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The choice of the electronic representation in on-the-fly quantum dynamics is crucial. The adiabatic representation is appealing since adiabatic states are readily available from quantum chemistry packages. The nuclear wavepackets are then expanded in a basis of Gaussian functions, which follow trajectories to explore the potential energy surfaces and approximate the potential using a local expansion of the adiabatic quantities. Nevertheless, the adiabatic representation is plagued with severe limitations when conical intersections are involved: the diagonal Born–Oppenheimer corrections (DBOCs) are non-integrable, and the geometric phase effect on the nuclear wavepackets cannot be accounted for unless a model is available. To circumvent these difficulties, the moving crude adiabatic (MCA) representation was proposed and successfully tested in low energy dynamics where the wavepacket skirts the conical intersection. We assess the MCA representation in the case of non-adiabatic transitions through conical intersections. First, we show that using a Gaussian basis in the adiabatic representation indeed exhibits the aforementioned difficulties with a special emphasis on the possibility to regularize the DBOC terms. Then, we show that MCA is indeed able to properly model non-adiabatic transitions. Tests are done on linear vibronic coupling models for the bis(methylene) adamantyl cation and the butatriene cation. This article is part of the theme issue ‘Chemistry without the Born–Oppenheimer approximation’.

The time-dependent reliability analysis aims at estimating the probability of failure, occurring within a specified time period, of a structure subjected to stochastic and dynamic loads or stochastic degradation of performance. Development of efficient numerical algorithms with accuracy assurance for solving this problem, although has been investigated with, e.g., Gaussian Process Regression (GPR)-based active learning procedures, keeps being a bottleneck. Inspired by the concept of up-crossing rate used in the first-passage methods, a new acquisition function (also called learning function) is developed with the consideration of the temporal correlation information across each sample trajectory. It measures the (subjective) probability of mis-judging the occurrence of the up-crossing event within each time sub-interval. With this new acquisition function, the classical active learning procedure is improved. Considering the necessity for estimating small failure probability, the proposed active learning method is then combined with the subset simulation for multi-stage learning. With this method, a series of intermediate surrogate failure surface is actively updated with the target of approaching the true failure surface with pre-specified error tolerance. The effectiveness of the proposed methods are demonstrated with numerical and engineering examples.

The central limit theorem has been found to apply to random vectors in complex Hilbert space. This amounts to sufficient reason to study the complex–valued Gaussian, looking for relevance to quantum mechanics. Here we show that the Gaussian, with all terms fully complex, acting as a propagator, leads to Schrödinger’s non-relativistic equation including scalar and vector potentials, assuming only that the norm is conserved. No physical laws need to be postulated a priori. It thereby presents as a process of irregular motion analogous to the real random walk but executed under the rules of the complex number system. There is a standard view that Schrödinger’s equation is deterministic, whereas wavefunction “collapse” is probabilistic (by Born’s rule)—we have now a demonstrated linkage to the central limit theorem, indicating a stochastic picture at the foundation of Schrödinger’s equation itself. It may be an example of Wheeler’s “It from bit” with “No underlying law”. Reasons for the primary role of C are open to discussion. The present derivation is compared with recent reconstructions of the quantum formalism, which have the aim of rationalizing its obscurities.

The problem of escape times from a region confined by two time-dependent boundaries is considered for a class of Gauss-Markov processes. Asymptotic approximations of the first exit time probability density functions in case of asymptotically constant and asymptotically periodic boundaries are obtained firstly for the Ornstein-Uhlenbeck process and then extended to the class of Gauss-Markov processes that can be obtained by a specified transformation. Some examples of application to stochastic dynamics and estimations of involved parameters by using numerical approximations are provided.

We define a time-dependent empirical process based on n i.i.d. fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for this process.

We define a time dependent empirical process based on n independent fractional Brownian motions and describe strong approximations to it by Gaussian processes. They lead to strong approximations and functional laws of the iterated logarithm for the quantile or inverse of this empirical process. They are obtained via time dependent Bahadur–Kiefer representations.

We propose a pseudospectral hybrid algorithm to approximate the solution of partial differential equations (PDEs) with non-periodic boundary conditions. Most of the approximations are computed using Fourier expansions that can be efficiently obtained by fast Fourier transforms. To avoid the Gibbs phenomenon, super-Gaussian window functions are used in physical space. Near the boundaries, we use local polynomial approximations to correct the solution. We analyze the accuracy and eigenvalue stability of the method for several PDEs. The method compares favorably to traditional spectral methods, and numerical results indicate that for hyperbolic problems a time step restriction of O (1/ N ) is sufficient for stability.

Goodness-of-fit tests are proposed for the skew-normal law in arbitrary dimension. In the bivariate case the proposed tests utilize the fact that the moment-generating function of the skew-normal variable is quite simple and satisfies a partial differential equation of the first order. This differential equation is estimated from the sample and the test statistic is constructed as an L₂-type distance measure incorporating this estimate. Extension of the procedure to dimension greater than two is suggested whereas an effective bootstrap procedure is used to study the behaviour of the new method with real and simulated data.

A simple exposition is given of Edgeworth and saddle-point approximations for some univariate, multivariate and conditional distributions. The application of these approximations to some problems of conditional statistical inference within the exponential family is illustrated. Examples connected with the time-dependent Poisson process, the von Mises distribution and with bioassay are among those studied in a little detail. Some general results are derived about conditional likelihoods and about the distribution of the maximum likelihood ratio test statistic. An appendix discusses regularity conditions.

In this paper, we will give an h-version finite-element method for a two-dimensional nonlinear elasto-plasticity problem. A family of admissible constitutive laws based on the so-called gauge-function method is introduced first, and then a high-order h-version semidiscretization scheme is presented. The existence and uniqueness of the solution for the semidiscrete problem are guaranteed by using some special properties of the constitutive law, and finally we will show that as the maximum element size h → 0, the solution of the semidiscrete problem will converge to the solution of the continuous problem. The high-order h-version discretization scheme introduced here is unusual. If the partition of the spatial space only has rectangles or parallelograms involved, then there would not be any limit on the element degree. However, if the partition of the spatial space has some triangular elements, then only certain combinations of finite-element spaces for displacement and stress functions can be used. The discretization scheme also provides a useful idea for applications of hp-version or high-order h-version finite-element methods for two-dimensional problems where the elasto-plastic body is not a polygon, such as a disk or an annulus.