Institut de Mathématiques de Bordeaux

ABSTRACT This paper is focused on the statistical analysis of data consisting of a collection of multiple series of probability measures that are indexed by distinct time instants and supported over a bounded interval of the real line. By modeling these time‐dependent probability measures as random objects in the Wasserstein space, we propose a new auto‐regressive model for the statistical analysis of multivariate distributional time series. Using the theory of iterated random function systems, results on the second‐order stationarity of the solution of such a model are provided. We also propose a consistent estimator for the autoregressive coefficients of this model. Due to the simplex constraints that we impose on the model coefficients, the proposed estimator that is learned under these constraints naturally has a sparse structure. The sparsity allows the application of the proposed model in learning a graph of temporal dependency from multivariate distributional time series. We explore the numerical performances of our estimation procedure using simulated data. To shed some light on the benefits of our approach for real data analysis, we also apply this methodology to two data sets, respectively made of observations from age distribution in different countries and those from the bike sharing network in Paris.

Abstract We consider the problem of finding the minimum of inhomogeneous Gaussian lattice sums: Given a lattice $L \subseteq \mathbb {R}^n$ and a positive constant $\alpha $ , the goal is to find the minimizers of $\sum _{x \in L} e^{-\alpha \|x - z\|^2}$ over all $z \in \mathbb {R}^n$ . By a result of Bétermin and Petrache from 2017 it is known that for steep potential energy functions—when $\alpha $ tends to infinity—the minimizers in the limit are found at deep holes of the lattice. In this paper, we consider minimizers which already stabilize for all $\alpha \geq \alpha _0$ for some finite $\alpha _0$ ; we call these minimizers stable cold spots. Generic lattices do not have stable cold spots. For several important lattices, like the root lattices, the Coxeter-Todd lattice, and the Barnes-Wall lattice, we show how to apply the linear programming bound for spherical designs to prove that the deep holes are stable cold spots. We also show, somewhat unexpectedly, that the Leech lattice does not have stable cold spots.

The transition matrix of a Markov chain (Xk,k≥0) on a finite or infinite rooted tree is said to be almost upper-directed if, given Xk, the node Xk+1 is either a descendant of Xk or the parent of Xk. It is said to be almost lower-directed if given Xk, Xk+1 is either an ancestor of Xk or a child of Xk. These models include nearest neighbor Markov chains on trees. Under an irreducibility assumption, we show that every almost upper-directed transition matrix on infinite (locally finite) trees has some invariant measures. An invariant measure π is expressed thanks to a determinantal formula. We give general explicit criteria for recurrence and positive recurrence. An efficient algorithm (the leaf addition algorithm) of independent interest allows π to be computed on many trees, without resorting to linear algebra considerations. Flajolet, in a series of papers (starting from [12]), provided some relations between continuous fractions, generating functions of weighted Mötzkin paths, and used them in connection with the analysis of birth and death processes. These fruitful representations made it possible to establish many formulae for continuous fractions. Analogous considerations appear here: this type of study can be extended to weighted paths on trees, whose generating functions can also be expressed, this time in terms of multicontinuous fractions.

Electroanatomic mapping systems measure the spread of activation and recovery over the surface of the heart. Propagation in cardiac tissue is complicated by the tissue architecture which produces a spatially varying anisotropic conductivity, leading to complex wavefronts. Curvature of the wavefront is known to affect both conduction velocity (CV) and action potential duration (APD). In this study, we sought to better define the impact of wavefront curvature on these properties, as well as the influence of conductivity, in order to recover intrinsic tissue properties. The dependence of CV and APD on curvature were measured for positive and negative curvatures for several ionic models, and then verified in realistic 2D and 3D simulations. Clinical data were also analysed. Results indicate that the effects of APD and CV are well described by simple formulae, and if the structure of the fibre is known, the intrinsic propagation velocities can be recovered. Geometrical curvature, as determined strictly by wavefront shape and ignoring the fibre structure, leads to large regions of spurious high curvature. This is important for determining pathological zones of slow conduction. In the simulations studied, curvature modulated APD by at most 20 ms.

This work explores water hydrolysis using magnesium as a decentralized dihydrogen source for off-grid households. A dedicated reactor design enabled on-demand dihydrogen generation, coupled with a Proton Exchange Membrane Fuel Cell (PEMFC) for electricity and heat production. Different energy management strategies were compared, highlighting the limitations of single-purpose approaches and the benefits of converting surplus electricity to heat. The integration of photovoltaic generation further reduced magnesium demand by 30%, thus reducing storage requirements to close to 1565 kg of magnesium powder per year, i.e., a volume of 0.9 m3 to cover the heat and electricity needs of a four-person household. Results demonstrate that combining water hydrolysis with magnesium and renewables provides a feasible and sustainable solution for autonomous energy supply in isolated sites.