For a slope in a regression with an intercept, the variance inflation factor is $1/(1-R_j^2)$, where $R_j^2$ comes from regressing its predictor on all the others. With one other predictor it is $1/(1-r^2)$, quantifying the increase in conditional estimation variance.

For a full Newtonian liquid film in the lubrication approximation, a cylinder translating parallel to a stationary plane with minimum gap $h_0\ll a$ exerts tangential force $2\pi \mu U\sqrt{2a/h_0}$ per unit axial length on the plane. The leading normal force vanishes by the odd symmetry of excess pressure.

For a sphere falling with speed $U$ through a tube of radius $a$, let $Q$ be the downward laboratory volume flux. In the sphere frame the wall moves upwards and the narrow annulus carries upward flux per unit circumference $q=(\pi a^{2}U-Q)/(2\pi a)$. Couette-Poiseuille flow in a thin gap gives $q=Uh/2-h^3{p}_z/(12\mu )$. Thus the pressure below minus above the sphere is $\Delta P=6\mu (2q{I}_3-U{I}_2)$, using lubrication resistance integrals for a nearly occluding sphere. A nearly piston-like laboratory flux can coexist with a nonzero relative gap flux.

For a smoothly moving circuit and a divergence-free magnetic field, the magnetic flux derivative is $d\Phi /dt={\int }_S{\partial }_t\mathbf{B}\cdot d\mathbf{S}-{\oint }_C(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{r}$. The minus sign follows by orienting the swept side surface compatibly with the two spanning surfaces. Combining this transport identity with Faraday's law yields the moving-loop electromotive-force law.

A morphism of sheaves of modules is surjective when the induced map on every stalk is onto. Equivalently every target section can locally be lifted. It need not be onto on global sections; the long exact sequence in sheaf cohomology describes that obstruction.

For a continuous map $f:X\to Y$, an f-morphism from a sheaf on $Y$ to a sheaf on $X$ is a family of the displayed maps commuting with restrictions. It is equivalently a morphism of sheaves $\mathcal{G}\to f_{\ast }\mathcal{F}$. For each $x\in X$ it induces a map ${\mathcal{G}}_{f(x)}\to {\mathcal{F}}_x$ on stalks, by pulling back the neighbourhood representing a germ.

A commuting isomorphism between two module extensions that is the identity on both end modules. Classifying middle modules up to arbitrary isomorphism is a different and generally coarser problem.

Pushing an inclusion $i:A\to B$ along $f:A\to A\prime$ gives a new module extension with kernel $A\prime$ and the same quotient. This pushout realizes the covariance of the degree-one Ext functor in its second argument.

With trivial action on both end modules, an extension has underlying abelian group ${\mathbb{Z}}^r\oplus {\mathbb{Z}}^s$ and generator action $\left(\begin{array}{cc}I& T\\ 0& I\end{array}\right)$. The integer matrix $T$ is invariant under equivalences fixing the ends. The one-element projective resolution with differential $z-1$ computes the same classification over the group ring.

In a planar force-driven encounter with a fixed equal sphere and positive impact parameter $b\gg a$, the mobility correction from a fixed distant sphere gives $Y-b=27a^{2}b/[32(X^2+b^2)]+O(a^4/b^3)$. The moving sphere reaches maximum deflection $27a^2/(32b)$ and rotates clockwise through $9\pi a^2/(32b^2)$ to leading order. The spin is obtained by integrating half the reflected Stokeslet vorticity, as prescribed by Faxén rotation law. These numerical formulas require a distant encounter.

For a finite Markov chain with transition matrix $P$, define $\bar{d}(t)={\max }_{x,y}\Vert P^t(x,\cdot )-P^t(y,\cdot ){\Vert }_{\mathrm{TV}}$. If $\pi$ is a stationary distribution and $d(t)={\max }_x\Vert P^t(x,\cdot )-\pi {\Vert }_{\mathrm{TV}}$, then $d(t)\le \bar{d}(t)\le 2d(t)$. The first bound follows by writing $\pi ={\sum }_y\pi (y)P^t(y,\cdot )$ and using the triangle inequality; the second uses the triangle inequality through $\pi$.

