Sensitivity analysis of the asymptotic behavior of a model for the environmental movement of radionuclides

The asymptotic behavior of a linear compartment model for the environmental movement of radionuclides is investigated. Here, the expression asymptotic behavior is used to designate the behavior of q(t) as t → ∞, where q is the solution of a vector differential equation of the form dq/dt = h + Kq. The asymptotic behavior of such equations is described. For the model and conditions under consideration, each element of q converges monotonically to a steady-state value. A hydrologic system is defined and used to illustrate this behavior. An approach to sensitivity analysis employing Latin hypercube sampling, rank transformations and stepwise regression is presented and then applied to this system. A total of 20 independent variables is introduced and the following dependent variables are investigated for the various components of the system: amount of radionuclide present at steady state. concentration of radionuclide at steady state, and time required to reach 90% of steady state. Finally, an application of asymptotic behavior in the analysis of a hypothetical site for the geologic isolation of high-level radioactive waste is described and a brief discussion of differential sensitivity analysis is given.     

Eventual saturation of the climate–carbon cycle feedback studied with a conceptual model

A saturation of climate–carbon cycle feedback was found earlier in the simulations with the IAP RAS climate model of intermediate complexity. Here, this eventual saturation is interpreted by using a conceptual linearised coupled model. It is shown that this saturation is due to weak, logarithmic, dependence of the carbon dioxide radiative forcing on its atmospheric concentration. This eventual saturation leads to the non-monotonic behaviour of climate–carbon cycle parameter f   in time. If the time scale of the atmospheric CO₂${\text{CO}}_2$ build up is _tp$t_{\text{p}}$ then, starting from an initial equilibrium, f   approaches maximum in time ?_tp$?t_{\text{p}}$. Afterwards, climate–carbon cycle parameter decreases and eventually tends to unity. The time scale of the latter decrease is _t1=(1−5)_tp$t_1=(1-5)t_{\text{p}}$. A dependence of _tm$t_{\text{m}}$ and _t1$t_1$ on governing parameters of the conceptual model is studied. It is argued that an eventual saturation of the climate–carbon cycle feedback is expected to occur also in the other integrations of sufficient length with coupled climate–carbon cycle models.     

A new growth model for fishes and the estimation of optimum age of fish populations

Evidence is presented that a simple power equation of the formX _t –X _m =±b(|tࢤt _m|)^B can describe growth in length and weight of fishes, whereX _t denotes fish length or weight at aget, X _m is length or weight (L _m orW _m ) at a reference aget _m , andb andB are parameters to be estimated by the least squares. The optimum age of fish populations (t _y ) may be estimated by the equationMW _m /b=±y ^B (B/y-M) whereM denotes natural mortality and wherey=t _y -t _m .     

Analytical methods for a stochastic mainland-island metapopulation model

We study a class of chain-binomial metapopulation models, giving special attention to the ‘mainland-island’ configuration, where patches receive immigrants from an external source. We evaluate the distribution of the number n_t of occupied patches at any census time t and establish a law of large numbers that identifies a deterministic trajectory which can be used to approximate the process when the number of patches is large. We also establish a central limit law, which shows that the fluctuations about this trajectory are approximately normally distributed. We describe briefly much finer results that can be used for model calibration.     

A reliability index for models

Since the prediction of environmental changes depends heavily on mathematical models, environmental planners have a definite need for a general test that can be easily applied to determine the reliability of a model or to choose between available models. In this paper a ‘reliability index’ for a model is defined. This index is a number k ? 1 which is determined from a set x₁,x₂,…,x_n of model predictions and a corresponding set y₁,y₂,…,y_n of observations. One interpretation of the index k is that the model predictions agree with observations, in some sense, within a factor of k. The index k is defined using an intuitive, geometric approach and is justified through agreement with another index developed using statistical techniques. Examples are given for a model which predicts the atmospheric movement of radon-222 originating from a concentrated source of radium-226, such as a uranium tailings pile.     

