An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.
There is a mismatch between levels of crime and its fear and often, cities might see an increase or a decrease in crime over time while the fear of crime remains unchanged. A model that considers fear of crime as an opinion shared by simulated individuals on a network will be presented, and the impact that different distributions of crime have on the fear experienced by the population will be explored. Results show that the dynamics of the fear is sensitive to the distribution of crime and that there is a phase transition for high levels of concentration of crime.
Many scientists, and in particular mathematicians, report difficulty in understanding thermodynamics. So why is thermodynamics so difficult? To attempt an answer, we begin by looking at the components in an exposition of a scientific theory. These include a mathematical core, a motivation for the choice of variables and equations, some historical remarks, some examples and a discussion of how variables, parameters, and functions (such as equations of state) can be inferred from experiments. There are other components too, such as an account of how a theory relates to other theories in the subject.
It will be suggested that theories of thermodynamics are hard to understand because (i) many expositions appear to argue from the particular to the general (ii) there are several different thermodynamic theories that have no obvious logical or mathematical equivalence (iii) each theory really is subtle and requires intense study (iv) in most expositions different theories are mixed up, and the different components of a scientific exposition are also mixed up. So, by presenting one theory at a time, and by making clear which component is being discussed, we might reduce the difficulty in understanding any individual thermodynamic theory. The key is perhaps separation of the mathematical core from the physical motivation. It is also useful to realise that a motivation is not generally the same as a proof, and that no theory is actually true.
By way of illustration we will attempt expositions of two of the simplest thermodynamic theories – reversible and then irreversible thermodynamics of homogeneous materials – where the mathematical core and the motivation are discussed separately. In conclusion we’ll relate these two simple theories to other, foundational and generalised, thermodynamic theories.
We will talk about the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations. When viscosity coefficients are given as a constant multiple of density's power, based on some analysis of the nonlinear structure of this system, by introducing some new variables and the initial layer compatibility conditions, we identify the class of initial data admitting a local regular solution with far field vacuum and finite energy in some inhomogeneous Sobolev spaces, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier (2006, Anal. Simi. Fluid Dynam.), Jiu-Wang-Xin (2014, JMFM) and so on. Moreover, in contrast to the classical well-posedness theory in the case of the constant viscosity, we show that one can not obtain any global classical solution whose $L^\infty$ norm of $u$ decays to zero as time $t$ goes to infinity under the assumptions on the conservation laws of total mass and momentum.
Novel machine learning techniques based on deep learning, i.e., the data-driven manipulation of neural networks, have reported remarkable results in many areas such as image classification, game intelligence, or speech recognition. Driven by these successes, many scholars have started using them in areas which do not focus on traditional machine learning tasks. For instance, more and more researchers are employing neural networks to develop tools for the discretisation and solution of partial differential equations. Two reasons can be identified to be the driving forces behind the increased interest in neural networks in the area of the numerical analysis of PDEs. On the one hand, powerful approximation theoretical results have been established which demonstrate that neural networks can represent functions from the most relevant function classes with a minimal number of parameters. On the other hand, highly efficient machine learning techniques for the training of these networks are now available and can be used as a black box. In this talk, we will give an overview of some approaches towards the numerical treatment of PDEs with neural networks and study the two aspects above. We will recall some classical and some novel approximation theoretical results and tie these results to PDE discretisation. Afterwards, providing a counterpoint, we analyse the structure of network spaces and deduce considerable problems for the black box solver. In particular, we will identify a number of structural properties of the set of neural networks that render optimisation over this set especially challenging and sometimes impossible. The talk is based on joint work with Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok, Felix Voigtlaender, and Mones Raslan
Matrix completion is an area of great mathematical interest and has numerous applications, including recommender systems for e-commerce. The recommender problem can be viewed as follows: given a database where rows are users and and columns are products, with entries indicating user preferences, fill in the entries so as to be able to recommend new products based on the preferences of other users. Viewing the interactions between user and product as links in a bipartite graph, the problem is equivalent to approximating a partially observed graph using clusters. We propose a divide and conquer algorithm inspired by the work of [1], who use recursive rank-1 approximation. We make the case for using an LP rank-1 approximation, similar to that of [2] by a showing that it guarantees a 2-approximation to the optimal, even in the case of missing data. We explore our algorithm's performance for different test cases.
