A Feynman integral via higher normal functions

We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of K3 surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the K3 family. We prove a conjecture by David Broadhurst that at a special kinematical point the Feynman integral is given by a critical value of the Hasse-Weil L-function of the K3 surface. This result is shown to be a particular case of Deligne's conjectures relating values of L-functions inside the critical strip to periods.Comment: Latex. 70 pages. 3 figures. v3: minor changes and clarifications. Version to appear in Compositio Mathematic

The NATO III 5 MHz Distribution System

A high performance 5 MHz distribution system is described which has extremely low phase noise and jitter characteristics and provides multiple buffered outputs. The system is completely redundant with automatic switchover and is self-testing. Since the 5 MHz reference signals distributed by the NATO III distribution system are used for up-conversion and multiplicative functions, a high degree of phase stability and isolation between outputs is necessary. Unique circuit design and packaging concepts insure that the isolation between outputs is sufficient to quarantee a phase perturbation of less than 0.0016 deg when other outputs are open circuited, short circuited or terminated in 50 ohms. Circuit design techniques include high isolation cascode amplifiers. Negative feedback stabilizes system gain and minimizes circuit phase noise contributions. Balanced lines, in lieu of single ended coaxial transmission media, minimize pickup

Control and stabilization of systems with homoclinic orbits

In this paper we consider the control of two physical systems, the near wall region of a turbulent boundary layer and the rigid body, using techniques from the theory of nonlinear dynamical systems. Both these systems have saddle points linked by heteroclinic orbits. In the fluid system we show how the structure of the phase space can be used to keep the system near an (unstable) saddle. For the rigid body system we discuss passage along the orbit as a possible control manouver, and show how the Energy-Casimir method can be used to analyze stabilization of the system about the saddles