Discrete triangular distributions are introduced, in order to serve as kernels in the non-parametric estimation for probability mass function. They are locally symmetric around every point of estimation. Their variances depend on the smoothing bandwidth and establish a bridge between Dirac and discrete uniform distributions. The boundary bias related to the discrete triangular kernel estimator is solved through a modification of the kernel near the boundary. The mean integrated squared errors and then the optimal bandwidth are investigated. We also study the adequate bandwidth for excess zeros. The performance of the discrete triangular kernel estimator is illustrated using simulated count data. An application to count data from football is described and compared with a binomial kernel estimator.

One of the most intriguing questions in plant ecology is which evolutionary strategy allows widely distributed species to increase their ecological range and grow in changing environmental conditions. Phenotypic plasticity and local adaptations are major processes governing species range margins, but little is known about their relative contribution for tree species distribution in tropical forest regions. We investigated the relative role of phenotypic plasticity and local adaptation in the ecological distribution of the widespread palm Euterpe edulis in the Brazilian Atlantic Forest. Genetic sampling and experiments were performed in old‐growth remnants of two forest types with higher (Seasonal Semideciduous Forests vs. Submontane Rainforest) and lower biogeographic association and environmental similarities (Submontane Rainforest vs. Restinga Forest). We first assessed the molecular genetic differentiation among populations, focusing on the group of loci potentially under selection in each forest, using single‐nucleotide polymorphism (SNPs) outliers. Further, we looked for potential adaptive divergence among populations in a common garden experiment and in reciprocal transplants for two plant development phases: seedling establishment and sapling growth. Analysis with outlier loci indicated that all individuals from the Semideciduous Forest formed a single group, while another group was formed by overlapping individuals from Submontane Rainforest and Restinga Forest. Molecular differentiation was corroborated by reciprocal transplants, which yielded strong evidence of local adaptations for seedling establishment in the biogeographically divergent Rainforest and Semideciduous Forest, but not for Restinga Forest and Submontane Rainforest. Phenotypic plasticity for palm seedling establishment favors range expansion to biogeographically related or recently colonized forest types, while persistence in the newly colonized ecosystem may be favored by local adaptations if climatic conditions diverge over time, reducing gene flow between populations. SNPs obtained by next‐generation sequencing can help exploring adaptive genetic variation in tropical trees, which impose several challenges to the use of reciprocal transplants.

Asymmetric discrete triangular distributions are introduced in order to extend the symmetric ones serving for discrete associated-kernels in the nonparametric estimation for discrete functions. The extension from one to two orders around the mode provides a large family of discrete distributions having a finite support. Establishing a bridge between Dirac and discrete uniform distributions, some different shapes are also obtained and their properties are investigated. In particular, the mean and variance are pointed out. Applications to discrete kernel estimators are given with solution to a boundary bias problem.

Multinomial responses frequently occur in dose level experiments. For example, in a study of the influence of gamma radiation on the emergence of house flies (Musca domestica L., 1758), three disjoint outcomes occurred: death before the pupae opened, death during emergence, and life after emergence. Although the flies are easy to breed, this sort of bioassay is, in general, very expensive since it requires the use of a gamma radiation source. Experiments therefore need to be designed to involve the minimum number of different doses. Here the theory of optimum experimental design is applied to provide efficient experiments to estimate the parameters of those multinomial logistic models that are a special case of the multivariate logistic models of Glonek and McCullagh (1995, Journal of the Royal Statistical Society, Series B 57, 533-546). The purpose is to reduce the overall experimental cost. The general equivalence theorem (Fedorov, 1972, Theory of Optimal Experiments) is adapted to this class of models, providing an effective method of generating and checking the optimality of designs. One example on flies demonstrates the method, which can be easily implemented.

Papaya (Carica papaya L.) is a tropical and climacteric fruit with antioxidant, anticarcinogenic and anti-mutagenic properties, containing carotenoids and high nutritional value.• Brazil is ranked second in the world for papaya production with 17.5% of global production.• One of the most important criteria for determining the ripeness stage over time is the peel color.

