A two-scale Stefan problem arising in a model for tree sap exudation

The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze-thaw cycles, gives rise to an interesting multi-scale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.Comment: 35 pages, 8 figures. arXiv admin note: text overlap with arXiv:1411.303

Bosonic behavior of entangled fermions

Two bound, entangled fermions form a composite boson, which can be treated as an elementary boson as long as the Pauli principle does not affect the behavior of many such composite bosons. The departure of ideal bosonic behavior is quantified by the normalization ratio of multi-composite-boson states. We derive the two-fermion-states that extremize the normalization ratio for a fixed single-fermion purity P, and establish general tight bounds for this indicator. For very small purities, P<1/N^2, the upper and lower bounds converge, which allows to quantify accurately the departure from perfectly bosonic behavior, for any state of many composite bosons.Comment: 9 pages, 5 figures, accepted by PR

Computation and Palaeography: Potentials and Limits

This manifesto documents the program and outcomes of Dagstuhl Seminar 12382 ‘Perspectives Workshop: Computation and Palaeography: Potentials and Limits’. The workshop focused on the interaction of palaeography, the study of ancient and me- dieval documents, with computerised tools, particularly those developed for analysis of digital images and text mining. The goal of this marriage of disciplines is to provide e cient solutions to time and labor consuming palaeographic tasks. It furthermore attempts to provide scholars with quantitative evidence to palaeographical arguments, consequently facilitating a better understanding of our cultural heritage through the unique perspective of ancient and medieval documents. The workshop provided a vital opportunity for palaeographers to interact and discuss the potential of digital methods with computer scientists specialising in machine vision and statistical data analysis. This was essential not only in suggesting new directions and ideas for improving palaeographic research, but also in identifying questions which scholars working individually, in their respective elds, would not have asked without directly communicating with colleagues from outside their research community

Computational polyconvexification of isotropic functions

Based on the characterization of the polyconvex envelope of isotropic functions by their signed singular value representations, we propose a simple algorithm for the numerical approximation of the polyconvex envelope. Instead of operating on the $d^2$-dimensional space of matrices, the algorithm requires only the computation of the convex envelope of a function on a $d$-dimensional manifold, which is easily realized by standard algorithms. The significant speedup associated with the dimensional reduction from $d^2$ to $d$ is demonstrated in a series of numerical experiments.Comment: 17 pages, 7 figure