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Attach two statistical flux quanta ${\Phi }_0=h/e$ per electron at density $n_s$. For $B>0$, $B^{\ast }=B-2n_s{\Phi }_0=B(1-2\nu )$ and the signed effective filling is ${\nu }^{\ast }=n_s{\Phi }_0/B^{\ast }=\nu /(1-2\nu )$. Inversion gives $\nu ={\nu }^{\ast }/(2{\nu }^{\ast }+1)$. Integer ${\nu }^{\ast }=p$ gives $p/(2p+1)$, while ${\nu }^{\ast }=-p$ gives $p/(2p-1)$. Using nonnegative effective filling instead requires explicitly tracking the effective-field sign. At half filling the mean effective field vanishes.

Let $D=t_1+t_2$, $a=t_1{\rho }_L+t_2{\rho }_N$ and $b=t_1{\rho }_H+t_2{\rho }_N$. A square of four layers has total thickness $2D$. For parallel moments its channel sheet resistances are $a/(2D^2)$ and $b/(2D^2)$; for antiparallel moments both are $(a+b)/(4D^2)$. Adding parallel spin conductances gives the title. The factor of four-layer thickness is essential; changing the normalization changes the prefactors. The nonmagnetic bulk resistivity would be ${\rho }_N/2$ when ${\rho }_N$ is this model's per-spin value.

From the four-layer CIP sheet resistance, $R_P=ab/[2D^2(a+b)]$ and $R_{AP}=(a+b)/(8D^2)$. Subtraction gives $R_{AP}-R_P=(a-b{)}^2/[8D^2(a+b)]\ge 0$ for positive channels. The ratio in the title normalizes by the low-resistance state; normalizing by the high-resistance state instead gives $(a-b{)}^2/(a+b{)}^2$. Common spacer scattering adds equally to $a,b$, diluting the contrast. The result assumes identical magnetic layers; arbitrary materials need not have the same sign of spin asymmetry.

The Bose-gas phase-only action gives a thermal infrared integral $\int d^{d}q/q^2$ and a zero-temperature equal-time integral $\int d^{d}q/|q|$. Thus an infinite uniform short-range interacting gas has no true finite-temperature condensate in $d\le 2$, and the one-dimensional ground state also lacks true condensate long-range order. This does not exclude local density coherence, a finite-size condensate description or two-dimensional algebraic order and superfluidity.^[ref]

A monoclinic zirconolite polytype with a two-layer repeat. Its zirconium site has sevenfold oxygen coordination.

A monoclinic zirconolite polytype with a four-layer repeat, observed in some actinide-substituted composition ranges.

A Middle Triassic stem relative of turtles combining modified ribs and ventral skeletal elements with temporal anatomy supporting a diapsid origin.

On either branch $q_1\ne 0$, all local canonical momenta for these coordinates have the form $P_1=(p_1-p_2)/(2q_1)+{\partial }_{Q_1}S$ and $P_2=p_2+{\partial }_{Q_2}S$, with arbitrary smooth $S(Q_1,Q_2)$. The mixed coordinate–momentum brackets fix the linear momentum terms, while the momentum–momentum bracket requires the remaining shift one-form to be closed. It is locally a gradient. The transformation is singular at $q_1=0$ and is not globally one-to-one across both signs of $q_1$.

Solving $p(F_{1/2})=1/2$ for survival under a linear force ramp gives the displayed expression. At large $f/(k_0{F}_0)$, its slope against $\log f$ approaches $F_0$; at low loading rate the median is approximately $f\log 2/k_0$. A full fit can infer activation distance $k_{B}T/F_0$ and zero-force escape rate $k_0$. An absolute barrier additionally needs an attempt prefactor, and equilibrium binding strength additionally needs reverse-state information.

Differentiate the logarithm of the rupture-force probability density $k(F)p(F)/f$. Its interior stationary point satisfies $e^{F/F_0}=f/(k_0{F}_0)$. The physical domain is $F\ge 0$, giving the displayed mode. This differs from the median rupture force under a linear ramp; replacing one by the other shifts a fitted intercept and can bias the inferred zero-force rate.

In the normal skin-effect regime of a good conductor, $\delta =\sqrt{2/(\mu \sigma \omega )}$ and the surface resistance is $R_s=1/(\sigma \delta )=\sqrt{\mu \omega /(2\sigma )}$. For identical geometries, resistance ratios depend on the square root of magnetic permeability divided by conductivity.

A coating substantially thinner than the skin depth does not screen the substrate as a bulk conductor would. Thickness of a few skin depths suppresses the field reaching the substrate exponentially. A unique required thickness additionally needs an allowed residual field or loss criterion.

For potential $V(x)=\alpha x$, momentum follows a characteristic with $\dot{p}=-\alpha$. The retarded quantum-mechanical propagator translates its momentum argument and accumulates the kinetic action along that characteristic. Its apparent division by $\alpha$ is removable on the delta-function support; at zero force it becomes the free diagonal momentum kernel. Fourier transform gives the usual constant-force position kernel with a cubic-time action term.

A positive density curvature $g$ makes uniform static density fluctuations costly. It does not imply an additional independent gapped excitation: the first-order term makes density and phase conjugate. For a simple nonrelativistic weakly interacting Bose gas, their coupled quadratic dynamics yields one positive Bogoliubov spectrum, which is gapless at zero momentum. The word massive for the density field describes its static curvature in this setting.

An event changing the phase winding of a condensate around a loop. A continuous nonzero order parameter cannot change its winding integer; a phase-slip path must pass through a locally vanishing amplitude or an equivalent vortex-crossing process. This use concerns a superfluid field, rather than only the relative phase of two classical oscillators.