For power spectral density $P(f)=C/(f^2+f_c^2)$, the low-frequency plateau is $P_0=C/f_c^2$ and the high-frequency asymptote is $C/f^2$. Their intersection on a log-log plot gives the displayed corner frequency. Equivalently $P(f_c)=P_0/2$. Use the range unaffected by additional low-frequency noise or high-frequency filtering; a measurement chain can otherwise shift an apparent corner.

For the same massless spring obeying Hooke's law, the frequency is $\nu =(2\pi {)}^{-1}\sqrt{k/m}$. The displayed mass ratio follows. For small independent frequency errors with common fractional standard uncertainty $ϵ$, monomial uncertainty propagation gives ${\sigma }_R/R\simeq \sqrt{8}ϵ$. If $ϵ$ instead describes individual measurements and each sample mean uses $n$ independent trials, its standard error is smaller by $\sqrt{n}$, giving ${\sigma }_R/R\simeq \sqrt{8/n}ϵ$. Four times as many trials halve this statistical uncertainty; a common calibration error needs its covariance included.

A quantum Hall edge state is confined across a boundary but can extend along it and carry electric current. B. I. Halperin related these current-carrying boundary states to quantized Hall transport in a disordered two-dimensional system.^[ref]

For filled, separated, noninteracting magnetic Bloch bands, the Berry curvature integral over the magnetic Brillouin zone is an integer Chern number $C$, giving $|{\sigma }_{xy}|=|C|e^2/h$. Patches of Berry connection meet through phases with integer windings; continuous gap-preserving changes cannot change $C$. D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs developed this band argument; it does not establish the interacting fractional case.^[ref]

Choose conventional surface current density along $+x$, magnetic field along $+z$, and negative electron charge. An isotropic in-plane response has the following sheet conductivity and sheet resistivity tensors for $s\ge 0$, $k>0$:These are matrix inverses, rather than componentwise reciprocals. The sheet conductivity has units siemens, the sheet resistivity ohms. With $K_y=0$, the sign is $E_y/K_x<0$, agreeing with the negative Hall coefficient of a single electron species. Contact polarity can reverse the measured voltage sign, so quote the Hall magnitude. For a homogeneous strip of width $W$ and longitudinal probe separation $L$, $I=W{K}_x$ givesAt $s=0$, $k=N$, these reduce to $R_K/N$ and zero. A nonzero transverse sheet resistivity creates no local Joule heating: the antisymmetric contribution cancels in $\mathbf{E}\cdot \mathbf{K}=(e^2/h)s|\mathbf{E}{|}^2$.

For an original nonideal example, choose $k=2$, $s=0.01$ and $L/W=2$. Then $V_{xx}/I\simeq 129.061\Omega$, while the Hall magnitude is below $R_K/2$ by the relative amount $s^2/(k^2+s^2)\simeq 25.0$ parts per million. An unwanted admixture $0.02V_{xx}$ in the Hall readout would instead introduce approximately $2.58\Omega$. The chosen calculation shows why a small intrinsic tensor correction does not by itself guarantee an accurate contact measurement.

The degeneracy of a Landau level is $eBA/h$ per resolved internal branch in area $A$. The filling factor $f=n_{s}h/(eB)$ varies continuously. Additional localized eigenstates need not carry bulk transport, allowing an integer plateau over an interval of $f$; a mobility gap need not be an empty spectral gap.^[ref]

The plot is an original phenomenological construction, not measured data or a calculation of Anderson localization. It stipulates dimensionless transverse sheet conductivity $k(f)=1+\frac{1}{2}{\sum }_{m=1}^5[1+\tanh ((f-m-1/2)/w)]$ and longitudinal sheet conductivity $s(f)=a{\sum }_{m=1}^5{\mathrm{sech}}^2((f-m-1/2)/w)$, with chosen $w=0.06$, $a=0.08$. Both are measured in units $e^2/h$. The quantum Hall conductivity tensor inversion produces $|{\rho }_{yx}|/R_K=k/(s^2+k^2)$ and ${\rho }_{xx}/R_K=s/(s^2+k^2)$. Regions near $k=N$, $s=0$ yield plateaus near $R_K/N$; the model alone does not explain their microscopic robustness. Each step represents one resolved branch, without assuming unresolved spin or valley multiplicities.

Klaus von Klitzing, G. Dorda and M. Pepper reported a silicon MOSFET measurement in 1980. Its gate controlled an semiconductor inversion layer; transverse contacts measured the Hall voltage and separate longitudinal contacts measured dissipation. One sweep held $B=18\mathrm{T}$, $T=1.5\mathrm{K}$ and $I=1\mu \mathrm{A}$ fixed while varying gate voltage. Hall plateaus accompanied small longitudinal voltages. Device-geometry comparisons and high-input-impedance voltage measurements helped test whether contact placement or electrical loading produced the result.^[ref]

This original functional diagram separates transverse and longitudinal readouts; it does not reproduce the historical device's dimensions. The gate inset explains how a field-effect transistor provides an adjustable two-dimensional electron gas. A longitudinal voltage close to zero is a useful check because accidental longitudinal separation between Hall contacts can otherwise contaminate the transverse measurement.

The von Klitzing constant is $R_K=h/e^2=25812.807459\dots \Omega$. The current SI fixes the Planck constant $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ and elementary charge $e=1.602176634\times 10^{-19}\mathrm{C}$ exactly, so $R_K$ is exact; the displayed decimal is truncated. It differs from the historical conventional value $R_{K\text{-}90}=25812.807\Omega$.^[ref]

As an original ideal example, $N=2$ and electric current $I=1\mu \mathrm{A}$ give $|V_H|=I{R}_K/2\simeq 12.9064\mathrm{mV}$. Changing the Hall bar's width changes the transverse electric field required for this electric potential difference, while the ideal ratio $|V_H|/I$ remains $R_K/2$.

On a regular level set $F(x,y,z)=0$, holding $z$ fixed gives $F_{x}dx+F_{y}dy=0$. If $F$ is continuously differentiable and both $F_x$ and $F_y$ are nonzero, the implicit function theorem gives $(\partial y/\partial x{)}_z=-F_x/F_y$ and $(\partial x/\partial y{)}_z=-F_y/F_x$. Multiplication proves the identity. The same held-fixed variable is essential, and a zero denominator invalidates the corresponding quotient rather than creating an infinite factor that may be cancelled formally.