydrodynamic interaction of spherical particles under Quincke rotation", Physical Review E 87, 043014 (2013).

Classification of teas (types, quality grades, region of origin etc) has been examined by many researchers, using both chemical composition and more recently digital image analysis techniques to extract features from the image that are useful for classification.

Factors affecting the success of classification include the choice of features, the classifier and the imaging modality. Building on previous work at Strathclyde in collaboration with the EEE department, this project will allow the student to examine the choice of any of these to achieve optimal results. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of both R and Matlab.

**Second supervisors: George Gettinby, Magnus Peterson**

Worldwide losses of honey bee colonies have attracted considerable media attention in recent years and a huge amount of research. Researchers at Strathclyde have experience since 2006 of carrying out a series of surveys of beekeepers in Scotland (http://personal.strath.ac.uk/a.j.gray/) and now have 5 years of data arising from these surveys. These data have been used to estimate colony loss rates in Scotland and to provide a picture of beekeepers’ experience and management practices.

This project will examine the data in more detail than has been done so far, and is likely to involve data modelling and multivariate methods to identify risk factors. Part of the project will involve establishing the spatial distribution of various bee diseases.

This project will build on links with the Scottish Beekeepers’ Association and membership of COLOSS, a network linking honey bee researchers in Europe and beyond. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of R.

**Introduction**

In this project we shall look at how stochastic models can be used to describe how infectious diseases spread. We will start off by looking at one of the simplest epidemic models, the SIS (susceptible-infected-susceptible) model. In this model a typical individual starts off susceptible, at some stage catches the disease and after a short infectious period becomes susceptible again.

These models are used for diseases such as pneumococcus amongst children and sexually transmitted diseases such as gonorrhea amongst adults (Bailey, 1975). Previous work has already looked at introducing stochastic noise into this model via the disease transmission term (Gray et al. 2011). This is called environmental stochasticity which means introducing the random effects of the environment into how the disease spreads. This results in a stochastic differential equation (SDE) model which we have analysed. We have derived an expression for a key epidemiological parameter, the basic reproduction number.

In the deterministic model this is defined as the expected number of secondary cases caused by a single newly-infected individual entering the disease-free population at equilibrium. The basic reproduction number is different in the stochastic model than the deterministic one, but in both cases it determines whether the disease dies out or persists. In the stochastic SIS SDE model we have shown the existence of a stationary distribution and that the disease will persist if the basic reproduction number exceeds one and die out if it is less than one.

**SDE Models with Environmental Stochasticity**

We have also looked at other SDE models for environmental stochasticity. One of these took a simple deterministic model for the effect of condom use on the spread of HIV amongst a homosexual population and introduced environmental stochasticity into the disease transmission term. Again we found that a key parameter was the basic reproduction number which determined the behaviour of the system.

As before this was different in the deterministic model than the stochastic one. Indeed it was possible for stochastic noise to stabilise the system and cause an epidemic which would have taken off in the deterministic model to die out in the stochastic model (Dalal et al., 2007). Similar effects were observed in a model for the internal viral dynamics of HIV within an HIV-infected individual (Dalal et al., 2008).

**Demographic Stochasticity**

The real world is stochastic, not deterministic, and it is difficult to predict with certainty what will happen. Another way to introduce stochasticity into epidemic models is demographic stochasticity. If we take the simple homogeneously mixing SIS epidemic model with births and deaths in the population we can derive a stochastic model to describe this situation by defining p(i, j, t) to be the probability that at time t there are exactly i susceptible and j infected individuals and deriving the differential equations satisfied by these probabilities.

Then we shall look at how stochastic differential equations can be used to approximate the above set of equations for p(i, j, t). This is called demographic stochasticity and arises from the fact that we are trying to approximate a deterministic process by a stochastic one (Allen, 2007). Although the reasons for demographic and environmental stochasticity are quite different the SDEs which describe the progress of the disease are similar. The first project which we shall look at is analysis of the SIS epidemic model with demographic stochasticity along the lines of our analysis of the SIS epidemic model with environmental stochasticity. **Further Work**

After this we intend to look at other classical epidemiological models, in particular the SIR (susceptible-infected-removed) model in which an individual starts off susceptible, at some stage he or she catches the disease and after a short infectious period he or she becomes permanently immune. These models are used for common childhood diseases such as measles, mumps and rubella (Anderson and May, 1991). We would look at introducing both environmental and demographic stochasticity into this model.

