C3

In this talk we will discuss a problem that was worked on during MISGSA 2020, a Study Group held in January at The University of Zululand, South Africa.

We look at a communication network with two types of users - Primary users (PU) and Secondary users (SU) - such that we reduce the network to a set of overlapping sub-graphs consisting of SUs indexed by a specific PU. Within any given sub-graph, the PU may be communicating at a certain fixed frequency F. The respective SUs also wish to communicate at the same frequency F, but not at the expense of interfering with the PU signal. Therefore if the PU is active then the SUs will not communicate.

In an attempt to increase information throughput in the network, we instead allow the SUs to communicate at a different frequency G, which may or may not interfere with a different sub-graph PU in the network, leading to a multi-objective optimisation problem.

We will discuss not only the problem formulation and possible approaches for solving it, but also the pitfalls that can be easily fallen into during study groups.

Stress perfusion cardiac magnetic resonance (CMR) imaging has been shown to be highly accurate for the detection of coronary artery disease. However, a major limitation is that the accuracy of the visual assessment of the images is challenging and thus the accuracy of the diagnosis is highly dependent on the training and experience of the reader. Quantitative perfusion CMR, where myocardial blood flow values are inferred directly from the MR images, is an automated and user-independent alternative to the visual assessment.

This talk will focus on addressing the main technical challenges which have hampered the adoption of quantitative myocardial perfusion MRI in clinical practice. The talk will cover the problem of respiratory motion in the images and the use of dimension reduction techniques, such as robust principal component analysis, to mitigate this problem. I will then discuss our deep learning-based image processing pipeline that solves the necessary series of computer vision tasks required for the blood flow modelling and introduce the Bayesian inference framework in which the kinetic parameter values are inferred from the imaging data.

Many manufacturing processes require the use of robots to transport parts around a factory line. Some parts, which are very thin (e.g. car doors)
are prone to elastic deformations as they are moved around by a robot. These should be avoided at all cost. A problem that was recently raised by
F.E.E. (Fleischmann Elektrotech Engineering) at the ESGI 158 study group in Barcelona was to ascertain how to determine the stresses of a piece when
undergoing a prescribed motion by a robot. We present a simple model using Kirschoff-Love theory of flat plates and how this can be adapted. We
outline how the solutions of the model can then be used to determine the stresses. 

Numerous mathematical models have been proposed for modelling cancerous tumour invasion (Gatenby and Gawlinski 1996), angiogenesis (Owen et al 2008), growth kinetics (Wang et al 2009), response to irradiation (Gao et al 2013) and metastasis (Qiam and Akcay 2018). In this study, we attempt to model the qualitative behavior of growth, invasion, angiogenesis and fragmentation of tumours at the tissue level in an explicitly spatial and continuous manner in two dimensions. We simulate the effectiveness of radiation therapy on a growing tumour in comparison with immunotherapy and propose a novel framework based on vector fields for modelling the impact of interstitial flow on tumour morphology. The results of this model demonstrate the effectiveness of employing a system of partial differential equations along with vector fields for simulating tumour fragmentation and that immunotherapy, when applicable, is substantially more effective than radiation therapy.

C*-algebras constructed from topological groupoids allow us to study many interesting and a priori very different constructions
of C*-algebras in a common framework. Moreover, they are general enough to appear intrinsically in the theory. In particular, it was recently shown
by Xin Li that all C*-algebras falling within the scope of the classification program admit (twisted) groupoid models.
In this talk I will give a gentle introduction to this class of C*-algebras and discuss some of their structural properties, which appear in connection
with the classification program.
 

A particularly active area of research in modern algebraic number theory is the study of a class of numbers, called periods. In their simplest form, periods are integrals of rational functions over domains defined by rational in equations. They form a ring, which encompasses all algebraic numbers, logarithms thereof and \pi. They arise in the study of modular forms, cohomology and quantum field theory, and conjecturally have a sort of Galois theory.

We will take a whirlwind tour of these numbers, before discussing non-periods. In particular, we will sketch the construction of an explicit non-period, often forgotten about.

We study the stability properties of the Eulerian mean flow generated by monochromatic surface-gravity waves propagating in a rotating frame. The wave averaged equations, also known as the Craik-Leibovich equations, govern the evolution of the mean flow. For propagating waves in a rotating frame these equations admit a steady depth-dependent base flow sometimes called the Ekman-Stokes spiral, because of its resemblance to the standard Ekman spiral. This base flow profile is controlled by two non-dimensional numbers, the Ekman number Ek and the Rossby number Ro. We show that this steady laminar velocity profile is linearly unstable above a critical Rossby number Roc(Ek). We determine the threshold Rossby number as a function of Ek using a numerical eigenvalue solver, before confirming the numerical results through asymptotic expansions in the large/low Ek limit. These were also confirmed by nonlinear simulations of the Craik-Leibovich equations. When the system is well above the linear instability threshold, Ro >> Roc, the resulting flow fluctuates chaotically. We will discuss the possible implications in an oceanographic context, as well as for laboratory experiments.

Automatic structures are algebraic structures, such as graphs, groups
and partial orders, that can be presented by automata. By varying the 
classes of automata (e.g. finite automata, tree automata, omega-automata) 
one varies the classes of automatic structures. The class of all automatic 
structures is robust in the sense that it is closed under many natural
algebraic and model-theoretic operations.  
In this talk, we give formal definitions to 
automatic structures, motivate the study, present many examples, and
explain several fundamental theorems.  Some results in the area
are deeply connected  with algebra, additive combinatorics, set theory, 
and complexity theory. 
We then motivate and pose several important  unresolved questions in the
area.

In a robust decision, we are pessimistic toward our decision making when the probability measure is unknown. In particular, we optimise our decision under the worst case scenario (e.g. via value at risk or expected shortfall).  On the other hand, most theories in reinforcement learning (e.g. UCB or epsilon-greedy algorithm) tell us to be more optimistic in order to encourage learning. These two approaches produce an apparent contradict in decision making. This raises a natural question. How should we make decisions, given they will affect our short-term outcomes, and information available in the future?

In this talk, I will discuss this phenomenon through the classical multi-armed bandit problem which is known to be solved via Gittins' index theory under the setting of risk (i.e. when the probability measure is fixed). By extending this result to an uncertainty setting, we can show that it is possible to take into account both uncertainty and learning for a future benefit at the same time. This can be done by extending a consistent nonlinear expectation  (i.e. nonlinear expectation with tower property) through multiple filtrations.

At the end of the talk, I will present numerical results which illustrate how we can control our level of exploration and exploitation in our decision based on some parameters.