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Fix right coset transversals $R_{+},R_{-}$ for $A,B$, containing $1$. In the convention $ta{t}^{-1}=\varphi (a)$, every element has a unique normal form $g_0{t}^{{ϵ}_1}{r}_1\cdots t^{{ϵ}_k}{r}_k$, with $r_i\in R_{{ϵ}_i}$ and $r_i\ne 1$ whenever ${ϵ}_{i+1}=-{ϵ}_i$. A permutation action on the set of these formal sequences proves uniqueness and embeds the base group. Transporting subgroup coefficients across stable letters converts general words to these normal forms.

A one-dimensional particle suspension settling without particle diffusion or overturning obeys the scalar conservation law ${\varphi }_t+{\partial }_{z}j(\varphi )=0$. The upward flux is negative for downward settling. Characteristic curves, compressive shocks and rarefaction waves describe concentration transport rather than individual particle paths.

Start with finite real linear combinations of $1$, $\cos (\lambda x)$ and $\sin (\lambda x)$ with arbitrary positive real frequencies, and use the inner product $⟨f,g{⟩}_M={\lim }_{R\to \infty }{R}^{-1}{\int }_{-R}^{R}fg$. Product-to-sum identities give existence of these averages and mutual orthogonality of distinct frequencies. The squared norm is $2a_0^2+{\sum }_{\lambda }(a_{\lambda }^2+b_{\lambda }^2)$. Its Hilbert space completion contains an uncountable orthonormal set, so is not a separable Hilbert space. This construction does not equip all locally square-integrable functions with an inner product: compactly supported nonzero functions have zero mean square, and other functions have divergent or nonexistent averages.

The Hilbert-Schmidt norm of a Hilbert-Schmidt operator is the square root of the displayed sum. The value is independent of the orthonormal basis, by the Parseval identity. For a finite matrix, it is the Frobenius norm.

For $(Tg)(v)=\int k(v,w)g(w)dw$, the Cauchy-Schwarz inequality gives $\Vert Tg{\Vert }_2\le \Vert k{\Vert }_{L^2(v,w)}\Vert g{\Vert }_2$. The kernel norm is the Hilbert-Schmidt norm, which bounds the operator norm. Uniform kernel bounds apply also to spatially parametrized operators after integrating over position.

Suppose an irreducible monic integral cubic has discriminant $-p$ for a prime $p>3$, and let $L$ be its splitting field. Its Galois group is $S_3$ and its quadratic subfield is $\mathbb{Q}(\sqrt{-p})$. The extension over that subfield is cyclic cubic and everywhere unramified. Away from $p$, separable reduction and Hensel lemma put the roots in an unramified local extension. At $p$, discriminant valuation one gives a double residue root and a separate simple root; lift the simple root, and the remaining quadratic's discriminant has square class $-p$. Its splitting field is already the quadratic subfield's completion. The imaginary quadratic base has no real places.

The splitting field of $T^3+T-1$ has discriminant square class $-31$ and Galois group $S_3$. It is the Hilbert class field of $K=\mathbb{Q}(\sqrt{-31})$: the relative extension is unramified cyclic cubic, and the ideal class group of Q of square root minus thirty-one has order three. Locally at $31$, the simple residue root $28$ lifts and the remaining quadratic splits over ${\mathbb{Q}}_{31}(\sqrt{-31})$, so the prime of $K$ above $31$ splits completely in its Hilbert class field. The inverse generator satisfies $T^3-T^2-1=0$.

Gas in a collapsing halo is heated toward a virial temperature. If its cooling time is short compared with the dynamical time, thermal support is removed and rapid condensation can form a galaxy. If cooling is slower, a hot atmosphere can persist or condense only gradually. This criterion links baryonic galaxy formation to the dark-matter assembly hierarchy without making every halo into one luminous object.

The baryon conversion efficiency of a halo is the fraction of its baryonic mass incorporated into stars. If the halo retains the cosmic baryon fraction $f_b$, its stellar-to-total-mass ratio is $f_b{f}_{\ast }$. This efficiency depends on halo mass, epoch, gas supply and feedback and must not be confused with a star-formation efficiency per free-fall time.

