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A uniform cover is irreducible when it is not the disjoint union, as a multiset, of two nonempty uniform covers. There are only finitely many irreducible uniform covers of a fixed finite set: their multiplicity vectors form an antichain in ${\mathbb{N}}^{2^n}$, and Dickson lemma forbids an infinite antichain there.

If $\mathcal{C}$ is a $k$-uniform cover of $[n]$ and $S\subseteq {\mathbb{R}}^n$ is a Euclidean body, thenSlicing in one coordinate, applying the theorem inductively to each slice, and then applying Hölder's inequality to the $k$ projected slice functions proves the inequality.

Exact covering is essential for unrestricted real volumes. Finite subsets of a Cartesian product satisfy the analogous cardinality inequality, and in that discrete version covering at least $k$ times suffices.

Let $(x_n)$ lie in a normed space $X$ and suppose ${\sum }_n|f(x_n)|<\infty$ for every $f\in X^{\ast }$. Then some $C$ satisfiesApply the uniform boundedness principle on the Banach space $X^{\ast }$ to the finite signed sums $f\mapsto {\sum }_n{a}_{n}f(x_n)$ with $|a_n|\le 1$.

For equal permeabilities on the two sides of an interface, tangential-field continuity and Snell law for electromagnetic waves givefor consistently signed electric amplitudes normal to the plane of incidence.

For a jet of velocity $U$ occupying $|y|<h$ in stationary fluid of the same density, disturbances proportional to $e^{ikx+\sigma t}$ satisfyfor a varicose mode of a planar jet, andfor a sinuous mode of a planar jet. Both have unstable growth rate

In a sinuous mode, the upper and lower jet interfaces move in the same direction, so the jet centreline bends while its thickness is unchanged to first order.

In a varicose mode, the upper and lower jet interfaces move in opposite directions, so the jet thickness varies while its centreline remains fixed.

The symmetric algebra is the quotient of $T(V)$ by the ideal generated by $v\otimes w-w\otimes v$. It is graded by symmetric powers:

An equivariant supertrace inserts a symmetry $f$ as well as fermion parity, $\mathrm{Tr}((-1{)}^{F}fA)$. In a path integral, $f$ produces boundary conditions twisted by its action.

A primary decomposition expresses an ideal or submodule as an intersection of primary components.

A functor between small categories is a strong epimorphism precisely when its image generates its codomain as a subcategory. It is a regular epimorphism precisely when it is surjective on objects and full. Hence a functor can be strong without being regular when its image arrows generate additional composites that have no single preimage.

Suppose a category has pullbacks and coequalizers and every pullback of a regular epimorphism is epic. Coequalizing the kernel pair of any morphism produces a factorization as a regular epimorphism followed by a monomorphism. Consequently every strong epimorphism is regular.

The Itô integral ${\int }_0^t{H}_{s}d{W}_s$ integrates a predictable square-integrable process against Brownian motion. It is defined first for predictable step processes and then completed using the Itô isometry.

For the Bockstein associated with $0\to \mathbb{Z}/2\to \mathbb{Z}/4\to \mathbb{Z}/2\to 0$,For a real line bundle $L$, one first has $\beta (w_1(L))=w_1(L{)}^2$. The general identity follows from the splitting principle for real vector bundles, the Bockstein derivation rule, and the elementary-symmetric-polynomial identity obtained by separating whether an index lies in a chosen $j$-element subset.

If a measurable set $E\subseteq [0,L]$ has measure $m(E)=L/2+\alpha$, then for $|t|<2\alpha$ the sets $E$ and $E+t$ lie in an interval of length $L+|t|$, soThus $(-2\alpha ,2\alpha )\subseteq E-E$. This quantitative overlap argument is a bounded form of the Steinhaus theorem.

A spinor field is a section of a spinor bundle. Clifford multiplication by tangent vectors acts fiberwise on spinor fields.

The SU(2) group is the group of two-by-two unitary matrices of determinant one. It is a compact three-dimensional Lie group whose underlying manifold is a three-sphere.

A three-variable tensor is diagonal when it vanishes unless its three arguments are equal.

Finkelstein-Rubinstein constraints impose the allowed signs under combined spatial and isospin rotations of a quantized Skyrmion. They encode the fermionic or bosonic exchange behavior of topological solitons.

If $g$ simplicially approximates $f$, then $f(x)$ and $|g|(x)$ lie in a common simplex for every $x$. The affine formulatherefore remains in the realization and gives a homotopy from $f$ to $|g|$.