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When $\beta N>\nu$ and $f>0$, the positive equilibrium isWriting $a=\beta I^{\ast }>0$, the reduced $(S,I)$ Jacobian has characteristic polynomialBoth roots have negative real part. Its discriminant is negative for sufficiently small $f$ and positive for sufficiently large $f$, giving respectively a stable focus and a stable node.

When $N>N_c=(\mu +\nu )/\beta$, the positive equilibrium isOn the invariant plane $X+Y+Z=N$, the $(X,Y)$ Jacobian at this equilibrium has negative trace $-(\mu +\beta Y^{\ast })$ and positive determinant $\beta Y^{\ast }(\mu +\nu )$, so both eigenvalues have negative real part and the equilibrium is locally asymptotically stable.

The singular values of a matrix measure stretching of Euclidean norm. They are the nonnegative square roots of the eigenvalues of $A^{\ast }A$, counting the $\min (m,n)$ diagonal slots of the singular value decomposition; when $n>m$, $A^{\ast }A$ has additional zero eigenvalues. Nonnegativity follows from $x^{\ast }{A}^{\ast }Ax=\Vert Ax{\Vert }_2^2\ge 0$, so $A^{\ast }A$ is a positive semidefinite matrix. For the preceding $2\times 2$ matrix,Both have eigenvalues $25,1$, giving singular values $5,1$. In particular,Here the operator norm uses Euclidean norm: no unit vector stretches more than fivefold. Zero singular values identify lost directions; the number of positive singular values equals the matrix rank. This summarizes both amplification and information loss with nonnegative numbers.^[ref]

Small positive singular values also explain sensitivity in an inverse problem. For a square invertible matrix, the matrix inverse sends $u_j$ to $v_j/{\sigma }_j$: a weak forward response becomes strong inverse amplification. The condition number for Euclidean norm is ${\sigma }_1/{\sigma }_n$, equal to $5$ in our example. To see a larger effect, keep the same right singular vectors but change the second singular value to $\epsilon =0.01$:The output perturbation has Euclidean norm $0.001$, but the recovered input changes by $0.1$. Its hundredfold amplification follows from the direction of the perturbation, not just its size. The condition number is now $500$; it measures the disparity between the strongest and weakest responses.^[ref]

A partial function is lambda-definable when one lambda term maps Church numerals in its domain to the numeral of the output and produces no numeral on inputs outside its domain.

At type $\sigma \to \sigma \to \sigma$, the Church Booleans are $\top =\lambda x.\lambda y.x$ and $\perp =\lambda x.\lambda y.y$.

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 is the convex hull of $n+1$ affinely independent points. Every point has a unique expression $x={\sum }_{i=0}^n{\lambda }_i{v}_i$ with ${\lambda }_i\ge 0$ and ${\sum }_i{\lambda }_i=1$, its barycentric coordinates. A zero-simplex is a point, a one-simplex a line segment, a two-simplex a filled triangle, and a three-simplex a tetrahedron. Its faces are obtained by taking subsets of the vertices. This uniform description allows simplicial complexes to build spaces from elementary pieces in every dimension.

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 set $A$ of positive integers and a set of prime numbers $\mathcal{P}$, the sifting function counts elements divisible by no prime number in $\mathcal{P}$ strictly below $z$:The strict inequality is the endpoint convention used in the Buchstab identity here. With an empty product, $P(z)=1$ and every element survives.

For a finitely supported nonnegative weighted sequence $\mathcal{A}=(a_n)$, use the same notation for $S(\mathcal{A},\mathcal{P},z)={\sum }_{\gcd (n,P(z))=1}{a}_n$. Choosing $a_n$ to be the indicator function of $A$ recovers the set version. For $A=\{1,\dots ,10\}$, all prime numbers allowed, and $z=5$, only $2$ and $3$ are removed as divisors, leaving $1,5,7$ and giving $S=3$.