Other epidemiological models which could be analysed include the SIRS (susceptible-infected-removed-susceptible) epidemic model, which is similar to the SIR epidemic model, except that immunity is not permanent, the SEIS (susceptible-exposed-infected-susceptible) model which is similar to the SIS model, but includes an exposed or latent class, and the SEIR (susceptible-exposedinfected- removed) model, which similarly extends the SIR model.

We would also aim to look at other population dynamic models such as the Lotka-Volterra predator-prey model. There is also the possibility of developing methods for parameter estimation in all of these epidemiological and population dynamic models, and we have started work on this with another Ph. D. student (J. Pan).

**References**

1. E. Allen, Modelling with Itˆo Stochastic Differential Equations, Springer-Verlag, 2007.

2. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.

3. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.

4. A.J. Gray, D. Greenhalgh, L. Hu, X.

5. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for AIDS and condom-use. J. Math. Anal. Appl. 325, 36-53, 2007.

6. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics. J. Math. Anal. Appl. 341, 1084-1101, 2008.

Differential equation models are commonly used to model infectious diseases. The population is divided up into compartments and the flow of individuals through the various classes such as susceptible, infected and removed is modelled using a set of ordinary differential equations (Anderson and May, 1991, Bailey, 1975). A basic epidemiological parameter is the basic reproduction number. This is defined as the expected number of secondary cases produced by a single newly infected case entering a disease-free population at equilibrium (Diekman and Heesterbeek). Typically the disease takes off if R0 > 1 and dies out if R0≤1.

However media awareness campaigns are often used to influence behaviour and if successful can alter the behaviour of the population. This is an area which has not been studied much until recently. The student would survey the existing literature on media awareness models in the literature and with the supervisor formulate mathematical models using differential equations for the effect of behavioural change on disease incidence. These would be examined using both analytical methods and computer simulation with parameters drawn from real data where appropriate. The mathematical techniques used would be differential equations, equilibrium and stability analyses and computer simulation.

**References**

1. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.

2. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.

3. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction number R0 for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365-382.

4. A. K. Misra, A. Sharma and J. B. Shukla, Modelling and analysis of effects of awareness programs by media on the spread of infectious diseases. Math. Comp. Modelling 53, 1221- 1228.

5. A. K. Misra, A. Sharma, V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, J. Biol. Systems, 19(2), 389-402,

In recent years there has been much work on reaction-diffusion equations in which the diffusion mechanism is not the usual Fickian one. Examples are integro-differential equations, porous media type equations, pseudodifferential equations, p-Laplacian type equations and prescribed curvature type (saturating flux) equations.

The motivation for this work comes from material science and mathematical ecology. However, there are applied contexts where these diffusion mechanisms have never been considered. One is in the area of combustion and the other is in the area of regularised conservation laws and shock propagation. This project, which would build on the work I did through the years on integrodifferential models and recently with M. Burns on the prescribed curvature equations, will use PDE, asymptotic, and topological methods to explore the dynamics of blowup and of shock propagation in canonical examples of reaction equations and nonlinear scalar conservation laws regularised by non-Fickian diffusion terms.

References:

[1] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

[2] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

Recently, a new class of model has been developed to describe, for example, phase separation in materials such as binary alloys. These take the form of integrodifferential equations. Coarsening, that is, creation of large scale patterns in such models is poorly understood.

There are partial results [1, 2] that use the maximum principle, while for most interesting problems such a tool is not available. This will be a mixture of analytic and numerical work and will need tools of functional analysis and semigroup theory.

References:

[1] D. B. Duncan, M. Grinfeld, and I. Stoleriu, Coarsening in an integro-differential model of phase transitions, Euro. J. Appl. Math. 11 (2000), 561-572.

[2] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math. 17 (2006), 221-232.

Nematic liquid crystals are perhaps one of the most classical and widely-used examples of soft matter, materials that combine directionality with fluidity and have special material directions, referred to as "nematic directors". Consequently, nematics have a direction-dependent response to external light, temperature, mechanical stress, electric fields etc., resulting in directional physical, optical and mechanical properties. In fact, nematics are the working material of choice for the multibillion-dollar liquid crystal display industry. Contemporary research in nematics has shifted from conventional displays to altogether new areas such as sensors, actuators, photonics and generally information-rich technologies. Scientifically, these advanced applications require a well-defined framework connecting fundamental physics to programmable materials science in fields such as biology and nanoscience and finally to commercial applications. Mathematics is the crucial link between physics and applications, which has been underexploited to date.