Diffusional mass transfer coefficient at the water–sediment interface for wind-induced flow in very shallow lagoons

Very shallow lagoons that are a few centimeters deep are common in the arid Andes of Northern Chile, Argentina, Bolivia and Perú. The dynamics of these lagoons are dominated by the water–sediment interface (WSI) and strong afternoon winds. Although many studies have examined the diffusional mass transfer coefficients (k _t ) of open channel flows, estimates for wind-induced flows are still unknown. The aim of this article is to propose and validate an analytical expression for computing k _t at the WSI for wind-induced flow. The laboratory measurements were conducted in a wind tunnel with a water tank of variable depth located at its downwind end. Natural muddy sediments were placed in the middle of the tank so that the dissolved oxygen (DO) was consumed in the sediments. The diffusional mass transfer coefficient that characterizes the DO uptake in the sediment was obtained from DO micro-profiles measured with an OX-25 Unisense microelectrode. Water velocity profiles were measured with a 2D side-view Sontek acoustic doppler velocimetry (ADV), and the wind shear velocity was computed based on wind velocity profiles that were measured with an Extech hot-wire anemometer. A total of 16 experiments were conducted with different water depths and wind shear stresses. The constants required by the model were determined from these experiments, and the analytical expression was successfully validated by the laboratory observations. The analytical expression obtained for computing k _t was also validated with field observations that were conducted in October, 2012, in Salar del Huasco, Northern Chile (20.274° S, 68.883° W, 3800 m above sea level). The comparison between the observed and predicted values of k _t provides a determination coefficient of r ² = 0.48 and a p value < 0.01. The results show that the value of k _t for wind-induced flow is proportional to the wind shear velocity and the inverse of the Reynolds number of the wind-induced current.     

Estimation of some parameters of large fish populations by capture-recapture experiments

A certain number of tagged fish is liberated and assumed to be distributed randomly among a natural fish population. The fish are subjected to a number of fishing experiments within relatively short periods, and lie between equal intervening periods of durationT. Untagged fish are retained, while tagged fish are released during the fishing experiments. Denoting the catchability of untagged fish byq _u and that for tagged fish byq _t , it is assumed that they are related by the equation “q _u =cq _t ” wherec is a constant. Denoting the survival rates of tagged fish and the effective fishing effort of commerical fisheries per unit time from the (k-1)^th to thek ^th experiments by _t S _k andf _k , respectively, it is assumed that they vary from period to period. Assuming that during thek ^th experiment, the number of untagged fish captured and the experimental fishing rate of tagged fish are denoted by _u X _k and _t P _k , respectively, then $$\begin{array}{*{20}c} {\frac{{(_u X_k )^2 }}{{[_u X_{(k - 1)} ][_u X_{(k + 1)} ]}} = \frac{{_t S_k }}{{_t S_{(k + 1)} }} \cdot \frac{{e^{ - (1 - c)q_t f_{(k + 1)} T} }}{{e^{ - (1 - c)q_t f_k T} }} \cdot } \\ {\frac{{1 - c_t P_{(k - 1)} }}{{1 - c_t P_k }} \cdot \frac{{(_t P_k )^2 }}{{[_t P_{(k - 1)} ][_t P_{(k + 1)} ]}}.} \\ \end{array}$$ The above equation containsc as a single unknown, while all other terms are supplied by the capture-recapture experiments, exceptf _k andf _(k+1) which may be obtained from fisheries statistics. A number of the above equations are obtained from several experiments and can be combined into a single equation to obtain an overall estimate forc which can be used to derive estimates for experimental fishing rates, abundance, and instantaneous natural and fishing mortality rates for natural fish populations. These estimates are free from type (A) tagging errors, and have the advantage of taking into consideration the probable different behaviour of tagged and untagged fish.     

Impact of temperature and species interaction on filamentous cyanobacteria may be more important than salinity and increased pCO2 levels

A future business-as-usual scenario (A1FI) was tested on two bloom-forming cyanobacteria of the Baltic Proper, Nodularia spumigena and Aphanizomenon sp., growing separately and together. The projected scenario was tested in two laboratory experiments where (a) interactive effects of increased temperature and decreased salinity and (b) interactive effects of increased temperature and elevated levels of pCO₂ were tested. Increased temperature, from 12 to 16 °C, had a positive effect on the biovolume and photosynthetic activity (F _v/F _m) of both species. Compared when growing separately, the biovolume of each species was lower when grown together. Decreased salinity, from 7 to 4, and elevated levels of pCO₂, from 380 to 960 ppm, had no effect on the biovolume, but on F _v/F _m of N. spumigena with higher F _v/F _m in salinity 7. Our results suggest that the projected A1FI scenario might be beneficial for the two species dominating the extensive summer blooms in the Baltic Proper. However, our results further stress the importance of studying interactions between species.     

