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The principle of epsilon induction says that every progressive class is universal: ifthen $\phi (x)$ holds for every set $x$. Over the other axioms of ZF, it is equivalent to the Axiom of foundation.

For any sets $X,Y$, either $X$ injects into $Y$ or $Y$ injects into $X$. Applying this to $X$ and its Hartogs ordinal proves the well-ordering theorem, so over ZF cardinal comparability is equivalent to choice.

Every set admits a well-order. In ZF this statement is equivalent to the axiom of choice.

A choice function on a family $\mathcal{A}$ of nonempty sets is a function $f$ with $f(A)\in A$ for every $A\in \mathcal{A}$.

Forthe zero-input stable states are $0$ and $(k+\sqrt{k^2-4})/2$. The switching threshold is the saddle-node bifurcation determined by $f(g_c,s_c)={\partial }_{g}f(g_c,s_c)=0$ on the low-concentration branch.

Even if ${\varphi }_n\sim {\psi }_n$ for every $n$, an expansion in the scale $({\varphi }_n)$ need not induce one in $({\psi }_n)$. A perturbation of ${\psi }_0$ that is smaller than ${\varphi }_0$ but larger than ${\varphi }_1$ supplies a counterexample.

For ${\chi }_0={\varphi }_0$ and ${\chi }_n=\sqrt{{\varphi }_{n-1}{\varphi }_n}$, the sequence $({\chi }_n)$ is asymptotic, but ${\chi }_n/{\varphi }_n\to \infty$ for every $n\ge 1$.

For a positive asymptotic sequence $({\varphi }_n)$, define ${\psi }_0={\varphi }_0$ andThen ${\psi }_n\sim {\varphi }_n$ term by term and $({\psi }_n)$ is again an asymptotic sequence.

For an independent sample from $\mathrm{Uniform}[-\theta ,\theta ]$, the maximum likelihood estimator is $M_n={\max }_i|X_i|$ andThe parameter-dependent support makes the convergence rate $n$ rather than $\sqrt{n}$, so the regular maximum-likelihood central limit theorem does not apply.

If $\sqrt{n}(T_n-\theta )$ is asymptotically normal, differentiability gives $\sqrt{n}(g(T_n)-g(\theta ))$ the variance multiplied by $g^{\prime }(\theta {)}^2$.

For estimating equations, asymptotic covariance often has the form $A^{-1}B{A}^{-1}$, called a sandwich covariance.

Regression heteroskedasticity-consistent standard errors use a specific sandwich covariance estimate; general estimating equations can produce sandwich covariances outside regression.

Slutsky's theorem combines convergence in distribution with convergence in probability through continuous algebraic operations.

Forrepeated integration by parts gives

For fixed $x$ and $y\to \infty$,

If an algebra is defined over a parameter ring $R$, a ring homomorphism $R\to k$ produces a specialized $k$-algebra by extension of scalars. Defining relations specialize by replacing every parameter by its image in $k$.

A finite-dimensional associative algebra is semisimple when its left regular module is a direct sum of simple modules. Equivalently, every left ideal is a direct summand. In particular, a semisimple algebra has no nonzero square-zero left ideal: if $L^2=0$ and a module projection $A\to L$ exists, its value $e$ at $1$ satisfies $e\in L$ and $e^2=e$, forcing $e=0$.

A left ideal $L$ of an associative algebra $A$ is a vector subspace satisfying $AL\subseteq L$. Equivalently, it is a submodule of the left regular $A$-module.

The group algebra $k[G]$ is the vector space with basis $G$ and multiplication obtained by extending the group law $k$-bilinearly.

A commutative Noetherian ring is Artinian if and only if every prime ideal is maximal. Equivalently, it is Artinian exactly when its Krull dimension is zero.

The Condorcet paradox is a cycle in collective pairwise preferences even though each individual voter's preference is transitive.