In this project, we will study two-dimensional (2D) and three-dimensional (3D) nematic systems in terms of their complex solution landscapes, an umbrella term used to describe the plethora of admissible NLC configurations in different settings. We will model the experimentally observable nematic configurations, how to switch between different configurations, and crucially how to use geometrical properties to control and steer processes of importance. We will work with experimentalists to use the theory for the design and optimisation of 2D and 3D nematic systems and test their potential for different applications. This is an exciting project at the interface of mathematical modelling, analysis, and scientific computation in a broad interdisciplinary setting.

**Qualification Details:** A bachelor’s degree (upper second-class honours or higher) in Mathematics. Applicants should have some experience with partial differential equations, advanced calculus, mechanics and ideally some knowledge of basic coding in any programming language e.g. MATLAB.

**Studentship details: **This 42-month studentship is funded by a Leverhulme Research Project grant awarded to Professor Apala Majumdar. There are two project partners: Dr Canevari (Verona), and an experimentalist, Professor Lagerwall (Luxembourg), and the project includes scientific collaborations with researchers at the University of Oxford and Peking University.

The student will join the Majumdar research group. This group has one postdoctoral researcher (Dr Han, Newton Fellow), who will be involved with the project, and three graduate students. More details about the Majumdar research group can be found at **https://themajumdargroup.wordpress.com/**** and** **https://www.strath.ac.uk/staff/majumdarapalaprofessor/.**

More generally, the student will also be part of the Continuum Mechanics and Industrial Mathematics Research Group at the University of Strathclyde. This is a vibrant research group with several faculty members working on liquid crystals, continuum mechanics and fluid mechanics.

The studentship covers home fees and stipend. All candidates are eligible, but international candidates would need to pay the fees difference between Home and Overseas rates; this needs to be discussed further with the Apala Majumdar. There are also funds to visit the project partners, Dr Canevari (Verona) and Professor Lagerwall (Luxembourg).

**Application**: Please contact Apala Majumdar at apala.majumdar@strath.ac.uk and send your CV with a brief statement of interest.

Nematic liquid crystals (NLCs) are classical examples of materials that are intermediate between conventional solids and liquids. NLCs are directional materials with preferred directions of the averaged molecular alignment. NLCs have direction-dependent responses to external electric fields and light, making them the working material of choice for a range of electro-optic applications, notably the thriving liquid crystal display industry. However, NLCs have weak responses to external magnetic fields, so that magnetic fields are not widely used for NLC-based applications. In the 1970’s, de Gennes and Prost, Rault, Cladis and Burger undertook pioneering experimental and theoretical work on ferronematics – suspensions of magnetic (nano)particles in a nematic host. The magnetic (nano)particles induce a spontaneous magnetisation and consequently, ferronematics have much stronger responses to external magnetic fields compared to conventional NLCs. Experimental research in ferronematics is booming, driven by the need to understand the fundamental physics of ferronematics and how we can model ferronematics and their potential applications in photonics and new materials technologies.

In this project, we will develop new mathematical models for ferronematics to account for both the nematic and magnetic order, and how they couple to each other. We will also develop new numerical methods for ferronematic systems and analyse the numerical schemes, including error and convergence analysis. We will apply the mathematical models and numerical schemes to study the experimentally observable states in prototype ferronematics and how we can switch between them, with a long-term goal of designing and controlling ferronematic systems for tailor-made applications.

Qualification Details: A bachelor’s degree (upper second-class honours or higher) in Mathematics. Applicants should have some experience with partial differential equations, advanced calculus, mechanics, numerical methods and ideally some knowledge of basic coding in any programming language e.g. MATLAB.

**Studentship details**: This 36-month studentship is funded by the University of Strathclyde. The supervisory team comprises Professor Apala Majumdar, an expert in liquid crystal modelling and Dr Michele Ruggeri, a numerical analyst.

The student will join the Majumdar research group. More details about the Majumdar research group can be found at https://themajumdargroup.wordpress.com/ and https://www.strath.ac.uk/staff/majumdarapalaprofessor/ More generally, the student will also be part of the Continuum Mechanics and Industrial Mathematics Research Group at the University of Strathclyde. This is a vibrant research group with several faculty members working on liquid crystals, continuum mechanics and fluid mechanics.