Mechanisms of moisture stress in a mid-latitude temperate forest: Implications for feedforward and feedback controls from an irrigation experiment

While it is well established that stomata close during moisture stress, strong correlations among environmental (e.g., vapor pressure deficit, soil moisture, air temperature, radiation) and internal (e.g., leaf water potential, sap flow, root-shoot signaling) variables obscure the identification of causal mechanisms from field experiments. Models of stomatal control fitted to field data therefore suffer from ambiguous parameter identification, with multiple acceptable (i.e., nearly optimal) model structures emphasizing different moisture status indicators and different processes. In an effort to minimize these correlations and improve parameter and process identification, we conducted an irrigation experiment on red maples (Acer rubrum L.) at Harvard Forest (summers of 2005 and 2006). Control and irrigated trees experienced similar radiative and boundary layer forcings, but different soil moisture status, and thus presumably different diurnal cycles of internal leaf water potential. Measured soil moisture and atmospheric forcing were used to drive a transient tree hydraulic model that incorporated a Jarvis-type leaf conductance in a Penman–Monteith framework with a Cowan-type (resistance and capacitance) tree hydraulic representation. The leaf conductance model included dependence on both leaf matric potential, Ψ_L (so-called feedback control) and on vapor pressure deficit, D (so-called feedforward control). Model parameters were estimated by minimizing the error between predicted and measured sap flow. The whole-tree irrigation treatment had the effect of elevating measured transpiration during summer dry-downs, demonstrating the limiting effect that subsurface resistance may have on transpiration during these times of moisture stress. From the best fitted model, we infer that during dry downs, moisture stress manifests itself in an increase of soil resistance with a resulting decrease in Ψ_L, leading to both feedforward and feedback controls in the control trees, but only feedforward control for the irrigated set. Increases in the sum-of-squares error when individual model components were disabled allow us to reject the following three null hypotheses: (1) the f(D) stress is statistically insignificant (p = 0.01); (2) the f(Ψ_L) stress is statistically insignificant (p = 0.07); and (3) plant storage capacitance is independent of moisture status (p = 0.07).     

A peaked function for modeling temperature dependence of plant productivity

We suggests that temperature response of plant productivity can be modeled by the Arrhenius function modified to describe the effect of temperature on enzyme activity: G_A(T) = 2f(T)/(1 + f²(T)), where f(T) = exp(E_a/RT_opt − E_a/RT), R the universal gas constant, E_a the activation energy and T_opt is the optimal temperature. In common with other functions used for modeling the temperature response of plant productivity, the curve of function G is almost symmetrical and bell-shaped. The special convenience of G_A is that it relates the width of the “bell” to thermodynamic concepts, such as activation energy of chemical reactions converting carbon dioxide and water to carbohydrates.     

A constrained least-squares approach to combine bottom-up and top-down CO2 flux estimates

Terrestrial CO₂ flux estimates are obtained from two fundamentally different methods generally termed bottom-up and top-down approaches. Inventory methods are one type of bottom-up approach which uses various sources of information such as crop production surveys and forest monitoring data to estimate the annual CO₂ flux at locations covering a study region. Top-down approaches are various types of atmospheric inversion methods which use CO₂ concentration measurements from monitoring towers and atmospheric transport models to estimate CO₂ flux over a study region. Both methods can also quantify the uncertainty associated with their estimates. Historically, these two approaches have produced estimates that differ considerably. The goal of this work is to construct a statistical model which sensibly combines estimates from the two approaches to produce a new estimate of CO₂ flux for our study region. The two approaches have complementary strengths and weaknesses, and our results show that certain aspects of the uncertainty associated with each of the approaches are greatly reduced by combining the methods. Our model is purposefully simple and designed to take the two approaches’ estimates and measures of uncertainty at ‘face value’. Specifically, we use a constrained least-squares approach to appropriately weigh the estimates by the inverse of their variance, and the constraint imposes agreement between the two sources. Our application involves nearly 18,000 flux estimates for the upper midwest United States. The constrained dependencies result in a non-sparse covariance matrix, but computation requires only minutes due to the structure of the model.     

