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

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

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

Surface-Wave Dispersion Retrieval Method and Synthesis Technique for Bianisotropic Metasurfaces

We propose a surface-wave dispersion retrieval method and synthesis technique that applies to bianisotropic metasurfaces that are embedded in symmetric or asymmetric environments. Specifically, we use general zero-thickness sheet transition conditions to relate the propagation constants of surface-wave modes to the bianisotropic susceptibility components of the metasurface, which can themselves be directly related to its scattering parameters. It is then possible to either obtain the metasurface dispersion diagram from its known susceptibilities or, alternatively, compute the susceptibilities required to achieve a desired surface-wave propagation. The validity of the method is demonstrated by comparing its results to those obtained with exact dispersion relations of well known structures such as the propagation of surface plasmons on thin metallic films. In particular, this work reveals that it is possible to achieve surface-wave propagation only on one side of the metasurface either by superposition of symmetric and asymmetric modes in the case of anisotropic metasurfaces or by completely forbidding the existence of the surface wave on one side of the structure using bianisotropic metasurfaces