High order curvature flows of plane curves with generalised Neumann boundary conditions

We consider the parabolic polyharmonic diffusion and the L 2 {L^{2}} -gradient flow for the square integral of the m-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L 2 {L^{2}} , then the evolving curve converges exponentially in the C ∞ {C^{\infty}} topology to a straight horizontal line segment. The same behaviour is shown for the L 2 {L^{2}} -gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.

XXXVI. A new method of constructing sun-dials, for any given latitude, without the assistance of dialing scales or logarithmic calculations.

Draw the straight horizontal line Bad Tab. XVI. (fig. 1.) of any convenient length, and on the end D thereof raise the perpendicular DE. Bisect Bad at A, and draw the right line Ace, making the angle Ead equal to the latitude of the place for which the dial is to serve, as suppose 51° ½ for the latitude of London.