The ratio of stable buoyancy destruction to shear production of turbulent kinetic energy. With down-gradient transport and turbulent Prandtl number ${\mathrm{Pr}}_t$, it obeys ${\mathrm{Ri}}_f={\mathrm{Ri}}_g/{\mathrm{Pr}}_t$, rather than automatically equaling the gradient Richardson number. Unstable buoyancy production gives a negative value in this convention.

On the matchings in a graph, a lazy Markov chain proposes adding an edge, deleting an edge, or exchanging an edge for the unique matching edge incident with it. A proposal incident with two matching edges is rejected. Select the proposed edge uniformly and use the Metropolis–Hastings acceptance probability for the displayed weights. Symmetric proposals give detailed balance; deletions connect every state to the empty matching in a graph. The normalizing constant is the matching generating function.

A polynomial regression birth move can introduce a coefficient $z$ while shifting the intercept by $-cz$, with $c$ the sample average of the new monomial. The fitted-value increment is $z(x^{k+1}-c)$, whose sample average vanishes. The Jacobian determinant of the coefficient transformation is one. A reversible-jump Markov chain Monte Carlo acceptance ratio must still include the normalized dimension-dependent prior distributions, the model-order prior distribution, the birth/death selection probabilities and the proposal density for $z$.

For nested regression models, append one coefficient $u$ drawn from density $g_k$ while retaining the variance $v$ and existing coefficients $a$. The dimension-matching map $(a,v,u)\mapsto (a,u,v)$ has absolute Jacobian one. With joint model/parameter target $t_k$, birth-selection probability $b_k$, and reverse death-selection probability $d_{k+1}$, the displayed acceptance probability and its reciprocal death rule enforce detailed balance. Include normalized parameter-prior densities and a model-order prior in $t_k$: constants that depend on model order cannot be dropped from a cross-model ratio.

To minimize $E_j(\theta )=2k_j-2{\ell }_j(\theta )$ over models with different parameter dimensions, sample from a target proportional to $w_j{e}^{-E_j(\theta )/(2T)}$ and cool $T$. A dimension-matching proposal augments the smaller model by auxiliary variables, transforms them bijectively into the larger parameter vector, and includes the proposal probability density function and absolute Jacobian determinant in the Metropolis–Hastings acceptance probability. For a positive Poisson mean $\lambda$ and normal parameters $(\mu ,v)$, one convenient transformation isIts absolute Jacobian determinant is one. If the forward and reverse model-jump selection probabilities are $b$ and $d$, the Poisson-to-normal ratio isThe reverse ratio is its reciprocal at the inverse map. This enforces detailed balance. When the target is proper and the annealing chain explores it adequately, the low-temperature laws concentrate on the minimum Akaike information criterion, rather than the maximum marginal model evidence. Likelihoods compared across models must describe the same observed data with a compatible reference measure.

For $N◃G$ and $\theta \in \mathrm{Irr}(N)$, set ${\theta }^g(n)=\theta (g^{-1}ng)$. Its inertia group is its stabilizer under this conjugation action and contains $N$. Write $\mathrm{Irr}(G\mid \theta )$ for the irreducible characters of $G$ whose restriction to $N$ contains $\theta$.

An extension of $\theta \in \mathrm{Irr}(N)$ from $N◃G$ is a character $\chi$ of $G$ whose restriction is exactly $\theta$. It is necessarily irreducible, because a nontrivial invariant subspace for $G$ would also be one for $N$. An ordinary extension can fail to exist even when $\theta$ is invariant under $G$.

A central composite design combines a factorial portion with axial points $\pm \alpha e_j$ and center runs. For a full $2^d$ factorial design, $d$ quantitative factors and $n_0\ge 1$ center runs, its run count is $2^d+2d+n_0$. The factorial portion supplies main effects and interaction terms, while the axial runs allow the individual squared terms of a full quadratic regression model to be estimated. Additional center replicates estimate pure error.

For two factors and $n_0=5$, this gives $4+4+5=13$ runs. The four corners are $(\pm 1,\pm 1)$, the four axial points are $(\pm \alpha ,0)$ and $(0,\pm \alpha )$, and the remaining five runs repeat $(0,0)$. Choosing $\alpha =\sqrt{2}$ makes this full-factorial design rotatable. Choosing $\alpha =1$ keeps the axial points on the faces and uses only three levels per factor, but does not give that rotatability in dimension at least two.

Circumscribed designs place the axial points outside the original factorial cube; inscribed designs rescale the cube inward to respect physical factor limits. A fractional factorial portion can reduce runs, but must retain sufficient resolution of a fractional factorial design to estimate the desired interactions: axial points have zero cross-products and cannot separate interactions that remain aliased in the factorial portion. The institutional handbook describes augmentation from a full or resolution V factorial portion.^[ref]

A rotatable design for a specified regression model has variance of a fitted regression mean depending on $x$ only through its Euclidean distance from the design center. It concerns prediction precision, not rotational symmetry of the mean response itself.

