A shearing box solves the shearing sheet equations in a finite domain with periodic azimuthal and vertical faces and a shearing-periodic boundary condition on its radial faces. It follows a small region rotating at a reference orbital frequency. The model can sustain stress and exchange energy with its imposed differential rotation, but its identified radial faces do not represent a global disk mass sink.

A sheaf of topological rings takes values in topological rings, with continuous restriction maps. For an open cover, the sheaf condition is an equalizer of the product of sections on the cover and the product on pairwise overlaps, in the category of topological rings. The structure sheaf of a formal scheme is an inverse limit of discrete structure sheaves with this topology.

// Target: algebra.bigb

A rational section is a section on a dense open subset, with two representatives identified when they agree on a common dense open subset. Intersecting representatives' domains defines addition. Multiplication by a similarly represented rational function on an algebraic variety makes these classes a module over the ring $\mathrm{Rat}(X)$. For an irreducible variety, this ring is its function field.

A local system is a locally constant sheaf of modules, for example the integral cohomology of fibres of a smooth proper family of compact complex manifolds. Parallel continuation around loops gives its monodromy representation.

For sheaves of modules $\mathcal{F},\mathcal{G}$ on a ringed space, the internal Hom sheaf assigns to an open $U$ the module of sheaf morphisms $\mathcal{F}{|}_U\to \mathcal{G}{|}_U$ respecting ${\mathcal{O}}_U$. Compatible morphisms glue, so this is a sheaf. Its global sections are precisely the global module-sheaf morphisms. For a locally free sheaf $\mathcal{F}$ of finite rank it is ${\mathcal{F}}^{\vee }\otimes \mathcal{G}$.

For a normal vector $\xi$, this identity defines the corresponding shape operator using the vector-valued second fundamental form. It is self-adjoint because of the symmetry of the second fundamental form. Differentiating $⟨\xi ,Y⟩=0$ shows that $A_{\xi }X=-(D_X\xi {)}^{\top }$. In higher codimension there is a different operator for each normal direction.

For a steady, slowly varying rotating channel, negligible transverse velocity and zero shallow-water potential vorticity give $u_y=f$. Cross-channel geostrophic balance gives $g{h}_y=-fu$, producing the displayed profiles. The specific energy $E=h+u^2/(2g)$ is constant across the channel, and $E+H$ is constant along it. Zero potential vorticity is not the value in an ordinary finite-depth reservoir at rest in the rotating frame; it requires vanishing absolute vorticity or a deep-reservoir approximation.

For an upstream irrotational current on an f-plane in a rigid layer of height $h$, a circular region of height $h(1-ϵ)$ has the displayed relative vorticity if its fluid retains upstream potential vorticity $f/h$. To leading order in the height contrast, take $u=-{\Psi }_y$, $v={\Psi }_x$ and $\Psi =-Uy+\psi$. Solving the Poisson equation for $\psi$ gives $\psi =-C{r}^2/2$ inside and $\psi =C{a}^2[\log (a/r)-1/2]$ outside, where $C=fϵ/2$. The disturbance is clockwise for positive $fϵ$. Variable-depth transport and the transition across the rim introduce higher-order corrections to this leading columnar inversion.

A set-valued mapping $F:X\rightrightarrows Y$ is upper hemicontinuous at $x$ when every open set $U$ containing $F(x)$ also contains $F(x^{\prime })$ for all $x^{\prime }$ in some neighborhood of $x$. Empty values are permitted. For a single-valued map this is the usual neighborhood definition of continuity.

If $Y$ is compact and the graph of a set-valued mapping is closed, upper hemicontinuity follows without a sequential-countability assumption. For each $y\in Y\setminus U$, the point $(x,y)$ is outside the closed graph, so there is a product neighborhood $N_y\times V_y$ disjoint from it. Finitely many $V_y$ cover the compact set $Y\setminus U$. Intersect their $N_y$: every nearby value is then contained in $U$.

Compact control is essential for this implication. The graph of $f(x)=1/x$ for $x\ne 0$, $f(0)=0$, is closed in ${\mathbb{R}}^2$, but the corresponding singleton-valued map is not upper hemicontinuous at zero. Its nearby outputs escape every bounded neighborhood of zero.

The union of a sequence of sets: an element belongs to it exactly when it belongs to at least one $A_n$.

A finite union is a union of finitely many sets. The indexing set is finite; its member sets need not be finite sets. The union of an empty indexed family is the empty set.

The countable intersection of a sequence of sets contains exactly the elements belonging to every set. De Morgan's laws identify its relative complement with the union of their complements.

Two sets are disjoint when their intersection is the empty set: $A\cap B=\varnothing$.

A finite intersection is an intersection of finitely many sets. The indexing set is finite, regardless of the size of the intersected sets. For an empty indexed family of subsets of a specified ambient set $X$, the convention is that the intersection equals $X$.

A real-valued set function $F$ on an algebra of sets is finitely additive whenfor disjoint members of the algebra. Taking both sets empty gives $F(\varnothing )=0$, and induction extends the identity to every finite disjoint union.

On the full power set of a finite set $V$, every subset is a disjoint union of singletons, soThus arbitrary signed weights on the points describe exactly these finitely additive set functions. Summing the pointwise indicator function identity for the inclusion-exclusion principle against those weights gives the corresponding identity for $F$.

The full-power-set qualification matters: on the smaller algebra $\{\varnothing ,\{a,b\}\}$, the value $F(\{a,b\})$ does not specify $F(\{a\})$ or $F(\{b\})$, because the singletons are outside the domain. One must not apply the singleton formula to undefined values.

A queue serves customers in arrival order. For multiple classes, retaining this order generally requires an ordered multiclass single-server queue state, rather than only a vector of class counts.

A nonempty server divides its capacity equally among all customers present. With class-$r$ exponential service parameter ${\mu }_r$, population $n_r$, and total population $n$, the class departure rate is ${\mu }_r{n}_r/n$. The resulting class-count process is a continuous-time Markov chain.

In an ordered queue, let ${\gamma }_i(n)$ be the fraction of service capacity assigned to position $i$ when $n$ customers are present. A symmetric discipline inserts an arrival into position $i$ with probability ${\gamma }_i(n+1)$. The same position rule for service and insertion makes transitions satisfy detailed balance for the word weights ${\prod }_i{\alpha }_{c_i}/{\mu }_{c_i}$. It accommodates class-dependent exponential service rates; processor sharing is an example.

For modular forms $f,g$ of weights $k,\ell$, respectively, the displayed combination cancels the anomaly in derivative transformation of a weak modular form and is a weight-$k+\ell +2$ cusp form. Equivalently, for weights $k,\ell$ in the order $f,g$, a conventional first bracket is $kfDg-\ell gDf$; its negative is equally modular. Differentiation kills each constant Fourier coefficient, proving cusp vanishing.