Tracing the minimum-energy path on the free-energy surface

The free energy profile of a reaction can be estimated in a molecular-dynamics approach by imposing a mechanical constraint along a reaction coordinate (RC). Many recent studies have shown that the temperature can greatly influence the path followed by the reactants. Here, we propose a practical way to construct the minimum energy path directly on the free energy surface (FES) at a given temperature. First, we follow the blue-moon ensemble method to derive the expression of the free energy gradient for a given RC. These derivatives are then used to find the actual minimum energy reaction path at finite temperature, in a way similar to the Intrinsic Reaction Path of Fukui on the potential energy surface [K Fukui J. Phys. Chem. 74, 4161 (1970)]. Once the path is know, one can calculate the free energy profile using thermodynamic integration. We also show that the mass-metric correction cancels for many types of constraints, making the procedure easy to use. Finally, the minimum free energy path at 300 K for the addition of the 1,1'-dichlorocarbene to ethylene is compared with a path based on a simple one-dimensional reaction coordinate. A comparison is also given with the reaction path at 0 K.Comment: Minor revisions: Citation and Equation numbers corrected. 26 pages, 6 figures, to appear in J. Chem. Phy

Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates

We present an efficient rigorous computational method which is an extension of the work Analytic Estimates and Rigorous Continuation for Equilibria of Higher-Dimensional PDEs (M. Gameiro and J.-P. Lessard, J. Differential Equations, 249 (2010), pp. 2237-2268). The idea is to generate sharp one-dimensional estimates using interval arithmetic which are then used to produce high-dimensional estimates. These estimates are used to construct the radii polynomials which provide an efficient way of determining a domain on which the contraction mapping theorem is applicable. Computing the equilibria using a finite-dimensional projection, the method verifies that the numerically produced equilibrium for the projection can be used to explicitly define a set which contains a unique equilibrium for the PDE. A new construction of the polynomials is presented where the nonlinearities are bounded by products of one-dimensional estimates as opposed to using FFT with large inputs. It is demonstrated that with this approach it is much cheaper to prove that the numerical output is correct than to recompute at a finer resolution. We apply this method to PDEs defined on three- and four-dimensional spatial domains

Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation

In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combination of the theory introduced in Global smooth solution curves using rigorous branch following (van den Berg et al., Math. Comput. 79(271):1565-1584, 2010) and in Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs (Gameiro and Lessard, J. Diff. Equ. 249(9):2237-2268, 2010). Using this method, one can obtain proofs of existence of global smooth solution curves of equilibria for large (continuous) parameter ranges and about local uniqueness of the solutions on the curve. As an application, we compute several smooth branches of equilibria for the three-dimensional Cahn-Hilliard equation

Toward a Scalable Upper Bound for a CVaR-LQ Problem

We study a linear-quadratic, optimal control problem on a discrete, finite time horizon with distributional ambiguity, in which the cost is assessed via Conditional Value-at-Risk (CVaR). We take steps toward deriving a scalable dynamic programming approach to upper-bound the optimal value function for this problem. This dynamic program yields a novel, tunable risk-averse control policy, which we compare to existing state-of-the-art methods.Comment: accepted by IEEE Control Systems Letters, June 202

Rigorous numerics in floquet theory: Computing stable and unstable bundles of periodic orbits

In this paper, a rigorous method to compute Floquet normal forms of fundamental matrix solutions of nonautonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution F(t) is a canonical decomposition of the form F(t) = Q(t)eRt, where Q(t) is a real periodic matrix and R is a constant matrix. To rigorously compute the Floquet normal form, the idea is to use the regularity of Q(t) and to simultaneously solve for R and Q(t) with the contraction mapping theorem in a Banach space of rapidly decaying coefficients. The explicit knowledge of R and Q can then be used to construct, in a rigorous computer-assisted way, stable and unstable bundles of periodic orbits of vector fields. The new proposed method does not require rigorous numerical integration of the ODE

Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form

We present an efficient numerical method for computing Fourier-Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field. Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold. The method requires only that some mild nonresonance conditions hold. The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier-Taylor expansion. A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional. We also discuss computations of cycle-to-cycle connecting orbits which exploit these manifolds

