\input ncs445hdr
% this is nh1cs445.tex of January 6, 1983 8:55 AM

\def\rtitle{Homework 1}
\def\header{Homework 1}
\setcount1 1

\sect{First Problem Set, due Thursday, January 13}

{\caveat The following two problems use the informal language of car trips for talking about convolutions.}\par

\prob A car going around a circle can be moving in two directions, clockwise ($-$) or counterclockwise ($+$), and independently, it can be facing clockwise ($-$) or counterclockwise ($+$). We indicate these four possibilities by $++, +-, -+$ and $-{-}$, where the first sign refers to the direction of motion and the second to the direction the car is facing. By convention, in pictures that we draw we represent motion direction by an arrowhead, and facing direction by a solid triangle. Now consider convolving two circles, a big one of radius $R$, and a little one of radius $r$ ($R>r$). There are sixteen possible situations, depending on the (direction of motion, direction of facing) state on each circle. The outcome of one particualr case is shown below. Give a table that has the results of the convolution in all sixteen cases and defend your answers.\par

\vskip 1.5in

\prob Draw on a piece of graph paper the convolution of the parabola $y=x↑2$ traversed twice (once east-to-west and once west-to-east) with a circle of radius 4 oriented counterclockwise. In both cases the traversing car is facing in the direction it is moving, and in the case of the parabola it turns around at infinity by rotating counterclockwise. Attempt, within reasonable time limits, to draw your picture quite accurately and to give coordinates for points on the convolution that you consider especially interesting. (This may require you to remember from analytic geometry the formula for the curvature of a curve at a point.) Don’t forget to specify the {\it direction the car is facing} while it traverses the convolution.\par

{\caveat The next three problems elaborate various properties of convolution and the multiplication of tracings we discussed.}\par

\prob Let $\Ascr$, $\Bscr$, and $\Cscr$ be three polygonal tracings so that each pair of two distinct ones of them is transverse. Show that then the pairs $\Ascr\ast\Bscr$ with $\Cscr$, and $\Ascr$ with $\Bscr\ast\Cscr$ are also transverse. We took care in the definition of convolution to guarantee that it is a commutative operation: $\Ascr\ast\Bscr = \Bscr\ast\Ascr$. But is it also {\it associative\/}? That is, do we have $(\Ascr\ast\Bscr)\ast\Cscr = \Ascr\ast(\Bscr\ast\Cscr)$? Give a proof or a counterexample.\par

\prob The convolution notion arose out of considering paint brushes moving along trajectories in the plane, so as to generate solid figures. Actually the framework that we have is flexible enough to allow us also brushes that function as {\it erasers}. Describe this phenomenon in more precise mathematical terms using our framework and give one or more examples where it occurs.\par

\prob In the main theorem on convolutions, namely that $\omega(p, \Ascr\ast\Bscr) = \delta(\Ascr\cdot\Bscr↑{p-I})$, the left hand side is symmetric in $\Ascr$ and $\Bscr$, but not the right hand side. How do you explain this anomaly?\par

\vfill\eject\end