A long wire with uniform current density $I/(\pi a^2)$ generates uniform heat per volume $H=I^2/({\pi }^2{a}^4\sigma )$. Radial steady conduction obeys $(r{T}_r{)}_r=-Hr/k$. Regularity at the centre removes a logarithmic solution, leaving the displayed parabolic temperature excess relative to the surface. Conductivity, electrical conductivity and current density are assumed constant.

A living representative of the rhynchocephalian lineage. Including this branch helps test alternatives for turtle placement rather than equating all lepidosaur diversity with lizards.

For dilute, independently migrating carriers of number density $n$, charge $q$ and diffusion coefficient $D$, balancing drift with diffusion gives mobility $D/(k_{B}T)$ per applied force. Multiplying drift velocity by charge density yields $\sigma =n{q}^{2}D/(k_{B}T)$. State which species defines $n$ and $D$: a vacancy diffusivity must be combined with vacancy density. Correlated motion requires a correction rather than automatic use of a tracer diffusivity.

The activation free-energy barrier for an ion to hop between sites. ionic conductivity also depends on the concentration of mobile defects and correlations between successive hops.

With the explicitly stated convention $H_R=D_{\mathrm{tr}}/D_{\sigma }$, the tracer and collective diffusion coefficients are related by $D_{\sigma }=D_{\mathrm{tr}}/H_R$. The Nernst-Einstein relation then gives $\sigma =n{q}^2{D}_{\mathrm{tr}}/(H_R{k}_{B}T)$. State the convention, since reciprocal definitions can otherwise cause confusion.

The orbital speed near transit differs from the circular value by approximately $(1+e\sin \omega )/\sqrt{1-e^2}$. At otherwise fixed geometry, duration is shortened by the inverse of this factor. Without eccentricity and periapsis constraints, a measured duration need not select one unique impact parameter.

Wick rotation of the free-line propagator gives the displayed coincident-endpoint kernel for positive Euclidean duration $T$. Equivalently it is the path integral over Dirichlet fluctuations about the straight classical path, with its time-sliced Gaussian measure included. It is a propagator density, not a full thermal partition function: that trace additionally integrates position and, on a multiply connected configuration space, sums the winding sectors. For an angular coordinate replace $M$ by moment of inertia.

An emergent description of correlated electrons in which an even number of statistical flux quanta is attached to each electron. The resulting effective field can turn selected fractional electronic fillings into integer effective fillings. Flux attachment encodes correlations; it is not attachment of literal external magnetic solenoids to individual electrons.

For uniform superconducting carrier density, the displayed combination of enclosed magnetic flux and the circulating supercurrent is quantized on a closed contour where the order parameter does not vanish. It follows by integrating the gauge-invariant phase gradient. Magnetic flux alone is quantized only when the current contribution on that contour is negligible.

For an instantaneous rotation about $x$ followed by free precession about $z$, the just-before-kick one-period operator is $U=e^{-i\Omega T{S}^z}{e}^{-i\Phi S^x}$. Sampling just after kicks reverses the displayed order. The spin generators are dimensionless and the pulse area $\Phi$ is an angle.

A two-solid microstructure formed by the coupled transformation of liquid at a eutectic point. Lamellae and rods are common morphologies. A hypoeutectic alloy can contain both primary crystals and a eutectic microstructure; those microconstituents are not the same accounting categories as its total phase fractions.

The invariant liquid-plus-two-solids equilibrium of a binary eutectic system at fixed pressure. It specifies both the eutectic temperature and the eutectic composition.

The transformation $L\to \alpha +\beta$ in which one liquid produces two solid phases at a binary eutectic composition and temperature under fixed pressure. The resulting two-phase constituent must be distinguished from either individual thermodynamic phase.

With $H_0=-\hslash {\omega }_0{\sigma }_z/2$ and a co-rotating transverse magnetic drive of phase $\phi$, an initial spin-up state becomes, up to common phase, $\cos (\Omega \tau /2)|\uparrow ⟩+i{e}^{-i(\phi +{\omega }_0\tau )}\sin (\Omega \tau /2)|\downarrow ⟩$. A quarter-turn pulse therefore prepares laboratory $+x$ when ${\phi }_{\mathrm{end}}=\pi /2$ and $+y$ when ${\phi }_{\mathrm{end}}=0$. Dropping the free-precession phase instead specifies targets in rotating axes. Transverse laboratory targets subsequently precess if the static field is retained.

For an initial pole of the Bloch sphere, one fixed-amplitude harmonic pulse detuned by $\Delta$ transfers at most ${\Omega }^2/({\Omega }^2+{\Delta }^2)$ of the population. A pure equatorial state requires half transfer, so some pulse duration reaches the equator exactly when $|\Delta |\le \Omega$. Adjusting drive phase then selects its transverse azimuth. Exact inversion to the opposite pole is unavailable for nonzero detuning with that single fixed pulse. This is a restriction on the specified controls, rather than on composite-pulse or frequency-swept state preparation.

For a rotating-frame two-state Hamiltonian operator $H=\hslash \mathbf{W}\cdot \mathbf{\sigma }/2$, the Bloch vector obeys $\dot{\mathbf{R}}=\mathbf{W}\times \mathbf{R}$. Thus $\mathbf{W}$ specifies a rotation axis and angular speed. With detuning defined as drive frequency minus transition frequency and the lower state at positive $R_z$, a phase-zero transverse drive can give $\mathbf{W}=(\Omega ,0,\delta )$. Changing basis or detuning convention requires changing the corresponding signs consistently.

Take a linear polarizer $P=\mathrm{diag}(1,0)$ and a reciprocal quarter-wave plate $Q=2^{-1/2}\left(\begin{array}{cc}1& i\\ i& 1\end{array}\right)$ in fixed physical transverse axes. A near-normal isotropic mirror has $M=-I_2$ up to global phase. If the source/detector side encounters P before Q, the round-trip matrix is $PQMQP=-P{Q}^{2}P=0$. Reversing the filter gives $QPMPQ=-QPQ$, which transmits half an initially unpolarized intensity. The result is double-pass rejection, not optical nonreciprocity; changing direction without changing the input polarization does not preserve the same physical input state.

A digital repeater can recover the symbol sequence and generate new pulses with restored amplitude, shape and timing. This resets waveform degradation between spans, provided decisions remain reliable. An amplifier alone increases the broadened pulse amplitude and also adds noise; it does not undo group velocity dispersion. optical dispersion compensation can instead reverse an accumulated frequency-dependent delay within its design bandwidth.

Different ensemble members accumulating different deterministic phases can shorten their averaged Bloch vector even when each member remains in a pure state. Averaging static frequency offsets gives an effective dephasing channel. It need not represent irreversible phase loss in each individual system; an appropriate refocusing pulse can reverse static offsets. Changing offsets during the sequence requires a temporal noise model.