Chan Wang

Point orthogonal projection onto a spatial algebraic curve plays an important role in computer graphics, computer-aided geometric design, etc. We propose an algorithm for point orthogonal projection onto a spatial algebraic curve based on Newton’s steepest gradient descent method and geometric correction method. The purpose of Algorithm 1 in the first step of Algorithm 4 is to let the initial iteration point fall on the spatial algebraic curve completely and successfully. On the basis of ensuring that the iteration point fallen on the spatial algebraic curve, the purpose of the intermediate for loop body including Step 2 and Step 3 is to let the iteration point gradually approach the orthogonal projection point (the closest point) such that the distance between them is very small. Algorithm 3 in the fourth step plays an important double acceleration and orthogonalization role. Numerical example shows that our algorithm is very robust and efficient which it achieves the expected and ideal result.

This paper examines optimal monetary policy rules in a model of vertical production and trade with reference currency. As evidenced by empirical findings, we assume that final-goods prices are sticky, but intermediate-goods prices are flexible. We find that even if intermediate-goods prices are flexible, monetary authorities need to respond to the shocks at the stage of intermediate-goods production. We also find that, when a shock occurs at the stage of final-goods production, monetary responses are independent of the expenditure share of final-goods producers on intermediate goods. For the first time in the literature, our model gives a condition under which both countries are willing to participate in monetary cooperation. Thus the gains from cooperation are real. In addition, we compare the volatility of the nominal exchange rate in Nash case with that in cooperative case, and compare the volatility of the nominal exchange rate in our model with that in a model without vertical production and trade as well. We also extend the model to consider a case of dual price stickiness. We find that the change in solution methods completely alters the conclusions of the model.