The logic $L_{{\omega }_1,\omega }$ extends first-order logic by countable logical conjunctions and logical disjunctions. Each formula has finitely many free variables, and quantifier strings are finite. In a countable first-order language, every countable structure has a Scott sentence in this logic that characterizes its isomorphism class among countable structures.

Write a small planar displacement in arclength coordinates as $X=s+X_2$ and $Y=y$. The inextensibility relation $X_s^2+Y_s^2=1$ gives $X_{2,s}=-y_s^2/2$ to quadratic order. Thus longitudinal material velocity enters one order later than the transverse velocity. For a periodic material stroke with no mean axial drift, its mean velocity vanishes, although its instantaneous contribution is needed for exact local force balance.

For an $H$-representation $V$ and $g\in G$, the conjugate representation ${}^{g}V$ of $gH{g}^{-1}$ is defined by $(gh{g}^{-1})v=hv$. This transports the acting subgroup; it is distinct from conjugating complex matrix entries. It occurs in the Mackey restriction formula.

If $n$ points and $m$ curves have the property that each pair of distinct points lies on at most $\lambda$ curves, then double counting gives ${\sum }_{\gamma }{k}_{\gamma }(k_{\gamma }-1)\le \lambda n(n-1)$, where $k_{\gamma }=|P\cap \gamma |$. The Cauchy-Schwarz inequality implies $I^2\le mI+\lambda mn(n-1)$ and hence $I\le m+\sqrt{\lambda mn(n-1)}$. The statement needs only this combinatorial multiplicity condition, not algebraicity.

An importance sampling proposal $q$ must be positive wherever the target-weighted integrand can contribute. The usual sufficient condition is that the target probability is absolutely continuous with respect to $q$. Otherwise multiplying observed draws by $f/q$ cannot recover a missing part of the integral. A strict rejection envelope supplies this condition for the rejected-proposal distribution.

For $\mu =\int hf$ and $0<A=\int |h|f<\infty$, the proposal $q_{\ast }=|h|f/A$ minimizes the variance of the ordinary unbiased importance sampling estimator. The Cauchy-Schwarz inequality gives $\int h^2{f}^2/q\ge A^2$, so the minimum variance for $n$ independent samples is $(A^2-{\mu }^2)/n$. The proposal need only cover the nonzero integrand.

If $f\le Mg$, the importance weight $w=f/g$ is bounded by $M$, and ${\mathbb{E}}_g[{\varphi }^2{w}^2]\le M{\mathbb{E}}_f{\varphi }^2$. A finite target second moment therefore guarantees finite variance and the iid central limit theorem for the unnormalized importance-sampling average. Its proposal-budget variance is ${\mathbb{E}}_g[{\varphi }^2{w}^2]-({\mathbb{E}}_f\varphi {)}^2$.

For rejection indicators with conditional acceptance $f(X_i)/(Mg(X_i))$, the conditional expectation of $(M/n){\sum }_i{I}_i\varphi (X_i)$ given all proposals is the unnormalized importance estimator. The Rao-Blackwell theorem reduces variance relative to that fixed-denominator estimator. It gives no such automatic comparison with the accepted-sample mean, whose denominator is the random count.

If a regression response $Y$ is observed but covariate $X$ is sometimes missing, likelihood inference requires a joint model: for example $f_{\beta }(y\mid x)g_{\eta }(x)$. An incomplete record contributes $\int f_{\beta }(y\mid x)g_{\eta }(x)dx$ to the observed-data likelihood. Missing at random with distinct parameters allows the missingness mechanism to be omitted; it does not remove the need to integrate over $X$.

The mean free path is the average distance travelled between collisions. For dilute targets with number density $n$ and effective collision cross-section ${\sigma }_c$, $\ell \sim 1/(n{\sigma }_c)$; identical-particle conventions introduce factors such as $\sqrt{2}$.

For a negative fundamental discriminant $D$, primitive positive definite integral binary quadratic forms of discriminant $D$, modulo proper equivalence, correspond bijectively to the ideal class group of $\mathbb{Q}(\sqrt{D})$. The displayed ideal gives one conventional choice of this correspondence. Reduced positive definite binary quadratic forms give unique class representatives with the usual boundary sign convention, making class-number computations finite.

The ideal-form correspondence for imaginary quadratic fields and the reduction bound $a\le \sqrt{31/3}$ give exactly the forms $[1,1,8]$, $[2,1,4]$ and $[2,-1,4]$. Thus the class number is three and the ideal class group is cyclic of order three.

The narrow ideal class group of a number field is the group of nonzero fractional ideals modulo principal fractional ideals generated by totally positive elements. It is the ray class group for the modulus consisting of all real Archimedean places and trivial finite part. The forgetful map to the ordinary ideal class group is surjective, with kernel $\{\pm 1{\}}^{r_1}/\mathrm{sgn}({\mathcal{O}}_K^{\times })$ under the unit signature map.

A hypergraph is $r$-uniform if each hyperedge has exactly $r$ vertices. Ordinary simple graphs are 2-uniform hypergraphs. Its average hypergraph vertex degree is $r|E|/|V|$.

The degree of a vertex $u$ in a hypergraph is the number of hyperedges containing $u$. For a uniform hypergraph, summing degrees counts each hyperedge once for each of its vertices, so ${\sum }_u{d}_H(u)=re(H)$. In a multihypergraph, incidences are counted with multiplicity.

A subset $I$ of the vertices of a hypergraph is independent when no hyperedge lies wholly inside $I$. This generalizes an independent set in an ordinary graph. If a hypergraph has ordinary graph edges as its vertices and forbidden copies as its hyperedges, these independent sets encode graphs without those forbidden copies.

A multihypergraph has a multiset of hyperedges, allowing the same subset of vertices to appear more than once. Its hypergraph vertex degrees and hypergraph codegrees count multiplicity. Such multiplicity is important when several original hyperedges produce the same link in a hypergraph container theorem.

Extremal hypergraph theory studies the largest or smallest possible size of a hypergraph under prescribed forbidden-subhypergraph or covering conditions. It generalizes extremal graph theory. The Turán density describes the asymptotic maximum edge proportion for a fixed uniform forbidden hypergraph.

A polychromatic coloring of a hypergraph assigns $k$ colors to vertices so that every hyperedge contains all $k$ colors. It is stronger than merely requiring each hyperedge to be nonmonochromatic. Independently random colors and the Lovász local lemma can construct such colorings when edges are sufficiently large relative to their overlap degrees.

Three corresponding side lengths determine a hyperbolic triangle up to an isometry of the hyperbolic plane. First match an ordered pair of vertices by an isometry. The third vertex lies at the intersection of two prescribed hyperbolic distance circles. There are at most two choices, interchanged by reflection in the geodesic through the matched pair; either choice can therefore be matched.