Cylindrospermopsis raciborskii (Woloszynska) Seenayya et Subba Raju (Ordem Nostocales) is one of the most troublesome bloom-forming species in Brazil. Understanding the population dynamics of the different morphotypes of C. raciborskii (straight and coiled) could assist in the prediction of favourable conditions for the proliferation of this potentially toxin-producing species. The aim of the present study was to assess the effects of two different light intensities and temperatures on the growth rate and morphology of the trichomes of the straight and coiled morphotypes. For such, two non-toxin producing strains of C. raciborskii were used -one with a coiled trichome (ITEP31) and another with a straight trichome (ITEP28). The strains were cultured in BG-11 medium in a climatic chamber under controlled conditions. Two light intensities (30 and 90 µmol.m -2 .s -1 ) were combined at temperatures of 21 and 31 °C and the growth rate and morphological changes were analysed. The morphotypes responded differently to the different temperatures and light intensities. Both strains exhibited faster growth velocities when submitted to higher light intensity and temperature. The lower temperature and higher luminosity hampered the development of both strains. Variations in cellular morphology and an absence of akinetes in both strains were related to the lower temperature (21 °C). The coiled morphotype demonstrated considerable phenotype plasticity, changing the morphology of trichome throughout its growth curve. Although molecular analysis does not sustain the separation of the morphotypes as distinct species, their different eco-physiological responses should be considered further knowledge of extreme importance for the population control of these potentially toxic organisms.Keywords: cyanobacteria, morphotype, growth curve, culture. Efeitos da intensidade luminosa e temperatura em Cylindrospermopsis raciborskii(Cyanobacteria) com tricomas retos e espiralados: taxa de crescimento e morfologia ResumoCylindrospermopsis raciborskii (Woloszynska) Seenayya et Subba Raju (Ordem Nostocales) é uma das espécies formadoras de florações mais prejudiciais no Brasil. Entender a dinâmica populacional dos diferentes morfotipos de C. raciborskii (reto e espiralado) pode auxiliar a previsão de condições favoráveis à proliferação e ao estabelecimento dessa espécie potencialmente produtora de saxitoxinas e cilindrospermopsina. O objetivo deste estudo foi avaliar os efeitos de duas diferentes intensidades luminosas e temperaturas na velocidade de crescimento e na morfologia dos tricomas de morfotipos reto e espiralado. Para tanto, foram utilizadas duas linhagens de C. raciborskii, não produtoras de toxinas, sendo uma com tricoma espiralado (ITEP31) e outra com tricoma reto (ITEP28), cultivadas em câmaras climáticas sob condições controladas em meio BG-11. Duas intensidades luminosas, 30 e 90 µmol.m -2 .s -1 , foram BIOLOGYBraz.

SUMMARY In dose level experiments on insects males and females in general respond differently but may not be readily separable at the time of the experiment, although dead insects may be sexed. Each application of a specified dose therefore yields data on the response of males and of females. The theory of optimum experimental design is extended to this novel situation and applied to give designs for parameter estimation and for the estimation of the dose for a required probability of death, typically the LD95‐value. Monte Carlo sampling is used to incorporate prior information into the design criteria.

The aim of this study was to evaluate the effect of the addition of xanthan gum and glycerol to the starch of green pea with high content of AM (cv. Utrillo) in the preparation of films and their physical characteristics. Filmogenic solution (FS) with different levels of pea starch (3, 4, and 5%), xanthan gum (0, 0.05, and 0.1%), and glycerol (glycerol–starch ratio of 1:5 w/w) were studied. The FS was obtained by boiling (5 min), followed by autoclaving for 1 h at 120°C. The films were prepared by casting. Films prepared only with pea starch were mechanically resistant when compared to other films, prepared with corn, cassava, rice, and even other pea cultivars (yellow, commercial). The tensile strength of these films is comparable to synthetic films prepared with high‐density polyethylene and linear low‐density polyethylene. However, they are films of low elasticity when compared to other films, such as rice starch films, and especially when compared to polyethylene films. The increased concentration of starch in the solution increased the puncture force. The increased concentration of glycerol slightly decreased the film crystallinity and interfered in the mechanical properties of the films, causing reduction of the maximum values of tensile strength, strain at break, and puncture force. The plasticizer also caused an increase of elongation at break. Xanthan gum was important to formation of films; however, it did not affect their mechanical properties.