For complete lattices, a monotone map $f:A\to B$ preserves arbitrary joins exactly when it has a right adjoint. In that case

A Galois connection between posets consists of monotone maps $f:A\to B$ and $g:B\to A$ satisfying $f(a)\le b$ exactly when $a\le g(b)$; equivalently, $f$ is left adjoint to $g$ in the poset-enriched sense.

If $f:[0,\infty )\to \mathbb{R}$ is continuous and is an additive function, then $f(t)=tf(1)$. Additivity proves this first for nonnegative rational numbers; because $f$ is continuous, the identity extends to every nonnegative real number.

M-separation extends d-separation to mixed graphs: a path is open given a conditioning set when every noncollider is unconditioned and every collider has a conditioned descendant.

A district is a maximal set of vertices connected by a path consisting entirely of bidirected edges.

An additive group is an abelian group whose operation is written as addition, with identity $0$ and inverse of $x$ written $-x$.

A sequence is exact at an object when the image of the incoming morphism equals the kernel of the outgoing morphism. It is exact when this holds everywhere.

With $K_{\mu \nu }=-{\perp }^{\rho }{}_{\mu }{\nabla }_{\rho }{n}_{\nu }$ and $a_{\mu }=n^{\rho }{\nabla }_{\rho }{n}_{\mu }$, spatial projection of ${\nabla }_{\mu }{T}^{\mu \nu }=0$ gives

The Gibbs--Thomson relation shifts the equilibrium chemical potential or composition at a curved interface by an amount proportional to its mean curvature. Small droplets therefore have a larger equilibrium solubility than flat interfaces.

If a four-dimensional Euclidean connection is independent of one coordinate and that connection component is renamed $\Phi$, the self-dual Yang-Mills equation reduces to the three-dimensional Bogomolny-Prasad-Sommerfield monopole equation $B_i=\pm D_i\Phi$.

A multiresolution analysis organizes a function into successively finer approximation spaces. In the usual orthonormal construction on $L^2(\mathbb{R})$, these are closed linear subspaces $(V_j{)}_{j\in \mathbb{Z}}$ withRescaling relates the levels: $f(x)\in V_j$ if and only if $f(2x)\in V_{j+1}$. Integer translates of a scaling function form an orthonormal basis of $V_0$. The detail space is the orthogonal complement of $V_j$ inside $V_{j+1}$, giving $V_{j+1}=V_j\oplus W_j$. Coarse information and new details therefore fit together without overlap in squared L2 norm.

For the Haar wavelet, $V_j$ consists of square-integrable functions constant on cells of length $2^{-j}$. Splitting every cell into two halves creates one new contrast per old cell, the Haar refinement identity. On the bounded example $[0,1]$, the first three spaces have dimensions $1,2,4$: a whole-interval mean, two half-interval means, and four quarter-interval means. The Haar projection diagram displays those three resolutions. This bounded construction starts at level zero; the displayed intersection condition refers to the full family on the real line.

The usefulness is a controlled way to increase resolution. Earlier coarse information remains available while new wavelet coefficients identify what the finer description adds. The wavelet approximation complexity connects this extra resolution to both approximation error and the number of coefficients being estimated. ^[ref]

An orthonormal wavelet $\psi$ generates an orthonormal basis of $L^2(\mathbb{R})$ throughThe factor $2^{j/2}$ compensates for the shrinking support of a function, keeping the L2 norm equal to one. Orthogonality makes different coefficients separate squared-norm contributions, and completeness allows the wavelet series to recover any square-integrable function in the L2 norm.

For the Haar wavelet, wavelets at one scale have disjoint supports. At different scales their dyadic intervals are either disjoint or nested. On the support of the finer wavelet, the coarser one is constant, while the finer one has zero integral, so their L2 inner product is zero. Together with the unit L2 norm, this explains orthonormality. On the bounded interval $[0,1]$, add the constant Haar scaling function and use $j\ge 0$, $0\le k<2^j$; the constant function is not an $L^2(\mathbb{R})$ basis element on the whole line.

A semilinear wave equation has the formso its highest-order part is linear while the forcing may depend nonlinearly on the solution and its lower derivatives.

The d'Alembert operator is the Lorentzian wave operator $\square ={\eta }^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }$ in flat spacetime, or $\square ={\nabla }^{\mu }{\nabla }_{\mu }$ on scalar fields in curved spacetime.

A unitary irreducible representation is both unitary and irreducible. Such representations are the building blocks in the harmonic analysis of groups and in quantum symmetry classifications.

In ultra-slow-roll inflation, $z=a\sqrt{2ϵ}\propto a^{-2}\propto {\eta }^2$. The comoving curvature mode equation isand its Bunch-Davies mode is proportional to $(1+ik\eta )e^{-ik\eta }/{\eta }^3$. Unlike ordinary slow roll, the superhorizon curvature perturbation grows as $a^3$.

The Eliassen–Palm flux is a wave-activity flux whose divergence forces the transformed mean flow. Its vertical component combines vertical eddy momentum flux with a correction from meridional buoyancy transport.

The torsion subgroup of an abelian group consists of all elements of finite order. For a finitely generated abelian group, it is the finite direct summand complementary to the free abelian part.

A closed orientable surface of genus $g$ is homeomorphic to a sphere with $g$ handles. Its Euler characteristic is $2-2g$.

An $n$-manifold has zero singular homology in every degree greater than $n$. One proof uses a countable exhaustion by subsets that retract onto $n$-dimensional CW complexes and the compact support of every singular cycle.