Environmental modeling of uranium interstitial compositions of non-stoichiometric oxides: experimental and theoretical analysis

Study of uranium interstitial compositions of non-stoichiometric oxides UO_2+x (x ∈ 0.1–0.02) in gas and condense phases has been presented, using various soft-ionization mass spectrometric methods such as ESI-, APCI-, and MALDI-MS at a wide dynamic temperature gradient (∈ 25–300 °C). Linearly polarized vibrational spectroscopy has been utilized in order to assign unambiguously, the vibrational frequencies of uranium non-stoichiometric oxides. Experimental design has involved xUO_2.66·yUO_2.33, xUO_2.66·yUO_2.33/SiO₂, xUO_2.66·yUO_2.33/SiO₂ (NaOH) and SiO₂/x′NaOH·y′UO₂(NO₃)₂·6H₂O, multicomponent systems (x = 1, y ∈ 0.1–1.0 and x′ = 1, y′ ∈ 0.1–0.6) as well as phase transitions UO₂(NO₃)₂·6H₂O → {U₄O₉(UO_2.25)} → U₃O₇(UO_2.33) → U₃O₈(UO_2.66) → {UO₃}, thus ensuring a maximal representativeness to real environmental conditions, where diverse chemical, geochemical and biochemical reactions, including complexation and sorption onto minerals have occurred. Experimental factors such as UV-irradiation, pH, temperature, concentration levels, solvent types and ion strength have been taken into consideration, too. As far as uranium speciation represents a challenging analytical task in terms of chemical identification diverse coordination species, mechanistic aspects relating incorporation of oxygen into UO _2+x form the shown full methods validation significantly impacts the field of environmental radioanalytical chemistry. UO₂ is the most commonly used fuel in nuclear reactors around the globe; however, a large non-stoichiometric range ∈ UO_1.65–UO_2.25 has occurred due to radiolysis of water on UO₂ surface yielding to H₂O₂, OH^·, and more. Each of those compositions has different oxygen diffusion. And in this respect enormous effort has been concentrated to study the potential impact of hazardous radionuclide on the environment, encompassing from the reprocessing to the disposal stages of the fuel waste, including the waste itself, the processes in the waste containers, the clay around the containers, and geological processes. In a broader sense, thereby, this study contributes to field of environmental analysis highlighting the great ability of various soft-ionization MS methods, particularly, MALDI-MS one, for direct assay of complex multicomponent heterogeneous mixtures at fmol–attomol concentration ranges, along with it the great instrumental features allowing, not only meaningful quantitative, but also structural information of the analytes, thus making the method indispensable for environmental speciation of radionuclides, generally.     

A simple and precise method for fitting a von Bertalanffy growth curve

The parameter K of the von Bertalanffy equation, as developed by Beverton and Holt (1957), is first estimated by the relation $$\log _e \left( {dL_t /dt} \right) = A - Kt$$ where dLt/dt denotes growth increments per a unit of age, t denotes age, and A is a constant. The K estimate is used to evaluate L∞; $$L_\infty = \left( {e^K \sum\limits_2^n {L_t - \sum\limits_1^{n - 1} {L_t } } } \right)/\left( {n - 1} \right)\left( {e^K - 1} \right)$$ The L∞ estimate is used to estimate t _o, and to obtain a better estimate for K; $$\log _e \left( {1 - L_t /L_\infty } \right) = - Kt + Kt_0 $$ The K estimate may be used to obtain another estimate for L∞. Solved examples show that a single iteration is sufficient to obtain fitted equations which are, on the average, as precise as equations fitted by the least squares method shown by Tomlinson and Abramson (1961). This new method can be used, with a slight modification, for the second equation given above, if growth data have unequal age intervals. The variance of K, t _o and log _e L∞ can be estimated by applying the simple methods used in the case of straight-line relationships.     

