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Dielectric breakdown of physical vacuum (Schwinger effect) is the textbook demonstration of compatibility of Relativity and Quantum theory. Although, the observation of this effect in its original static formulation is practically unachievable, it has been shown that the requirements on field strengths can be significantly reduced for dynamical generalizations of Schwinger effect. Here, we report on observation of an analog dynamical Schwinger effect in gapped Dirac semiconductor lead-halide perovskite MAPbBr$_3$. Specifically, we observe strong photoluminescence of a lead-halide perovskite driven by deep sub-gap irradiation, and use the quasi-adiabatic Landau-Dykhne approach to interpret our data in terms of the dynamical Schwinger effect. Further, the exponential sensitivity of the Schwinger effect to driving fields allows us to measure the local frozen-in fields in a nominally cubic single perovskite crystal at room temperature. Finally, we demonstrate an AC analogue of biasing in our system -- the non-perturbative cooperation between two time-dependent fields simultaneously driving the sample. Our results establish lead-halide perovskites as an excellent platform for simulating effects of strong fields on Dirac fields. In addition, they contribute to the on-going discussion about inversion-breaking in MAPbBr$_3$ single crystal and pave the way for a mid-infrared light detection with lead-halide perovskites.

In this paper, the construction of $C^{1}$ cubic quasi-interpolants on a three-direction mesh of $\RR^{2}$ is addressed. The quasi-interpolating splines are defined by directly setting their Bernstein-B\'{e}zier coefficients relative to each triangle from point and gradient values in order to reproduce the polynomials of the highest possible degree. Moreover, additional global properties are required. Finally, we provide some numerical tests confirming the approximation properties.

Spatio-temporal models for count data are required in a wide range of scientific fields and they have become particularly crucial nowadays because of their ability to analyse COVID-19-related data. Models for count data are needed when the variable of interest take only non-negative integer values and these integers arise from counting occurrences. Several R-packages are currently available to deal with spatiotemporal areal count data. Each package focuses on different models and/or statistical methodologies. Unfortunately, the results generated by these models are rarely comparable due to differences in notation and methods. The main objective of this paper is to present a review describing the most important approaches that can be used to model and analyse count data when questions of scientific interest concern both their spatial and their temporal behaviour and we monitor their performance under the same data set. For this review, we focus on the three R-packages that can be used for this purpose and the different models assessed are representative of the two most widespread methodologies used to analyse spatiotemporal count data: the classical approach (based on Penalised Likelihood or Estimating Equations) and the Bayesian point of view. A case study is analysed as an illustration of these different methodologies. In this case study, these packages are used to model and predict daily hospitalisations from COVID-19 in 24 health regions within the Valencian Community (Spain), with data corresponding to the period from 28 June to 13 December 2020. Because of the current urgent need for monitoring and predicting data in the COVID-19 pandemic, this case study is, in itself, of particular importance and can be considered the secondary objective of this work. Satisfactory and promising results have been obtained in this second goal.

The aim of this paper is to model an ordinal response variable in terms of vector-valued functional data included on a vector-valued RKHS. In particular, we focus on the vector-valued RKHS obtained when a geometrical object (body) is characterized by a current and on the ordinal regression model. A common way to solve this problem in functional data analysis is to express the data in the orthonormal basis given by decomposition of the covariance operator. But our data present very important differences with respect to the usual functional data setting. On the one hand, they are vector-valued functions, and on the other, they are functions in an RKHS with a previously defined norm. We propose to use three different bases: the orthonormal basis given by the kernel that defines the RKHS, a basis obtained from decomposition of the integral operator defined using the covariance function, and a third basis that combines the previous two. The three approaches are compared and applied to an interesting problem: building a model to predict the fit of children's garment sizes, based on a 3D database of the Spanish child population.

This paper focuses on the application of Discriminant Analysis to a set of geometrical objects (bodies) characterized by currents. A current is a relevant mathematical object to model geometrical data, like hypersurfaces, through integration of vector fields along them. As a consequence of the choice of a vector-valued Reproducing Kernel Hilbert Space (RKHS) as a test space to integrate hypersurfaces, it is possible to consider that hypersurfaces are embedded in this Hilbert space. This embedding enables us to consider classification algorithms of geometrical objects. A method to apply Functional Discriminant Analysis in the obtained vector-valued RKHS is given. This method is based on the eigenfunction decomposition of the kernel. So, the novelty of this paper is the reformulation of a size and shape classification problem in Functional Data Analysis terms using the theory of currents and vector-valued RKHS. This approach is applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children's wear.

Object classification according to their shape and size is of key importance in many scientific fields. This work focuses on the case where the size and shape of an object is characterized by a current}. A current is a mathematical object which has been proved relevant to the modeling of geometrical data, like submanifolds, through integration of vector fields along them. As a consequence of the choice of a vector-valued Reproducing Kernel Hilbert Space (RKHS) as a test space for integrating manifolds, it is possible to consider that shapes are embedded in this Hilbert Space. A vector-valued RKHS is a Hilbert space of vector fields; therefore, it is possible to compute a mean of shapes, or to calculate a distance between two manifolds. This embedding enables us to consider size-and-shape classification algorithms. These algorithms are applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children's wear.