The studentship covers home fees and stipend. All candidates are eligible, but international candidates would need to pay the fees difference between Home and Overseas rates; this needs to be discussed further with Apala Majumdar.** **

**Application**: Please contact Apala Majumdar (apala.majumdar@strath.ac.uk) and Michele Ruggeri (michele.ruggeri@strath.ac.uk) and send your CV with a brief statement of interest.

Up to 2002, most of the existing strong convergence theory for numerical methods requires the coefficients of the SDEs to be globally Lipschitz continuous [1]. However, most SDE models in real life do not obey the global Lipschitz condition. It was in this spirit that Higham, Mao and Stuart in 2002 published a very influential paper [2] (Google citation 319) which opened a new chapter in the study of numerical solutions of SDEs---to study the strong convergence question for numerical approximations under the local Lipschitz condition.

Since the classical explicit Euler-Maruyama (EM) method has its simple algebraic structure, cheap computational cost and acceptable convergence rate under the global Lipschitz condition, it has been attracting lots of attention.

Although it was showed that the strong divergence in finite time of the EM method for SDEs under the local Lipschitz condition, some modified EM methods have recently been developed these SDEs. For example, the tamed EM method was developed in 2012 to approximate SDEs with one-sided Lipschitz drift coefficient and the linear growth diffusion coefficient. The stopped EM method was developed in 2013. Recently, Mao [3] initiated a significantly new method, called the truncated EM method, for the nonlinear SDEs. The aim of this PhD is to develop the truncated EM method. The detailed objectives are:

(1) To study the strong convergence of the truncated EM method in finite-time for SDEs under the generalised Khasminskii condition and its convergence rate.

(2) To use the truncated EM method to investigate the stability of the nonlinear SDEs. Namely to study if the numerical method is stochastically stable when the underlying SDE is stochastically stable and to study if we can infer that the underlying SDE is stochastically stable when the numerical method is stochastically stable for small stepsize.

A PhD studentship might be available for the project.

References:

[1] Mao X., Stochastic Differential Equations and Applications, 2nd Edtion, Elsevier, 2007.

[2] Higham D., Mao X., Stuart A., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40(3) (2003), 1041--1063.

[3] Mao X., The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), 370--384.

Income needs seem to be a top priority at the moment and with low savings rates and top UK equity income funds yielding less than 4\%, it is perhaps easy to understand why. With savings rates continuing at their record lows, some savers are turning to alternatives. It is in this spirit that many financial institutions are offering stock market linked savings plans to those looking to combine a high yield opportunity with some protection against a falling stock market. That is why the stock market linked savings accounts have recently become very popular. The returns of these accounts are random so the returns, even the initial capital, are not guaranteed. They are very much different from the familiar fixed-term-fixed-rate savings accounts.

The aim of this PhD project is to perform the stochastic and numerical analysis on the stock market linked savings accounts in order to establish the theory on the mean percentage of return (MPR). The MPR depends mainly on two factors:

(1) The structure of the saving accounts, namely the terms of the portfolios (plans);

(2) The behaviour of the stock market linked.

There are various portfolios in the market. This PhD project will analyze a number of typical plans. On the stochastic modelling of the stock market, there are various stochastic differential equation (SDE) models. In this project, some of these SDE models will be used. The key objective here is to establish the explicit formulas for the MPRs of the underlying stock marker linked saving accounts if possible; otherwise develop some new techniques to establish better approximation schemes.

The project will then develop the techniques of the stochastic and numerical analysis to deal with other more complicated financial derivatives including bond, fund.

A PhD studentship might be available for the project.

Cholesteric liquid crystals are chiral systems which possess a spontaneously formed helical structure with the pitch in micron range which is important for various applications in optics and nanophotonics. In recent years the interest has shifted in the direction of lyotropic cholesterics which are the solutions of various chiral macromolecules, viruses or chiral nanocrystals. These systems are important for biology (for example, cholesteric states of DNA) and also provide some very useful natural anisotropic chiral materials.

Among the most interesting resources to explore are cellulose and chitin, key biopolymers in the plant and animal world, respectively. Both have excellent mechanical properties and can be extracted as nanorods with high degree of crystallinity.Both are also chiral. Molecular chirality of such nanorods is amplified into a helically modulated long-range ordered cholesteric liquid crystal phase when they are suspended in water.

