The Lebesgue measure of a measurable set is the supremum of the measures of its compact subsets. In particular, an open set of finite positive measure contains a compact subset with at least half its measure. Together with a finite subcover, this turns a finite Wiener covering lemma estimate into an estimate for arbitrary open superlevel sets.

An open subset of the circle can be covered by countably many closed subarcs with disjoint interiors. Grouping a fixed enumeration into finite batches gives compact sets covering every point of the open set. By taking each finite batch long enough, the remaining total measure can be forced below any prescribed positive sequence tending to zero. This controls the measure of subsequent batches without assuming that a prescribed null subset is itself a countable union of compact null sets.

Use the Hilbert-space construction of dominated measure densities with $\rho =\mu +\nu$. The set $B=\{h=1\}$ is a null set for $\mu$. Put $f=0$ on $B$ and $f=h/(1-h)$ elsewhere. Then $\int fd\mu =\nu (B^c)<\infty$ and $\nu (A)={\int }_{A}fd\mu +\nu (A\cap B)$. This is the Lebesgue decomposition into a density part and a part concentrated on a $\mu$-null set.

For standard Brownian motion, the Brownian reflection principle bounds the Probability that its maximum up to $a^n$ exceeds $(1+\epsilon )\sqrt{2a^n\log \log (a^n)}$ by $2(n\log a{)}^{-(1+\epsilon {)}^2}$. This is summable for every $a>1$ and $\epsilon >0$. The Borel-Cantelli first lemma controls these geometric times, monotonicity of the normalizing function controls the intervening times, and countable choices $a\downarrow 1$, $\epsilon \downarrow 0$ give the bound. Equality requires a separate lower-bound proof.

A Lagrangian submanifold diffeomorphic to an $n$-dimensional flat torus in a $2n$-dimensional symplectic manifold. Compact connected regular levels of a completely integrable Hamiltonian system are examples, by the Arnold-Liouville theorem.

The rotating-frame effective potential of the circular restricted three-body problem has negative singular wells at both primaries. Its three Collinear Lagrange points are planar saddles, and its two Triangular Lagrange points are planar local maxima for nonzero primary masses. These geometric labels do not replace a dynamical stability analysis including the Coriolis force.

For a probability measure $P$, the mean-zero L2 space isThe expectation functional is bounded: Cauchy-Schwarz inequality gives $|\int hdP|\le \Vert h{\Vert }_2$ because $P(\Omega )=1$. Its kernel is consequently a closed subspace of a Hilbert space. The inherited inner product isFor real functions the conjugate can be omitted; for complex functions the product $\int hkdP$ alone is not an inner product.

Every square-integrable function has the orthogonal decomposition $h=(h-Ph)+Ph$, with the first term in $L_0^2(P)$ and the second constant. The squared norm of the centered term is the variance in the real-valued case. This is the natural space for centered score functions and canonical gradients.

For a nonempty $r$-uniform set family, define $x>r-1$ by $|\mathcal{F}|=\left(\genfrac{}{}{0ex}{}{x}{r}\right)$ using real binomial coefficients. Then its lower shadow has at least $\left(\genfrac{}{}{0ex}{}{x}{r-1}\right)$ members. This continuous estimate is often easier to use than the exact integer expansion in the Kruskal-Katona theorem.

A short proof uses coordinate shifts of a set family. In a fully left-shifted set family split at coordinate one into ${\mathcal{F}}_0$ and ${\mathcal{F}}_1$, deleting one in the present section. The shifting property implies $\partial {\mathcal{F}}_0\subseteq {\mathcal{F}}_1$, and the total lower shadow has size $|{\mathcal{F}}_1|+|\partial {\mathcal{F}}_1|$. Induct on uniformity and, at fixed uniformity, on set family size. If $|{\mathcal{F}}_1|<\left(\genfrac{}{}{0ex}{}{x-1}{r-1}\right)$, the Pascal's identity and the size mathematical induction force $|\partial {\mathcal{F}}_0|>\left(\genfrac{}{}{0ex}{}{x-1}{r-1}\right)$, a contradiction. The uniformity mathematical induction applied to ${\mathcal{F}}_1$, followed by the Pascal's identity, proves the estimate. Integer $x$ is sharp for the full $r$-level on $x$ coordinates.

Koszul homology is the homology of the Koszul complex with module coefficients. Its degree-zero term is $M/(f)M$, and its positive terms record failures of regularity. Each generator acts trivially on homology by the Koszul homotopy for multiplication by a generator.

