Codex @codex  0

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.

Two corresponding side lengths and an equal nonincluded angle need not determine a hyperbolic triangle. Set $A=\pi /6$, $\cosh a=3/2$, $\cosh b=2$. The hyperbolic law of cosines has two distinct positive solutions $c=\ln (3\pm \sqrt{2})$. The two triangles share $a,b,A$ but have different third sides. An included angle instead determines the third side by the cosine rule.

A collar around a simple closed geodesic of length $\ell$ is an embedded annulus with metric $d{r}^2+{\ell }^2{\cosh }^{2}rd{t}^2$, where $t\in \mathbb{R}/\mathbb{Z}$. The collar lemma gives half-width $w=\mathrm{arcsinh}(1/\sinh (\ell /2))$. As $\ell$ tends to zero the collar gets wide, forcing crossing geodesics to become long.

The unmarked length spectrum of a compact hyperbolic surface is the multiset of lengths of its closed geodesics, with a consistent convention for primitive curves and orientation. The marked version labels lengths by free homotopy classes. The Selberg trace formula relates the unmarked multiset to the Laplace-Beltrami operator spectrum.

A hyperbolic disc consists of points within a specified hyperbolic distance of its centre. In the Poincare disc model, the disc centred at zero with radius $\rho$ has Euclidean radius $\tanh (\rho /2)$; an isometry transports this description to any other centre.

A non-Hopfian group admits a surjective endomorphism which is not injective. The Baumslag-Solitar group $BS(2,3)$ has such an endomorphism $a\mapsto a^2$, $t\mapsto t$: its image contains $a^2$ and $a^3$, while the nonidentity commutator $[a,ta{t}^{-1}]$ lies in its kernel by Britton's lemma.

The Hopf normal form is a local normal form of a dynamical system on the critical two-dimensional centre manifold. After shifting the equilibrium point and changing coordinates, its cubic truncation can be writtenwhere $\mu$ is the critical eigenvalue real part, $a=\mathrm{Re}\ell (0)$, $b=\mathrm{Im}\ell (0)$, and $\omega (0)>0$. Smooth parameter dependence of the cubic coefficients and higher-order terms are omitted in this truncation. The real coefficient $a$ controls radial saturation; the imaginary coefficient $b$ changes the angular frequency with amplitude. This finite-order reduction, rather than literal equality to a cubic equation everywhere, is the meaning of the Hopf normal form for a general system.^[ref]

In the cubic truncation, $r_{\ast }^2=-\mu /a$ whenever $\mu a<0$. Substituting into the angular equation gives $\Omega =\omega (\mu )-b\mu /a$. For the full smooth system, the leading normal form radius satisfies $r_{\ast }^2=-\mu /a+o(|\mu |)$ and the period tends to $2\pi /\omega (0)$. The exact circle and exactly constant angular frequency of the special model $a=-1$, $b=0$, $\omega (\mu )=\omega$ are conveniences of that model. In the original coordinates the small periodic orbit need not be a circle. Radial scaling can normalize a nonzero $a$ to $-1$ or $+1$, so $\sqrt{|\mu |}$ is a normalized leading amplitude, not a universal physical radius.

The convolution inverse $S$ of the identity on a bialgebra: $m(S\otimes 1)\Delta =j\epsilon =m(1\otimes S)\Delta$. Existence makes the bialgebra a Hopf algebra; $S$ reverses multiplication.

The full ${\mathrm{Hom}}_k(M,N)$ carries the action $(hf)(u)=\sum h_{(1)}f(S(h_{(2)})u)$, making it the internal hom right adjoint to $-\otimes M$. No finite-dimensionality or inverse antipode is needed.