Explore the vast range of titles available.

We investigate a class of stochastic chemical reaction networks with [Formula omitted] chemical species [Formula omitted], ..., [Formula omitted], and whose complexes are only of the form [Formula omitted], [Formula omitted],..., n, where [Formula omitted] are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter N. A natural hierarchy of fast processes, a subset of the coordinates of [Formula omitted], is determined by the values of the mapping [Formula omitted]. We show that the scaled vector of coordinates i such that [Formula omitted] and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as N gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.

We investigate a class of stochastic chemical reaction networks withn≥1chemical speciesS₁ ,S_(n) , and whose complexes are only of the formkᵢSᵢ ,i=1 ,n , where(kᵢ)are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameterN . A natural hierarchy of fast processes, a subset of the coordinates of(Xᵢ(t)) , is determined by the values of the mappingi↦kᵢ . We show that the scaled vector of coordinatesisuch thatkᵢ=1and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit asNgets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.

We investigate a class of stochastic chemical reaction networks with n ≥ 1 chemical species S 1 , ..., S n , and whose complexes are only of the form k i S i , i = 1 ,..., n , where ( k i ) are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter N . A natural hierarchy of fast processes, a subset of the coordinates of ( X i ( t ) ) , is determined by the values of the mapping i ↦ k i . We show that the scaled vector of coordinates i such that k i = 1 and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as N gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.

The asymptotic properties of some Markov processes associated to stochastic chemical reaction networks (CRNs) driven by the kinetics of the law of mass action are analyzed. The scaling regime introduced in the paper assumes that the norm of the initial state is converging to infinity. The reaction rate constants are kept fixed. The purpose of the paper is of showing, with simple examples, a scaling analysis in this context. The main difference with the scalings of the literature is that it does not change the graph structure of the CRN or its reaction rates. Several CRNs are investigated to illustrate the insight that can be gained on the qualitative properties of these networks. A detailed scaling analysis of a CRN with several interesting asymptotic properties, with a bi-modal behavior in particular, is worked out in the last section. Additionally, with several examples, we also show that a stability criterion due to Filonov for positive recurrence of Markov processes may simplify significantly the stability analysis of these networks.

We present a chemical reaction network that is unstable under deterministic mass action kinetics, exhibiting finite-time blow-up of trajectories in the interior of the state space, but whose stochastic counterpart is positive recurrent. This provides an example of noise-induced stabilization of the model's dynamics arising due to noise perturbing transversally the divergent trajectories of the system that is independently of boundary effects. The proof is based on a careful decomposition of the state space and the construction of suitable Lyapunov functions in each region.

We investigate a class of stochastic chemical reaction networks with \\(n{\\ge}1\\) chemical species \\(S_1\\), \\ldots, \\(S_n\\), and whose complexes are only of the form \\(k_iS_i\\), \\(i{=}1\\),\\ldots, \\(n\\), where \\((k_i)\\) are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter \\(N\\). A natural hierarchy of fast processes, a subset of the coordinates of \\((X_i(t))\\), is determined by the values of the mapping \\(i{\\mapsto}k_i\\). We show that the scaled vector of coordinates \\(i\\) such that \\(k_i{=}1\\) and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as \\(N\\) gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.

In this paper we study a class of stochastic chemical reaction networks (CRNs) for which chemical species are created by a sequence of chain reactions. We prove that under some convenient conditions on the initial state, some of these networks exhibit a discrete-induced transitions (DIT) property: isolated, random, events have a direct impact on the macroscopic state of the process. If this phenomenon has already been noticed in several CRNs, in auto-catalytic networks in the literature of physics in particular, there are up to now few rigorous studies in this domain. A scaling analysis of several cases of such CRNs with several classes of initial states is achieved. The DIT property is investigated for the case of a CRN with four nodes. We show that on the normal timescale and for a subset of (large) initial states and for convenient Skorohod topologies, the scaled process converges in distribution to a Markov process with jumps, an Additive Increase/Multiplicative Decrease (AIMD) process. This asymptotically discontinuous limiting behavior is a consequence of a DIT property due to random, local, blowups of jumps occurring during small time intervals. With an explicit representation of invariant measures of AIMD processes and time-change arguments, we show that, with a speed-up of the timescale, the scaled process is converging in distribution to a continuous deterministic function. The DIT analyzed in this paper is connected to a simple chain reaction between three chemical species and is therefore likely to be a quite generic phenomenon for a large class of CRNs.

We investigate the asymptotic properties of Markov processes associated to stochastic chemical reaction networks (CRNs) driven by the kinetics of the law of mass action. Their transition rates exhibit a polynomial dependence on the state variable, with possible discontinuities of the dynamics along the boundary of the state space. As a natural choice to study stability properties of CRNs, the scaling parameter considered in this paper is the norm of the initial state. Compared to existing scalings of the literature, this scaling does not change neither the topology of a CRN, nor its reactions constants. Functional limit theorems with this scaling parameter can be used to prove positive recurrence of the Markov process. This scaling approach also gives interesting insights on the transient behavior of these networks, to describe how multiple time scales drive the time evolution of their sample paths for example. General stability criteria are presented as well as a possible framework for scaling analyses. Several simple examples of CRNs are investigated with this approach. A detailed stability and scaling analyses of a CRN with slow and fast timescales is worked out.

Le présent ouvrage vise à démystifier le fonctionnement de la formule québécoise de financement des universités. Il formalise mathématiquement les règles existantes et observe comment, à travers cette formule, l’importance relative de différents facteurs est prise en compte. Ce livre interroge également les effets des modifications possibles à la structure de la formule de financement et les réformes susceptibles d’être endossées par différents établissements.

Introduction of clonal reproduction through seeds （apomixis） in crops has the potential to revolutionize agricul- ture by allowing self-propagation of any elite variety, in particular F1 hybrids. In the sexual model plant Arabidopsis thaliana synthetic clonal reproduction through seeds can be artificially implemented by （i） combining three muta- tions to turn meiosis into mitosis （MiMe） and （ii） crossing the obtained clonal gametes with a line expressing modified CENH3 and whose genome is eliminated in the zygote. Here we show that additional combinations of mutations can turn Arabidopsis meiosis into mitosis and that a combination of three mutations in rice （Oryza sativa） efficiently turns meiosis into mitosis, leading to the production of male and female clonal diploid gametes in this major crop. Suc- cessful implementation of the MiMe technology in the phylogenetically distant eudicot Arabidopsis and monocot rice opens doors for its application to any flowering plant and paves the way for introducing apomixis in crop species.