For a full quadratic regression model in $d\ge 2$ variables, consider all $2^d$ factorial corners, the $2d$ axial points $\pm \alpha e_i$, and center runs giving a nonsingular information matrix of an experimental design. Sign symmetry cancels odd moments and permutation symmetry makes the second coordinate moments equal. Fourth-order rotational balance requires, for distinct $i,j$,The factorial corners contribute $2^d$ to both sums, axial runs contribute $2{\alpha }^4$ only to the first, and center runs contribute zero. Thus $2^d+2{\alpha }^4=3\cdot 2^d$, givingFor $d=2$, $\alpha =\sqrt{2}$; for $d=3$, $\alpha =8^{1/4}$. Center replication can estimate pure error but does not change this fourth-moment equation. If a fractional factorial portion is used, its own moments and estimability must be checked; simply substituting the original factor count into the full-factorial formula is not justified. In one dimension rotational invariance only means reflection symmetry, so the distinct-coordinate fourth-moment equation does not impose the displayed choice.

A coded experimental variable rescales a physical setting ${\xi }_j$ around its center $c_j$, dividing by a chosen positive half-range $h_j$. The endpoints $c_j\pm h_j$ become $\pm 1$, giving dimensionless factorial design coordinates. The choice of scale affects the metric used for steepest ascent in response surface methodology.

For a fitted local first-order response $\hat{m}(x)={\hat{\beta }}_0+{\hat{b}}^{T}x$, the unit direction maximizing its directional derivative in coded Euclidean distance is $d=\hat{b}/\Vert \hat{b}\Vert$, by the Cauchy-Schwarz inequality. Run sequential feasible settings $x_0+hd$, increase $h$ while measured response improves, and refit near promising settings. Physical increments are $h_{j}h{d}_j$, so coded and unscaled physical gradients need not give the same experimental path. A near-zero gradient calls for further modeling rather than division by zero.

For a two-factor factorial design with coded corners $(\pm 1,\pm 1)$, observe each of the four corners once and add $n_0\ge 2$ independent center runs at $(0,0)$. Assume independent Gaussian errors with common variance ${\sigma }^2>0$. Write ${\overline{Y}}_F$ for the corner average, ${\overline{Y}}_0$ for the center average, and $S_0^2$ for the unbiased center-run sample variance. Then the center-point curvature contrast isFor the full quadratic regression mean ${\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_{12}{x}_1{x}_2+{\beta }_{11}{x}_1^2+{\beta }_{22}{x}_2^2$, the contrast has expected value ${\beta }_{11}+{\beta }_{22}$. The corner average cancels the linear and cross-product terms, while both squared coordinates equal one at every corner and zero at the center.

Under the null ${\beta }_{11}+{\beta }_{22}=0$,The independent normal contrast and center residuals justify this exact F-distribution; a first-order mean model alone would not imply it without the error assumptions. The center observations supply pure error, while the extra corner-to-center comparison detects one aggregate curvature direction.^[ref]

Zero contrast does not exclude curvature. The mean $x_1^2-x_2^2$ has a saddle shape but gives the same mean at all four corners and the center. A central composite design adds axial runs that can distinguish the separate squared terms.

For a complex unital Banach algebra and a circle enclosing its element's Banach algebra spectrum, the displayed identity holds for $n\ge 0$. On a sufficiently large circle, expand the Neumann series and integrate it term by term. At any resolvent point the local series $R({\lambda }_0+h)={\sum }_{j\ge 0}(-h{)}^{j}R({\lambda }_0{)}^{j+1}$ proves norm analyticity. Apply continuous complex linear functionals and the scalar Cauchy theorem to deform the large circle to any radius greater than the maximum spectral modulus; the Hahn-Banach theorem separates points and restores the vector integral identity. Its norm bound $\Vert x^n\Vert \le R^{n+1}{\max }_{|\lambda |=R}\Vert R(\lambda )\Vert$ gives the reverse inequality in the spectral radius formula.

For a closed subalgebra of a Banach algebra $B\subseteq A$ with common identity, the algebra resolvent set ${\rho }_B(x)$ is clopen in ${\rho }_A(x)$. A point in ${\rho }_A(x)$ has a small connected disc on which membership of ${\rho }_B(x)$ is constant, so it cannot lie in $\partial {\sigma }_B(x)$. Also ${\sigma }_A(x)\subseteq {\sigma }_B(x)$; a point in the interior of the former is in the interior of the latter. Together these observations prove the boundary inclusion, without assuming a spectral-boundary theorem.

