Global bifurcation of homoclinic solutions of hamiltonian systems

We provide global bifurcation results for a class of nonlinear hamiltonian systemsComment: 25 page

Strong convergence rates of probabilistic integrators for ordinary differential equations

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2017), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.Comment: 25 page

Data Assimilation: A Mathematical Introduction

These notes provide a systematic mathematical treatment of the subject of data assimilation

Existence and stability of TE modes in a stratified non-linear dielectric

Starting from Maxwell's equations for a stratified optical medium with a non-linear refractive index, we derive the equations for monochromatic planar TE modes. It is then shown that TE modes in which the electromagnetic fields are travelling waves correspond to solutions of these reduced equations in the form of standing waves. The equations of the paraxial approximation are then formulated and the stability of the travelling waves is investigated in that contex

On the spectral theory of a tapered rod

This paper establishes the main features of the spectral theory for the singular two-point boundary-value problem and which models the buckling of a rod whose cross-sectional area decays to zero at one end. The degree of tapering is related to the rate at which the coefficient A tends to zero as s approaches 0. We say that there is tapering of order p ≥ 0 when A C([0, 1]) with A(s) > 0 for s (0, 1] and there is a constant L (0, ∞) such that lims→0A(s)/sp = L. A rigorous spectral theory involves relating (1)−(3) to the spectrum of a linear operator in a function space and then investigating the spectrum of that operator. We do this in two different (but, as we show, equivalent) settings, each of which is natural from a certain point of view. The main conclusion is that the spectral properties of the problem for tapering of order p = 2 are very different from what occurs for p < 2. For p = 2, there is a non-trivial essential spectrum and possibly no eigenvalues, whereas for p < 2, the whole spectrum consists of a sequence of simple eigenvalues. Establishing the details of this spectral theory is an important step in the study of the corresponding nonlinear model. The first function space that we choose is the one best suited to the mechanical interpretation of the problem and the one that is used for treating the nonlinear problem. However, we relate this formulation in a precise way to the usual L2 setting that is most common when dealing with boundary-value problem

12.—Spectral Theory of Rotating Chains

Two eigenvalue problems associated with steady rotations of a chain are considered. To compare the spectra of these two problems, let σK(n) denote the set of all angular velocities with which a chain of unit length with one end fixed and the other free can rotate in a vertical plane so as to have exactly n nodes on thevertical axis (including the fixed end). In a linearised theory σK(n) is a single point, i.e. In the full non-linear theory σK(n) is an infinite interval lying to the right of ⍵n. Indeed, This is established in [1]. Next, let σM(β, n) denote the set of all angular velocities with which a chain, having ends fixed at unit distance apart on the vertical axis, can rotate in a vertical plane so as to have exactly n+l nodes on the vertical axis (including the ends) and so that the tension takes the value β at the lower end. This problem, in which the length of the chain is not prescribed is a model for a spinning process in which the ‘chain' is continuously created in a rotating configuration. For β>0, we again have in the linearised theory that σM(β, n) is a singleton, i.e. In the full non-linear theory σM(β, n) lies to the left of λn(β). Although unable to determine exactly σM(β, n) for β>0, we have where ⍵n λn(β), and are all characterised as the nth zeros of known combinations of Bessel function

Spectrum of a Self-Adjoint Operator and Palais-Smale Conditions

The spectrum and essential spectrum of a self-adjoint operator in a real Hilbert space are characterized in terms of Palais-Smale conditions on its quadratic form and Rayleigh quotient respectivel

Helium ion microscopy and energy selective scanning electron microscopy – two advanced microscopy techniques with complementary applications

Both scanning electron microscopes (SEM) and helium ion microscopes (HeIM) are based on the same principle of a charged particle beam scanning across the surface and generating secondary electrons (SEs) to form images. However, there is a pronounced difference in the energy spectra of the emitted secondary electrons emitted as result of electron or helium ion impact. We have previously presented evidence that this also translates to differences in the information depth through the analysis of dopant contrast in doped silicon structures in both SEM and HeIM. Here, it is now shown how secondary electron emission spectra (SES) and their relation to depth of origin of SE can be experimentally exploited through the use of energy filtering (EF) in low voltage SEM (LV-SEM) to access bulk information from surfaces covered by damage or contamination layers. From the current understanding of the SES in HeIM it is not expected that EF will be as effective in HeIM but an alternative that can be used for some materials to access bulk information is presented