A spectral characterization of nonlinear normal modes

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity.Belgian Network DYSCO (Dynamical Systems, Control, and Optimization) funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy OfficeThis is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jsv.2016.05.01

Analysis and design of nonlinear resonances via singularity theory

Bifurcation theory and continuation methods are well-established tools for the analysis of nonlinear mechanical systems subject to periodic forcing. We illustrate the added value and the complementary information provided by singularity theory with one distinguished parameter. While tracking bifurcations reveals the qualitative changes in the behaviour, tracking singularities reveals how structural changes are themselves organised in parameter space. The complementarity of that information is demonstrated in the analysis of detached resonance curves in a two-degree-of-freedom system.G. Habib would like to acknowledge the financial support of the Belgian National Science Foundation FRS-FNRS (PDR T.0007.15). The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n.670645

The desmosome and pemphigus

Desmosomes are patch-like intercellular adhering junctions (“maculae adherentes”), which, in concert with the related adherens junctions, provide the mechanical strength to intercellular adhesion. Therefore, it is not surprising that desmosomes are abundant in tissues subjected to significant mechanical stress such as stratified epithelia and myocardium. Desmosomal adhesion is based on the Ca2+-dependent, homo- and heterophilic transinteraction of cadherin-type adhesion molecules. Desmosomal cadherins are anchored to the intermediate filament cytoskeleton by adaptor proteins of the armadillo and plakin families. Desmosomes are dynamic structures subjected to regulation and are therefore targets of signalling pathways, which control their molecular composition and adhesive properties. Moreover, evidence is emerging that desmosomal components themselves take part in outside-in signalling under physiologic and pathologic conditions. Disturbed desmosomal adhesion contributes to the pathogenesis of a number of diseases such as pemphigus, which is caused by autoantibodies against desmosomal cadherins. Beside pemphigus, desmosome-associated diseases are caused by other mechanisms such as genetic defects or bacterial toxins. Because most of these diseases affect the skin, desmosomes are interesting not only for cell biologists who are inspired by their complex structure and molecular composition, but also for clinical physicians who are confronted with patients suffering from severe blistering skin diseases such as pemphigus. To develop disease-specific therapeutic approaches, more insights into the molecular composition and regulation of desmosomes are required

Evaluation of appendicitis risk prediction models in adults with suspected appendicitis

Background Appendicitis is the most common general surgical emergency worldwide, but its diagnosis remains challenging. The aim of this study was to determine whether existing risk prediction models can reliably identify patients presenting to hospital in the UK with acute right iliac fossa (RIF) pain who are at low risk of appendicitis. Methods A systematic search was completed to identify all existing appendicitis risk prediction models. Models were validated using UK data from an international prospective cohort study that captured consecutive patients aged 16–45 years presenting to hospital with acute RIF in March to June 2017. The main outcome was best achievable model specificity (proportion of patients who did not have appendicitis correctly classified as low risk) whilst maintaining a failure rate below 5 per cent (proportion of patients identified as low risk who actually had appendicitis). Results Some 5345 patients across 154 UK hospitals were identified, of which two‐thirds (3613 of 5345, 67·6 per cent) were women. Women were more than twice as likely to undergo surgery with removal of a histologically normal appendix (272 of 964, 28·2 per cent) than men (120 of 993, 12·1 per cent) (relative risk 2·33, 95 per cent c.i. 1·92 to 2·84; P < 0·001). Of 15 validated risk prediction models, the Adult Appendicitis Score performed best (cut‐off score 8 or less, specificity 63·1 per cent, failure rate 3·7 per cent). The Appendicitis Inflammatory Response Score performed best for men (cut‐off score 2 or less, specificity 24·7 per cent, failure rate 2·4 per cent). Conclusion Women in the UK had a disproportionate risk of admission without surgical intervention and had high rates of normal appendicectomy. Risk prediction models to support shared decision‐making by identifying adults in the UK at low risk of appendicitis were identified

Decompositions of Hilbert spaces, stability analysis and convergence probabilities for discrete-time quantum dynamical semigroups

We investigate convergence properties of discrete-time semigroup quantum dynamics, including asymptotic stability, probability and speed of convergence to pure states and subspaces. These properties are of interest in both the analysis of uncontrolled evolutions and the engineering of controlled dynamics for quantum information processing. Our results include two Hilbert space decompositions that allow for deciding the stability of the subspace of interest and for estimating of the speed of convergence, as well as a formula to obtain the limit probability distribution for a set of orthogonal invariant subspaces

Uncovering detached resonance curves in single-degree-of-freedom systems

In the last decade, the problem of detached resonant curves (DRCs) has received a growing attention. However, most of recent research works seem to ignore papers from the sixties and the seventies, which already investigated the subject. With the aim of recovering this "forgotten" literature, we briefly review the majority of the studies on the topic. Furthermore, implementing singularity theory in conjunction with a classical harmonic balance procedure, we analyze DRCs in a single-degree-of-freedom system, possessing nonlinear damping force. A geometrical interpretation of the phenomenon is provided

Isolated resonances and nonlinear damping

We analyze isolated resonance curves (IRCs) in single-degree-of-freedom systems possessing nonlinear damping. Through the combination of singularity theory and the averaging method, the onset and merging of IRCs, which coincide to isola and simple bifurcation singularities, respectively, can be analytically predicted. Numerical simulations confirm the accuracy of the analytical developments. Another important finding of this paper is that we unveil a geometrical connection between the topology of the damping force and IRCs. Specifically, we demonstrate that extremas and zeros of the damping force correspond to the appearance and merging of IRCs. Considering a damping force possessing several minima and maxima confirms the general validity of the analytical result. It also evidences a very complex scenario for which different IRCs are created, co-exist and then merge together to form a super IRC which eventually merges with the main resonance peak

Analysis and design of nonlinear resonances via singularity theory

Bifurcation theory and continuation methods are well-established tools for the analysis of nonlinear mechanical systems subject to periodic forcing. We illustrate the added value and the complementary information provided by singularity theory with one distinguished parameter. While tracking bifurcations reveals the qualitative changes in the behaviour, tracking singularities reveals how structural changes are themselves organised in parameter space. The complementarity of that information is demonstrated in the analysis of detached resonance curves in a two-degree-of-freedom system

A spectral characterization of nonlinear normal modes

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity