Mathematics & StatisticsSeminars and colloquia

28AUG2024
Prof Zhuoyi Song (Fudan University of China)  Stochastic Analysis
Title: Mathematical Analysis of Refractory Period Distribution and the Underlying Molecular Regulation Mechanisms for Signal Transduction SystemsLocation: LT908Time: 3.00pm

04SEPT2024
Dr Jorge Carneiro (NOVA University of Lisbon)  Health Ecology Modelling
Title: TBCLocation: LT908Time: 1.00pm
Nonlinear evolutionary processes, operator theory for the study of differential and integral equations. Enumerative, bijective and algebraic combinatorics.
Title: Mesh pattern occurrence in random permutations
Date: 3.00pm Tuesday 3rd October
Venue: LT907
Abstract: We say that the likelihood of a mesh pattern is the asymptotic probability that a random permutation contains an occurrence of the pattern. In this talk we will investigate the likelihood of a variety of patterns, determining their values for every vincular pattern. For bivincular patterns, the Small Anchors Theorem distinguishes between those patterns whose likelihood equals zero, those whose likelihood is positive but less than 1, and those whose likelihood equals 1. We also know how to determine the (rational) likelihood of any bivincular pattern formed of what we call anchored trees. Other bivincular patterns, such as the small ascent and small descent, have irrational likelihoods, whose values can be established using the Chen–Stein method. This is joint work with Jason Smith.
Title: On semitransitivity of (extended) Mycielski graphs
Date: 3.00pm Tuesday 17th October 2023
Venue: LT908
Abstract: An orientation of a graph is semitransitive if it is acyclic, and for any directed path v0 → v1 → · · · → vk either there is no arc between v0 and vk, or vi → vj is an arc for all 0 ≤ i < j ≤ k. An undirected graph is semitransitive if it admits a semitransitive orientation. Semitransitive graphs generalize several important classes of graphs, and they are precisely the class of wordrepresentable graphs studied extensively in the literature.
The Mycielski graph of an undirected graph is a larger graph, constructed in a certain way, that maintains the property of being trianglefree but enlarges the chromatic number. These graphs are important as they allow to prove the existence of trianglefree graphs with arbitrarily large chromatic number. An extended Mycielski graph is a certain natural extension of Mycielski graphs.
In this talk, I will discuss a complete characterization of semitransitive extended Mycielski graphs and comparability Mycielski graphs, as well as a conjectured complete characterization of semitransitive Mycielski graphs. My results are a farreaching extension of the result of Kitaev and Pyatkin on nonsemitransitive orientability of the Mycielski graph µ(C5) of the cycle graph C5. Also, I will mention how to use a recent result of Kitaev and Sun to shorten the length of the original proof of nonsemitransitive orientability of µ(C5) from 2 pages to a few lines.
Title: Eigenvalues of canonical systems
Date: 3.00pm Tuesday 31st October 2023
Venue: LT907
Abstract: In this talk I shall consider eigenvalues of 2x2 canonical systems, which cover onedimensional Schrödinger equations, SturmLiouville equations, Jacobi operators, Dirac systems and (generalised) Krein strings as special cases. I am particularly interested in the asymptotic behaviour and the density of eigenvalues. This behaviour is connected with the growth and the smoothness of the coefficients. It turns out that the results change substantially when one moves from dense to sparse spectrum. I will also mention a trace formula for the eigenvalues.
Title: Beyond the Hodge Theorem: curl and asymmetric pseudodifferential projections
Date: 3.00pm Tuesday 7th November 2023
Venue: LT907
Abstract: Consider a single photon living in curved space. It is described by Maxwell’s equations. We seek solutions harmonic in time. This reduces to the spectral problem for the operator curl, whose spectrum is, in general, asymmetric about zero (think particle/antiparticle).
