Premise: Seed germination over time is characterized by a sigmoid curve, called a germination curve, in which the percentage (or absolute number) of seeds that have completed germination is plotted against time. A number of individual coefficients have been developed to characterize this germination curve. However, as germination is considered to be a qualitative developmental response of an individual seed that occurs at one time point, but individual seeds within a given treatment respond at different time points, it has proven difficult to develop a single index that satisfactorily incorporates both percentage and rate. The aim of this paper is to develop a new coefficient, the continuous germination index (CGI), which quantifies seed germination as a continuous process, and to compare the CGI with other commonly used indexes.Methods: To create the new index, the germination curves were smoothed using nondecreasing splines and the CGI was derived as the area under the resulting spline. For the comparison of the CGI with other common indexes, a regression model with functional response was developed.Results: Using both an experimentally obtained wild pea (Pisum sativum subsp. elatius) seed data set and a hypothetical data set, we showed that the CGI is able to characterize the germination process better than most other indices. The CGI captures the local behavior of the germination curves particularly well.Discussion: The CGI can be used advantageously for the characterization of the germination process. Moreover, B‐spline coefficients extracted by its construction can be employed for the further statistical processing of germination curves using functional data analysis methods.

With large-scale database systems, statistical analysis of data, occurring in the form of probability distributions, becomes an important task in explorative data analysis. Nevertheless, due to specific properties of density functions, their proper statistical treatment of these data still represents a challenging task in functional data analysis. Namely, the usual $L^2$ metric does not fully accounts for the relative character of information, carried by density functions; instead, their geometrical features are captured by Bayes spaces of measures. The easiest possibility of expressing density functions in an $L^2$ space is to use centred logratio transformation, even though this results in functional data with a constant integral constraint that needs to be taken into account in further analysis. While theoretical background for reasonable analysis of density functions is already provided comprehensively by Bayes spaces themselves, preprocessing issues still need to be developed. The aim of this paper is to introduce optimal smoothing splines for centred logratio transformed density functions that take all their specific features into account and provide a concise methodology for reasonable preprocessing of raw (discretized) distributional observations. Theoretical developments are illustrated with a real-world data set from official statistics and with a simulation study.

Seed dormancy and timing of its release is an important developmental transition determining the survival of individuals, populations, and species in variable environments. Medicago truncatula was used as a model to study physical seed dormancy at the ecological and genetics level. The effect of alternating temperatures, as one of the causes releasing physical seed dormancy, was tested in 178 M. truncatula accessions over three years. Several coefficients of dormancy release were related to environmental variables. Dormancy varied greatly (4–100%) across accessions as well as year of experiment. We observed overall higher physical dormancy release under more alternating temperatures (35/15 °C) in comparison with less alternating ones (25/15 °C). Accessions from more arid climates released dormancy under higher experimental temperature alternations more than accessions originating from less arid environments. The plasticity of physical dormancy can probably distribute the germination through the year and act as a bet-hedging strategy in arid environments. On the other hand, a slight increase in physical dormancy was observed in accessions from environments with higher among-season temperature variation. Genome-wide association analysis identified 136 candidate genes related to secondary metabolite synthesis, hormone regulation, and modification of the cell wall. The activity of these genes might mediate seed coat permeability and, ultimately, imbibition and germination.

Contact problem for a large deformed beam with an elastic obstacle is formulated, analyzed, and numerically solved. The beam model is governed by a nonlinear fourth-order differential equation developed by Gao, while the obstacle is considered as the elastic foundation of Winkler’s type in some distance under the beam. The problem is static without a friction and modeled either using Signorini conditions or by means of normal compliance contact conditions. The problems are then reformulated as optimal control problems which is useful both for theoretical aspects and for solution methods. Discretization is based on using the mixed finite element method with independent discretization and interpolations for foundation and beam elements. Numerical examples demonstrate usefulness of the presented solution method. Results for the nonlinear Gao beam are compared with results for the classical Euler-Bernoulli beam model.