A Gamma–Poisson hierarchical model uses $Y_i\mid {\theta }_i\sim \mathrm{Poisson}(t_i{\theta }_i)$, ${\theta }_i\mid b\sim \mathrm{Gamma}(a,b)$, and $b\sim \mathrm{Gamma}(c,r)$, with gamma distributions in shape-rate convention. Conditional independence gives full conditional distributions ${\theta }_i\mid b,y\sim \mathrm{Gamma}(a+y_i,b+t_i)$ and $b\mid \theta ,y\sim \mathrm{Gamma}(c+na,r+{\sum }_i{\theta }_i)$.

Partial pooling estimates related local parameters jointly through shared hyperparameters. Each local Bayesian posterior balances its own likelihood function against a group prior distribution, with stronger shrinkage when its data are weak or the group variation is small. It lies between forcing all local parameters to be equal and estimating them independently. Random-effects meta-analysis is an important application.

In a screening misclassification model, true disease states are latent and screen results are emissions. A cancer-free state may emit a positive result with a false positive rate, while a perfectly sensitive pre-clinical state always emits a positive result. Conditional independence of emissions given latent states is an additional model assumption, not a consequence of a marginal false-positive rate.

The standard projective resolution of a quiver representation has length one, even for quivers with oriented cycles. Thus every path algebra is a hereditary ring, and higher extension groups vanish. Applying a long exact sequence of Ext groups to $0\to I\to K\to C\to 0$ gives a surjection ${\mathrm{Ext}}^1(K,W)\twoheadrightarrow {\mathrm{Ext}}^1(I,W)$, since the next term ${\mathrm{Ext}}^2(C,W)$ is zero. This is the hereditary step in the Ringel lemma on bricks.

Start at base two. At each nonzero step, change every base occurrence in the hereditary base representation to the next base and subtract one. Although the natural-number terms can grow, their ordinal ranks of a Goodstein term decrease strictly. Keep zero fixed once it is reached.

For a nonzero vector $u$, the open hemisphere of $S^d$ with direction $u$ is $\{x\in S^d:u\cdot x>0\}$. Its boundary is excluded, so opposite open hemispheres are disjoint.

The Hellinger affinity of dominated probability measures is the integral of the geometric mean of their probability density functions. It lies in $[0,1]$ by Cauchy-Schwarz inequality. The unnormalized squared Hellinger distance is $2-2$ times this affinity; this identity makes the relation with relative entropy particularly simple.

The scalar inequality $-\log u\ge 1-u$, applied to $u=\sqrt{q/p}$ and integrated against $P$, gives $K(P,Q)\ge 2-2\int \sqrt{pq}d\mu$. The right side is the unnormalized squared Hellinger distance. Infinite divergence and zero density regions are handled by the usual extended-integral conventions.

For a matter-era superhorizon tensor with $h=h_{\mathrm{prim}}[1-(k\eta {)}^2/10+\cdots ]$, the tensor CMB line-of-sight source gives ${\Theta }_{22}\propto h_{\mathrm{prim}}(k{\eta }_0{)}^2$ and ${\Theta }_{32}/{\Theta }_{22}=-ik{\eta }_0/(3\sqrt{7})$ when ${\eta }_{\ast }\ll {\eta }_0$. The extra spatial phase makes the octupole one order smaller.

The height of a prime ideal is the supremum of lengths of strict prime chains ending at it, equivalently the Krull dimension of its localization at a prime ideal. The Krull height theorem bounds the height of a minimal prime over an $n$-generated ideal by $n$.

For level-one weight-$k$ modular forms, $T_m{T}_n={\sum }_{d\mid \gcd (m,n)}{d}^{k-1}{T}_{mn/d^2}$. Thus coprime indices multiply directly, and $T_p{T}_{p^r}=T_{p^{r+1}}+p^{k-1}{T}_{p^{r-1}}$. The Fourier coefficients of a composite-index Hecke operator prove these relations by regrouping divisors. They show that the Hecke algebra of modular forms is commutative and generated by prime operators.