Capital Budgeting and State Approval of Power Plants

John A. Helmuth is an Associate Professor of Finance in the Department of Accounting and Finance, College of Business Administration at Rochester Institute of Technology. Donald Kent is an Assistant Professor of Accounting at Robert Wesleyan College, Rochester, New York. Jeffrey P. Lessard is an Assistant Professor of Finance in the Department of Accounting and Finance, College of Business Administration at Rochester lnstitute of Technology

Influence possible des protozoaires sur le taux de mortalité des bactéries autotrophes nitrifiantes

Le modèle de l'IAWQ du processus de boues activées représente les mécanismes endogènes de la biomasse nitrifiante par le décès des micro-organismes (équation d'ordre 1 par rapport à la biomasse). La constante de décès, ou taux de mortalité bA, est aujourd'hui encore mal connue, et en particulier les facteurs influants sur sa valeur. De récentes études ont montré que la prédation par la microfaune pourrait être un facteur déterminant sur la valeur de bA. Cette étude se propose donc de quantifier l'effet de la prédation sur la valeur de bA. Deux réacteurs maintenus sans alimentation en substrat ont été caractérisés en parallèle: l'un a reçu une dose d'antibiotique spécifique aux eucaryotes (cycloheximide) afin de diminuer la quantité d'organismes de la microfaune, alors que l'autre n'a reçu aucun antibiotique (témoin). Les résultats obtenus montrent que le cycloheximide inhibe la plupart des organismes de la microfaune sauf les amibes; celles-ci semblent plutôt stimulées par cet antibiotique. En ce qui concerne la nitrification, un ralentissement de la production de nitrate dans le réacteur traité à l'antibiotique est observé à partir du sixième jour. Cette diminution de production de nitrate est probablement causée par une réduction de l'azote nitrifiable (qui est mobilisé par les amibes) couplée à une prédation des organismes nitrifiants par les amibes. D'ailleurs, l'augmentation de la prédation par les amibes à partir du jour 6 a diminué l'activité nitrifiante également mesurée par respirométrie (rO2 Nmax). Cette diminution du taux de respiration indique une augmentation du taux de mortalité (bA) des organismes nitrifiants. En effet, la valeur du taux de mortalité mesurée dans le réacteur témoin est de 0.08 d-1 alors que selon la microfaune présente dans le réacteur inhibé au cycloheximide, la valeur de ce taux de mortalité a varié entre 0.05 d-1 et 0.15 d-1.Designing biological wastewater treatment plants with the aid of the model developed by the IAWQ requires the knowledge of biological kinetic parameters. For nitrifying activated sludge, these parameters are related to nitrifying bacteria: maximum autotrophic growth rate µAmax, yield coefficient YA and the autotrophic decay rate bA. Although variables influencing µAmax and YA values are well known, this is not the case for bA. MARTINAGE and PAUL (2000) have recently shown that the bA value is strongly influenced by the influent quality, leading to the assumption that influent quality has a strong effect on microfauna composition, and consequently on the grazing rate of microfauna on nitrifying bacteria. In fixed-film processes, protozoan grazing reduces the bacterial population considerably (NATUSCKA and WELANDER, 1994). However, although many data are available concerning the grazing rates of different protozoa, the effect of microfaunal grazing on nitrification is still a matter of debate (RATSAK et al., 1994) and its effect on the bA value is still unknown. These two topics are investigated here.Nitrifying activated sludges were grown in two identical batch reactors, but in one, cycloheximide was added to inhibit eucaryotic growth (MAURINES CARBONEILL et al., 1998). Microfauna organism numbers were quantified in both reactors by microscopic observations of flagellated protozoa (>8 µm), amoebae, ciliates, rotifers and higher invertebrates (Fig. 3). Microbial counts were then correlated with the bA value. The latter was determined using the procedure proposed by SALZER (1992) which consists of characterising the time behaviour of the maximum nitrification rate measured by respirometry of activated sludge under substrate starvation. Under these conditions bacteria die and organic nitrogen is released into the bulk phase. This nitrogen is ammonified, and nitrifying bacteria use this substrate to produce nitrate, and then autotrophic bacterial growth occurs. This method takes this growth into account by characterising nitrate production during the experiment (Fig. 2).The effect of cycloheximide on nitrification was first determined to make sure that this compound is not inhibitory toward nitrifiers. Results obtained (Table 1) show that cycloheximide was not inhibitory toward nitrate production or the maximum nitrification oxygen uptake rate (rO2 N) after 4 hours of contact with nitrifying biomass. Cycloheximide addition in the activated sludge had an important impact on rotifers and flagellates but no effect on ciliates; it also seemed to stimulate amoebae growth. In both reactors, flagellates were mainly Peranama, attached ciliates were mainly Opercularia and Epistylis and a few Vorticella. Free ciliates like Aspidisca and Euplotes were found in both reactors.Variation with time of the abundance of microfauna organisms is shown in Figures 4 and 5 for both reactors. In the reference reactor the number of microfauna organisms decreased with time (Fig. 4) probably due to substrate starvation. Microfauna composition remained however diversified. For the inhibited reactor (Fig. 5), three periods were observed. During period I, the microfauna was mainly composed of ciliates and the number of microfaunal organisms decreased rapidly. During period II, an important growth of amoebae was observed. Cycloheximide was then added during this period to reduce their number. This growth of amoebae seems to be caused by the resistance of these micro-organisms toward inhibiting compounds (SRIKANTH et BERK, 1993). During period III, the number of microfaunal organisms was lower than during period II, and microfauna was mainly composed of ciliates.Nitrate concentration behaviour, necessary for bA calculation, is shown on Figure 6. In the reference reactor, nitrate concentrations varied linearly. For the inhibited reactor, the linear pattern was not observed during period II. This result was probably caused by an important nitrogen assimilation need of amoebae (ELDRIGE and JACKSON, 1993). Because organic nitrogen released by bacterial decay is consumed by amoebae assimilation, less nitrogen is available for the ammonification process and therefore for nitrification. Ammonia concentrations remained below 0.2 mg N·l-1 during all the experiment for both reactors. When amoebae disappeared from the inhibited reactor (period III) nitrate concentration varied linearly again.Variations of the maximum nitrification oxygen uptake rate (rO2 Nmax) with time are presented in Figure 7 (A&B) for both reactors. Two curves are plotted on each figure. Empty squares represent the measured rO2 N and black points represent the maximum nitrification rate that would have been measured if no growth on ammonification products had occurred. For the reference reactor (Fig. 7A), a value for bA of 0.08 d-1 can be calculated and can be considered constant for a constant microfauna composition.Three bA values can be estimated for the reactor inhibited with cycloheximide (Fig. 7B), corresponding to the three periods observed for microfauna composition. During period I, the bA value is 0.05 d-1 : a decrease in the microfaunal organism numbers implies a decrease of the bA value. During period II, when a development of amoebae is observed, the bA value increases and reaches 0.15 d-1. During period III with reduced grazing, the bA value is 0.13 d-1. Since during periods I and III the microfauna is mainly composed of ciliates, this difference between bA values is likely due to the observed difference in floc size between periods I and III.The results obtained during this study tend to prove (1) that the use of cycloheximide reduces microfaunal populations but can lead to a development of amoebae, and (2) that microfauna grazing seems to have an influence on the bA value, which can vary from 0.05 to 0.15 d-1 depending on microfaunal composition and abundance

