Pollen and achene characters of natural interspecific hybrid Solidago ×niederederi Khek were analyzed and compared with putative parental species S. virgaurea L. and S. canadensis L. to estimate the level of disturbances in generative reproduction resulting from its hybrid nature. Pollen viability (stainability) of Solidago ×niederederi from one newly discovered locality in NE Poland was evidently reduced to ~65% in both viability tests (acetocarmine and Alexander). The diameter of viable pollen (median 21.11 µm) fell between S. canadensis (median 19.52 µm) and S. virgaurea (median 23.48 µm). Both parental species produced normally developed achenes with high frequency (~90%) whereas in the hybrid, the seed set was dramatically low (6%). The results clearly indicated that sexual reproduction of hybridogenous taxon S. ×niederederi is disturbed, and its potential impact as an invasive species depends mainly on vegetative propagation.

Many Asteraceae species have been introduced into horticulture as ornamental or interesting exotic plants. Some of them, including Solidago and Galinsoga, are now aggressive weeds; others such as Ratibida are not. Special modifications of the ovule tissue and the occurrence of nutritive tissue have been described in several Asteraceae species, including invasive Taraxacum species. This study examined whether such modifications might also occur in other genera. We found that the three genera examined - Galinsoga (G. quadriradiata), Solidago (S. canadensis, S. rigida, S. gigantea) and Ratibida (R. pinnata) - differed in their nutritive tissue structure. According to changes in the integument, we identified three types of ovules in Asteraceae: “Taraxacum” type (recorded in Taraxacum, Bellis, Solidago, Chondrilla), with well-developed nutritive tissue having very swollen cell walls of spongy structure; “Galinsoga” type (in Galinsoga), in which the nutritive tissue cells have more cyto-plasm and thicker cell walls than the other integument parenchyma cells, and in which the most prominent character of the nutritive tissue cells is well-developed rough ER; and “Ratibida” type (in Ratibida), in which the nutritive tissue is only slightly developed and consists of large highly vacuolated cells. Our study and future investigations of ovule structure may be useful in phylogenetic analyses.

Studies concerning the ultrastructure of the periendothelial zone integumentary cells of Asteraceae species are scarce. The aim was to check whether and/or what kinds of integument modifications occur in Onopordum acanthium. Ovule structure was investigated using light microscopy, scanning electron microscopy, transmission electron microscopy and histochemistry. For visualization of calcium oxalate crystals, the polarizing microscopy was used. The periendothelial zone of integument in O. acanthium is well developed and composed of mucilage cells near the integumentary tapetum and large, highly vacuolated cells at the chalaza and therefore they differ from other integumentary cells. The cells of this zone lack starch and protein bodies. Periendothelial zone cells do not have calcium oxalate crystals, in contrast to other integument cells. The disintegration of periendothelial zone cells was observed in a mature ovule. The general ovule structure of O. acanthium is similar to other members of the subfamily Carduoideae, although it is different to "Taraxacum", "Galinsoga" and "Ratibida" ovule types.

Acta Technologica Agriculturae 1/2016Dušan Páleš et al.The most effective way for determination of curves for practical use is to use a set of control points. These control points can be accompanied by other restriction for the curve, for example boundary conditions or conditions for curve continuity (Sederberg, 2012). When a smooth curve runs only through some control points, we refer to curve approximation. The B-spline curve is one of such approximation curves and is addressed in this contribution. A special case of the B-spline curve is the Bézier curve Rédl et al., 2014). The B-spline curve is applied to a set of control points in a space, which were obtained by measurement of real vehicle movement on a slope (Rédl, 2007(Rédl, , 2008. Data were processed into the resulting trajectory (Rédl, 2012;Rédl and Kučera, 2008). Except for this, the movement of the vehicle was simulated using motion equations (Rédl, 2003;Rédl and Kročko, 2007). B-spline basis functionsBézier basis functions known as Bernstein polynomials are used in a formula as a weighting function for parametric representation of the curve (Shene, 2014). B-spline basis functions are applied similarly, although they are more complicated. They have two different properties in comparison with Bézier basis functions and these are: 1) solitary curve is divided by knots, 2) basis functions are not nonzero on the whole area. Every B-spline basis function is nonzero only on several neighbouring subintervals and thereby it is changed only locally, so the change of one control point influences only the near region around it and not the whole curve.These numbers are called knots, the set U is called the knot vector, and the half-opened interval 〈u i , u i + 1 ) is the i-th knot span. Seeing that knots u i may be equal, some knot spans may not exist, thus they are zero. If the knot u i appears p times, hence u i = u i + 1 = ... = u i + p -1 , where p >1, u i is a multiple knot of multiplicity p, written as u i (p). If u i is only a solitary knot, it is also called a simple knot. If the knots are equally spaced, i.e. (u i + 1 -u i ) = constant, for every 0 ≤ i ≤ (m -1), the knot vector or knot sequence is said uniform, otherwise it is non-uniform.Knots can be considered as division points that subdivide the interval 〈u 0 , u m 〉 into knot spans. All B-spline basis functions are supposed to have their domain on 〈u 0 , u m 〉. We will use u 0 = 0 and u m = 1.To define B-spline basis functions, we need one more parameter k, which gives the degree of these basis functions. Recursive formula is defined as follows:This definition is usually referred to as the Cox-de Boor recursion formula. If the degree is zero, i.e. k = 0, these basis functions are all step functions that follows from Eq. (1). N i, 0 (u) = 1 is only in the i-th knot span 〈u i , u i + 1 ). For example, if we have four knots u 0 = 0, u 1 = 1, u 2 = 2 and u 3 = 3, knot spans 0, 1 and 2 are 〈0, 1), 〈1, 2) and 〈2, 3), and the basis functions of degree 0 are N 0, 0 (u) = 1 on interval 〈0, 1) Acta In this co...