A standard platinum resistance thermometer uses very pure, annealed platinum with strain-limiting construction. An industrial platinum resistance thermometer commonly uses supported wire or a film. Their response curves differ.^[ref]

The word standard describes the instrument's metrological role. Calibrated SPRTs realize specified interpolation intervals of the International Temperature Scale of 1990 and transfer reference temperatures to other thermometers.^[ref] They are primary standards for realizing that practical scale, but their electrical resistance alone does not determine absolute thermodynamic temperature without calibration. Primary thermometry such as the corrected gas thermometer obtains thermodynamic temperature from a physical law. A reference in a comparison bath may be an SPRT; an individually calibrated industrial platinum resistance thermometer can also serve as a working reference with an appropriate uncertainty.

The plane-parallel Cowling displacement-pressure equations admit this stellar oscillation for every compatible equilibrium density and adiabatic sound speed profile. In the upward-displacement convention its horizontal amplitude is $i\mathbf{k}\xi /k$, its Eulerian pressure amplitude is $\rho g\xi$, and its Lagrangian density and pressure perturbations vanish. Its displacement decays into the star but grows upward without bound, so a pure linear continuation to infinite height is not physical.

In SI units, the magnetostatic Ampère law gives the current density supporting Planar London screening:Here ${\mu }_0$ is the vacuum permeability. The sign follows from $(\nabla \times \mathbf{B}{)}_y={\partial }_z{B}_x$ in the chosen coordinates. Thus $B_0>0$ requires electric current along $-\hat{\mathbf{y}}$. Its own magnetic field opposes the imposed magnetic field inside the material. The electric current is distributed over a finite layer; its depth integral is the effective surface current densityThe units are $\mathrm{A}{\mathrm{m}}^{-2}$ for $\mathbf{J}$ and $\mathrm{A}{\mathrm{m}}^{-1}$ for $\mathbf{K}$. Shrinking ${\lambda }_L$ at fixed $B_0$ increases the surface current density, while this integrated surface current density stays fixed. The current sheet approximation is the large-length-scale limit of this layer, not a claim that an actual sample carries an infinitely thin electric current. It reproduces the magnetic-field jump across a surface current between the exterior $B_0\hat{\mathbf{x}}$ and a field-free interior, with outward unit normal $-\hat{\mathbf{z}}$.

A photon hitting a resting mass $m$ has squared total energy invariant $m^2{c}^4+2m{c}^{2}E$. Producing a larger mass $M$ and an outgoing photon requires this invariant to exceed $M^2{c}^4$ if the photon has positive energy. The displayed threshold is therefore the limiting infimum as the final photon becomes soft, not an attained physical state with a zero-energy photon. Above threshold the outgoing photon energy in the centre-of-momentum frame is positive.

For a photon incident on a stationary electron, producing an additional electron and positron leaves three massive particles in the final state. Four-momentum conservation gives $W^2=m_e^2{c}^4+2m_e{c}^2{E}_{\gamma }$ for the centre-of-momentum energy. Its minimum is $3m_e{c}^2$, so the incident threshold is $4m_e{c}^2$. At threshold all final particles are at rest in their centre-of-momentum frame, which moves at $4c/5$ relative to the target laboratory frame. This differs from a heavy-target threshold because the electron target's recoil cannot be neglected.

A damped mode $\delta =e^{-k^2/k_D^2}(A\cos q+B\sin q)$ does not alone imply oscillations in its ensemble power spectrum. Averaging its modulus squared gives the displayed formula. Equal independent initial variances, $P_A=P_B$ and $P_{AB}=0$, erase the oscillatory variance through ${\cos }^{2}q+{\sin }^{2}q=1$. A coherent initial phase, as supplied by a regular growing adiabatic mode, selects one common oscillator phase and retains acoustic peaks. Its leading density and velocity oscillations are a quarter-period apart, and angular projection also preserves their correlated contribution.

An ionization-bounded nebula contains enough absorbing material to trap the relevant ionizing photons, unlike a density-bounded region that ends before its ionization front. In dust-free hydrogen at steady photoionization equilibrium, the absorbed stellar photon rate equals the integrated Case B recombination rate. Escape, absorption by dust, and competition from other species must be accounted for before identifying the absorbed hydrogen rate with the entire stellar ionizing luminosity.

