Two connected pistons apply Pascal's law to convert force and displacement. For an ideal incompressible fluid, $A_1{s}_1=A_2{s}_2$ and $F_1/A_1=F_2/A_2$, hence $F_1{s}_1=F_2{s}_2$: output force gain is not an energy gain.

A partition of an integer has only odd hook lengths exactly when it is a staircase. Absence of a hook of length two forces the last row to have length one, prohibits equal consecutive rows by examining the last boxes of an equal-length block, and prohibits gaps at least two by examining a penultimate row box. Thus consecutive row lengths differ by one. Conversely a staircase has $h(i,j)=2k-2i-2j+3$, always odd. Its size is $k(k+1)/2$. The hook-length formula then gives $v_2(f^{\lambda })=v_2(n!)$, and no other partition of the same size has that full two-adic divisibility, since it would also have an odd hook product.

Given a partially ordered set $B$ and a set $A$, the pointwise order on the function space of functions $A\to B$ compares their values at every argument. It is a partial order: reflexivity and transitivity hold at every argument, and antisymmetry follows because two functions equal at every argument are equal. Even if $B$ has a total order, this function space need not: the two functions on $\{0,1\}$ with values $(0,1)$ and $(1,0)$ are incomparable when $B=\mathbb{R}$.

An irreflexive transitive comparison whose induced incomparability relation is transitive. Incomparability is then an equivalence relation, and its classes are totally ordered by the comparison. A comparison-based sorting algorithm can treat incomparable elements as ties; a merge algorithm that takes from the left input on ties preserves stable sorting. Ordinary ordering on real numbers is an example; machine NaN values require separate handling.

A subset $U$ of a partially ordered set is an upper set when $x\in U$ and $x\le y$ imply $y\in U$. The upward closure $\uparrow A=\{y:\exists a\in A,a\le y\}$ is the smallest upper set containing $A$. Both union and intersection preserve this property.

The partial order on nonempty closed integer intervals given by subset inclusion. The least upper bound of two such intervals is their interval hull $[\min (l,l^{\prime }),\max (u,u^{\prime })]$. The hull can contain points absent from both original intervals, so it need not equal their union.

In a partially ordered set, $x\ll y$ means that every nonempty directed set $D$ whose least upper bound exists and satisfies $y\le \sup D$ contains some $d$ with $x\le d$. Taking $D=\{y\}$ shows that $x\ll y$ implies $x\le y$; the converse need not hold. Some formulations quantify over the empty directed set as well; this changes the behaviour at the least element and must be specified.

A partially ordered set in which, for every $x$, the elements way-below $x$ form a directed set with least upper bound $x$. Directed least upper bounds other than these approximating ones need not exist.

Every subset having an upper bound has a least upper bound. If the empty subset is included, a nonempty such partially ordered set has a least element. This condition differs from a directed-complete partial order: a directed set need not have an upper bound.

A weakly linear partial function on the real numbers has a domain closed under rational scalar multiplication and addition, and it preserves addition on that domain. A nonempty domain is a vector space over the rational numbers, and additivity implies rational homogeneity of an additive function by integer multiplication and division. The empty function also satisfies the defining conditions vacuously. This terminology does not assert homogeneity under arbitrary real scalars.

The union of a chain of partial-function graphs is a partial function: two values at the same input occur together in a larger chain member and therefore agree. Any closure or algebraic identity involving finitely many inputs passes to the union if it holds in every chain member, because one member contains all those inputs.

A complex normed vector space has an inducing inner product exactly when its norm satisfies the parallelogram law. Define $B(u,v)=[\Vert u+v{\Vert }^2-\Vert u-v{\Vert }^2]/4$. Subtract two parallelogram identities to obtain $B(u+w,v)+B(u-w,v)=2B(u,v)$. Taking $w=u$ gives doubling, and then replacing $u,w$ by the half-sum and half-difference gives additivity. Integer scaling and division give rational homogeneity; continuity gives real homogeneity. Thus $B$ is a symmetric real-bilinear form. The complex formula $P(u,v)=B(u,v)+iB(u,iv)$ satisfies $P(iu,v)=iP(u,v)$, because $B(iu,v)=-B(u,iv)$ and $B(iu,iv)=B(u,v)$. Symmetry gives conjugate symmetry of $P$, and $P(u,u)=\Vert u{\Vert }^2$ gives positive definiteness. The reverse implication follows by expanding squared norms. Completeness is needed only for the resulting inner product space to be a Hilbert space.

An outer measure is regular in this sense when every set admits a measurable hull of equal outer measure. This is different from merely specifying inner or outer approximation of already measurable sets by compact or open sets. Lebesgue outer measure has a hull that is a Borel set: for finite outer measure choose open covers of total volumes within $1/j$ of its value and intersect their unions. Monotonicity gives equality of the hull's outer measure with that of the original set. For infinite outer measure the whole Euclidean space is a hull. This regularity is useful for turning numerical equality of inner and outer sizes into measurability.

An outer measure on a metric space is metric when positively separated sets satisfy the displayed equality. Every Borel set is then measurable under Caratheodory measurability criterion. Lebesgue outer measure is metric: refine every box of a covering into finitely many boxes of diameter smaller than the separation. Each refined box meets at most one of the two sets, splitting the cover without changing its total volume. Taking infima proves the additive lower bound; outer subadditivity gives equality.

A real $m\times k$ matrix whose columns are an orthonormal set, equivalently $Q^{T}Q=I_k$. In the complex case use the conjugate transpose instead. This requires $k\le m$; only in the square case does the transpose also provide a two-sided inverse, making $Q$ an orthogonal matrix.

An orthonormal basis of the polynomial space ${\Pi }_n$ need not be indexed by polynomial degree. Permuting the normalized Legendre polynomials, for example, preserves orthonormality but changes the degrees attached to the labels. Results about the number of polynomial roots of $p_k$ require the extra condition $\mathrm{deg}{p}_k=k$, or that the polynomials arise by applying the Gram-Schmidt process to $1,x,x^2,\dots$ in that order. Under this degree ordering, $p_k$ is orthogonal to every polynomial of lower polynomial degree because those earlier basis members span ${\Pi }_{k-1}$.

For a positive integration weight on a nondegenerate interval, this implies that $p_k$ has $k$ distinct polynomial roots in the open interval. If fewer than $k$ roots change sign there, multiply $p_k$ by the product of their linear factors. The product has constant nonzero sign except at its finitely many zeros, so its weighted integral cannot vanish, contradicting orthogonality to this lower-degree polynomial. The usual Gaussian quadrature construction therefore uses the degree-ordered family; arbitrary relabelling is not enough.

A finite collection of nonzero pairwise orthogonal vectors is linearly independent. Given ${\sum }_i{a}_i{q}_i=0$, take the inner product with $q_j$ to obtain $a_j\Vert q_j{\Vert }^2=0$. Positive definiteness and $q_j\ne 0$ force $a_j=0$ for every index. The nonzero condition matters: adjoining a zero vector preserves pairwise orthogonality but destroys linear independence.