2014/15 semester 2 events in Mathematics

Cardiac electrophysiology models have been used for over 50 years to understand the mechanisms by which the heart beat is synchronised and adapts to changes in physiological demands. The success of the field has led to an explosion of different models of ion currents and cardiac cells. For just one particular potassium current (the IKr current carried through the hERG channel) we have counted over 30 mathematical models (or parameterisations of a handful of structures). We discuss how we went back to the beginning and attempted to design experiments to select the correct model structure, as well as to identify each parameter in the models as uniquely as practical. We will show how this is allowing us to quantify variability between cells for the first time.

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The Kepler satellite's primary goal has been to search for planets orbiting other stars. To do this it has stared at a fixed patch of sky containing over 100000 stars for several years in the hope that it will catch tiny dips in their light caused by planetary transits. However, this wealth of data enables us to look for other interesting features on these stars, in particular stellar flares. I will present a method we developed to try and efficiently detect flares in this large data set. I will discuss the issues that arise when looking at Kepler data, such as dealing with the variability of stellar light curves and problems with other transient artefacts. I will present some results from looking at a short stretch of Kepler data. Finally, I will talk about what the future plans for the search method are.

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In this talk we will introduce a model that arises in the theory of non-equilibrium phase transitions, in particular in the description of the growth of self-affine surfaces. We will briefly comment on the role that this model is meant to play in this physical theory. Furthermore, we will mention some open questions of physical nature related to it. Subsequently we will start with the rigorous analysis of our model, which is a fourth order partial differential equation. We will describe our progress in building an existence theory for the full model, of parabolic nature, and for its stationary counterpart. For the latter case existence and multiplicity results are provided, and for the former one we will show local in time existence of the solution, that can be made global for small enough data, and cannot if these are large enough. We will show how these results fit into the physical theory, and what open questions are left for the future.

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Sunspots occurring on the solar surface following a typical pattern during the activity cycle. At the beginning of the cycle they appear at high latitude, whereas at the end they appear close to the equator. This is associated with an underlying strong toroidal field which migrates equatorward. Since a few years this behavior has been reproduced in global convective dynamo simulations. I will present results from our simulations of global convective dynamos. All of these simulations produce cyclic and migrating mean magnetic fields. Through detailed comparisons, we show that the migration direction can be clearly explained by an alpha-Omega dynamo wave following the Parker-Yoshimura rule. This lead to the conclusion, that the equatorward migration in this and other work is due to a positive (negative) alpha-effect in the northern (southern) hemisphere and a negative radial gradient of rotation outside the inner tangent cylinder of these models. This idea is supported by a strong correlation between negative radial shear and toroidal field strength in the region of equatorward propagation. In the Sun the only region, where the rotation rate possesses a negative radial gradient, is in the near-surface shear layer. A positive alpha-effect there would lead to an equatorward propagating dynamo wave. Furthermore, I will present results of combined simulation of solar dynamo and a coronal envelope. These simulation give us indication, that the latitudinal temperature variations play an important role in generating the differential rotation profile of the Sun through turbulent Reynold stresses. I will show, that the meridional component of the Reynolds stress can explain the formation of a near-surface shear layer in the Sun. This lead to the conclusion, that in Sun the magnetic field is generated and stored just below the surface, and sunspot are generated by local mechanisms of concentration flux.

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The finite element solution of the Stokes problem is subject to the well-known inf-sup condition. This condition usually depends on the domain and the degree of the polynomial spaces used in the discretisation. Now, when anisotropic finite elements are used, this inf-sup condition usually degenerates with the aspect ratio. In this talk I will present results identifying the part of the pressure space that is responsible for this degeneration, and present a way to solve that. In the second part of the talk, this technique will be applied to some, non inf-sup stable this time, pairs of spaces for the Stokes and Oseen equations. The work presented in this talk is in collaboration with Mark Ainsworth (Brown) and Andreas Wachtel (Strathclyde).