Spectral asymmetry is a classical subject in analysis and geometry, whose origins lie in the papers of Atiyah, Patodi and Singer. In this talk I will discuss a new approach to the study of spectral asymmetry based on the use of pseudodifferential techniques developed in a series of recent joint papers by Dmitri Vassiliev and myself.
Emphasis will be placed on ideas and motivation; the talk will include a brief historical overview of the development of the subject area.
Title: Travelling waves in the RosenauKdV equation
Date: 3.00pm Tuesday 28th November 2023
Venue: LT907
Abstract: We consider the existence of monotone travelling waves in the RosenauKdV equation, which exhibits phenomena that cannot be seen in the wellknown BurgersKdV equation. Many problems are still open. This is joint work with N. Bedjaoui and G. Maypaokha (UPJV).
Title: Mathematical analysis of fracture and related phenomena in atomistic modelling of crystalline materials
Date: 3.00pm, Tuesday 5th December 2023
Venue: LT907
Abstract: The atomistic modelling of fracture and related phenomena in crystalline materials poses a string of mathematically nontrivial and exciting challenges, both on a theoretical and a practical level. At the heart of the problem lies a discrete domain of atoms (a lattice), which exhibits spatial inhomogeneity induced by the crack surface, particularly pronounced in the vicinity of the crack tip. Atoms interact in a highly nonlinear way, resulting in a severely nonconvex energy landscape facilitating nontrivial behaviour of atoms such as (i) crack propagation; (ii) nearcrack tip plasticity  emission and movement of defects known as dislocations in the vicinity of the crack tip; (iii) surface effects  atoms at the crack surface relaxing or possibly attaining an altogether different crystalline structure. On the practical side, the richness of possible phenomena renders the task of setting up numerical simulations particularly tricky  numerical artefacts, e.g. induced by prescribing a particular boundary condition, can lead to inconsistent results.
In this talk I will aim to summarise ongoing efforts aimed at putting the atomistic modelling of fracture on a rigorous mathematical footing. I will begin by introducing a mathematical framework, based in part on bifurcation theory, giving rise to welldefined models for which regularity and stability of solutions can be discussed, followed by describing how the theory can be used to set up numerical continuationbased simulations, e.g. for Mode I crack propagation in silicon on the (111) cleavage plane, using stateoftheart interatomic potentials. Subsequently I will outline how this framework can be used to rigorously derive upscaled models of nearcracktip plasticity.
Finally, I will also touch upon recent work on proposing a much more general framework in which a wide range of defect nucleation & migration phenomena in the atomistic modelling of crystalline materials can be treated as bifurcation events.
Title: Thresholds for patterns in random permutations
Date: 3.00pm Tuesday 12th December 2023
Venue: LT420
Abstract: In this talk we will investigate thresholds for the appearance and disappearance of consecutive patterns occurring within large random permutations as the number of inversions increases. We establish these lower and upper thresholds for any fixed consecutive pattern. We also consider thresholds for classical and vincular patterns. To do so, we work with inversion sequences, which we consider to be weak integer compositions. As a result, we introduce a model of random integer compositions which we call the geometric random composition. This talk will focus on how we transfer thresholds for patterns in the geometric random composition to get thresholds for patterns in the uniform random permutation.
Title: Distribution of mesh patterns
Date: 2.00pm Tuesday 23rd January 2024
Venue: LT907
Abstract: The notion of a mesh pattern, generalizing several classes of permutation patterns, was introduced in 2011 by Branden and Claesson to provide explicit expansions for certain permutation statistics as, possibly infinite, linear combinations of (classical) permutation patterns. There is a long line of research papers dedicated to the study of mesh patterns and their generalizations.
In this talk, I will discuss a systematic study of avoidance and distribution of mesh patterns of short length, and also some other more general results
Title: Coagulation and Combinatorics
Date: 2.00pm on Tuesday 20th February 2024
Venue: LT907
Abstract: I will review the recent combinatorics approach to discrete coagulation equations due to Łepek and coworkers and the open problems that arise in that area.
