Implicit descriptions of piecewise‐linear configurations Available to Purchase

By a polyhedral or piecewise‐linear configuration in Rⁿ we mean the finite union of finite intersections of open or closed halfspaces. It is proved that every such configuration can be characterized by the single algebraic‐like equation involving only addition, subtraction, multiplication, absolute value and signum operators. Next, all such simplest implicit characterizations are found for idealized diodes, the ideal step function and the hysteresis function. Finally, the one‐dimensional infinite potential trough is analysed and the obtained equations are conjectured to be the simplest possible. All this allows among others to describe systems containing piecewise‐linear elements in a compact form suitable for further analytic studies.