An Early Cretaceous mammal initially interpreted as a metatherian; later comparative analysis supported a eutherian placement, illustrating uncertainty in fossil assignment. ^[ref]

A Middle Jurassic mammaliaform with aquatic adaptations and preserved integument. The fossil supplies a minimum occurrence for a fur-bearing covering, not the exact origin date of endothermy. ^[ref]

The surface flux of a uniformly heated cylinder is $Ha/2$. Equating it to the Newton-cooling flux $\alpha [T(a)-T_{\mathrm{air}}]$ gives the displayed surface temperature. The full temperature profile adds the internal parabolic excess to this surface value. Electrical power per length equals the surface heat loss per length in steady state.

The fraction of an alloy that solidifies through the eutectic reaction, equal to the remaining liquid fraction immediately before that reaction in a simple equilibrium solidification path. It differs from the total fraction of either solid phase, and later precipitation can change the phase inventory.

A member of the theropod lineage of dinosaurs, within which birds are nested. The hands and integument of nonavian relatives inform comparisons of avian origins.

A family of fractional quantum Hall effect fillings obtained by attaching $2p$ flux quanta and filling $q$ effective Landau levels of composite fermions. Positive integers $p,q$ and the relative effective-field direction specify the branch.

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.

Colatitude is the angle from the positive polar axis, usually denoted $\theta$ with $0\le \theta \le \pi$. On a sphere it complements latitude $\lambda$: $\theta =\pi /2-\lambda$. Thus the north pole, equator and south pole have colatitudes zero, $\pi /2$ and $\pi$.

With longitude $\varphi$, the standard spherical coordinates are $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$. The surface element is $r^2\sin \theta d\theta d\varphi$, which shows why an integral over the sphere must include the sine weight. Some disciplines exchange the names of $\theta$ and $\varphi$, so formulas should accompany symbol conventions.

Longitude is an angular coordinate around a chosen polar axis, measured from a reference half-plane. On Earth the conventional reference is the Greenwich meridian. In a Cartesian coordinate convention it is $\varphi =atan2(y,x)$, with a specified range such as $(-\pi ,\pi ]$.

Together with latitude and a radius it describes a point using spherical coordinates. Longitude is undefined at the poles because every meridian meets there. Its numerical jump at the chosen meridian cut is a coordinate discontinuity, not a discontinuity of the sphere itself. On an oblate reference ellipsoid, geographic latitude needs an additional geodetic convention.

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.