The phase diagram for a lattice ring polymer under applied force, with variable solvent quality, for different topological knot states, is determined for the first time.In addition to eliminating pseudophases where the polymer is flattened into a single layer, it is found that non-trivial knots result in additional pseudophases under tensile force conditions.

Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation algorithms designed to update the estimation of the state in an on-line fashion, as data is acquired sequentially. For linear problems subject to Gaussian noise, filtering can be performed exactly using the Kalman filter. For nonlinear systems filtering can be approximated in a systematic way by particle filters. However in high dimensions these particle filtering methods can break down. Hence, for the large nonlinear systems arising in applications such as oceanography and weather forecasting, various ad hoc filters are used, mostly based on making Gaussian approximations. The purpose of this work is to study the accuracy and stability properties of these ad hoc filters. We work in the context of the 2D incompressible Navier-Stokes equation, although the ideas readily generalize to a range of dissipative partial differential equations (PDEs). By working in this infinite dimensional setting we provide an analysis which is useful for the understanding of high dimensional filtering, and is robust to mesh-refinement. We describe theoretical results showing that, in the small observational noise limit, the filters can be tuned to perform accurately in tracking the signal itself (filter accuracy), provided the system is observed in a sufficiently large low dimensional space; roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics. The tuning corresponds to what is known as variance inflation in the applied literature. Numerical results are given which illustrate the theory. The positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters can perform poorly in reproducing statistical variation about the true signal.

We consider an elliptic optimal control problem where the objective functional contains evaluations of the state at a finite number of points. In particular, we use a fidelity term that encourages the state to take certain values at these points, which means our problem is related to ones with state constraints at points. The analysis and numerical analysis differs from when the fidelity is in the L 2 norm because we need the state space to embed into the space of continuous functions. In this paper we discretise the problem using two different piecewise linear finite element methods. For each discretisation we use two different approaches to prove a priori L 2 error estimates for the control. We discuss the differences between these methods and approaches and present numerical results that agree with our analytical results.