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The Church numeral $c_n=\lambda f.\lambda x.f^{n}x$ encodes the natural number $n$ by iteration.

This theorem establishes strong normalization in the normal form sense. Every well-typed simply typed lambda term admits no infinite beta-reduction sequence.

An untyped lambda term $Y$ is a fixed-point combinator when $Yf{\equiv }_{\beta }f(Yf)$ for every term $f$.

If a lambda term beta-reduces to both $M$ and $N$, then $M$ and $N$ beta-reduce to a common term. Consequently two beta-equivalent beta-normal forms are alpha-equivalent.

Every well-typed term in the simply typed lambda calculus has some finite beta-reduction sequence ending in a beta-normal form.

Every holomorphic function $f$ on a simply connected domain has a single-valued antiderivative. Fixing $z_0$ and settinggives a path-independent function with $F^{\prime }=f$, because the Cauchy integral theorem makes the integral around every closed path zero.

The geometric realization is a pseudomanifold without boundary in the convention used here.

A pure $n$-dimensional simplicial complex is a pseudomanifold when every codimension-one simplex belongs to exactly two top-dimensional simplices and the top simplices are connected through shared codimension-one faces.

An $n$-simplex has $n+1$ affinely independent vertices.

For $0\le x_i\le 1$, $x_1+x_2+x_3\le 5/2$, and objective $x_1+2x_2+4x_3$, the oriented edge graph has eight improving paths from the origin to the unique optimum $(1/2,1,1)$. Their lengths range from three to five pivots.

With $u=-{\psi }_y$, $v={\psi }_x$, $\eta =y/\Delta$, and $\psi =U\Delta f(\eta )$, the plume boundary-layer equation reduces to

If real matrices satisfy $C=SD{S}^{-1}$ for an invertible complex $S=X+iY$, then $CS=SD$, so both real matrices $X,Y$ intertwine $C$ and $D$. Since $\det (X+tY)$ is a nonzero real polynomial, some real $t$ makes $X+tY$ invertible and supplies a real similarity.

Every square matrix over a field is similar to its transpose.

A similarity transformation replaces a square matrix by a similar matrix. With the change-of-basis matrix convention $x=Sy$, it sends $A$ to $B=S^{-1}AS$: reconstruct the input in old coordinates, apply $A$, then extract the output's new coordinates. These three operations explain the order of the factors.

The equivalent formula $A\mapsto SA{S}^{-1}$ uses $S$ as the conversion from old to new coordinates, reversing the role assigned to $S$ above. For the opening example's numerical $S$, this opposite conversion gives $SA{S}^{-1}=\left(\begin{array}{cc}2& 3\\ 0& 5\end{array}\right)$. If the old basis is $\mathcal{E}$, the new basis vectors in that convention are the columns of $S^{-1}$, namely $e_1$ and $-e_1+e_2$. Both formulas are similarity transformations; the direction of the coordinate conversion decides which one describes a specified basis.

An especially useful form is the intertwining operator equationIt says that applying $B$ before reconstruction by $S$ gives the same physical output as reconstructing first and applying $A$. Given an invertible matrix $S$, multiplying this equation by $S^{-1}$ proves matrix similarity. Without invertibility, the equation alone does not prove matrix similarity.

An upper-bound sieve bounds the size of a sifted set from above. In a dimension-one problem with one forbidden class modulo each relevant prime $p$, its main density factor is comparable to ${\prod }_p(1-1/p)$.

For a finite integer set $A$ and a set of primes $\mathcal{P}$, the sifting function counts elements of $A$ divisible by no prime in $\mathcal{P}$ below $z$:

For a nonnegative weighted sequence $\mathcal{A}=(a_n)$, the same notation means $S(\mathcal{A},\mathcal{P},z)={\sum }_{(n,P(z))=1}{a}_n$.

For an algebraic number $\alpha$ of degree $d$ and a small $\kappa >0$, Siegel lemma constructs integer polynomials $P,Q$ of degree at most $n$ and exponentially bounded coefficients for which $P(X)+\alpha Q(X)$ has a zero at $X=\alpha$ of order close to $2n/d$. A Wronskian then prevents every specialization $P(X)+yQ(X)$ from having a high-order zero at a rational point. This is the auxiliary-polynomial mechanism in the Thue-Siegel method.

The $j$th normalized derivative isIf $P(X)={\sum }_r{a}_r{X}^r$ has integer coefficients, then $D_{j}P(X)={\sum }_{r\ge j}\left(\genfrac{}{}{0ex}{}{r}{j}\right)a_r{X}^{r-j}$ also has integer coefficients. For $\mathrm{deg}P\le n$, its naive polynomial height is at most $2^{n}H(P)$.

The VC dimension is the largest number of points on which a binary hypothesis class realizes every possible labeling.

For binary hypothesis classes,because the label vectors realized by the union form the union of the two sets of realized label vectors.

A period-three orbit forces an interval-covering graph with adjacency matrixIts number of closed pointed length-$n$ itineraries is $\mathrm{tr}(A^n)=L_n$. Möbius removal of lower periods and division by $n$ give four primitive cyclic classes at $n=7$ and five at $n=8$.