The aim of this project is to develop a molecular-statistical theory of chirality transfer in cholecteric nanorod phase, determined by steric and electrostatic chiral interactions, and quantitatively describe the variation of helix pitch as a function of rod length, concentration, dispersity and temperature. The theory will be built upon the previous results, obtained for different cholesteric liquid crystals ((see some references to our work below).

The project will include a collaboration with two experimental group at the University of Luxembourg and the University of Stuttgart. These groups have an enormous expertise in the field of lyotropic liquid crystals.

[1] Honorato-Rios, C., Lehr, C., Sch¨utz, C., Sanctuary, R., Osipov, M. A., Baller, J. and Lagerwall, J. P.F.., Fractionation of cellulose nanocrystals: enhancing liquid crystal ordering with- out promoting gelation, *Asia Materials, *10, 455–465 (2018).

[2]. Dawin, Ute C., Osipov, Mikhail A. and Giesselmann, F. Electrolyte effects on the chiral induction and on its temperature dependence in a chiral nematic lyotropic liquid crystal J *of Phys. Chem. B*, 114 (32). 10327-10336 (2010)

[3] A. V. Emelyanenko, M. A. Osipov and D. A. Dunmur, ] Molecular theory of helical sense inversions in chiral nematic liquid crystals *Phys. Rev. E, *62, 2340 (2000)

Elastic constants of nematic liquid crystals, which describe the energy associated with orientational deformation of such anisotropic fluids, are among the most important parameters for various applications of liquid crystal materials. The elastic constants of nematic liquid crystals have been well investigated both experimentally and theoretically in the past. During the past decade a number of novel liquid crystals materials with unconventional molecular structure have been investigated and it has been found that these systems are characterised by the anomalous values and behaviour of the elastic constants. In particular, it has been shown that in the nematic phase exhibited by the so-called V-shaped bent-core liquid crystals two of the three elastic constants decrease nearly to zero with the decreasing temperature. This behaviour is still very poorly understood.

It should be noted that bent-core liquid crystals attract a very significant attention at present because they also exhibit a number of unusual novel phases with a nanoscale helical structure. It is now generally accepted that a transition into these unusual phases may be driven by the dramatic reduction of the elastic constants.

The aim of this project is the generalise the existing molecular-statistical theory of elasticity of nematic liquid crystals to the case of bent-core nematics, composed of biaxial and polar molecules, using the preliminary results obtained in recent years (see, for example, the references to some of our recent papers given below). Another aim is to explain the existing experimental data on the temperature variation of the elastic constants of bent-core nematic liquid crystals.

The project will include a collaboration with the experimental group at the University of Leeds and the theoretical group from Russian Academy of Sciences. These collaborations are very important for the success of the project.

[1] M. A. Osipov and G. Pajak, Effect of polar intermolecular interactions on the elastic constants of bent-core nematics and the origin of the twist-bend phase, The European Physical Journal E 39, 45 (2016).

[2] M. A. Osipov and G. Pajak, Polar interactions between bent-core molecules as a stabilising factor for inhomogeneous nematic phases with spontaneous bend deformations, Liquid Crystals 44, 58 (2016).

[3] S. Srigengan, M. Nagaraj, A. Ferrarini, R. Mandle, S.J. Cowling, M.A. Osipov, G. Pająk, J.W. Goodby and H.F. Gleeson, J. Mater. Chem. C, 2013, 6, 980

We wish to develop innovative methods for modelling high-dimensional time series. Practical time series data, including both continuous-valued and discrete-valued data such as climate record data, medical data, and financial and economic data, are used for empirical analysis.

Models for forecasting multivariate conditional mean and multivariate conditional variance (volatility) are concerned. Techniques for dimension reduction, such as dynamic factor analysis, are used.

The estimation of models for panel data analysis and the option valuation with co-integrated asset prices is discussed.

Nowadays people often meet problems in forecasting a functional. A functional may be a curve, a spatial process, or a graph/image. In contrast to conventional time series analysis, in which observations are scalars or vectors, we observe a functional at each time point; for example, daily mean-variance efficient frontiers of portfolios, yield curves, annual production charts and annual weather record charts.

Our goal is to develop new models, methodology and associated theory under a general framework of Functional Time Series Analysis for modelling complex dynamic phenomena. We intend to build functional time series models and to do forecasting.

