Proceedings Volume 2010 (format .pdf) - SimpBTH

quicker translocation of the raw sap and assimilates, ensuring a faster flow of theminerals and leading to an increase of the chlorophyll content. It also stimulatesplant-rooting, seed and pollen germination, shoot growth and proliferation, flowerfertility and fruit development; it stimulates the multiplication of micro-organismsin the soil, accelerates the decomposition of the organic matter, indirectlyimproving soil fertility. Atonik is different from the hormonal stimulators, avoidingthe tissue over-saturation in water.Immediately after using Atonik, the fungicide Ronilan 50 DE is used forcontrolling grapevine diseases. The former is used for growth stimulation, the latterfor destroying different fungus-caused diseases (downy mildew, powdery mildew,grey rot).We are interested by to determine the optimal quantities of two factors:Atonik and Ronilan, so the main shoots reach a well-balanced growth obviouslyinfluencing on the grape yield. We try to solve this problem, applying themultivariate approximation methods. For this, we use the papers indicated inreferences with [2] and [6].A general problem of two dimensional stochastic approximation can beformulated mathematically as it follows:We consider an experiment and two families of random variables{ Yi( x)}, i = 1, 2 , (1)with corresponding distribution functions:Fi( Y ( x)), i = 1, 2 , (2)the regression functions:and the variances:+∞Mi ( x) = ∫ YdFi( Y ( x)), i = 1, 2 , (3)2iWe suppose there exists:−∞( ) 2i( ) ( )uniquely determined, by θ1 θ2 θsM1( x)and M2 ( )σ x = E Y − M x , i = 1, 2(4)θs[ θ1,θ2]+ = and M ( θ ) M ( θ )= , (5)= , with the means1 1 2 2x are strictly monotone increasing, continuous and theintersection of their ranges is an interval of positive length.Sequential sets of two experiments are to be performed in the following manner.To initial levels, we consider a vectorX = X , X , (6)[ ]1 11 21329