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In this talk I would like to present some ideas on bifurcation analysis for a very simple model of a gene regulatory network. This will be a short story about the long way from a simple idea on a sunny morning in Lyon to a beautiful functional analysis.

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Experiments and numerical simulations provide plenty of data describing entangled systems arising from dynamics or otherwise. For example, there are recent experiments studying entangled hair, and magnetic fields are often modelled as entangled lines. But what is a good measure of entanglement, and how much entanglement should we expect? We look at modeling the entanglement of purely random orbits such as Brownian motion. This is joint work with Marko Budisic and Huanyu Wen.

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The growth of solid tumors can be described at a number of different scales from the cell to the organ scales. For a large number of cells, the ’fluid mechanical’ approach has been advocated recently by many authors in mathematics or biophysics. Several levels of mathematical descriptions are commonly used, including only elastic effects, nutrients, active movement, surrounding tissue, vasculature remodeling and several other features.
We will focus on the links between two types of mathematical models. The ‘microscopic’ description is at he cell population density level and a more macroscopic, description is based on a free boundary problem close to the classical Hele-Shaw equation. Asymptotic analysis is a tool to derive these Hele-Shaw free boundary problems from cell density systems in the stiff pressure limit. This modeling also opens other questions as circumstances in which instabilities develop.
This work is a collaboration with F. Quiros and J.-L. Vazquez (Universidad Autonoma Madrid), M. Tang (SJTU) and N. Vauchelet (LJLL).

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The derivation of continuum models which represent underlying discrete or microscale phenomena is emerging as an important part of mathematical biology: integration between subcellular, cellular and tissue-level behaviour is crucial to understanding tissue growth and mechanics. I will present a new macroscale description of nutrient-limited tissue growth, which is formulated as a microscale free-boundary problem within a porous medium. A multiscale homogenisation method is employed to enable explicit accommodation of the influence of the underlying microscale tissue structure, and its evolution, on the macroscale dynamics.

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Recent observations started revealing the potential diversity in compositions of extrasolar planets. With the expectation of the further constraints in the future, it is crucial to understand how the final composition of planets is related to planet formation processes. A few groups have estimated the elemental abundances of simulated Solar System planets and successfully reproduced a global trend of the compositions of terrestrial planets. However, most studies were limited to refractory species and did not consider volatile ones that include biogenic elements such as C and N. In this talk, we discuss two potential pathways to deliver biogenic species to terrestrial planets – one of them is the delivery from the outer region of a protostellar disk and the other is the incorporation of these species in the inner region of the disk as a result of the ice line evolution. We test these scenarios by assuming that the initial elemental abundances of embryos and planetesimals are determined by the equilibrium condensation model, and by simulating the formation of terrestrial planets.

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Plant growth typically occurs through the coordinated anisotropic expansion of plant cells. Growth is regulated by hormones and is driven by high intracellular pressures generated by osmosis. This machinery allows a plant primary root, for example, to penetrate soil in a direction guided by gravity, while seeking out nutrients and avoiding obstacles. I will describe the biomechanical aspects of a computational multiscale model for root gravitropism that incorporates descriptions of cell walls as fibre-reinforced viscoelastic polymer networks and adopts upscaling approaches to describe the growth of multicellular tissues.

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Many space plasmas are in a partially ionised state, where neutrals, heavy ions and electrons are treated as a separate fluids, however they behave in a collective manner. Usually the standard theory considers that the plasma is in ionisation equilibrium, i.e. the rates of ionisation and recombination are equal and collisions occurring in the plasma are always elastic However this is an idealistic restriction, the two rates are not always equal and a different approach is needed. In my presentation I will consider the ionisation non-equilibrium of plasmas (with special emphasis on prominences) due to inelastic collisions and construct the rate equation for the excited of atoms and ion/electrons. Assuming a simple form for the ionisation and excitation rate, I will discuss briefly what changes such considerations will bring into the equations describing the nature of magnetic and magnetoacoustic waves.