When the true economic system consists of many equations, or our economic observations have a very high dimension, one may meet the ``curse of dimensionality" problem. We try to impose a common factor structure to reduce dimension for the parametric and nonparametric stability analysis of a large system. Replacing unobservable common factors by principle components in parametric and nonparametric estimation will be justified.

In contrast to conventional factor models which focuses on reducing dimensions and modeling conditional first moment, the proposed project devotes attention to dimensional reduction and statistical inference for conditional second moments (covariance matrices). The direct motivation lies in the increasing need to model and explain risk and uncertainty of a large economic system.

The other distinctive point is that the proposed project considers factor models for high frequency data. A key application is the analysis of high dimensional and high-frequency financial time series, although the potential uses are much wider.

Marine ecosystems are generally composed of large numbers of species of widely varying sizes, ranging from unicellular species, through zooplankton and up to large fish and whales. The distribution of the total biomass of all species by size is known as the biomass spectrum.

In addition to the size variation due to differing characteristic sizes of the different species, and unlike the case in terrestrial systems, individuals themselves often undergo increases in body size of several orders of magnitude, from small eggs and larvae at a few millimetres in length up to large adults at the metre length scale.

At different parts of its life cycle an individual will be present at different parts of the biomass spectrum. Despite this apparent complexity it has long been known from field observations that biomass spectra show many regularities. In particular the logarithm of biomass density is approximately linearly related to the logarithm of body length with negative slope. Moreover, the slope of the spectrum is potentially sensitive to environmental and anthropogenic perturbations, for example the removal of large fish due to commercial fishing.

For these reasons biomass spectra have gained currency in recent years as a tool for studying the integrated ecosystem impacts of climate change and human exploitation of the seas.

The relative simplicity of representing the entire ecosystem as a size-structured spectrum has also permitted the development of mathematical representations in terms of partial differential equations, and these can be used to make predictions about ecosystem level responses to fishing and environmental change.

To date most modelling efforts have focused on steady state analyses of biomass spectra, which may be compared to annually averaged size spectra from field observations. In temperate shelf seas such as the North Sea, however, the biomass spectrum is not static but subject to seasonal impulses caused by increased primary production from phytoplankton in the spring bloom, and by seasonal reproduction by zooplankton and fish. This is known from both observational and mathematical models to induce annual ripples in the biomass distribution that propagate up the spectrum, gradually attenuating at larger sizes.

Fish larvae can exploit this by growing in size at a rate that allows them to feed near the peak of the wave, a phenomenon that has been dubbed by John Pope and his co-workers “surf-riding the biomass spectrum”. This project will focus on developing existing models to better represent these processes. In order to test the models the parameters will be estimated using Bayesian inference methods by fitting to a variety of data sets, including a long term (multi-annual) high temporal resolution (weekly) data from the North Sea.

Many fish populations worldwide have been heavily exploited and there is accumulating evidence from both observational and theoretical studies that this harvesting can induce evolutionary changes. Such responses can affect the stock sustainability and catch quality, and so there is a recognized need for new management strategies that minimise these risks. Most results suggest that high mortality on larger fish favours early maturation.

However, recent theoretical work has shown that trade-offs between growth and maturation can lead to more complex evolutionary responses. Surprisingly, harvesting large fish can select for either late or early maturation depending on the effect of maturation on growth rate. To date most theoretical studies have used evolutionary invasion analyses on simple age-based discrete-time models or on continuous-time coupled ODE representations of size structure.

In common with many generic models of fish population dynamics, population control occurs by unspecified density-dependence at settlement. While these simplifications carry the advantage of analytical tractability, the analysis assumes steady state populations. This, together with the stylised life-histories precludes comparing model results with field data on secular changes in size-distributions an sexual maturity.

The work in this project will develop a new generation of testable model for fisheries-induced adaptive changes with the potential to inform future management decisions. This will involve developing a consumer-resource model which a length-structured fish population feeding on a dynamic biomass spectrum.

Differently-sized fish will compete for food by exploiting overlapping parts of the food size spectrum. The population will be partitioned by length at maturity, and this will be the heritable trait under selection. The model will be used to explore how changes in mortality and food abundance affect the evolutionarily stable distribution of maturation lengths.

