The Fourth Element

This tutorial clarifies the axiomatic definition of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">( #FF0000v^#000000(#FF0000α#000000)#000000,#FF0000i^ #000000(#FF0000β#000000)#000000)</tex></formula> circuit elements via a lookup table dubbed an A-pad, of admissible <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">(#FF0000v#000000,#FF0000i#000000)</tex></formula> signals measured via Gedanken probing circuits. The <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">(#FF0000v^ #000000(#FF00FFα#000000)#000000,#FF0000i^#000000( #FF00FFβ#000000)#000000)</tex></formula> elements are ordered via a complexity metric. Under this metric, the memristor emerges naturally as the fourth element, characterized by a state-dependent Ohm's law. A logical generalization to memristive devices reveals a common fingerprint consisting of a dense continuum of pinched hysteresis loops whose area decreases with the frequency <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">ω</tex></formula> and tends to a straight line as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">ω →∞</tex></formula> , for all bipolar periodic signals and for all initial conditions. This common fingerprint suggests that the term memristor be used henceforth as a moniker for memristive devices.