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For treatment groups Control, LD, and SD and binary outcomes, equal LD and SD efficacy can be imposed while retaining separate group totals by using separate treatment main effects but one shared treated-versus-control interaction with outcome. Comparing this five-parameter model with the saturated six-parameter model gives a one-degree-of-freedom analysis of deviance for nested generalized linear models.

A saturated log-linear model has enough parameters to reproduce every observed contingency-table cell count and therefore has zero residual deviance.

Localization at a prime ideal $\mathfrak{p}$ means inverting $R\setminus \mathfrak{p}$. The resulting local ring $R_{\mathfrak{p}}$ has maximal ideal $\mathfrak{p}{R}_{\mathfrak{p}}$.

Every localization of a Noetherian ring is Noetherian. Each ideal $J\subseteq S^{-1}R$ is the extension of its contraction to $R$, whose finite generating set consequently generates $J$.

Extension and contraction give inverse inclusion-preserving bijections between prime ideals of $S^{-1}R$ and prime ideals of $R$ disjoint from $S$:

The localization map $\iota :R\to S^{-1}R$ sends every element of $S$ to a unit. If a ring homomorphism $f:R\to T$ also sends every element of $S$ to a unit, there is a unique homomorphism $\tilde{f}:S^{-1}R\to T$ with $\tilde{f}(r/s)=f(r)f(s{)}^{-1}$ and $\tilde{f}\iota =f$.

A valuation ring is an integral domain $R$ such that, for every nonzero element $x$ of its fraction field $K$, either $x\in R$ or $x^{-1}\in R$. Equivalently, its ideals are totally ordered by inclusion.

Quantum teleportation transfers an unknown qubit using one shared Bell pair, a Bell-basis measurement by the sender, two classical bits, and a Pauli correction by the receiver.

Two parties can distinguish $|{\Phi }^{+}⟩$ from $|{\Phi }^{-}⟩$ by both measuring in the Hadamard basis and comparing outcomes: equal outcomes identify $|{\Phi }^{+}⟩$, while unequal outcomes identify $|{\Phi }^{-}⟩$.

The maximal abelian extension $K^{\mathrm{ab}}$ of a field $K$ is the compositum in a fixed separable closure of all finite abelian extensions of $K$.

The roots of unity in a finite extension $K/{\mathbb{Q}}_p$ form a finite group. Each element of their prime-to-$p$ part is a Teichmuller representative, while any $p$-power part lies among the principal units.

A p-adic unit is an invertible element of the ring ${\mathbb{Z}}_p$. For a p-adic integer $x$, this is equivalent to $v_p(x)=0$, $|x{|}_p=1$, or a nonzero initial p-adic digit $b_0$. A rational number $a/b$ is a p-adic unit when neither its reduced numerator nor its denominator is divisible by $p$.

Thus $3$ and $1/3$ are p-adic units at $p=2$, while $2$ is a p-adic integer with no inverse inside ${\mathbb{Z}}_2$. The modular inversion converges to a p-adic inverse construction computes the inverse of an integer unit to arbitrary finite precision.

For a finite extension $L/K$ of non-Archimedean local fields, the ramification index $e(L/K)$ is the index of the value group of $K$ in that of $L$. Normalized valuations satisfy $v_L{|}_K=e(L/K)v_K$.

The Newton polygon of a polynomial over a valued field is the lower convex hull of the points formed from coefficient indices and valuations. Its slopes determine the valuations of roots, counted with multiplicity.

For a discrete valuation ring with uniformizer $\pi$, an Eisenstein polynomial is a monic polynomial whose nonleading coefficients are divisible by $\pi$ and whose constant coefficient is not divisible by ${\pi }^2$. It is irreducible, and adjoining one of its roots gives a totally ramified extension.

A finite extension of local fields is unramified when its ramification index is one and its residue-field degree equals the field degree. A local field has a unique unramified extension of each positive degree inside a fixed algebraic closure.

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 defined by $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$ when $|x{|}_p<1$ in a complete extension of ${\mathbb{Q}}_p$. The terms tend to zero in the p-adic absolute value, giving convergence even though the corresponding real series may behave differently.

On a sufficiently small group of principal units, the logarithm is a group homomorphism into the additive field and is inverse to the p-adic exponential function. For $p=2$, it converges on $1+2{\mathbb{Z}}_2$ but becomes injective only after restricting to $1+4{\mathbb{Z}}_2$, since $\log (-1)=0$. It converts multiplicative congruence questions into additive ones while retaining these domain restrictions.

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.