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">$( \color{#FF0000}v^{\color{#000000}(\color{#FF0000}\alpha\color{#000000})}\color{#000000},\color{#FF0000}i^{ \color{#000000}(\color{#FF0000}\beta\color{#000000})}\color{#000000})$</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">$(\color{#FF0000}v\color{#000000},\color{#FF0000}i\color{#000000})$</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">$(\color{#FF0000}v^{ \color{#000000}(\color{#FF00FF}\alpha\color{#000000})}\color{#000000},\color{#FF0000}i^{\color{#000000}( \color{#FF00FF}\beta\color{#000000})}\color{#000000})$</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">$ \omega$</tex></formula> and tends to a straight line as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\omega \rightarrow\infty$</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.