Comparisons with survey data on North Sea demersal fish will be used to assess whether the historical harvest rates are sufficient to explain growth rate changes as an evolutionary response. Finally the evolutionarily stable optimal harvesting strategies will be identified.

The behaviour of sessile droplets is an area of very active international research, with new publications appearing on an almost daily basis and entire conferences now dedicated to the topic. Over the last decade, Professor Stephen Wilson has collaborated very successfully with Professor Khellil Sefiane from the School of Engineering at the University of Edinburgh (and a Visiting Professor at the University of Strathclyde) of a variety of practically important fluid-dynamical problems, including evaporating droplets, bubble dynamics, self-rewetting fluids, and anti-surfactants.

The aim of the present project is to build on the proposed supervisors’ previous work on evaporating sessile droplets to explore two exciting new aspects of this scientifically and practically important problem.

Very recent work by Sefiane and his collaborators have provided the first comprehensive experimental investigation of vapour absorption by sessile droplets of a desiccant liquid (i.e. one which draws moisture from the air). The mathematical modelling and analysis of such systems is an exciting and challenging open problem.

In practice, droplets almost never occur singly, but, due to the inherent complexity of the multiple-droplet problem, so far very little work has been done on the interactions between evaporating droplets. The aim of the proposed work is to investigate the fascinating but virtually unexplored subject of the collective behaviour of large arrays of small sessile droplets. Interactions between the droplets (in particular, so-called “shielding” effects) are expected to lead to very different collective behaviour compared to that of isolated droplets.

The student will join a lively and mutually supportive cohort of fellow PhD students within the Continuum Mechanics and Industrial Mathematics (CMIM) research group.

You should have (or expect to have) a UK Honours Degree (or equivalent) at 2.1 or above in Mathematics, Mathematics and Physics, Physics or a closely related discipline with a high mathematical content. Knowledge of Continuum Mechanics and mathematical methods (such as asymptotic methods) for the solution of partial differential equations is desirable (but not essential, particularly for overseas applicants).

Informal enquiries can be made to the lead supervisor, Professor Stephen K. Wilson, Department of Mathematics and Statistics, University of Strathclyde, Glasgow at s.k.wilson@strath.ac.uk and/or +44(0)141 548 3820. However, formal applications must be made via the online application procedure which can be found at

https://www.strath.ac.uk/courses/research/mathematicsstatistics/

remembering to list the title of the project as “Mathematical Modelling and Analysis of Sessile Droplets” and Professor Stephen K. Wilson as the first supervisor. There is no need to include a detailed research plan, but a brief outline of your relevant experience (if any) and your motivation for choosing this project would assist with the selection procedure.

In recent years there has been an explosive growth of interest in the behaviour and control of fluids at small (typically sub-millimetre) scales motivated by a range of novel applications including ink-jet printing and lab-on-a-chip technologies.

Much of the most exciting current research concerns the interaction between fluids and both rigid and flexible structures at small scales, and so the aim of the present project is to use a judicious combination of asymptotic methods and judiciously chosen numerical calculations to bring new insight into the behaviour of a variety of novel fluid-structure interaction problems in microfluidics.

In the last decade or so there has been an explosion of interest in droplet evaporation, driven by new technological applications as diverse as crop spraying, printing, cooling technologies such as heat pipes, and DNA micro-array analysis.

One particularly interesting aspect of this problem which has thus far received relatively little attention is that of fluids whose surface tension exhibits a local minimum with temperature, known as self-rewetting fluids, a property that can have a profound effect on the dynamics of droplets on heated substrates.

The aim of the project is to build on the existing literature on conventional surface-tension-gradient driven spreading and droplet drying (see, for example, the references to some of our recent work on these problems given below) to bring new physical insight into this challenging scientific problem, and hence to harness the novel properties of self-rewetting droplets in a range of applications.

The project will be a collaboration colleagues at the University of Edinburgh who will be undertaking a parallel series of experimental investigations on this problem which will be key to the successful outcome of the project.

Dunn, G.J., Wilson, S.K., Duffy, B.R., David, S., Sefiane, K. “The strong influence of substrate conductivity on droplet evaporation” J. Fluid Mech. 623 329-351 (2009)

Dunn, G.J., Duffy, B.R., Wilson, S.K., Holland, D. “Quasi-steady spreading of a thin ridge of fluid with temperature-dependent surface tension on a heated or cooled substrate” Q. Jl. Mech. appl. Math. 62 (4) 365-402 (2009)