Multivariate Juggling Probabilities

We consider refined versions of Markov chains related to juggling introduced by Warrington. We further generalize the construction to juggling with arbitrary heights as well as infinitely many balls, which are expressed more succinctly in terms of Markov chains on integer partitions. In all cases, we give explicit product formulas for the stationary probabilities. The normalization factor in one case can be explicitly written as a homogeneous symmetric polynomial. We also refine and generalize enriched Markov chains on set partitions. Lastly, we prove that in one case, the stationary distribution is attained in bounded time.Comment: 28 pages, 5 figures, final versio

Bumping sequences and multispecies juggling

Building on previous work by four of us (ABCN), we consider further generalizations of Warrington's juggling Markov chains. We first introduce "multispecies" juggling, which consist in having balls of different weights: when a ball is thrown it can possibly bump into a lighter ball that is then sent to a higher position, where it can in turn bump an even lighter ball, etc. We both study the case where the number of balls of each species is conserved and the case where the juggler sends back a ball of the species of its choice. In this latter case, we actually discuss three models: add-drop, annihilation and overwriting. The first two are generalisations of models presented in (ABCN) while the third one is new and its Markov chain has the ultra fast convergence property. We finally consider the case of several jugglers exchanging balls. In all models, we give explicit product formulas for the stationary probability and closed form expressions for the normalisation factor if known.Comment: 25 pages, 9 figures (v3: final version, several typos and figures fixed

Input Redundancy under Input and State Constraints (Extended version of the submission accepted to Automatica)

For a given unconstrained dynamical system, input redundancy has been recently redefined as the existence of distinct inputs producing identical output for the same initial state. By directly referring to signals, this definition readily applies to any input-to-output mapping. As an illustration of this potentiality, this paper tackles the case where input and state constraints are imposed on the system. This context is indeed of foremost importance since input redundancy has been historically regarded as a way to deal with input saturations. An example illustrating how constraints can challenge redundancy is offered right at the outset. A more complex phenomenology is highlighted. This motivates the enrichment of the existing framework on redundancy. Then, a sufficient condition for redundancy to be preserved when imposing constraints is offered in the most general context of arbitrary constraints. It is shown that redundancy can be destroyed only when input and state trajectories lie on the border of the set of constraints almost all the time. Finally, those results are specialized and expanded under the assumption that input and state constraints are linear

Implicit FEM and Fluid Coupling on GPU for Interactive Multiphysics Simulation

International audienceWe present a method to implement on the GPU an implicit FEM solver which is fast and stable enough to handle interactions and collisions. We combine this method with GPU-based fluids and collision detection to achieve interactive multiphysics simulations entirely running on the GPU

Structure Transition in PSS/Lysozyme Complexes: A Chain-Conformation-Driven Process, as Directly Seen by Small Angle Neutron Scattering

Measurements of chain conformation in proteins/polyelectrolytes complexes (lysozyme and PSSNa) show that the crossover observed between an open structure -a chain network crosslinked by the proteins, and a globular one - dense globules of ~ 10 nm aggregated in a fractal way, results from a conformation modification prior to the transition. Before showing this, we have widened the parameters range for the observation of the transition. We had shown before that the two structures can be formed depending on chain length (for a given [PSS]/[lysozyme] ratio): gel for large chains, globules for short chains. We show here that the crossover between these two regimes can also be reached as a function of chains concentration or salinity of the buffer. Since all these crossover parameters act on chains overlapping concentration c*, we reinforce the idea of a transition from the dilute to the semi-dilute regime, but c* is shifted compared to pure PSS solutions. In order to understand this, we have measured by SANS the conformation of a single chain of PSS in presence of proteins within the complexes. This is achieved by a specific labeling trick where we take advantage of the fact that lysozyme and hydrogenated PSS chains have the same neutron scattering length density. In the gel structure, the PSS chains keep a wormlike structure as in pure solutions, but their persistence length is strongly reduced, from 50 {\AA} without proteins to 20 {\AA} in average with lysozyme. With this value of 20 {\AA}, we calculate new overlapping thresholds (concentration, mass, ionic strength) in agreement with observed ones. In a second stage, after the globular structure is formed, the PSS chains get a third conformation, no longer wormlike, but more collapsed, within the globules

A sparse grid approach to balance sheet risk measurement

In this work, we present a numerical method based on a sparse grid approximation to compute the loss distribution of the balance sheet of a financial or an insurance company. We first describe, in a stylised way, the assets and liabilities dynamics that are used for the numerical estimation of the balance sheet distribution. For the pricing and hedging model, we chose a classical Black & Scholes model with a stochastic interest rate following a Hull & White model. The risk management model describing the evolution of the parameters of the pricing and hedging model is a Gaussian model. The new numerical method is compared with the traditional nested simulation approach. We review the convergence of both methods to estimate the risk indicators under consideration. Finally, we provide numerical results showing that the sparse grid approach is extremely competitive for models with moderate dimension.Comment: 27 pages, 7 figures. CEMRACS 201