Future ocean acidification will be amplified by hypoxia in coastal habitats

Ocean acidification is elicited by anthropogenic carbon dioxide emissions and resulting oceanic uptake of excess CO₂ and might constitute an abiotic stressor powerful enough to alter marine ecosystem structures. For surface waters in gas-exchange equilibrium with the atmosphere, models suggest increases in CO₂ partial pressure (pCO₂) from current values of ca. 390 μatm to ca. 700–1,000 μatm by the end of the century. However, in typically unequilibrated coastal hypoxic regions, much higher pCO₂ values can be expected, as heterotrophic degradation of organic material is necessarily related to the production of CO₂ (i.e., dissolved inorganic carbon). Here, we provide data and estimates that, even under current conditions, maximum pCO₂ values of 1,700–3,200 μatm can easily be reached when all oxygen is consumed at salinities between 35 and 20, respectively. Due to the nonlinear nature of the carbonate system, the approximate doubling of seawater pCO₂ in surface waters due to ocean acidification will most strongly affect coastal hypoxic zones as pCO₂ during hypoxia will increase proportionally: we calculate maximum pCO₂ values of ca. 4,500 μatm at a salinity of 20 (T = 10 °C) and ca. 3,400 μatm at a salinity of 35 (T = 10 °C) when all oxygen is consumed. Upwelling processes can bring these CO₂-enriched waters in contact with shallow water ecosystems and may then affect species performance there as well. We conclude that (1) combined stressor experiments (pCO₂ and pO₂) are largely missing at the moment and that (2) coastal ocean acidification experimental designs need to be closely adjusted to carbonate system variability within the specific habitat. In general, the worldwide spread of coastal hypoxic zones also simultaneously is a spread of CO₂-enriched zones. The magnitude of expected changes in pCO₂ in these regions indicates that coastal systems may be more endangered by future global climate change than previously thought.     

Assessing spatial uncertainty of the Portuguese fire risk through direct sequential simulation

Forest fires have a significant economic, social, and environmental impact in Portugal. For that its fire risk was assessed through Bayes Formalism, where the main component of the risk of fire was assessed by the conditional probability of fire I(u,t) given a class of the daily severity rating (DSR) for a specific period of time—P[I(u,t)|R(u,t)]. The evaluation of this a posterior probability, P[I(u,t)|R(u,t)], was based on the update of marginal local probability of fire in each chosen region u (Durão, 2006).DSR values were used to calculate fire's risk, taking into account historical data, I(s,t), in a given region s, and also to define DSR's local thresholds in order to have P [I(u,t)|R(u,t)] ≥ 0.65.In this paper we characterize these posterior probabilities using direct sequential simulation models (DSS models) to obtain the spatial distribution of these probabilities over the entire Portugal, in order to assess the risk of fire and associated spatial uncertainty. Local probability density functions (pdfs) and spatial uncertainty are evaluated by a set of equiprobable simulated images of these posterior probabilities.Results are presented and discussed for the Portuguese fire seasons of the 2-year period, 2003-2004. The conditional probabilities reproduced reasonably well what was officially published for the studied fire seasons. We expect that a better understanding of both spatial and temporal patterns of fire in Portugal together with uncertainty measures constitutes an important tool for managers, helping to improve the effectiveness of fire prevention, detection and fire fighting resources allocation in critical social and environmental areas.     

Estimating dyad association probability under imperfect and heterogeneous detection

In animal behaviour studies, association indices estimate the proportion of time two individuals (i.e. a dyad) spend in association. In terms of dyads, all association indices can be interpreted as estimators of the probability that a dyad is associated. However, traditional indices rely on the assumptions that the probability to detect a particular individual (p) is either approximately one and/or homogeneous between associated and not associated individuals. Based on marked individuals we develop a likelihood based model to estimate the probability a dyad is associated (ψ) accounting for p < 1 and possibly varying between associated and not associated individuals. The proposed likelihood based model allows for both individual and dyadic missing observations. In addition, the model can easily be extended to incorporate covariate information for modeling p and ψ. A simulation study showed that the likelihood based model approach yield reasonably unbiased estimates, even for low and heterogeneous individual detection probabilities, while, in contrast, traditional indices showed moderate to strong biases. The application of the proposed approach is illustrated using a real data set collected from a population of Commerson's dolphin (Cephalorhynchus commersonii) in Patagonia Argentina. Finally, we discuss possible extensions of the proposed model and its applicability in animal behaviour and ecological studies.     

Mathematical analysis of change in forest carbon use efficiency with stand development: A case study on Abies veitchii Lindl.