Title: Vortex stretching in NavierStokes turbulence
Date: 3.00pm on Tuesday 27th February 2024
Venue: LT907
Abstract: Turbulence in the NavierStokes equations is a major nonlinear problem, that cannot be tackled with existing analytical methods. I discuss key characteristics of turbulence, explaining its physical content and the momentum balance it encodes. I indicate the key analytical difficulty and past failures to tackle it in the velocitypressure space. I introduce the vortex dynamical viewpoint and explain the association between the latter and key turbulent physics. By employing the vortex dynamical viewpoint, I compute key aspects of turbulent vortex stretching, including, among other, the Lyapunov exponents of the tangent system, and the correlations between strain rate eigendirections and coherent, filamentary vorticity.
Title: Coagulation, nonassociative algebras and combinatorial trees
Date: 2.00pm Tuesday 12th March 2024
Venue: LT907
Abstract: We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the nonassociative algebra generated by the convolution product of the coalescence term. The nonassociative solution expansion is equivalently represented by binary trees. We demonstrate that the existence of such solutions corresponds to establishing the compatibility of two binarytree generating procedures, by:
 grafting together the roots of all pairs of ordercompatible trees at preceding orders, or
 attaching binary branches to all free branches of trees at the previous order.
We then show that the solution represents a linearised flow, and also establish a new numerical simulation method based on truncation of the solution tree expansion and approximating the integral terms at each order by fast Fourier transform. In particular, for general separable frequency kernels, the complexity of the method is linearloglinear in the number of spatial modes/nodes.
Title: Asymptotic enumeration of monotone permutation grid classes
Date: 2.00pm Tuesday 26th March 2024
Venue: LT907
Abstract: A monotone grid class Grid(M) is a set of permutations whose shape satisfies constraints specifies by the matrix M, all of whose entries are in {1,−1, 0}. Each entry of the matrix corresponds to a cell in a gridding of a permutation. If the entry is 1, then the points in the cell must increase. If it is −1, they must decrease. If it is 0, the cell must be empty. To find the exact number of permutations of length n in Grid(M) is hard, because a permutation may have more than one gridding, so we only determine the asymptotic enumeration. To do this, we find the number of gridded permutations, the typical proportion of points in each cell, and the ways in which permutations can be gridded. Our focus is mainly on Lshaped, Tshaped, and Xshaped classes.
Title: Noninterpenetration of matter in lowerdimensional structures
Date: 3pm Wednesday 19th June 2024
Venue: LT907
Abstract: Noninterpenetration of matter is a wellknown challenge for solid elastic materials combining analytical and geometrical aspects. In the bulk model, at least on the conceptual level, noninterpenetration is quite understood even if many challenges still remain open. In lowerdimensional structures (plates, rods), the situation seems to be even less clear. Focusing on rods in the plane, we will introduce a possible concept of noninterpenetration and show density and Γlimit results in this case. This is a joint work in progress with B. Benešová, D. Campbell, and S. Hencl (all from Prague).
Liquid crystals, Droplet evaporation, Thinfilm flow, Complex fluids, Medical product design, Flows in porous & complex media, Nonlinear waves.
