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Several mixed-effects models for longitudinal data have been proposed to accommodate the non-linearity of late-life cognitive trajectories and assess the putative influence of covariates on it. No prior research provides a side-by-side examination of these models to offer guidance on their proper application and interpretation. In this work, we examined five statistical approaches previously used to answer research questions related to non-linear changes in cognitive aging: the linear mixed model (LMM) with a quadratic term, LMM with splines, the functional mixed model, the piecewise linear mixed model, and the sigmoidal mixed model. We first theoretically describe the models. Next, using data from two prospective cohorts with annual cognitive testing, we compared the interpretation of the models by investigating associations of education on cognitive change before death. Lastly, we performed a simulation study to empirically evaluate the models and provide practical recommendations. Except for the LMM-quadratic, the fit of all models was generally adequate to capture non-linearity of cognitive change and models were relatively robust. Although spline-based models have no interpretable nonlinearity parameters, their convergence was easier to achieve, and they allow graphical interpretation. In contrast, piecewise and sigmoidal models, with interpretable non-linear parameters, may require more data to achieve convergence.

Electromagnetic forming and perforations (EMFP) are complex and innovative high strain rate processes that involve electromagnetic-mechanical interactions for simultaneous metal forming and perforations. Instead of spending costly resources on repetitive experimental work, a properly designed numerical model can be effectively used for detailed analysis and characterization of the complex process. A coupled finite element (FE) model is considered for analyzing the multi-physics of the EMFP because of its robustness and improved accuracy. In this work, a detailed understanding of the process has been achieved by numerically simulating forming and perforations of Al6061-T6 tube for 12 holes and 36 holes with two different punches, i.e., pointed and concave punches using Ls-Dyna software. In order to shed light on EMFP physics, a comparison between experimental data and the formulated numerical simulation has been carried out to compare the average hole diameter and the number of perforated holes, for different types of punches and a range of discharge energies. The simulated results show acceptable agreement with experimental studies, with maximum deviations being less than or equal to 6%, which clearly illustrates the efficacy and capability of the developed coupled Multi-physics FE model.

For the stochastic heat equation with multiplicative noise we consider the problem of estimating the diffusivity parameter in front of the Laplace operator. Based on local observations in space, we first study an estimator that was derived for additive noise. A stable central limit theorem shows that this estimator is consistent and asymptotically mixed normal. By taking into account the quadratic variation, we propose two new estimators. Their limiting distributions exhibit a smaller (conditional) variance and the last estimator also works for vanishing noise levels. The proofs are based on local approximation results to overcome the intricate nonlinearities and on a stable central limit theorem for stochastic integrals with respect to cylindrical Brownian motion. Simulation results illustrate the theoretical findings.

The scale function holds significant importance within the fluctuation theory of Levy processes, particularly in addressing exit problems. However, its definition is established through the Laplace transform, thereby lacking explicit representations in general. This paper introduces a novel series representation for this scale function, employing Laguerre polynomials to construct a uniformly convergent approximate sequence. Additionally, we derive statistical inference based on specific discrete observations, presenting estimators of scale functions that are asymptotically normal.

Faraday's law on electromagnetic induction, one of the most fundamental laws of nature, indicates that a change of magnetic field through a coil wire induces a current in the wire. Electromagnetic induction has many paramount technological applications today, and provides the link between electric and magnetic fields, which is crucial to explain the existence of electromagnetic waves. In our quest to replicate Faraday's law for mechanical systems, we design an infinite mass-spring "helix-like structure", which consists of a helix and a central line, and implement Bloch-Floquet conditions to obtain travelling wave solutions to the proposed problem. The structure's geometrical chirality is considered in conjunction with a dynamic chirality, introduced by placing gyroscopes along its central line. It is shown that the interplay between these two chiralities acts as a mechanical analogue of Faraday's law, breaking the symmetry of the associated dispersion diagram.

We analyze and test a simple-to-implement two-step iteration for the incompressible Navier-Stokes equations that consists of first applying the Picard iteration and then applying the Newton iteration to the Picard output. We prove that this composition of Picard and Newton converges quadratically, and our analysis (which covers both the unique solution and non-unique solution cases) also suggests that this solver has a larger convergence basin than usual Newton because of the improved stability properties of Picard-Newton over Newton. Numerical tests show that Picard-Newton dramatically outperforms both the Picard and Newton iterations, especially as the Reynolds number increases. We also consider enhancing the Picard step with Anderson acceleration (AA), and find that the AAPicard-Newton iteration has even better convergence properties on several benchmark test problems.

We propose center-outward superquantile and expected shortfall functions, with applications to multivariate risk measurements, extending the standard notion of value at risk and conditional value at risk from the real line to $\mathbb{R}^d$. Our new concepts are built upon the recent definition of Monge-Kantorovich quantiles based on the theory of optimal transport, and they provide a natural way to characterize multivariate tail probabilities and central areas of point clouds. They preserve the univariate interpretation of a typical observation that lies beyond or ahead a quantile, but in a meaningful multivariate way. We show that they characterize random vectors and their convergence in distribution, which underlines their importance. Our new concepts are illustrated on both simulated and real datasets.

We investigate a filtered Lie-Trotter splitting scheme for the ``good" Boussinesq equation and derive an error estimate for initial data with very low regularity. Through the use of discrete Bourgain spaces, our analysis extends to initial data in $H^{s}$ for $0<s\leq 2$, overcoming the constraint of $s>1/2$ imposed by the bilinear estimate in smooth Sobolev spaces. We establish convergence rates of order $\tau^{s/2}$ in $L^2$ for such levels of regularity. Our analytical findings are supported by numerical experiments.

Literature is full of inference techniques developed to estimate the parameters of stochastic dynamical systems driven by the well-known Brownian noise. Such diffusion models are often inappropriate models to properly describe the dynamics reflected in many real-world data which are dominated by jump discontinuities of various sizes and frequencies. To account for the presence of jumps, jump-diffusion models are introduced and some inference techniques are developed. Jump-diffusion models are also inadequate models since they fail to reflect the frequent occurrence as well as the continuous spectrum of natural jumps. It is, therefore, crucial to depart from the classical stochastic systems like diffusion and jump-diffusion models and resort to stochastic systems where the regime of stochasticity is governed by the stochastic fluctuations of L\'evy type. Reconstruction of L\'evy-driven dynamical systems, however, has been a major challenge. The literature on the reconstruction of L\'evy-driven systems is rather poor: there are few reconstruction algorithms developed which suffer from one or several problems such as being data-hungry, failing to provide a full reconstruction of noise parameters, tackling only some specific systems, failing to cope with multivariate data in practice, lacking proper validation mechanisms, and many more. This letter introduces a maximum likelihood estimation procedure which grants a full reconstruction of the system, requires less data, and its implementation for multivariate data is quite straightforward. To the best of our knowledge this contribution is the first to tackle all the mentioned shortcomings. We apply our algorithm to simulated data as well as an ice-core dataset spanning the last glaciation. In particular, we find new insights about the dynamics of the climate in the curse of the last glaciation which was not found in previous studies.

The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a Caputo fractional Laplacian and a variable coefficient wave number $\mu$, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix-sequences. We report numerical evidences supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.