If $r={\max }_{j\ge 2}|{\lambda }_j|$ for a finite irreducible Markov chain with at least two states, then its worst-case total variation distance from its stationary distribution satisfies $d(t)\ge r^t/2$. Choose a possibly complex eigenfunction $f$ for an eigenvalue of modulus $r$, normalize ${\max }_x|f(x)|=1$, and start where the maximum is attained. Since $\pi (f)=0$, $|P^{t}f(x)-\pi (f)|=r^t$; the defining sum for total variation distance bounds this difference by $2d(t)$. For $0<r<1$, this yields $t_{\mathrm{mix}}(\epsilon )\ge (r/(1-r))\log (1/(2\epsilon ))$ for $0<\epsilon <1/2$, because $-r\log r\le 1-r$.

With $E=T-U<0$ and $T\ge 0$, the positive binding energy satisfies $U\ge |E|$. The minimum-separation binding-energy bounds imply $r\le A_0/|E|$. At least one pair stays close; this does not bound every particle's distance or exclude a negative-energy gravitational collision.

At fixed outer boundary velocity, extending the inclusion velocity rigidly through its interior makes the particle flow an admissible competitor for the particle-free Minimum-dissipation theorem for Stokes flow. Its interior rate-of-strain tensor vanishes. For a linear background, the increase in viscous dissipation is force times relative translation, torque times relative rotation, plus the contraction of the particle stresslet tensor with background strain.

For $X\sim \mathrm{Bin}(n,\theta )$, the posterior mean under $\mathrm{Beta}(\sqrt{n}/2,\sqrt{n}/2)$ has constant squared-error risk $1/[4(\sqrt{n}+1{)}^2]$. Its unique Bayes property and constant risk make it the unique minimax estimator.

The flux $F(p)=p/\sqrt{1+|p{|}^2}$ of the minimal surface equation for a graph has a positive definite symmetric matrix derivative. Its eigenvalues are $(1+|p{|}^2{)}^{-3/2}$ in the direction of $p$ and $(1+|p{|}^2{)}^{-1/2}$ orthogonally; at $p=0$ every eigenvalue is one. On $|p|\le M$ it satisfiesThus an averaged linearization of a nonlinear divergence-form equation is uniformly elliptic where the relevant gradients are bounded. Pointwise positivity alone does not supply a uniform bound on an unbounded gradient range.

The isothermal coordinates $(u,v)$ give the immersion $(u-u^3/3+u{v}^2,v-v^3/3+u^{2}v,u^2-v^2)$. Its coordinate functions are harmonic and its metric is $(1+u^2+v^2{)}^2(d{u}^2+d{v}^2)$, so it is a complete immersed minimal surface with self-intersections.

An outer area-minimizing surface is a closed surface in a Riemannian initial-data slice whose area is no greater than that of any enclosing competitor. This is a variational property, not a consequence of containment alone. It is important when choosing the horizon-area quantity in the Riemannian Penrose inequality and when examining an apparent-horizon area comparison.

Two successive integrations by parts of the standard normal density tail give positive remainder integrals establishing these strict inequalities for every $x>0$. The lower bound can be negative at small $x$ without invalidating the inequality.

A pseudo-marginal Metropolis–Hastings algorithm replaces an intractable likelihood function by a nonnegative unbiased likelihood estimator and includes its auxiliary randomness in the state. If $\mathbb{E}[\hat{L}(\theta ,U)]=L(\theta )$ for $U\sim m_{\theta }$, the extended target is proportional to $p(\theta )\hat{L}(\theta ,u)m_{\theta }(u)$. An independence proposal for new auxiliary randomness cancels the $m_{\theta }$ factors in the Metropolis–Hastings acceptance probability. On rejection the old estimate must be retained. Integrating the extended target gives the desired marginal posterior distribution exactly.