The kissing polynomials and their Hankel determinants

In this paper, we investigate algebraic, differential and asymptotic properties of polynomials $p_n(x)$ that are orthogonal with respect to the complex oscillatory weight $w(x)=\mathrm {e}^{\mathrm {i}\omega x}$ on the interval $[-1,1]$, where $\omega>0$. We also investigate related quantities such as Hankel determinants and recurrence coefficients. We prove existence of the polynomials $p_{2n}(x)$ for all values of $\omega \in \mathbb {R}$, as well as degeneracy of $p_{2n+1}(x)$ at certain values of $\omega $ (called kissing points). We obtain detailed asymptotic information as $\omega \to \infty $, using recent theory of multivariate highly oscillatory integrals, and we complete the analysis with the study of complex zeros of Hankel determinants, using the large $\omega $ asymptotics obtained before.

Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface

We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel–Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus $0$, then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.

Analysis of a model of a natural gas pipeline—a transfer function approach

A framework for natural gas pipelines is developed in a context similar to the theory of electric transmission lines. The system of semi-linear partial differential equations describing the time-dependent flow of natural gas is linearized around the steady-state flow. Additional approximations lead to a constant coefficient linear system that is equivalent to an electrical circuit that is analytically solvable and admits an ABCD matrix representation of input and output. The sinusoidal steady-state operation of natural gas pipelines is analysed including the distortion of waves. It is shown that the timing of the propagation of phases and other events is accurately represented in the approximation. The quantitative accuracy for flux and gas density of the approximation depending on different operating scenarios and depending on the frequency of the disturbances is documented.

Learning stochastic closures using ensemble Kalman inversion

Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore, researchers often seek simpler descriptions that describe complex phenomena without numerically resolving all the interacting components. Stochastic differential equations (SDEs) arise naturally as models in this context. The growth in data acquisition, both through experiment and through simulations, provides an opportunity for the systematic derivation of SDE models in many disciplines. However, inconsistencies between SDEs and real data at short time scales often cause problems, when standard statistical methodology is applied to parameter estimation. The incompatibility between SDEs and real data can be addressed by deriving sufficient statistics from the time-series data and learning parameters of SDEs based on these. Here, we study sufficient statistics computed from time averages, an approach that we demonstrate to lead to sufficient statistics on a variety of problems and that has the secondary benefit of obviating the need to match trajectories. Following this approach, we formulate the fitting of SDEs to sufficient statistics from real data as an inverse problem and demonstrate that this inverse problem can be solved by using ensemble Kalman inversion. Furthermore, we create a framework for non-parametric learning of drift and diffusion terms by introducing hierarchical, refinable parameterizations of unknown functions, using Gaussian process regression. We demonstrate the proposed methodology for the fitting of SDE models, first in a simulation study with a noisy Lorenz ’63 model, and then in other applications, including dimension reduction in deterministic chaotic systems arising in the atmospheric sciences, large-scale pattern modeling in climate dynamics and simplified models for key observables arising in molecular dynamics. The results confirm that the proposed methodology provides a robust and systematic approach to fitting SDE models to real data.

The Pantelides algorithm for delay differential-algebraic equations

We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

Robust chaos and the continuity of attractors

As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely, but that for many families of piecewise-smooth maps it provides a way to think about changing structures within parameter regions of robust chaos and form a stronger notion of robustness. We obtain conditions for the continuity of an attractor and demonstrate the results with coupled skew tent maps, the Lozi map and the border-collision normal form.

A PDE model for bleb formation and interaction with linker proteins

The aim of this paper is to further develop mathematical models for bleb formation in cells, including cell membrane interactions with linker proteins. This leads to nonlinear reaction–diffusion equations on a surface coupled to fluid dynamics in the bulk. We provide a detailed mathematical analysis and investigate some singular limits of the model, connecting it to previous literature. Moreover, we provide numerical simulations in different scenarios, confirming that the model can reproduce experimental results on bleb initiation.

Likely oscillatory motions of stochastic hyperelastic solids

Stochastic homogeneous hyperelastic solids are characterized by strain-energy densities where the parameters are random variables defined by probability density functions. These models allow for the propagation of uncertainties from input data to output quantities of interest. To investigate the effect of probabilistic parameters on predicted mechanical responses, we study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material, formulated as quasi-equilibrated motions where the system is in equilibrium at every time instant. Additionally, we study finite shear oscillations of a cuboid, which are not quasi-equilibrated. We find that, for hyperelastic bodies of stochastic neo-Hookean or Mooney–Rivlin material, the amplitude and period of the oscillations follow probability distributions that can be characterized. Further, for cylindrical tubes and spherical shells, when an impulse surface traction is applied, there is a parameter interval where the oscillatory and non-oscillatory motions compete, in the sense that both have a chance to occur with a given probability. We refer to the dynamic evolution of these elastic systems, which exhibit inherent uncertainties due to the material properties, as ‘likely oscillatory motions’.

Moving frames and Noether’s finite difference conservation laws I

We consider the calculation of Euler–Lagrange systems of ordinary difference equations, including the difference Noether’s theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We introduce the difference moving frame, a natural discrete moving frame that is adapted to difference equations by prolongation conditions. For any Lagrangian that is invariant under a Lie group action on the space of dependent variables, we show that the Euler–Lagrange equations can be calculated directly in terms of the invariants of the group action. Furthermore, Noether’s conservation laws can be written in terms of a difference moving frame and the invariants. We show that this form of the laws can significantly ease the problem of solving the Euler–Lagrange equations, and we also show how to use a difference frame to integrate Lie group invariant difference equations. In this Part I, we illustrate the theory by applications to Lagrangians invariant under various solvable Lie groups. The theory is also generalized to deal with variational symmetries that do not leave the Lagrangian invariant. Apart from the study of systems that are inherently discrete, one significant application is to obtain geometric (variational) integrators that have finite difference approximations of the continuous conservation laws embedded a priori. This is achieved by taking an invariant finite difference Lagrangian in which the discrete invariants have the correct continuum limit to their smooth counterparts. We show the calculations for a discretization of the Lagrangian for Euler’s elastica, and compare our discrete solution to that of its smooth continuum limit.