## PROJECTS

###### Disclaimer: The sole purpose of all the simulations and figures on this website is to demonstrate the range of expertise that MMCESP can currently offer, therefore, any contents from this list of projects or any figure or animation that you find on any part of this website is strictly not for sale, because some of these are collaborative work with academic partners and companies which may have different licence agreement to that which regulates commercial contents online. Therefore it is explicitly brought to your attention that MMCESP is only commercialising the problem-solving skills and not the contents that it uploads on this website.

##### Model for the dynamics of spot pattern on stingrays

This project is based on the application of reaction-diffusion theory to model the seasonal pattern transition of spots on stingrays. Some stingrays have an interesting dynamical evolution of spots on their skin, which transitions from season to season into different types of spatial structure, the video on the right is a reaction-diffusion model simulated to produce the evolution of spatial pattern on a disk-shape domain that is used as a two dimensional approximation for the stingray's body. Deterministic spatiotemporal evolution models have a great number of applications in the frontiers of understanding and predicting certain aspects of naturally evolving processes.

##### Model for serotonin release in brain

This is the simulation of a diffusive model for the concentration of serotonin pulse in brain. The domain is the r-z cross-section plane in a cylindrical approximated domain representing a single synapse and the evolving surface shows the decay of serotonin molecules as they undergo oxidation to release electrons and thus transform the chemical signal into electric pulse. The z-axis here represent the direction of the synaptic cleft height, which is positive in the direction of the pre-synaptic membrane and negative in the direction of post-synaptic membrane.

##### Pattern formation on the surface of Klein bottle

This is a numerical simulation of the activator-depleted reaction-diffusion model for Turing type pattern formation on the parametric representation of the surface of a Klein bottle. The reaction-kinetics together with diffusion manages to establish self-organised behaviour on the surface. Klein bottle is a surface that requires four spatial dimensions for it to exist without self-intersecting, in the same way that a mobius strip requires three spatial dimensions for its edge not to self-intersect.

##### Heisenberg uncertainty principle

This is a mathematical model that demonstrates how the uncertainty in the position of a quantum particle increases with its velocity. In principle it means that the degree of our knowledge about the position of a particle decides how precisely we can measure its velocity and vice versa. In particular the certainty of simultaneous knowledge on both the position and velocity is bounded above by a finite value given in terms of Plank's constant h and the irrational number Pi. If such bound is exceeded for example by increasing the accuracy of measurement about the velocity of a particle, then Heisenberg uncertainty principle claims that such gain of extra certainty comes at the price of loss for the equivalent amount of accuracy about the of position of that particle.

##### Temperature evolution on a spinning wheel

A model for aircraft wheel whilst spinning in which the evolution of heat due to friction on the surface of the wheel is governed by diffusion and transport due to spinning. The model shows both forward and backward spinning at varying velocities, which can be clearly visualised if observed on full screen. The wheel is modelled by a two dimensional parametric surface of a torus for which the major radius is a unit length and the minor radius is one third of the length of the major radius.

##### Model for an Oxygen sensor "The Clark Electrode"

This is a mathematical model to quantify the performance of Clark Electrode. This device was invented back in 1956 by a German chemist named Leland Clark. It was the first device that doctors used to obtain a quantitative measure of Oxygen concentration in blood. Prior to the use of the electrode doctors used to estimate Oxygen concentration in blood from the redness of ciliary muscles inside the patient's eye. Finite element method is used to quantify both the steady state and time-dependent models of Clark electrode. We used non-uniform mesh to account for the discontinuities (singular regions) at the material interface within the domain

##### BPEPDE

The name stands for Bayesian Parameter Estimation for Partial Differential Equations, which is a python based machine learning software. Our expertise developed a user-guide full documentation for the software that returns probability distributions for the parameters of a given mathematical model that is equipped with experimental data. It produces probability distributions for one or more parameters in the model. BPEPDE employs Bayesian statistics and the main computations are based on Metropolis-Hastings algorithm. The example on the right shows the output of BPEPDE for a simple harmonic oscillatory model in which the unknown parameter is the gravitational constant of earth. The performance accuracy of BPEPDE is demonstrated through experimental data of sample sizes from 100, 1000, 10000 and 1000000 for which BPEPDE recovers the earth's gravitational constant correct up to three decimal places which is 9.807 meters per second squared.

##### Image recognition

In this project we developed an efficient image analysis algorithm that is based on the use of singular value decomposition method (svd). The algorithm takes a number of images as input and builds from these a spectral characteristic criteria for deciding whether a newly encountered image is a member of the training class of the input images or not. This is a machine learning algorithm that extensively uses algebra through the analysis of the dominant eigenvalues to decide the specificity of image recognition criteria. In the second column within the image on the right, we have the orthogonal projection of the average face of each of the film-star onto the eigenvectors corresponding upto and including the first 64 dominant eigenvalues. The bar chart is the unique characteristic computed from the projection of the average of the training data set onto the eigenvectors of the dominant effect.

##### Model of damped harmonic motion

In this project we fit a set of data from damped harmonic motion with random noise to a model with stochasticity, in which we characterise and refine by parameter estimation the variance of noise such that the limiting case with the variance of the noise tends to zero the solution of the stochastic model exactly coincides with the deterministic model that is fitted through a black curve. We use the prior knowledge as given that the experimental data belongs to a damped harmonic motion. We use the standard assumptions of a normally distributed noise in the experimental measurement.

##### UK power sources and demand

This is a data-driven case-study to analyze time-series data of power demand in the UK. We studied five most predominant sources of power namely coal, wind, nuclear, oil and solar. Considering these power sources, we explored the overall power demand on the national scale. One of the many aims of the study was to explore the evolution of different power sources, in particular, coal with the wind. This was to predict the future trajectories of these sources from the perspective of overall power consumption. In turn, we applied time series forecasting techniques to predict the future behaviour of these trajectories and how numerous power sources interact with each other.