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For a set partition $\mathcal{P}$ of the vertices of a graph of order $n$, definewhere the sum is ordered, the adjacency indicator defines the density also on diagonal pairs, and exceptional vertices are treated as singleton parts. Refinement cannot decrease this energy, by the Cauchy-Schwarz inequality. An irregular pair of equal cells of size $m$, with witnesses of relative size at least $\epsilon$ and density discrepancy greater than $\epsilon$, increases the energy by more than ${\epsilon }^4{m}^2/n^2$ for each orientation. Splitting refined atoms into equal chunks and leftover singleton parts is a further refinement, so restoring equitability causes no energy loss.

Two compact connected symplectic surfaces are symplectomorphic exactly when their genus of a surface and symplectic area agree. The sufficiency follows from Moser theorem applied to a positive interpolation of their symplectic forms after an orientation-preserving diffeomorphism.

If a stretched variable is $x=ϵr$ and an outer expansion contains ${ϵ}^2\log x$, its expression in the fixed-$r$ region is ${ϵ}^2\log r+{ϵ}^2\log ϵ$. The parameter logarithm must therefore be present in the inner expansion, often multiplying a homogeneous solution selected by a boundary condition. An ansatz containing only integer powers with parameter-independent coefficients misses this switchback term. The same reasoning applies to other powers of $ϵ$ and other stretched-coordinate scalings.

For an island in a constant-depth ocean with Sverdrup balance, choose a counterclockwise contour running from the island's southern tip east to the basin wall, north along that wall, west to the northern tip, and south along the island's western coast. If that contour avoids dissipative boundary layers, the ocean transport streamfunction on the island isHere $\mathbf{W}$ is the wind acceleration appearing in the depth-averaged momentum equation. The basin-boundary streamfunction is zero. This integral fixes the otherwise undetermined constant on the island without requiring a friction law, as in Godfrey's original island-circulation model.

An ocean transport streamfunction defines depth-integrated horizontal transport. For constant depth $H$, it is $H$ times the velocity streamfunction. Its difference between two boundaries gives signed transport across a connecting line.

For signed training labels, a linear soft-margin support vector machine minimizeswith $C>0$. Its Karush-Kuhn-Tucker conditions give $w={\sum }_i{\alpha }_i{y}_i{x}_i$, ${\sum }_i{\alpha }_i{y}_i=0$, and $0\le {\alpha }_i\le C$.

For any field $F$ and vector subspace $W\subseteq F^m$, some $u\in W$ has at least $\dim W$ nonzero coordinates. Choose $u$ with largest support of a vector $S$. Restriction from $W$ to $F^S$ is injective: a nonzero vector in its kernel of a linear map vanishes on $S$, so adding it to $u$ would enlarge $S$. Thus $\dim W\le |S|$. The argument works over finite fields, where assuming a generic vector avoids finitely many hyperplanes would be invalid.

For ${\psi }_1=\cos (\pi x)\sin (\pi z)$, the first temperature mode is ${\theta }_1=(2\pi {)}^{-1}\sin (\pi x)\sin (\pi z)$. The streamfunction advection bracket is $J({\psi }_1,{\theta }_1)=(\pi /4)\sin (2\pi z)$, independent of $x$. The second-order equations therefore have ${\psi }_2=0$ andThis modifies the vertical temperature gradient. Its interaction with the critical roll produces the cubic term in the amplitude equation.

An SU(2) matrix is a two-by-two complex unitary matrix with determinant one. It has the form $\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)$ with $|a{|}^2+|b{|}^2=1$. Its action on the Riemann sphere is a rotational Möbius transformation; opposite matrices induce the same sphere rotation.

The units in the structure sheaf of a scheme form a multiplicative sheaf of abelian groups. On a smooth algebraic curve, a nonzero rational function is a unit at a point precisely when its discrete valuation there is zero. Its inclusion in the sheaf of nonzero rational functions on an irreducible variety relates divisor on an algebraic curve to the Picard group through the long exact sequence in sheaf cohomology.

For a continuous semimartingale $X$ and a sufficiently smooth $f$, the Stratonovich integral obeys $f(X_t)-f(X_0)={\int }_0^t{f}^{\prime }(X_s)\circ d{X}_s$. In general $f^{\prime }(X)$ is a semimartingale, so the definition must allow this larger class of integrands.

The latent-to-sensible heat ratio is the reciprocal of the Stefan number defined by $\mathrm{Ste}=c_p\Delta T/L$. Large $S$ means that phase change uses much more energy than changing temperature by $\Delta T$.

When the applied inner torque dominates the advective angular-momentum flux, the constant-kinematic viscosity inner profile resembles a decretion disk. This resemblance does not alone imply outward mass transport.

The content vector of a standard Young tableau $T$ is $(c_T(1),\dots ,c_T(n))$, where $c_T(r)$ is the Content of a Young-diagram cell containing $r$. Such vectors are exactly the integer vectors satisfying: the first coordinate is zero; each subsequent coordinate has an earlier neighbor differing by one; between two occurrences of $a$ both $a-1$ and $a+1$ occur. To reconstruct the tableau, insert each entry in the next available cell on its prescribed diagonal. The neighbor conditions supply its required predecessors, so each insertion is an addable node of a Young diagram.

For a unital associative algebra $A$ and prescribed $A$-bimodule $M$, an extension is $0\to M\to E\to A\to 0$ where $E$ is unital, $E\to A$ is unital, $M$ is a square-zero ideal, and the induced actions are the prescribed ones. Equivalences induce the identity on $A$ and $M$. A unital linear section produces a normalized Hochschild cocycle $\mu (a,b)=s(a)s(b)-s(ab)$. Conversely, $(a,m)(b,n)=(ab,an+mb+\mu (a,b))$ on $A\oplus M$ defines the extension. Changing section changes $\mu$ by a coboundary, giving classification by $H{H}^2(A,M)$.

Over an algebraically closed field, a smooth projective rational curve is isomorphic to ${\mathbb{P}}^1$. Its line bundles are determined by degree, its degree-zero line bundle is trivial, and $H^1({\mathbb{P}}^1,\mathcal{O})=0$. An integral projective curve of arithmetic genus zero is automatically smooth and rational by the normalization genus formula.

For $N\ge 2$ backlogged stations, the largest one-slot success probability is $(1-1/N{)}^{N-1}$, attained at $p=1/N$. Its limit is $e^{-1}$; the exact finite-$N$ maximum is larger, and for $N=1$ it is one.

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.