Short relaxation times but long transient times in both simple and complex reaction networks

When relaxation towards an equilibrium or steady state is exponential at large times, one usually considers that the associated relaxation time $\tau$, i.e., the inverse of that decay rate, is the longest characteristic time in the system. However that need not be true, and in particular other times such as the lifetime of an infinitesimal perturbation can be much longer. In the present work we demonstrate that this paradoxical property can arise even in quite simple systems such as a chain of reactions obeying mass action kinetics. By mathematical analysis of simple reaction networks, we pin-point the reason why the standard relaxation time does not provide relevant information on the potentially long transient times of typical infinitesimal perturbations. Overall, we consider four characteristic times and study their behavior in both simple chains and in more complex reaction networks taken from the publicly available database "Biomodels." In all these systems involving mass action rates, Michaelis-Menten reversible kinetics, or phenomenological laws for reaction rates, we find that the characteristic times corresponding to lifetimes of tracers and of concentration perturbations can be much longer than $\tau$

Ergodic Actions and Spectral Triples

In this article, we give a general construction of spectral triples from certain Lie group actions on unital C*-algebras. If the group G is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple which satisfies the first order condition of Connes' axioms. This provides a link between the "algebraic" existence of ergodic action and the "analytic" finite summability property of the unbounded selfadjoint operator. More generally, for compact G we carefully establish that our (symmetric) unbounded operator is essentially selfadjoint. Our results are illustrated by a host of examples - including noncommutative tori and quantum Heisenberg manifolds.Comment: 18 page

Global wake instabilities of low aspect-ratio flate-plates

This paper investigates the linear destabilization of three-dimensional steady wakes developing behind flate plates placed normal to the incoming flow. Plates characterized by low length-to-width ratio $L$ are considered here. By varying this aspect ratio in the range $1 \le L \le 6$ three destabilization scenarios are identified. For very low aspect ratio $1 \le L \le 2$, the flow is first destabilized, when increasing the Reynolds number,by a steady global mode that breaks the top/bottom planar reflectional symmetry. The symmetric steady flow bifurcates, via a pitchfork bifurcation, towards an asymmetric steady wakes, similarly to the case of axisymmetric wakes behind sphere and disks. For long aspect ratio, $2.5 \le L \le 6$, the first unstable mode also breaks the top/bottom symmetry but is unsteady. A Hopf bifurcation occurs, as for the wake developing behind a two-dimensional circular cylinder. Finally an intermediate regime $2 \le L \le 2.5$ is found for which the flow gets first unstable to an unsteady mode that breaks the left/right planar reflectional symmetry.Comment: 25 pages, 14 figure

All-pay auctions with endogenous rewards

This paper examines a perfectly discriminating contest (all-pay auction) with two asymmetric players. Valuations are endogenous and depend on the effort each player invests in the contest. The shape of the valuation function is common knowledge and differs between the contestants. Some key properties of R&D races, lobbying activity and sport contests are captured by this framework. Once the unique equilibrium in mixed strategies analyzed, we derive a closed form of the expected expenditure of both players. We characterize the expected expenditure by the means of incomplete Beta functions. We focus on unordered valuations.all-pay auctions, contests

All-Pay Auctions with Endogenous Rewards

This paper examines a perfectly discriminating contest (all-pay auction) with two asymmetric players. Valuations are endogenous and depend on the effort each player invests in the contest. The shape of the valuation function is common knowledge and differs between the contestants. Some key properties of R&D races, lobbying activity and sport contests are captured by this framework. Once the unique equilibrium in mixed strategies analyzed, we derive a closed form of the expected expenditure of both players. We characterize the expected expenditure by means of incomplete Beta functions. We focus on unordered valuations.All-pay auctions, contests