For phase $kx+mz-\omega t$ with $k\ne 0$, hold $z$ fixed: a contour of constant wave phase advances in $x$ at $\omega /k$. This is the speed at which a tilted wavefront crosses a horizontal line. It differs from the horizontal component of the vector phase velocity, $\omega k/(k^2+m^2)$, which is defined along the wavevector.

In a smooth interior region with locally constant shear ${\Omega }^{\prime }$, a wound azimuthal Fourier mode has radial wavenumber $k_s\simeq -m{\Omega }^{\prime }t$. Integrating its resistive damping rate $\eta k_s^2$ gives the exponential decay in the title. The characteristic time is ${\tau }_c=[3/(\eta m^2{\Omega }^{\prime 2}){]}^{1/3}$. For $m=1$ and ${\Omega }^{\prime }=-{\Omega }_0/d$, ${\Omega }_0{\tau }_c=(3\mathrm{Rm}{)}^{1/3}$, where $\mathrm{Rm}={\Omega }_0{d}^2/\eta$. This is a high-magnetic Reynolds number interior transient on a time long compared with the rotation time but short compared with $d^2/\eta$. Physical magnetic field components may have additional algebraic factors from taking derivatives of the wound magnetic vector potential. A maintained exterior field, an axial core and a circulation-edge boundary layer prevent interpreting this estimate as an exact, uniform, infinite-time solution.

The equilibrium pressure of a vapor in contact with its liquid or solid at a specified temperature. For pure liquid–gas phase coexistence with negligible interface effects, it is the pressure on their phase coexistence curve. In a gas mixture, distinguish the component's partial pressure from the total pressure; the pure-substance Integrated Clausius-Clapeyron relation does not describe the total pressure of arbitrary humid air.

A triple point is a temperature–pressure pair at which three distinct bulk thermodynamic phases of a one-component substance coexist in thermodynamic equilibrium. In the familiar solid–liquid–gas case, their three phase coexistence curves meet. Other combinations, such as two crystal polymorphs and a liquid, also qualify: “triple” counts thermodynamic phases, rather than states-of-matter categories.

For the same substance, the chemical potential in each thermodynamic phase must agree:There are two independent equality conditions for two controls, temperature and pressure. Generically they select an isolated pair $(T_t,p_t)$; the third pairwise equality follows from the first two. Equivalently, the phase rule gives $F=1-3+2=0$. This counts independent intensive controls, not phase fractions: changing the amounts of the three thermodynamic phases need not change the ideal bulk temperature and pressure, provided all three remain present. Different sets of crystal polymorphs can give different triple points for the same substance.^[ref]

At a liquid-gas critical point, the two fluid thermodynamic phases instead merge, and their mass density difference and latent heat vanish. It is the endpoint of the liquid–gas phase coexistence curve, rather than a meeting of three distinct thermodynamic phases. The regular independent-equality count used for a triple point cannot explain this merger. A triple point can retain finite mass density and entropy differences between all three thermodynamic phases.^[ref]

A hyperphantom map of spectra is a stable map $p:D\to E$ such that $pu=0$ for every $u:{\Sigma }^{-j}{\Sigma }^{\infty }X\to D$, for arbitrary based spaces $X$ and all nonnegative $j$. Equivalently it induces zero on the entire represented cohomology theory. This is a stronger condition than being a phantom map of spectra, which tests only finite spectra.

Choose one representative of each finite spectrum and form $X={\bigvee }_{(F,u:F\to D)}F$. Let $e:X\to D$ evaluate each summand and let $p:D\to \mathrm{cofib}(e)$. Every map from a finite spectrum factors through its corresponding summand, so $p$ is a phantom map of spectra. If $p=0$, exactness supplies a section of $e$, making $D$ a retract of $X$. Consequently a spectrum that is not a retract of any wedge of finite spectra has a nonzero such phantom.