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Mathematical models are being used increasingly to understand the roles of particular genes, proteins or signals in the gene regulation networks that govern the behaviour of bacteria. Using examples from Bacillus subtilis, Staphylococcus aureus and Clostridium difficile (ranging through sporulation, toxin production and cell-cell communication), we illustrate how numerical and asymptotic approaches to ODE models can elucidate individual roles and highlight the key aspects of a system. The use of asymptotic methods is particularly fruitful in the common situation where there is a lack of data available for model parameterisation. Where possible, we focus on the generation of hypotheses which can be tested in the laboratory.

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Magnetic helicity is a valuable quantity in MHD since has two unique properties: it is almost conserved and it cascades to large scales. However, observations provide only 1D or 2D cuts of magnetic field configurations while computation of helicity requires the knowledge of the 3D magnetic field and its vector potential. First, I will review how theory is rescuing observations to derive magnetic helicity from different types of magnetic data. Second, I will describe cases where helicity was tracked during eruptions. Third, I will focus on the helicity estimation in magnetic clouds where precise magnetic field measurements are available but only along the spacecraft trajectory. Finally, I will compare helicity budgets realized over one solar cycle.

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On October 22, 2011, a circular ribbon, shortly preceded by localized inner and remote compact brightening, was observed in AR11324 using SDO/AIA. The global evolution of such peculiarly shaped flares is believed to be caused by reconnection at a coronal null point, with the circular ribbon being the footprint of the dome-shaped fan, and the inner and remote brightening tracing the two photospheric anchoring of the spines. The high spatial resolution and time cadence of AIA observations give the unique opportunity to explore the details of this flare event, which is globally agreeing with, but not entirely explained by, the null/slipping reconnection model. In particular, pre-flare brightening and a three-phase time evolution of EUV signals do not have straightforward interpretations. Also, the non-eruptive character of the flare makes standard interpretations inadequate. In this talk I will briefly introduce the nonlinear-force-free field (NLFFF) extrapolation technique, and apply it to the interpretation of that event. In particular, I will show how the combination of the NLFFF extrapolation with a topological analysis based on quasi-separatrix layers (QSL) allows to interpret the AIA observation of the dynamic of the flare. In addition to the expected topological features of a standard circular-ribbon flare model, we find a flux rope and a complex QSL internal to the dome, which are connected with the brightening locations. Guided by the AIA observation, we are able to provide a coherent description of the specific dynamic of this event that, because of its non-eruptive nature, has a peculiar reconnection pattern. The different peaks in the EUV signals are then explained by the cooling of specific sets of reconnected field lines that are matching both the reconnection process evinced by our modeling and the multi-wavelengths observations. The combination of high-resolution observations with powerful modeling and analysis techniques allow us to interpret very complex observations with realistic models of the coronal field, helping us to deepen the theoretical understanding of flare dynamics.

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This talk will be in two parts. In the first part I will introduce recent work exploring the behaviour of braided magnetic flux ropes with contorted geometry. This work extends the work of several members of the Dundee MHD research group on straight braided flux tubes; where it has been demonstrated that the evolution of “braided” fields differ significantly to those of twisted tubes. The additional freedom of the tube to distort in the atmosphere is crucial here. The complexity of the braided fields implies the are more likely to be prevented form expanding, where their twisted counterparts are not, this is confirmed in the simulations. Much of the work relies on the geometry of tubes and ribbons commonly used in elasticity and biophysics... The second topic concerns an explanation for the mechanism by which thin ribbons can be sheared with a pair of scissors to create ornamental coils for gifts. I report on experiments in which we identify key parameters which dictated the final coiling radius, namely the blade sharpness, the force applied against the blade and the speed at which the ribbon is pulled over the blade. We develop a viscoplastic thin ribbon model to explain this phenomenon and find very reasonable fits to the data.

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