Title: Resonant freesurface water waves in closed basins
Date: 1.00pm Thursday 25th January 2024
Title: Datadriven design optimisation of chemical reactors
Date: 1.00pm Thursday 1st February 2024
Title: Oscillatory bodily flows: the eye and the brain
Date: 1.00pm Thursday 8th February 2024
Title: Determination of the index of refraction of antireflection coatings
Date: 1.00pm Thursday 22nd February 2024
Title: Electrostatics and variational perturbation theory
Date: 1.00pm Thursday 29th February 2024
Title: TBC
Date: 1.00pm Thursday 7th March 2024
Title: Interplay between zonal jets, waves and turbulence: application to gas giants
Date: 1.00pm Thursday 14th March 2024
Title: Growing in the wind  an interdisciplinary investigation of wind influence on plant growth
Date: 1.00pm Thursday 21st March 2024
Title: tbc
Date: 3.00pm Wednesday 7th February 2024
Venue: tbc
Abstract: tbc
Title: tbc
Date: 3.00pm Wednesday 28th February 2024
Venue: LT908
Abstract: tbc
Marine Population Modelling, Mathematical Biology, Epidemiology and Statistical Informatics
Title: The Role of Individuals in Determining the Impacts of Changing Environments on Population Level Dynamics
Date: Wednesday 7th February, 1.002.00pm
Venue: LT907
Abstract:
Climate change is having profound effects on the incidence of vector borne disease, such as dengue, chikungunya and West Nile virus. However, developing effective measures of disease risk on a global scale are challenged by the complex ways in which environmental variation acts in vectorhostpathogen systems. One way in which insect vectors, such as mosquitos, respond to environmental variation is to change their traits this can result in populations comprised of groups of individuals which differ in their traits (e.g. size, fecundity). For example, if food is scarce for juvenile mosquitos then when they become adults they are smaller, and lay fewer eggs to ensure there is less competition for food in the next generation. The environment of the juvenile determines the trait the individual has as an adult. In this way the individuals adapt to the environment as well as shape the environment for future generations.
Current models oversimplify the interaction between individuals, populations and the environment, so risk misestimating predictors of disease risk. Here, we derive a mathematical framework for capturing the interaction of individuals, their traits and the population dynamics. I will show how this new mathematical framework leads to both interesting mathematical questions and can be used to help explain the location, magnitude and timing of historical dengue outbreaks.
Title: Quantifying Performance and Resilience in Marine Assemblages with Complex LifeHistories
Date: Wednesday 21st February, 1.002.00pm
Venue: LT908
Abstract: Ongoing global change challenges our ability to predict how natural populations will both respond to novel climatic regimes and utilise available habitat space. Corals are critical to the functioning of coastal reef ecosystems and, yet, in spite of their intrinsic and economic value, are threatened by an increasing plethora of abiotic and biotic disturbances. Preventing the ensuing loss of coral coverage and diversity calls for a mechanistic understanding of resilience across coral species and populations that is currently lacking in coral reef science. Meanwhile, changing coastal conditions and our everexpanding exploitation of marine resources, has heralded a perceived increase in the abundance of coastal jellyfish assemblages. While jellyfish are also an important component of coastal marine communities, their public perception is often tainted by their proclivity for aggregating in vast numbers, known as jellyfish blooms, which can disrupt fishing and tourism activities. However, despite the socioeconomic ramifications associated with the formation of these jellyfish blooms, the complex and cryptic lifecycles exhibited by jellyfish species largely precludes accurate predictions into their timing and location, restricting our ability to manage and mitigate their ecological and economic impacts. Here, I will introduce research comprising statestructured population modelling, novel transient demographic approaches, and stateoftheart hydrodynamic simulations, that offers valuable insight into the performance and resilience of these complex and cryptic marine assemblages under future climate scenarios; frameworks that represent key decisionsupport tools for informing both the conservation of global coral reefs and our management of the socioeconomic impacts of jellyfish bloom formation.
Title: Some DataDriven Approaches to Surveillance of Covid19 in the UK
Date: Wednesday 28th February, 1.002.00pm
Venue: LT908
Abstract: Whilst much of the work in modelling transmission of the pandemic was conducted using mathematical transmission models, the quantity of data made available through open data portals, such as the Covid Dashboard, provided alternatives to understanding and intervening to improve public health outcomes. In this talk I will outline some statistical approaches using surveillance data from varying spatial scales to study underlying dynamics of transmission of Covid19 in the UK. In particular, multivariate flexible regression models and dimension spatial dimension reduction techniques will be used to estimate relative transmissibility of emerging variants of concern, as well as nowcasting current states of pandemic from noisy multivariate time series, respectively.
