Motivated by the heat flow and bubble analysis of biharmonic mappings, we study further regularity issues fourth order Lamm–Rivière system $$\begin{aligned} \Delta ^{2}u=\Delta (V\cdot \nabla u)+\mathrm{div}(w\nabla u)+(\nabla \omega +F)\cdot u+f \end{aligned}$$ in dimension four, with an inhomogeneous term f which belongs to some natural function space. We obtain optimal higher sharp Hölder co...

Lindenmayer systems (L-systems) are a grammar system that consists of string rewriting rules. The rules replace every symbol in parallel with successor to produce the next string, and this procedure iterates. In stochastic context-free L-system (S0L-system), may have one or more rule, each an associated probability selection. Properly constructed been found be useful for modeling simulating som...

conclusions our present study shows that sugar beet molasses is a source of plp and l-ser; and hence, suggests the general application of the approach to produce l-trp more economically. results consequently, the potassium phosphate buffer system was considered as the suitably buffered reaction medium. the concentration of l-trp produced in reaction medium was 0.3 mm and 0.12 mm for glucose and...