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For distinct reduced fractions $p/q$ and $u/v$,If $q>v$, this is strictly greater than $q^{-2}$.

For the degree-$d$ minimal polynomial $P\in \mathbb{Z}[X]$ of $\alpha$, the nonzero integer $q^{d}P(p/q)$ gives $|P(p/q)|\ge q^{-d}$. On a fixed neighbourhood of $\alpha$, the mean value theorem boundsOutside that neighbourhood the desired lower bound is immediate, giving $|\alpha -p/q|>c(\alpha )q^{-d}$ after decreasing the constant.

An unlink is a link whose components bound pairwise disjoint embedded disks in $S^3$.

The system $Ax=b$ is solvable exactly when $b$ is orthogonal to every vector in $\ker A^T$.

A linear equation is an equation in which the unknowns occur only to the first power and are combined linearly.

Starting at the same value, forward and backward Euler have opposite leading local errors. Their half-difference therefore estimates either local error:For a first-order method, a local-tolerance controller consequently scales the next step by $(\mathrm{tol}/E_n{)}^{1/2}$, usually with a safety factor.

If a linear system has negative eigenvalues with widely separated magnitudes, an explicit method can be forced to resolve the fastest decaying mode solely for stability. For eigenvalues $-1$ and $-100$, forward Euler requires $h\le 0.02$, whereas backward Euler is stable for every positive step size.

A method is A-stable when its stability domain contains the closed left half-plane.

For a real $2\times 2$ Jacobian matrix $J$, both eigenvalue have negative real part exactly whenThe linearization stability theorem then makes a hyperbolic equilibrium locally asymptotically stable.

A pulled front is selected by growth and diffusion in its small-amplitude leading edge rather than by nonlinear dynamics behind the front.

This criterion concerns travelling-front solutions of Fisher's equation.

For diffusion coefficient $D$ and leading-edge growth rate $r>0$, a Fisher-KPP-type pulled front has minimum speed $2\sqrt{Dr}$.

The transportation problem minimizes linear shipping cost while matching prescribed row supplies and column demands.

This is the finite discrete optimal transport problem with prescribed supply and demand marginals.

At a primal-dual optimum, each positive variable corresponds to a tight dual constraint and conversely for positive dual variables.

Every feasible dual objective value bounds every feasible primal objective value.

The dual pairsatisfies weak duality because $b^{T}y\le y^{T}Ax=x^T{A}^{T}y\le c^{T}x$ for every feasible pair.

For a primal maximization problem $Ax\le b$, $x\ge 0$, the dual minimizes $b^{T}y$ subject to $A^{T}y\ge c$, $y\ge 0$.

Rational canonical form expresses a linear map as a direct sum of companion matrices of invariant factors, and works over any field.

An invariant direct-sum decomposition splits a space into nonzero subspaces preserved by the operator.

A companion matrix represents multiplication by x in the power basis modulo a monic polynomial.

A cyclic vector has iterates under an operator that span the whole vector space.