For a phantom map of spectra $p:D\to E$ and a finite CW complex $K$, Spanier-Whitehead duality identifies $D_k(K)$ with maps from the finite spectrum ${\Sigma }^k({\Sigma }^{\infty }K{)}^{\vee }$. Hence $p_{\ast }=0$ there. An arbitrary CW complex is the filtered union of its finite subcomplexes, and represented homology theory commutes with that filtered union because spheres are compact. Thus $p$ induces zero on the entire represented homology theory.

Put $c_1^2=\gamma p_1/{\rho }_1$ and $v_{A1}^2=B_1^2/({\mu }_0{\rho }_1)$. The mass, momentum and energy jump conditions for a perpendicular magnetohydrodynamic shock factor into the unchanged state $x=1$ andFor the compressive physical branch $x>1$, positive downstream pressure implies $x<(\gamma +1)/(\gamma -1)$ when $\gamma >1$ and the upstream sound speed or Alfvén speed is nonzero. Indeed the polynomial impliesIt then gives $u_1^2>c_1^2+v_{A1}^2$: the upstream flow exceeds the perpendicular fast magnetosonic wave speed. The exactly cold unmagnetized strong-shock limit reaches the upper compression bound.

With $B_x=0$, a fast simple wave in magnetohydrodynamics has constant $u_y$ and $B_y/\rho =K$. Its longitudinal velocity satisfies $d{u}_x/d\rho =\pm c_f/\rho$, and ${\rho }_t+(u_x\pm c_f){\rho }_x=0$. Along this state curve the speed derivative is $\pm [c_f/\rho +K^2/(2{\mu }_0{c}_f)]$, which is nonzero. Compressive profiles therefore exhibit shock formation by characteristic intersection, unlike a linearly degenerate characteristic field.

Average neighboring periodogram ordinates with nonnegative weights summing to one. Let the number $L_T$ of averaged ordinates tend to infinity while their frequency span, of order $L_T/T$, tends to zero. For a smooth spectral density and suitable short-memory or fourth-order dependence conditions, the bias tends to zero and variance is of order $1/L_T$. Smooth second-order spectral density alone does not ensure this: a time-constant random scale multiplying independent white noise has a smooth unconditional spectrum but a nonergodic random limiting local power.

For a centered stationary real Gaussian sequence, let $A_T$ and $B_T$ be its cosine and sine coefficients with normalization $1/\sqrt{\pi T}$ at a nonzero Fourier frequency. Grouping the double covariance sum by lag and extending each lag sum to a whole record leaves at most $2k$ omitted bounded products. The full cosine-sine sum is zero, giving the bound. If $\sum k|{\gamma }_k|<\infty$, the two coefficients become asymptotically uncorrelated, and their limiting joint normal distribution consequently has independent components.

Let $X_t$ be a centered real stationary Gaussian time series with absolutely summable autocovariances and two-sided angular-frequency spectral density $f$. The parent periodogram uses $I_T(\omega )=(2\pi T{)}^{-1}|{\sum }_{t=1}^T{X}_t{e}^{-it\omega }{|}^2$. At an interior positive frequency define the one-sided quantitiesThus the doubling applies to both the statistic and its target spectral density.

For Fourier-grid frequencies ${\omega }_T=2\pi k_T/T\to \omega \in (0,\pi )$ with $f(\omega )>0$, setThen $I_T^{+}({\omega }_T)=A_T^2+B_T^2$. Summability makes their variances tend to $f(\omega )=s(\omega )/2$ and their covariance tend to zero. Joint Gaussianity therefore givesUnder these Gaussian assumptions the fourth moments also converge, so $\mathrm{var}(I_T^{+}({\omega }_T))\to s(\omega {)}^2$. An unsmoothed periodogram is therefore not a consistent estimator of a positive spectral density. At zero and the Nyquist frequency $\pi$, the sine coordinate vanishes and the real-only calibration differs; the interior doubling and two-coordinate argument do not apply there.^[ref]

A state can have no positive-probability return at any positive time, for example when its transition-matrix column is zero. Its return-time set is empty. The convention $\gcd \varnothing =0$ assigns period zero; conventions defining period only for a nonempty return set instead leave it undefined. Such a state must not be called aperiodic merely because its Communicating class is a singleton.