[1] Shen, B.H., Ji, S. and Ye, J., 2009, June. Mining discrete patterns via binary matrix factorization. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 757-766). ACM.
[2] Koyutürk, M. and Grama, A., 2003, August. PROXIMUS: a framework for analyzing very high dimensional discrete-attributed datasets. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 147-156). ACM.
Many protein interaction databases provide confidence scores based on the experimental evidence underpinning each in- teraction. The databases recommend that protein interac- tion networks (PINs) are built by thresholding on these scores. We demonstrate that varying the score threshold can re- sult in PINs with significantly different topologies. We ar- gue that if a node metric is to be useful for extracting bio- logical signal, it should induce similar node rankings across PINs obtained at different thresholds. We propose three measures—rank continuity, identifiability, and instability— to test for threshold robustness. We apply these to a set of twenty-five metrics of which we identify four: number of edges in the step-1 ego network, the leave-one-out dif- ference in average redundancy, average number of edges in the step-1 ego network, and natural connectivity, as robust across medium-high confidence thresholds. Our measures show good agreement across PINs from different species and data sources. However, analysis of synthetically gen- erated scored networks shows that robustness results are context-specific, and depend both on network topology and on how scores are placed across network edges.
In this work, study the mean first saturation time (MFST), a generalization to the mean first passage time, on networks and show an application to the 2015 Burundi refugee crisis. The MFST between a sink node j, with capacity s, and source node i, with n random walkers, is the average number of time steps that it takes for at least s of the random walkers to reach a sink node j. The same concept, under the name of extreme events, has been studied in previous work for degree biased-random walks [2]. We expand the literature by exploring the behaviour of the MFST for node-biased random walks [1] in Erdős–Rényi random graph and geographical networks. Furthermore, we apply MFST framework to study the distribution of refugees in camps for the 2015 Burundi refugee crisis. For this last application, we use the geographical network of the Burundi conflict zone in 2015 [3]. In this network, nodes are cities or refugee camps, and edges denote the distance between them. We model refugees as random walkers who are biased towards the refugee camps which can hold s_j people. To determine the source nodes (i) and the initial number of random walkers (n), we use data on where the conflicts happened and the number of refugees that arrive at any camp under a two-month period after the start of the conflict [3]. With such information, we divide the early stage of the Burundi 2015 conflict into two waves of refugees. Using the first wave of refugees we calibrate the biased parameter β of the random walk to best match the distribution of refugees on the camps. Then, we test the prediction of the distribution of refugees in camps for the second wave using the same biased parameters. Our results show that the biased random walk can capture, to some extent, the distribution of refugees in different camps. Finally, we test the probability of saturation for various camps. Our model suggests the saturation of one or two camps (Nakivale and Nyarugusu) when in reality only Nyarugusu camp saturated.
[1] Sood, Vishal, and Peter Grassberger. ”Localization transition of biased random walks on random
networks.” Physical review letters 99.9 (2007): 098701.
[2] Kishore, Vimal, M. S. Santhanam, and R. E. Amritkar. ”Extreme event-size fluctuations in biased
random walks on networks.” arXiv preprint arXiv:1112.2112 (2011).
[3] Suleimenova, Diana, David Bell, and Derek Groen. ”A generalized simulation development approach
for predicting refugee destinations.” Scientific reports 7.1 (2017): 13377.
The physics literature has for a long time posited a connection between the geometry of continuous random fields and discrete percolation models. Specifically the excursion sets of continuous fields are considered to be analogous to the open connected clusters of discrete models. Recent work has begun to formalise this relationship; many of the classic results of percolation (phase transition, RSW estimates etc) have been proven in the setting of smooth Gaussian fields. In the first part of this talk I will summarise these results. In the second I will focus on the number of excursion set components of Gaussian fields in large domains and discuss new results on the mean and variance of this quantity.
TBC