Title: Estimating the Size of Aedes Aegypti Populations from Dengue Incidence Data: Implications for the Risk of Yellow Fever, Zika Virus and Chikungunya Outbreaks
Date: Wednesday 17th April, 1.002.00pm
Venue: TBC
Abstract: In this talk I present a model to estimate the density of aedes mosquitoes in a community affected by dengue. The model is based on the fitting of a continuous function to the incidence of dengue infections, from which the density of infected mosquitoes is derived straightforwardly. Further derivations allows the calculation of the latent and susceptible mosquitoes' densities, the sum of the three equals the total mosquitoes' density. The model is illustrated with the case of the risk of urban yellow fever resurgence in dengue infested areas but the same methods apply for other aedestransmitted infections like Zika and chikungunya viruses.
Title: Using Minimalistic FoodWeb Models to Inform Fisheries Management
Date: Wednesday 1st May, 1.002.00pm
Venue: TBC
Abstract: Chance and Necessity (CaN) modelling is a minimalistic foodweb modelling framework integrating the existing knowledge about an entire or part of the ecosystem, the available data, and uncertainties. Unlike most of the large ecosystem models, the CaN framework does not aim at predicting the future state of commercial stocks, but rather to reconstruct possible past dynamics of a foodweb. Reconstructions result from the delimitation of the possible “statespace” of the foodweb based on ecological survey data and expert knowledge, and the exploration of this “statespace”. Here, I will present the underlying concepts of CaN modelling, present case studies of CaN modelling applications and provide examples on how the model outputs can be used to improve the management of commercial fisheries in the future.
Title: Modelling and Inference and Heterogeneity in Bacterial Growth
Date: Wednesday 22nd May, 1.002.00pm
Venue: LT908
Abstract: E. coli is a common bacterium found in the intestines of humans and animals. While many strains are harmless, some can cause serious illness, such as diarrhoea, urinary tract infections, and in severe cases, even kidney failure or death. Understanding its behaviour and mechanisms of infection is therefore vital for public health. Like many bacteria, E. coli can also develop resistance to antibiotics. Persister cells are a small subpopulation of bacterial cells which are slowgrowing, which allows them to survive in harsh conditions such as exposure to antibiotics or other stressful environments. These cells are distinct from regular bacterial cells because they are not killed by antibiotics. Studying how resistance emerges and spreads within bacterial populations helps in developing strategies to combat antibioticresistant strains. The aim of this talk is to discuss the range of modelling approaches to determine the impact noninheritable variation between individual cells has on population growth and population response to stressful environments. Various models will be considered including systems of ODEs, agestructured PDEs, renewal equations, and stochastic branching process. Where possible, comparisons and connections will be made between the different modelling approaches. We also discuss some initial work on model inference, where the objective is to indirectly determine the possibility of heterogenous subpopulations from total population data. These techniques will be applied to a probabilistic model of data generated from a microfluidic dynamic cytometer.
Title: Heterogeneity and Identifiability in Mathematical Biology
Date: Wednesday 29th May, 1.002.00pm
Venue: LT908
Abstract: Heterogeneity is a dominant factor in the behaviour of many biological processes and is often a significant source of the variation observed in biological data. Despite this, it is relatively rare for mathematical models of biological systems to incorporate variability in model parameters as a source of noise. In the first part of talk, I motivate and present a new computationally efficient method for inference and identifiability analysis of socalled random parameter models based on an approximate momentmatched solution constructed through a multivariate Taylor expansion.
Effective application of mathematical models to interpret biological data and make accurate predictions typically requires that model parameters are identifiable. Yet, there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture the spatial heterogeneities features inherent to many phenomena. In the second part of this talk, I provide an introduction to structural identifiability before presenting a new methodology applicable to a broad class of PDE models. I then conclude by discussing the future of identifiability analysis for the spatial, random parameter, and stochastic models that are fast becoming pervasive throughout mathematical biology.
