When backlog $N$ and attempt-control scale $S$ are large with $N/S\approx \kappa$, idle and success probabilities approach $e^{-\kappa }$ and $\kappa e^{-\kappa }$. Expected update increments then motivate a fluid model described by ordinary differential equations.

For each existential first-order formula $\exists y\phi (y,\mathbf{x})$ in a structure, a Skolem function selects a witness when one exists, with an arbitrary default when none exists. An expansion containing these functions satisfiesA subset closed under all such functions is elementary in the original language by the Tarski-Vaught test. Simultaneously choosing these functions is performed in the ordinary classical metatheory with choice.

Under edge adjacency, a skew Young diagram is totally disconnected when every component has one cell. This means that no two cells share an edge. It is stronger than being a horizontal strip, which excludes repeated columns but permits adjacent cells in one row. The trivial-constituent criterion for a skew representation of a symmetric group concerns horizontal strips, not total disconnection in this sense.

A horizontal strip is a skew Young diagram containing at most one cell in each column. Equivalently, $\lambda /\mu$ is a horizontal strip when ${\lambda }_i\ge {\mu }_i\ge {\lambda }_{i+1}$ for all $i$. The multiplicity of the trivial representation in the associated skew representation of a symmetric group is one for horizontal strips and zero otherwise. A column with two cells supplies a cyclic tableau vector on which an adjacent transposition acts by $-1$, excluding invariants.

A standard skew Young tableau fills the $k$ cells of a skew Young diagram bijectively with $1,\dots ,k$, increasing along each row and column. Equivalently it is a linear extension of a partially ordered set formed from the row and column inequalities. Interchanging consecutive labels preserves standardness exactly when their cells are incomparable in that partial order.

For the intersection of $m$ null hypotheses, reject when ${\min }_{i}m{p}_{(i)}/i\le \alpha$. The Simes inequality establishes a level-$\alpha$ test under independence of valid true-null p-values. The displayed quantity is its combined p-value.

A nonempty shortest conjugacy representative is a cyclically reduced word. If $w=sv{s}^{-1}$ with first and last letters inverse, $v$ is a shorter word representing a conjugate. Consequently $w^n$ has length exactly $n|w|$ as a freely reduced word, and any segment of the periodic word of length at most $|w|$ fits in a cyclic rotation of $w$. This observation turns a Dehn shortening segment crossing copy boundaries into a shortening of a conjugacy representative.

For limit ordinals $\alpha \le \beta$, the structures $V_{\alpha }$ and $V_{\beta }$ agree on whether any shared set is a cardinal number. They agree on being an ordinal. If $\delta <\alpha$ is not a cardinal number, a bijection from some $\gamma <\delta$ onto $\delta$ has rank of a set at most $\delta +3$, hence belongs to $V_{\alpha }$. Its defining properties are bounded formulas in set theory. Both structures therefore see the same failure witness. This full agreement is stronger than downward absoluteness of cardinalhood between arbitrary transitive models.

Given a coherent sheaf $\mathcal{F}$ and an ample Cartier divisor $A$ on a projective scheme, one integer $a_0$ ensures $H^i(X,\mathcal{F}(aA+N))=0$ for all $i>0$, all $a\ge a_0$, and every nef divisor $N$. The uniformity in $N$ is stronger than Serre vanishing. Fujino's note on Fujita vanishing states the theorem for arbitrary projective schemes over a field.

For ample line bundles $L,B$ and a coherent sheaf $\mathcal{F}$ on a projective scheme, one $M$ gives $H^i(X,\mathcal{F}\otimes L^m\otimes B^t)=0$ for every $i>0$, $m\ge M$ and $t\ge 0$. If $B$ is very ample, Serre vanishing for the finitely many fixed twists $\mathcal{F}(-iB)$ makes $\mathcal{F}\otimes L^m$ zero-regular. Persistence of Castelnuovo–Mumford regularity gives all nonnegative $B$-twists. For general ample $B$, use a very ample power and treat finitely many residues of $t$.

