Search

A number of numerical codes have been written for the problem of finding the circle of smallest radius in the Euclidean plane that encloses a finite set P of points, but these do not give much insight into the geometry of this circle. We investigate geometric properties of the minimal circle that may be useful in the theoretical analysis of applications. We show that a circle C enclosing P is minimal if and only if it is rigid in the sense that it cannot be translated while still enclosing P. We show that the center of the minimal circle is in the convex hull of P. We use this rigidity result and an analysis of the case of three points to find sharp estimates on the diameter of the minimal circle in terms of the diameter of P.

At the Georgia Institute of Technology, a computer program is used in freshman calculus which graphically illustrates upper and lower Riemann sums and generates values of their differences. The students often observe that the differences $\Delta _n $ seem to be proportional to ${1 / n}$, where n is the number of subdivisions; but this is only approximate.We make this rigorous by showing that \[ \Delta _n = \frac{V} {n} + O\left( {\frac{1} {{n^3 }}} \right)\quad {\text{ as }}n \to \infty ,\] for nice functions, where V is the total variation. The proof is simple, and is a nice illustration of the ideas of asymptotic analysis, and several other techniques of analysis.

Kronecker's theorem is used to show that the irrational flows on then-dimensional torus are globally observed by a large class of continuous functions. These results are used to study the observability of Riccati flows on the Grassman manifolds.

We consider the evolution equation (E) du/dt = au + f(t,u), u(0) = u0 in a Hilbert space H, where A is assumed to generate a C0 semigroup. The asymptotic-in-time behavior of solutions to (E) is studied together with the asymptotic-in-time behavior of methods for approximating (E).