Lance D. Drager

On the geometry of the smallest circle enclosing a finite set of points

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.

Global observability of a class of nonlinear discrete time systems

Abstract It is shown that discrete time systems whose dynamics are governed by piecewise monotone maps of the interval are globally observable under a general class of observations. The results rest on the fact that the dynamics are ergodic.

Global Observability of Flows on the Torus: An Application of Number Theory

Kroneckers 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.

Non-Linear Delay Differential Equations and Function Algebras

Publisher Summary This chapter describes some results concerning the qualitative properties of bounded solutions of the nonlinear (scalar) delay differential equation. The chapter extends these results and places them in a general framework by considering certain subalgebras of the algebra of bounded continuous functions on the line.

Observability of permutations, and stream ciphers

We study the observability of a permutation on a finite set by a complex-valued function. The analysis is done in terms of the spectral theory of the unitary operator on functions defined by the permutation. Any function f can be written uniquely as a sum of eigenfunctions of this operator; we call these eigenfunctions the eigencomponents of f. It is shown that a function observes the permutation if and only if its eigencomponents separate points and if and only if the function has no nontrivial symmetry that preserves the dynamics. Some more technical conditions are discussed. An application to the security of stream ciphers is discussed.

Observing ergodic translations with discontinuous functions: An example in global observability

Doug McMahon was killed in a climbing accident in late 1986. He was a member of the Department of Mathematics at Arizona State University. The first two authors feel fortunate to have known Doug and to have worked with him. We would also like to thank Cris Brynes and Clyde Martin for getting us together with Doug.

Global Observability of Ergodic Translations on Compact Groups

Harmonic analysis is used to show that an ergodic translation on a compact abelian group is observed by almost all continuous scalar functions.

Relations between ergodicity and observability

Aspects of global observability (GO) for some standard examples of chaotic systems are considered, the underlying philosophy being that such systems should be easy to observe and should be observed by large classes of scalar functions. Consideration is given to GO for piecewise monotone maps of the interval (a standard example of a chaotic system). Attention is then given to GO for ergodic translations on compact (metrizable) topological groups; the most familiar examples are the irrational translation flows on the n-torus.

Some Results on Non-Resonant Non-Linear Delay Differential Equations

We study the non-linear delay differential equation x′(t) + g(x(t), x (t−τ))= f(t) under a non-resonance condition which assures the existence of a unique bounded solution. Using the algebra structure of the space of bounded continuous functions we investigate the properties of this solution. We discuss some generalizations and the initial value problem.

Asymptotics of Numerical Methods for Nonlinear Evolution Equations

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).