Strong Approximation of Empirical Copula Processes by Gaussian Processes

We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm

Semiclassical approximations for Hamiltonians with operator-valued symbols

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter $\varepsilon\ll 1$ controls the separation of time scales and the limit $\varepsilon\to 0$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $\varepsilon\to 0$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $\varepsilon$-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn, coming from an \epsi-dependent classical Hamilton function and an $\varepsilon$-dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order $\varepsilon^2$. In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics. Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation.Comment: Final version to appear in Commun. Math. Phys. Results have been strengthened with only minor changes to the proofs. A section on the Hofstadter model as an application of the general theory was added and the previous section on other applications was remove

Nonlinear coherent states and Ehrenfest time for Schrodinger equation

We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.Comment: 27 page

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

In this paper, we model Rogue Waves as localized instabilities emerging from homogeneous and stationary background wavefields, under NLS dynamics. This is achieved in two steps: given any background Fourier spectrum P(k), we use the Wigner transform and Penrose’s method to recover spatially periodic unstable modes, which we call unstable Penrose modes. These can be seen as generalized Benjamin–Feir modes, and their parameters are obtained by resolving the Penrose condition, a system of nonlinear equations involving P(k). Moreover, we show how the superposition of unstable Penrose modes can result in the appearance of localized unstable modes. By interpreting the appearance of an unstable mode localized in an area not larger than a reference wavelength λ0 as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to δ spectra, where the standard Benjamin–Feir instability is recovered, and to more general spectra. In that context, we present a scheme for the numerical resolution of the Penrose condition and estimate the sharpest possible localization of unstable modes. Keywords: Rogue Waves; Wigner equation; Nonlinear Schrodinger equation; Penrose modes; Penrose conditio

A new MRI rating scale for progressive supranuclear palsy and multiple system atrophy: validity and reliability

AIM To evaluate a standardised MRI acquisition protocol and a new image rating scale for disease severity in patients with progressive supranuclear palsy (PSP) and multiple systems atrophy (MSA) in a large multicentre study. METHODS The MRI protocol consisted of two-dimensional sagittal and axial T1, axial PD, and axial and coronal T2 weighted acquisitions. The 32 item ordinal scale evaluated abnormalities within the basal ganglia and posterior fossa, blind to diagnosis. Among 760 patients in the study population (PSP = 362, MSA = 398), 627 had per protocol images (PSP = 297, MSA = 330). Intra-rater (n = 60) and inter-rater (n = 555) reliability were assessed through Cohen's statistic, and scale structure through principal component analysis (PCA) (n = 441). Internal consistency and reliability were checked. Discriminant and predictive validity of extracted factors and total scores were tested for disease severity as per clinical diagnosis. RESULTS Intra-rater and inter-rater reliability were acceptable for 25 (78%) of the items scored (≥ 0.41). PCA revealed four meaningful clusters of covarying parameters (factor (F) F1: brainstem and cerebellum; F2: midbrain; F3: putamen; F4: other basal ganglia) with good to excellent internal consistency (Cronbach α 0.75-0.93) and moderate to excellent reliability (intraclass coefficient: F1: 0.92; F2: 0.79; F3: 0.71; F4: 0.49). The total score significantly discriminated for disease severity or diagnosis; factorial scores differentially discriminated for disease severity according to diagnosis (PSP: F1-F2; MSA: F2-F3). The total score was significantly related to survival in PSP (p<0.0007) or MSA (p<0.0005), indicating good predictive validity. CONCLUSIONS The scale is suitable for use in the context of multicentre studies and can reliably and consistently measure MRI abnormalities in PSP and MSA. Clinical Trial Registration Number The study protocol was filed in the open clinical trial registry (http://www.clinicaltrials.gov) with ID No NCT00211224

Flexible Working in the UK and its Impact on Couples' Time Coordination

The ability to combine work with quality time together as a family is at the heart of the concept of work-life balance. Using previously unexploited data on couples work schedules we investigate the effect of flexible working on couples coordination of their daily work schedules in the UK. We consider three distinct dimensions of flexible working: flexibility of daily start and finish times (flexitime), flexibility of work times over the year (annualized hours), and generalized control of working hours. We show that having flexitime at work increases a couples amount of coordination of their daily work schedules by a half to one hour, which is double the margin of adjustment enjoyed by couples with no flexitime. The impact is driven by couples with children. In contrast to flexitime, the other two forms of flexible working do not seem to increase synchronous time. Our results suggest that having flexitime plays an important role in relaxing the work scheduling constraints faced by families with young children, and that effective flexible working time arrangements are those that increase the workers and not the employers flexibility