Changes in carbon use efficiency (CUE), which is defined as the ratio of net primary production (NPP) to gross primary production (GPP), were analyzed for Abies veitchii Lindl. forests with respect to stand development by developing a simple mathematical model incorporating data on physiological variables and leaf mass ratio. A decrease in CUE with stand development was successfully expressed as a function of stand biomass (y) based on the following three assumptions: (1) a power-law relationship between mean respiration and mean individual tree mass, (2) a power-functional relationship between mean gross primary production and mean individual tree mass, and (3) self-thinning relationship between stand biomass and density. Based on this model, a parameter of CUE–y relationship was defined, and it was clarified that CUE decrease with stand development is caused not by the ratio of specific respiration rate to specific gross photosynthetic rate, but by leaf mass ratio. Since CUE is high in young forests, helpful information on selecting woody species when planting seedlings was provided from the viewpoints of reducing CO₂ in the atmosphere and global warming.     

Theoretical analysis of change in forest carbon use efficiency with stand development: A case study on Hinoki Cypress (Chamaecyparis obtusa (Sieb. et Zucc.) Endl.) plantation from the seedling stage

Changes in carbon use efficiency (CUE), which is defined as the ratio of net primary production (NPP) to gross primary production (GPP), were estimated for the aerial parts of the Hinoki Cypress (Chamaecyparis obtusa (Sieb. et Zucc.) Endl.) with respect to stand development. The analysis incorporated previously published data from the early stages of stand development, namely the seedling stages of the cypress. For this analysis, a simple mathematical model to assess the changes in CUE was developed by incorporating data on physiological variables and mass of woody species. The CUE tended to increase with increases in the aboveground biomass of the stand, and then decreased gradually despite increases in the aboveground biomass. The CUE-value (0.28, 0.39) of the seedling stage was lower than that (0.33-0.58) of the young or mature trees. To examine the effect of physiological variables and mass on CUE, the ratios of the specific respiration rate to the specific photosynthetic rate (r/a) and the leaf biomass to the aboveground biomass or leaf mass ratio (y_L/y_T) were calculated. The low value of CUE at the seedling stage was due to the high ratio of specific respiration rate to specific photosynthetic rate r/a, but was not due to the high value of the leaf mass ratio y_L/y_T. In addition, the decline in CUE associated with older stages of stand development was due to the decreasing changes in y_L/y_T, and the r/a ratio did not influence the change in CUE.     

A general solution of Benjamin-type gravity current in a channel of non-rectangular cross-section

We consider the steady-state propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate x. The bottom and top of the channel are at z =?0, H, and the cross-section is given by the quite general form ?f ₁(z) ≤?y ≤?f ₂(z) for 0 ≤?z ≤?H, where f _1,2 are piecewise continuous functions and f ₁ +?f ₂ >?0 for ${z \in(0,H)}$ . The interface of the current is horizontal, the (maximum) thickness is h, its density is ρ _c . The reduced gravity g′ =?|ρ _c /ρ _a ? 1|g (where ${- g\hat{z}}$ is the gravity acceleration and ρ _a the density of the ambient) drives the current with speed U into the stationary ambient fluid. We show that the dimensionless Fr =?U/(g′ h)^1/2, the rate of energy dissipation (scaled with the rate of pressure work), and the dimensionless head-loss Δ/h, can be expressed by compact formulas which involve three integrals over the cross-section areas of the current and ambient. By some standard manipulations these integrals are simplified into quite simple line-integrals of the shape-function of the channel, f(z) =?f ₁(z) +?f ₂(z), and of z f(z). This theory applies to Boussinesq and non-Boussinesq currents of “heavy” (bottom) and “light” (top) type. The classical results of Benjamin (J Fluid Mech 31:209–248, 1968) for a rectangular channel are fully recovered. We also recover the Fr results of Marino and Thomas (J Fluid Eng 131(5):051201, 2009) for channels of shape y =?±b z ^α (where b, α are positive constants); in addition, we consider the energy dissipation of these flows. The results provide insights into the effect of the cross-section shape on the behavior of the steady-state current, in quite general cases, for both heavy-into-light and light-into-heavy fluid systems, Boussinesq and non-Boussinesq. In particular, we show that a very deep current displays ${Fr = \sqrt{2}}$ , and is dissipative; the value of Fr and rate of dissipation (absolute value) decrease when the thickness of the current increases. However, in general, energy considerations restrict the thickness of the current by a clear-cut condition of the form h/H ≤?a _max ?< 1.