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The Teichmuller representative of a nonzero residue class $\bar{a}\in {\mathbb{F}}_q^{\ast }$ is the unique lift $\omega$ satisfying ${\omega }^{q-1}=1$ and $\omega \equiv a$ modulo the maximal ideal.

The $p$-adic logarithm is the convergent series $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$ on a sufficiently small group of principal units.
It is locally inverse to the p-adic exponential function.

The Laurent series field $k((t))$ consists of formal series ${\sum }_{n\ge N}{a}_n{t}^n$ with finitely many negative-degree terms. Its $t$-adic valuation is the least exponent with nonzero coefficient.

A Lipschitz bound is any valid inequality of the displayed form with a finite constant $L$. It need not use the least Lipschitz constant. For $f(x)=3x-2$, both $L=3$ and $L=5$ are valid, but $L=2$ fails. The tighter bound gives a sharper error estimate: input error at most $0.01$ guarantees output error at most $0.03$ using $L=3$, compared with the valid but weaker $0.05$ using $L=5$. The bound constrains errors; it does not state that every pair achieves that amplification.

An entire function bounded by a polynomial in the radius is itself a polynomial, by Cauchy estimates.

If an entire function satisfies $|h(z)|\le |\mathrm{Re}z{|}^{-1/2}$ away from the imaginary axis, then $h=0$. Cauchy's formula on $|z|=R$ bounds every Taylor coefficient by a constant timeswhich tends to zero as $R\to \infty$.

The image of every nonconstant entire function is dense in $\mathbb{C}$. If a disc about $w$ were omitted, then $1/(f-w)$ would be bounded and entire, so Liouville's theorem would make $f$ constant.

A Riccati equation is a first-order equation quadratic in the unknown function.

Every convergent series of positive rational terms has a subseries whose rational partial sums $p_j/q_j$ satisfyChoose each new term small enough to enforce all previous tail bounds. Separation of reduced fractions first makes $\alpha$ irrational, and the Liouville approximation theorem then rules out every finite algebraic degree.

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 $2\sqrt{Dr}$.