In this talk, we shall introduce various identities among partitions of integers, and how these can be expressed via formal power series. In particular, we shall look at the Rogers Ramanujan identities of power series, and discuss possible combinatorial proofs using partitions and Durfree squares.
Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods, each of which highlights the properties of graphs in a unique way. I will discuss a novel approach to study graphs: the simplex geometry (a simplex is a generalized triangle). This perspective, proposed by Miroslav Fiedler, introduces techniques from (simplex) geometry into the field of graph theory and conversely, via an exact correspondence. We introduce the graph-simplex correspondence, identify a number of basic connections between graph characteristics and simplex properties, and suggest some applications as example.
Reference: https://arxiv.org/abs/1807.06475
In this talk I will introduce Hilbert's 10th Problem (H10) and the model-theoretic notions necessary to explore this problem from the perspective of mathematical logic. I will give a brief history of its proof, talk a little about its connection to decidability and definability, then close by speaking about generalisations of H10 - what has been proven and what has yet to be discovered.
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.
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.
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.
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.