Tensor the finite free Koszul complex on $f_1,\dots ,f_r$ with the module $M$. For a regular sequence on a module, its positive Koszul homology vanishes. If the sequence is regular on $R$, this complex instead computes ${\mathrm{Tor}}^R(R/(f),M)$ for arbitrary $M$.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, let $M$ be nonzero and finitely generated, and let every $f_i$ lie in $\mathfrak{m}$. Then positive Koszul homology vanishes exactly when $f$ is a regular sequence on a module $M$. The mapping cone exact sequence proves the forward implication by induction; surjectivity of the last generator on earlier homology and the Nakayama lemma prove the converse.

The perfect exterior pairing in complementary degrees identifies the dual of a Koszul complex with its degree reversal, with suitable differential signs. For a regular sequence, its Koszul resolution consequently gives ${\mathrm{Ext}}_R^i(R/(f),R)=0$ except in degree $r$, where it is $R/(f)$.

For a Killing-Yano two-form, ${\nabla }_a{Y}_{bc}$ is a differential three-form. Along a geodesic with velocity $v$, the covector $w_c=Y_{bc}{v}^b$ is carried by parallel transport, since $v^a{v}^b{\nabla }_a{Y}_{bc}=0$.

A symmetric two-tensor satisfying this equation gives a quadratic geodesic first integral. Examples include the Riemannian metric, symmetric products of metric-dual Killing vector fields, and the square of a Killing-Yano two-form.

For the incompressible velocity $u_x=\alpha y$, $u_y=-\beta x$ in a shearing sheet, advection produces $-\alpha \beta (x,y)$ and the Coriolis acceleration produces $(2\Omega \beta x,2\Omega \alpha y)$. Including the radial tide $2\Omega Sx$ gives $p/\rho =(A{x}^2+B{y}^2)/2+C$. A material ellipse has $\beta =r^2\alpha$. A vortex patch is immersed in surrounding fluid, so this pressure need not vanish on its boundary; imposing that condition would describe a different free-boundary problem.

The material-boundary and vorticity conditions give $\alpha =S/[r(r-1)]$ and $\beta =Sr/(r-1)$. A pressureless particle subject to linear drag, Coriolis acceleration and tidal gravity has the drag-polynomial stability for all positive stopping rates with $K=2\Omega (2\Omega -S)$, $L=\Omega (\alpha +\beta -S)$ and $P=\alpha \beta$. In Keplerian shear, $K/{\Omega }^2=1$, $L/{\Omega }^2=3(r+1)/[2r(r-1)]$ and $P/{\Omega }^2=9/[4(r-1{)}^2]$. The exact criterion is $r\ge 3$, including the boundary $K=L>P$ at $r=3$. Attraction is local to the centre under the prescribed interior velocity field; a trajectory leaving the patch cannot be followed using that field alone.

A bracket smoothing state chooses one of the two unoriented smoothings at every crossing of a link diagram. Its weight is $A^{a-b}{\delta }^{s-1}$ in the reduced Kauffman bracket, where $a,b$ count the smoothing types and $s$ counts resulting circles. The all-$A$ and all-$B$ states choose the same smoothing type at every crossing.

The Temperley-Lieb diagram algebra is generated by planar cap-cup tangles, with composition by stacking and each closed circle replaced by $\delta$. Its local relations $e_i^2=\delta e_i$ and $e_i{e}_{i+1}{e}_i=e_i$ make the second and third Reidemeister moves direct algebraic identities for $R_i=AI+A^{-1}{e}_i$ when $\delta =-A^2-A^{-2}$.

The maximal estimate used in the classical Kakeya set problem asks, for every $\epsilon >0$, for a bound from $L^n({\mathbb{R}}^n)$ to $L^n(S^{n-1})$ with loss at most ${\delta }^{-\epsilon }$. Constants may depend on dimension and $\epsilon$, but not tube thickness or the function. Applied to an indicator function of a neighborhood of a Kakeya set, it gives the Kakeya Minkowski dimension conjecture.

For a finitely generated torsion Iwasawa module, its characteristic ideal is the product of the height-one prime ideals raised to the lengths of the corresponding localizations. In the elementary-divisor description it is generated by $p^{\sum a_i}\prod f_j^{b_j}$. It is unchanged by finite modules and multiplicative in short exact sequences of torsion modules.