Shifts in Metabolic Demands in Growing Altricial Nestlings Illustrate Context-Specific Relationships between Basal Metabolic Rate and Body Composition

Basal metabolic rate (BMR) in animals is interpreted as reflecting the size and metabolic intensity of energy-consuming tissues. However, studies investigating relationships between the mass of specific organs and interindividual variation in BMR have produced inconsistent patterns with regard to which organs have the largest impact on BMR variation. Because of the known flexibility in organ mass and metabolic intensity within individual organs, relationships between BMR and body-composition variables are bound to be context specific. Altricial nestlings are excellent models to illustrate this phenomenon because of the extreme variation in body composition occurring during growth. Using European starlings at three age classes, we studied changes in body composition together with its effect on variation in resting metabolic rate (RMR) in order to highlight the context-specific nature of these relationships. Our data suggest a transition in metabolic costs during growth in starling nestlings. During the linear phase of growth, energy is mainly consumed by tissue-synthesis processes, with fast-growing organs having a large influence on RMR. In the plateau phase of growth, the energy expenditure is transferred to functional costs, with high-intensity organs having a predominant effect on RMR variation. Our data illustrates the context-specific nature of organ mass-metabolic rate correlations, which complicates inter- and intraspecific comparisons of BMR. In the future, such comparisons must be done while taking the physiological state of the study animal into account