For a constant-depth resting layer on an f-plane, the depth-scaled linear potential vorticity anomaly isIt satisfies $q_t=0$ under the linearized shallow water equations. The physical perturbation of $(f+\zeta )/(H+\eta )$ is $q/H$. The signed scalar $q$, not its absolute value, is used in potential-vorticity inversion and balanced energy pairings.

A set of pairwise incomparable nodes in a set-theoretic tree. A maximal tree antichain has a comparable member for every node of the tree.

A maximal chain with nodes at unbounded heights in a set-theoretic tree. Taking predecessor closure gives one node at each level below the tree height.

A set-theoretic tree with one root, extensions of every node at every higher level, at least two immediate successors for every node, and distinct limit-level nodes distinguished by their predecessor chains. If splitting occurs only at later levels, a continuous cofinal selection of levels can enforce immediate splitting.

A set-theoretic tree of height ${\omega }_1$ with countable levels and no cofinal branch. More generally, a $\kappa$-Aronszajn tree has height a regular uncountable $\kappa$, levels of size less than $\kappa$, and no cofinal branch.

A set-theoretic tree of height ${\omega }_1$ with countable levels and at least ${\aleph }_2$ distinct cofinal branches.

For a regular uncountable cardinal number $\kappa$, a set-theoretic tree of height $\kappa$ with nonempty levels of cardinality less than $\kappa$. The regular-height convention is important for the unbounded-extension kernel of a regular tree.

Every node has an extension at every higher level below the height of the set-theoretic tree. Equivalently, its extension heights are unbounded in the set-theoretic tree height. Having no terminal nodes alone is weaker.

At a limit level, two nodes with the same predecessor at every smaller level must be equal. This is a uniqueness condition; it need not provide an upper node for every cofinal branch through the lower levels.

For a map $f:X\to B$, let $E_f$ consist of points of $X$ and paths starting at their images, with projection $p(x,\gamma )=\gamma (1)$. Appending segments of a base homotopy proves the homotopy lifting property, so $p$ is a Serre fibration. The constant-path inclusion $s:X\to E_f$ satisfies $ps=f$ and is a homotopy equivalence: forget the path for an inverse, and shorten paths to prove the homotopy. The homotopy need not preserve the endpoint in $B$, so the statement concerns an equivalence of total spaces over a commuting triangle, not a fibrewise inverse. The fibre over $b_0$ is the homotopy fibre of $f$.

For a Serre fibration with this fiber over a simply connected CW base, the fiber identity class has only one possible nonzero outgoing differential in the cohomological Serre spectral sequence, landing in $H^{n+1}(B;G)$. If that group vanishes, the class extends to the CW total space. Its representing map, paired with projection to $B$, is an isomorphism on fiber and base homotopy groups and hence a weak homotopy equivalence to the product.

The class of all finitely generated abelian groups has the closure properties needed for the Hurewicz theorem modulo a Serre class. In a simply connected space, degreewise membership of homotopy groups and integral homology groups in this class is equivalent.

A semigroup presentation specifies generators $X$ and equalities between nonempty words. The resulting semigroup is the quotient of the free semigroup $X^{+}$ by the smallest congruence containing these equalities. Two words are equal precisely when one can be transformed into the other by finitely many replacements in contexts, using the defining equalities in either direction.

A smooth zero-interior-potential vorticity mode hasDecay for $z\to \infty$ gives $\hat{\psi }=C{e}^{-mz}$. Its penetration depth $1/m$ increases with horizontal wavelength and decreases with buoyancy frequency. Singular neutral interior sheets are a distinct continuous-spectrum class.

For basic shear $U(z)=\Lambda z$ with a resting lower boundary, the rigid-boundary buoyancy condition for quasi-geostrophic waves and the decaying vertical exponential giveThis is real. The edge wave propagates in the same laboratory direction as the basic flow above its resting lower wall. If the basic buoyancy is $M^{2}y+N^{2}z$, then $\Lambda =-M^2/f_0$.

For a forced sphere and a distant equal force-free, torque-free sphere, the leading incident Stokeslet strain is proportional to $(\mathbf{U}\cdot \mathbf{n})(I-3\mathbf{nn})$. Its reflected stresslet is axisymmetric about the line of centers and has zero vorticity on that line. The nominal fifth-order self-spin coefficient therefore vanishes. Finite-size incident-strain corrections and higher force-free scattering multipoles first supply a generic seventh-order angular velocity. Purely longitudinal forcing has zero spin exactly by axial symmetry. The leading translation correction is fourth order for a generic orientation, but sixth order for purely transverse forcing. These cancellations matter when using the method of reflections for Stokes flow to infer orders from far-field decay alone.

For an oriented closed surface smoothly embedded in an oriented four-manifold, a transverse section of the normal bundle gives a homologous small push-off. Its signed zeros are exactly the signed intersections with the original surface. This identifies the evaluation of the Euler class with the self-intersection number.

With orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge dt$, ${ϵ}_{123}=1$, and temporal gauge $A_t=0$, the mixed gauge curvature is $F_{ti}={\partial }_t{A}_i$. The self-dual Yang-Mills equations then have the displayed form. Placing $dt$ first in the orientation reverses the sign, so the metric alone does not specify it.

For a strictly feasible primal-dual conic optimization problem and a logarithmically homogeneous barrier, the central path consists of solutionsThe primal-dual gap is $⟨s,y⟩=\nu \mu$. Existence requires appropriate feasibility and boundedness hypotheses, rather than merely a full-rank constraint matrix. Linearizing these equations gives a central-path Newton system.

For a nondegenerate self-concordant barrier, the open local Hessian ballis contained in its barrier domain. This is the Dikin ellipsoid. It supplies a quantitative way to ensure that a damped Newton method stays interior, without relying on Euclidean distance to a possibly curved boundary.