For solenoidal velocity and magnetic field, with constant magnetic diffusivity $\eta$, the resistive induction equation impliesThe product rule gives $D_{t}P=\mathbf{x}\cdot D_t\mathbf{B}+\mathbf{u}\cdot \mathbf{B}$ and $\Delta P=\mathbf{x}\cdot \Delta \mathbf{B}+2\nabla \cdot \mathbf{B}$. Consequently the radial velocity, through $Q$, is the source term for the radial magnetic field.

// Target: fluid-mechanics.bigb

For a planar magnetic field written as $\mathbf{B}=\nabla \times (A\hat{\mathbf{z}})$ and a circular velocity $\mathbf{u}=s\Omega (s)\hat{\mathbf{\varphi }}$, the resistive induction equation reduces, after choosing the additive gauge transformation, to ${\partial }_{t}A+\Omega {\partial }_{\varphi }A=\eta {\nabla }^{2}A$. A Fourier mode $A=a(s,t)e^{im[\varphi -\Omega (s)t]}$ therefore obeys the equation in the title. Indeed, each radial derivative acting on the exponential replaces ${\partial }_s$ by ${\partial }_s-imt{\Omega }^{\prime }$, while the azimuthal Laplacian contributes $-m^2/s^2$. At zero magnetic diffusivity, $a$ is constant in time, but the physical magnetic field can still be amplified algebraically by differential rotation.

A probe's sheath and leads carry heat between the sensing region and the surroundings through thermal conduction. In a stable, sufficiently uniform heat bath, compare the indicated temperature as insertion depth increases, allowing settling each time. A plateau supports adequate immersion for that apparatus and required uncertainty; it does not prove that every depth or medium works. Keep the sensing element and reference thermometer at comparable depths, and check bath gradients independently, so moving the probe does not merely sample a different temperature. The documented platinum resistance thermometer comparison calibration includes these controls.

Measuring electric current creates Joule heating, raising the sensing temperature. At fixed bath temperature, compare settled readings at $I$ and $\sqrt{2}I$ and extrapolate the approximately linear $I^2$ dependence toward zero current.^[ref]

Under the local model $R(I)=R_{\mathrm{zero}}+k{I}^2$,For synthetic readings $109.629\Omega$ at $1\mathrm{mA}$ and $109.633\Omega$ at $\sqrt{2}\mathrm{mA}$, the extrapolated value is $109.625\Omega$. The lower-current excess $0.004\Omega$ corresponds to $0.01039\dots {}^{\circ }\mathrm{C}$ in the Pt100 local linear model. At $109.625\Omega$, $1\mathrm{mA}$ dissipates $109.625\mu \mathrm{W}$; power alone does not specify a temperature rise without a thermal coupling model. These stipulated values are not measurements.

Repeat the comparison after settling and test the $I^2$ approximation if necessary. Changing from liquid to air changes heat transfer; a self-heating correction need not transfer unchanged. The saturation-pressure equilibrium measurement provides an existing example of controlling this disturbance in resistance thermometry.

A resistance thermometer using platinum as its sensing material. It provides the temperature readout in the saturation-pressure equilibrium measurement.

Transduction turns electrical resistance into a temperature indication. A uniform wire has $R={\rho }_{\mathrm{el}}\ell /A$, with electrical resistivity ${\rho }_{\mathrm{el}}$, length $\ell$, and cross-sectional area $A$. Stronger atomic vibrations at higher temperature scatter electrons more, increasing platinum's electrical resistivity; impurities and crystallographic defects also contribute.^[ref]

A small known electric current $I$ and measured electric potential difference $V$ give $R=V/I$ through Ohm's law; a resistance bridge can instead compare the element with a reference resistor. Inverting the calibrated response then gives the sensing element's temperature. This makes electrical readout convenient, but the platinum resistance thermometer lead resistance error, resistance thermometer self-heating, and resistance thermometer immersion test explain why a stable display alone is insufficient. The gas thermometer supplies a contrasting route through an equation of state; the saturation-pressure equilibrium measurement shows why a trustworthy independent temperature readout matters.

Keep the circular portion of a Hankel contour at fixed nonzero radius. Its numerator $I(z)={\int }_H{t}^{z-1}{e}^{t}dt$ is an entire function of $z$, because the contour avoids zero and its tails decay exponentially, uniformly on compact parameter sets. At a positive integer $n$, the integrand is single-valued and entire in $t$, so the bank integrals cancel and the circular integral is zero. Thus $I(n)=0$, canceling the simple zero of $2i\sin (\pi z)$ in the Hankel representation of the gamma function. At a nonpositive integer $-n$, the circular integral instead gives $2\pi i/n!$, yielding residue $(-1{)}^n/n!$.