For a compact complex $n$-manifold and a holomorphic vector bundle $E$, analytic Serre duality givesThe pairing integrates the wedge product of bundle-valued Dolbeault cohomology representatives, contracting $E$ with its dual.

A sublevel set consists of points whose objective value is at most a chosen threshold. Coercivity makes finite sublevel sets bounded, while sequential lower semicontinuity makes them sequentially closed sets in the selected topology.

For $\varphi >0$ and nonnegative variances, $\gamma (h)={\tau }^2{1}_{\{h\ne 0\}}+{\sigma }^2(1-e^{-{\varphi }^2\Vert h{\Vert }^2})$ admits covariance $C(h)={\sigma }^2{e}^{-{\varphi }^2\Vert h{\Vert }^2}+{\tau }^2{1}_{\{h=0\}}$. The Gaussian kernel is positive definite because it is the Fourier transform of the density of a Gaussian distribution. Adding independent white noise supplies the nugget effect. Its scale is $a=1/\varphi$. For ${\sigma }^2>0$, its exact range of a semivariogram is infinite. If ${\sigma }^2=0$, it is a pure nugget effect with no nontrivial structured range.

If $W(s)$ is stationary, then $Z(s)=W(s)-W(0)$ has exactly the same increments and semivariogram, but $\mathrm{Var}Z(s)=2{\gamma }_W(s)$ is generally nonconstant. Thus existence of a stationary covariance model for a semivariogram does not imply stationarity of every process with that semivariogram. Adding an independent random constant also leaves the semivariogram unchanged while changing the covariogram.

The nugget is the discontinuity of a distance semivariogram at zero. Independent location-specific noise of variance ${\tau }^2$ contributes covariance ${\tau }^2$ at zero displacement and zero covariance at nonzero displacement, so its semivariogram is ${\tau }^2{1}_{\{h\ne 0\}}$.

The sill is the large-distance level of a semivariogram when that limit exists. For covariance tending to zero, the sill is the total marginal variance; a nugget plus a structured component has total sill ${\tau }^2+{\sigma }^2$. A nonzero constant covariance component is invisible to the semivariogram.

The exact range is a separation beyond which the semivariogram reaches its sill. A Gaussian semivariogram with nonzero structured variance approaches its sill asymptotically and therefore has infinite exact range. Software range parameters may instead specify a scale.

For a distance bin $B$, the empirical semivariogram is $\hat{\gamma }(B)=(2|B|{)}^{-1}{\sum }_{(i,j)\in B}(z_i-z_j{)}^2$. A nonconstant drift adds half its squared pairwise differences to the expected raw semivariogram. Residual semivariograms estimate spatial dependence after adjusting for the drift.

For a stationary internal gravity wave, the Taylor–Goldstein equation gives $m^2=N^2/U^2-U^{\prime \prime }/U-k^2$. In a propagating interval with slowly varying $m$, the WKB approximation has amplitude proportional to $m^{-1/2}$ and phase derivative $m$. The conditions $|m^{\prime }|\ll m^2$ and $|m^{\prime \prime }|\ll |m{|}^3$ fail at a regular turning level. With $U>0$, the upward branch has $m>0$ and negative intrinsic frequency; laboratory group velocity is $U(k^2,km)/(k^2+m^2)$ when the curvature term is negligible.

For a harmonic function in the quadrant with boundary derivatives $u_y(0,y)=g_1(y)$ and $u_x(x,0)=g_2(x)$, integrating the data to equal corner values and applying the Schwarz integral formula after the conformal map $w=z^2$ givesThese are tangential boundary derivatives, so uniqueness requires an additional normalization or growth restriction. A homogeneous contribution has the form $ih(z^2)/z$ with $h$ holomorphic in the complex upper half-plane and real on its boundary away from zero. For example, $u=xy$ has zero prescribed derivatives on both edges.