Title: How Lifestyle Differences Affect Epidemic Spread: Heterogeneous Density Dependence in Infectious Contact Rates
Date: Wednesday 10th July, 1.002.00pm
Venue: LT908
Abstract:
Title: Cascading and MultiStressor Effects in Coastal Ecosystems
Date: Monday 19th August, 3.004.00pm
Venue: LT511
Abstract: Coastal ecosystems are simultaneously exposed to a plethora of humaninduced stressors, such as climate warming, eutrophication, pollution, overfishing, and pathogens. These stressors interact with each other, driving counterintuitive responses and undesirable results in management efforts. For example, the reduction of nutrient loads is widely believed to be the solution for eutrophication problems in coastal ecosystems. However, we show that these efforts have not always resulted in, and may not in the future result in, the desired reduction of phytoplankton biomass. Instead, the effects of deeutrophication are overridden by climate warming, which intensifies temperaturedependent grazing of zooplankton by small carnivores, such as juvenile fish, leading to reduced herbivory (by zooplankton on phytoplankton) and thus increased standing stock of algae. This effect is especially strong in the shallow and turbid waters of coastal seas worldwide. High turbidity persistently limits the rates of photosynthesis, shifting bottomup control towards topdown control and a stronger influence of higher trophic levels. In another case study, we show the compounding effects of climate warming and marine viruses on food web dynamics in a coastal environment, leading to a decline in primary production and carbon export, and higher retention of nutrients in the upper water column. Our results highlight the importance of stressor interactions and cascading effects in understanding responses in coastal ecosystems, a benchmark for ecosystem modelling and the effective development of management and conservation strategies.
Title: TBC
Date: Wednesday 4th September, 1.002.00pm
Venue: LT908
Abstract:
Numerical solutions of PDEs, Stochastic computation, Numerical linear algebra, Computational physics & engineering
Title: DEM applied in the mining and mineral extraction industry
Date: 1pm Tuesday 19th March 2024
Venue: LT907
Abstract: As the world transitions towards net zero though electrification our reliance upon critical minerals and ores such as copper, iron, lithium, etc is ever increasing. The mining and mineral extraction industry currently uses 47% of the worlds available energy supply. Mining and mineral extraction companies are faced with decarbonising their impact on the planet while simultaneously increasing extraction from excavation sites to meet demand using traditional technology. Discrete Element Modelling (DEM) gives us insight into why products behave the way they do and process condition parameter exploration to ultimately drive towards new designs and a more sustainable future. The presentation will cover who the Weir Advanced Research Centre (WARC) is at UoS, our journey so far in using DEM for Weir products and the future direction of coupling such tools with other mathematical tools.
Title: Reducible networks of PrandtlIshlinskii operators with economic and financial applications
Date: 1pm Tuesday 26th March 2024
Venue: LT907
Abstract: If the nodes in a network have inputoutput responses satisfying a certain property (ie. they are PrandtlIshlinskii (PI) operators) then remarkable simplifications are possible. For arbitrary network topologies (under mild additional conditions) the entire network can be rigorously reduced to a single aggregated PI operator. This is true even if cascading behaviour, such as bubbles and crashes, can occur in the network.
Two applications will be presented. One is a financial market model incorporating momentum traders. The other is a macroeconomic model with the aggregated PI operator representing inflation expectations in the economy.
Title: Noninterpenetration of matter in lowerdimensional structures
Date: 3pm Wednesday 19th June 2024
Venue: LT907
Abstract: Noninterpenetration of matter is a wellknown challenge for solid elastic materials combining analytical and geometrical aspects. In the bulk model, at least on the conceptual level, noninterpenetration is quite understood even if many challenges still remain open. In lowerdimensional structures (plates, rods), the situation seems to be even less clear. Focusing on rods in the plane, we will introduce a possible concept of noninterpenetration and show density and Γlimit results in this case. This is a joint work in progress with B. Benešová, D. Campbell, and S. Hencl (all from Prague).
