A 2-adic approach of the human respiratory tree

We propose here a general framework to address the question of trace operators on a dyadic tree. This work is motivated by the modeling of the human bronchial tree which, thanks to its regularity, can be extrapolated in a natural way to an infinite resistive tree. The space of pressure fields at bifurcation nodes of this infinite tree can be endowed with a Sobolev space structure, with a semi-norm which measures the instantaneous rate of dissipated energy. We aim at describing the behaviour of finite energy pressure fields near the end. The core of the present approach is an identification of the set of ends with the ring Z_2 of 2-adic integers. Sobolev spaces over Z_2 can be defined in a very natural way by means of Fourier transform, which allows us to establish precised trace theorems which are formally quite similar to those in standard Sobolev spaces, with a Sobolev regularity which depends on the growth rate of resistances, i.e. on geometrical properties of the tree. Furthermore, we exhibit an explicit expression of the "ventilation operator", which maps pressure fields at the end of the tree onto fluxes, in the form of a convolution by a Riesz kernel based on the 2-adic distance.Comment: 22 page

Distributed synaptic weights in a LIF neural network and learning rules

Leaky integrate-and-fire (LIF) models are mean-field limits, with a large number of neurons, used to describe neural networks. We consider inhomogeneous networks structured by a connec-tivity parameter (strengths of the synaptic weights) with the effect of processing the input current with different intensities. We first study the properties of the network activity depending on the distribution of synaptic weights and in particular its discrimination capacity. Then, we consider simple learning rules and determine the synaptic weight distribution it generates. We outline the role of noise as a selection principle and the capacity to memorized a learned signal.Comment: Physica D: Nonlinear Phenomena, Elsevier, 201

A model for cost efficient Workforce Organizational Dynamics and its optimization

This paper presents a workforce planning model scalable to an entire hierarchical organization. Its main objective is to design a cost optimal target which leverages flexible workforce solutions while ensuring an efficient promotional flux. The value of this paper lies in its proposal of an adequate flexibility rate using various solution types and in its discussion about external hiring ratios. The mathematical structures of the models are analyzed and numerical simulations illustrate the theoretical background

Toward an integrated workforce planning framework using structured equations

Strategic Workforce Planning is a company process providing best in class, economically sound, workforce management policies and goals. Despite the abundance of literature on the subject, this is a notorious challenge in terms of implementation. Reasons span from the youth of the field itself to broader data integration concerns that arise from gathering information from financial, human resource and business excellence systems. This paper aims at setting the first stones to a simple yet robust quantitative framework for Strategic Workforce Planning exercises. First a method based on structured equations is detailed. It is then used to answer two main workforce related questions: how to optimally hire to keep labor costs flat? How to build an experience constrained workforce at a minimal cost

Dispersion and Strichartz estimates for the Liouville equation

AbstractWe consider the Liouville equation associated with a metric g of class C2 and we prove dispersion and Strichartz estimates for the solution of this equation in terms of geodesics associated with g. We introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the Euclidean case holds. To deal with the case of non-trapped long range perturbation of the Euclidean metric, we prove a global velocity moments effect on the solution. In particular, we obtain global in time Strichartz estimates for metrics such that the dispersion estimate is not satisfied

Adaptation and Fatigue Model for Neuron Networks and Large Time Asymptotics in a Nonlinear Fragmentation Equation

International audienceMotivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last discharge.In the linear setting, we extend an argument by Laurençot and Perthame to prove exponential decay to the steady state. This extension allows us to handle coefficients that have a large variation rather than constant coefficients. In another extension of the argument, we treat a weakly nonlinear case and prove total desynchronization in the network. For greater nonlinearities, we present a numerical study of the impact of the fragmentation term on the appearance of synchronization of neurons in the network using two "extreme" cases.Mathematics Subject Classification (2000)2010: 35B40, 35F20, 35R09, 92B20

Dynamics of a structured neuron population

We study the dynamics of assemblies of interacting neurons. For large fully connected networks,the dynamics of the system can be described by a partial differential equation reminiscent of age-structure models used in mathematical ecology, where the "age" of a neuron represents the time elapsed since its last discharge. The nonlinearity arises from the connectivity J of the network. We prove some mathematical properties of the model that are directly related to qualitative properties. On the one hand we prove that it is well-posed and that it admits stationary states which, depending upon the connectivity, can be unique or not. On the other hand, we study the long time behavior of solutions; both for small and large J, we prove the relaxation to the steady state describing asynchronous firing of the neurons. In the middle range, numerical experiments show that periodic solutions appear expressing re-synchronization of the network and asynchronous firing