We construct trivariate macro-elements of class Cr for any r ≥ 1 over the Alfeld refinement of any tetrahedral partition in R3. In our construction, the degree of polynomials used for these macro-elements is the lowest possible. We also give the dimension formula for the subspace of consisting of these macroelements.

Macro-elements of smoothness C r on Clough-Tocher triangle splits are constructed for all r 0. These new elements are improvements on elements constructed in 11] in that (disproving a conjecture made there) certain unneeded degrees of freedom have been removed. Numerical experiments on Hermite interpolation with the new elements are included. A bivariate macro-element deened on a triangle T con...

Macro-elements of smoothness C r on Powell-Sabin triangle splits are constructed for all r 0. These new elements are improvements on elements constructed in 10] in that certain unneeded degrees of freedom have been removed. A bivariate macro-element deened on a triangle T consists of a nite dimensional linear space S deened on T, and a set of linear functionals forming a basis for the dual of S...

Nowadays, the most rapid method for producing healthy and disease-free Lisiantus is micropropagation. With respect to high economic value of this plant which is regarded among the 10 top cutting flowers in the world, this research was carried out to suggest a suitable protocol for its in vitro propagation, using nodal sections as an explant. To carry out this object, the effects of the pH (5.5,...