Stochastic Differential Equations, Stochastic Computation, Time Series, Probability, Image Analysis
Title: Numerical Solutions of a MarkovSwitching OneFactor Volatility Model with NonGlobally Lipschitz Continuous Coefficients
Friday 25th January, 2024, 4.005.00pm
Venue: LT907
Abstract: We extend the onefactor stochastic volatility model to incorporate coefficient terms of superlinear growth under the Markovswitching framework. Since the proposed model is intractable analytically, we develop various mathematical techniques to investigate convergence in probability of the numerical solutions under the local Lipschitz condition. Finally, we perform simulation examples to demonstrate the theoretical results and justify the theoretical results for the valuation of some financial options.
Title: Explicit Convergence Rates for the M/G/1 Queue under Perturbation
Friday 19th April, 2024, 3.004.00pm
Venue: LT907
Abstract: Stochastically ordered Markov process is a topic of special concern to us. As is mentioned by Meyn and Tweedie, many Markov processes are stochastically ordered in their initial state. Thus, we established convergence rates for discretetime Markov chains on a countable state space that are stochastically ordered starting from a stationary distribution under perturbation. We investigate the explicit criteria to obtain the ordinary ergodicity, geometric ergodicity and polynomial ergodicity for the embedded M/G/1 queue under perturbation. The explicit geometric convergence rates for the original system and the system under perturbation are calculated. Our bounds in the geometric case and polynomial case are closely connected to the first hitting times. Two examples are provided to illustrate our result.
Title: Truncated EulerMaruyama Method for TimeChanged SDEs with SuperLinear State Variables and H/"older's Continuous Time Variables
Friday 3rd May, 2024, 3.004.00pm
Venue: LT907
Abstract: In this work, an explicit numerical method is developed for a class of nonautonomous timechanged stochastic differential equations, whose coefficients obey H\”older's continuity in terms of the time variables and are allowed to grow superlinearly in terms of the state variables. The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained. Simulations are provided to demonstrate the theoretical results.
Title: A New Criterion on Stability in Distribution for a Hybrid Stochastic Delay Differential Equation
Friday 17th May, 2024, 3.004.00pm
Venue: LT907
Abstract: A new sufficient condition for stability in distribution of a hybrid stochastic delay differential equation (SDDE) has been proposed in this work. Although the new criterion leads to stability for an SDDE, its main component only depends on the coefficients of a corresponding SDE without delay. The Lyapunov method is applied to find an upper bound, so that the SDDE is stable in distribution if the delay is less than the upper bound. Also, the criterion shows that delay terms can be an impetus toward the stability in distribution.
Title: Limit Theorems for Weakly Dependent Random Fields with Applications to HighDimensional Time Series
Friday 31st May, 2024, 2.003.00pm
Venue: LT908
Abstract: We establish limit theorems, law of large numbers (LLN) and central limit theorem (CLT), for weakly dependent arrays of random fields which are not necessarily stationary and may have asymptotically unbounded moments. The weak dependence condition is proved to be inherited through transformation, and this makes our results applicable to statistical inference of highdimensional time series models. Consistency and asymptotic normality of maximum likelihood estimation can be proved for high dimensional time series models which are checked to be weakly dependent as random fields, allowing for nonstationarity and unbounded trending moments, when sample size and/or dimension go to infinity. As an example for application of our general theory, asymptotic properties of estimation for network autoregression are obtained under mild conditions.
