Phenotypic variation of floral organs in Malus using frequency distribution functions

Background Phenotypic diversity of floral organs plays an important role in plant systematic taxonomy and genetic variation studies. Previous research have focused on the direction of variation but disregarded its degree. Phenotypic variation (including directions and degrees) of 17 floral traits from wild to cultivated crabapples were explored by comparing their distributions and deviations in three different dimensions: floral organ number, size, and the shape. Results Except for petal number, petal length / petal width, and sepal length / sepal width, the analyzed floral traits of cultivated crabapples all showed downward distributed box bodies in box plot analysis and left deviations of fitted curves in frequency distribution function analysis when compared to the wild, which revealed consistent variation directions of petaloid conversion (pistils or stamens → petals), size miniaturization (large → small), and shape narrowness (petal shape: circular → elliptic; sepal shape: triangular → lanceolate). However, only seven floral traits exhibited significant differences in box plot analysis, while all of the traits in frequency distribution function analysis were obviously offset. The variation degrees were quantitatively characterized by sizing traits > shaping traits > numbering traits and by horizontal dimensions > radial dimensions. Conclusions Frequency distribution function analysis was more sensitive than the box plot analysis, which constructed a theoretical basis for Malus flower type breeding and would provide a new quantitative method for future evaluation of floral variation among different groups of angiosperms at large.

Dramatic Effect of the Utilized Theory on Frequency Distribution Function of Carbon Nanotubes

In this study, we investigate the modal density of double-walled carbon nanotubes. Emphasis is placed on the effect of the utilized theory. Specifically, we compare the modal density obtained via classical Bernoulli–Euler theory with that obtained by employing the refined Bresse–Timoshenko theory with nonlocal and surface effects taken into account. We show that the effect of refinements is dramatic.