Description The motion of water masses by ocean currents is one of the main drivers of marine biological productivity. Being an inherently chaotic phenomenon, Lyapunov exponents have found important applications in characterizing and understanding this fluid transport process. In particular, local Lyapunov-exponent methods (finite-time and finite-size) have revealed to be useful to locate barriers to transport, attracting manifolds, and similar organizing structures in fluid flows. Here we review some recent applications of finite-size Lyapunov exponents (FSLEs) to characterize ocean transport and its impact on biological processes. The identification of the so-called Lagrangian coherent structures is addressed, and we also use FSLEs for the original goal they were introduced, namely quantifying the intensity of dispersion or mixing at a particular spatial scale. We will explore here the three-dimensional structure of Benguela eddies, of the Oxygen Minimum Zone in the Eastern Tropical Pacific, the relationship between mixing intensity and biological productivity in upwelling areas, and the role of Lyapunov structures as biological corridors for marine birds.

Description We will show some of the recent result in our group concerning dynamics in multiplex networks. On the one hand we consider multiplex networks as set of nodes in different layers. At each layer the set of nodes is the same but the connections among the nodes can be different in the layers. Furthermore the connections among the layers is described by a “network of layers”. For the most simple processes in the system we show how the eigenvalues and eigenvectors of the whole system depend on those of the individual layers and also on those of the network of layers. Additionally we have studied different processes across the layers (diffusion) and between the layers (reaction). In this case Turing patterns appear as an effect of different average connectivities in different layers. As a particular case of multiplex network, one can also analyze networks that change in time, since in this case each layer of the multiplex corresponds to a snapshot of the interaction pattern. For this situation, we have shown that there are different mechanisms that dominate the diffusion of information in the system depending on the relative effect of mobility and diffusion among the nodes.

Description In fashion retailing, product inventory is important to capture consumers' attention and increase sales. We study empirically the role of inventory on sales. We first develop an aggregate category demand model and find that inventory has a weak influence on sales compared with indirect effects of store heterogeneity and seasonality. We then describe a market share model where we show that product-level inventory has a large impact on its market share within the category. This supports the idea that inventory has a strong role in helping customers choose a particular product within the assortment. We finally describe how a retailer should optimally decide its inventory levels within a category and describe the properties of the optimal solution. Applying such optimization to our data set yields revenue improvements of 2.6% on average.

Description A 'flash crash' is a deep and rapid fall in stock prices, combined with a fast recovery to previous values. This phenomenon is also relevant regarding the mechanism of how a limit order book is built up. We will study the distribution of flash crashes in a Brownian framework via excursion theory. In particular, we derive a Laplace-Mellin transform of the joint excursion height and length in terms of the Riemann Xi function, and relate two different representations of the joint density by the transformation formula for Jacobi's Theta function