Title: Mathematical Analysis of Refractory Period Distribution and the Underlying Molecular Regulation Mechanisms for Signal Transduction Systems
Friday 28th August, 2024, 3.004.00pm
Venue: LT908
Abstract: Cellular decisions are governed by signal transduction pathways involving a series of chemical reactions. The refractory period (RP) represents the time it takes for the reaction system to regain responsiveness after a stimulus, making it a crucial factor in signal transduction pathways. Analytical expressions for RP distributions are essential for understanding its molecular regulation mechanisms. However, it depends on solving CME for systems with second or higherorder reactions, which remain open problems with traditional methods. We are developing new theories and methodologies to solve RP distributions for general timevariant signal transduction systems with secondorder reactions. Our recent research shows that using pathwise representations can bypass solving CMEs analytically. Using this method, we solved the RP distribution for a class of nonlinear timevariant systems with A+AC type of secondorder reactions. We will extend to more complicated systems with A+B — C type of systems.
Edinburgh Mathematics Society (EMS)
Title: Can We Rely On AI?
Date: 3.00pm Friday 19th January 2024
Venue: LT908
Abstract: Over the last decade, adversarial attack algorithms have revealed instabilities in deep learning tools. These algorithms raise issues regarding safety, reliability and interpretability in artificial intelligence (AI); especially in high risk settings. Mathematics is at the heart of this landscape, with ideas from optimization, numerical analysis and high dimensional stochastic analysis playing key roles. From a practical perspective, there has been a war of escalation between those developing attack and defence strategies. At a more theoretical level, researchers have also studied bigger picture questions concerning the existence and computability of successful attacks. I will present examples of attack algorithms in image classification and optical character recognition. I will also outline recent results on the overarching question of whether, under reasonable assumptions, it is inevitable that AI tools will be vulnerable to attack.
Tea/coffee will also be served in the staff common room (LT911) from 2:30pm.
For those of you who cannot attend in person, the talk will be streamed over zoom.
https://strath.zoom.us/j/87299195984
Meeting ID: 872 9919 5984
Password: 729852
Jointly Hosted Seminars
Title: Solution multiplicity and effects of data and eddy viscosity on NavierStokes solutions inferred by physicsinformed neural networks
Date: 10am Wednesday 6th December 2023
Venue: Zoom (https://liverpoolacuk.zoom.us/j/97615683533?pwd=SWVxVXNnRmRCaUVlUkNINlRxNm94Zz09) Meeting ID: 976 1568 3533 Passcode: m1LJJ$nU
Abstract: Physicsinformed neural networks (PINNs) have emerged as a new simulation paradigm for fluid flows and are especially effective for inverse and hybrid problems. However, vanilla PINNs often fail in forward problems, especially at high Reynolds (Re) number flows. Herein, we study systematically the classical liddriven cavity flow at Re=2,000, 3,000 and 5,000. We observe that vanilla PINNs obtain two classes of solutions, one class that agrees with direct numerical simulations (DNS), and another that is an unstable solution to the NavierStokes equations and not physically realizable. We attribute this solution multiplicity to singularities and unbounded vorticity, and we propose regularization methods that restore a unique solution within 1\% difference from the DNS solution. In particular, we introduce a parameterized entropyviscosity method as artificial eddy viscosity and identify suitable parameters that drive the PINNs solution towards the DNS solution. Furthermore, we solve the inverse problem by subsampling the DNS solution, and identify a new eddy viscosity distribution that leads to velocity and pressure fields almost identical to their DNS counterparts. Surprisingly, a single measurement at a random point suffices to obtain a unique PINNs DNSlike solution even without artificial viscosity, which suggests possible pathways in simulating high Reynolds number turbulent flows using vanilla PINNs.
Speaker Bio: Zhicheng Wang received Ph.D. degree in engineering thermal physics from University of Chinese Academy of Sciences in 2013. He is currently an associate professor at School of Energy and Power Engineering, Dalian University of Technology, Dalian, China. His main research interests include high order numerical methods and scientific machine learning for predicting turbulent flows and multiphase phase flows, as well as the fluid structure interactions. He has published more than 20 papers in Proc. Natl. Acad. Sci. U.S.A., J. Comput. Phys., J. Fluid Mech., Comput. Method Appl. M., J. Fluids Struct., Phys. Fluids.