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. In a Noetherian ring, 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.

A Hecke eigenform is a nonzero modular form that is an eigenvector for every Hecke operator. A cuspidal eigenform has nonzero first Fourier coefficient: $a_1(T_{n}f)=a_n(f)$ proves this. Normalize $a_1=1$, and then $T_{n}f=a_n(f)f$. The Hecke multiplication relations give multiplicativity and the prime-power recurrence of the coefficients.

A heavy symmetric top precesses uniformly when its inclination is constant and its azimuthal speed is constant. For a nonvertical inclination, the condition is $I_1\cos \theta {\Omega }^2-I_3{\omega }_3\Omega +Mgl=0$. The real-root condition depends on whether the body axis points above or below the horizontal.

For a homogeneous material, thermal inertia is $\Gamma =\sqrt{k\rho c_p}$, where $k$ is thermal conductivity, $\rho$ is density and $c_p$ is specific heat capacity. It measures the resistance of the surface temperature to periodic heating and controls the phase lag relevant to the Yarkovsky effect.

If $K$ is the zero-Dirichlet heat kernel on an interval for $w_t=w_{xx}-cw$, integration by parts givesThe opposite endpoint signs are essential. Nonzero boundary values are limits of the full formula; evaluating each homogeneous sine mode at the boundary prematurely loses the forcing.

For an equally weighted mixture of individual survivor functions $F_i$, the population hazard function is $\overline{h}(t)={\sum }_i{F}_i(t)h_i(t)/{\sum }_i{F}_i(t)$. Thus the weights are the surviving proportions, and the cumulative hazard function is $-\log (n^{-1}{\sum }_i{e}^{-H_i(t)})$, not generally the mean individual integrated hazard.

For a positive lapse function $\alpha$, shift vector ${\beta }^i$, and the convention $K_{ij}=-({\partial }_t{\gamma }_{ij}-{\mathcal{L}}_{\beta }{\gamma }_{ij})/(2\alpha )$ for extrinsic curvature of a spatial hypersurface, the time-coordinate harmonic slicing condition yields the displayed advective lapse equation. Combining ${\partial }_t(\sqrt{\gamma }/\alpha )={\partial }_i(\sqrt{\gamma }{\beta }^i/\alpha )$ with the spatial volume evolution identity proves it without setting the shift vector to zero.

If ${\mathcal{P}}_{\ell }$ is the space of degree-$\ell$ homogeneous polynomials and ${\mathcal{H}}_{\ell }$ its harmonic polynomials, then ${\mathcal{P}}_{\ell }={\mathcal{H}}_{\ell }\oplus |x{|}^2{\mathcal{P}}_{\ell -2}$. Under the Fischer inner product, the second summand is orthogonal to the kernel of the Laplacian. Iteration expresses every polynomial restricted to the unit sphere as a sum of spherical harmonics.

A bounded entire harmonic function satisfies $u(x)={\mathbb{E}}_{x}u(B_t)$ because $u(B)$ is a bounded local martingale. The Gaussian heat kernel has a directional derivative with $L^1$ norm $\sqrt{2/\pi }/\sqrt{t}$. Consequently $|u(x)-u(y)|\le \Vert u{\Vert }_{\infty }\sqrt{2/\pi }|x-y|/\sqrt{t}$; letting $t\to \infty$ proves the harmonic Liouville theorem in every dimension.

A handle decomposition builds an $n$-dimensional manifold by successively adjoining $D^h\times D^{n-h}$ along $S^{h-1}\times D^{n-h}$. The attaching data comprise an attaching sphere and its framing of an embedded sphere. Reading a decomposition backwards replaces index $h$ by index $n-h$ and interchanges its attaching spheres and belt spheres.

This Hamming distance on partitions identifies a group with its complement because exchanging the two group names does not change the partition. Complementation is an isometry of the ordinary Hamming distance, so minimizing over this two-element group action gives a quotient metric and preserves the triangle inequality. Negating a zero-one indicator is not complementation; with $\pm 1$ membership vectors, it is.

The Hamming cube is $\{-1,1{\}}^n$, or equivalently $\{0,1{\}}^n$, equipped with Hamming distance. Under its uniform probability measure, the distance from a fixed vertex has the binomial distribution with parameters $n,1/2$. This converts geometric ball sizes into probability tails.

A symplectic Lie group action is weakly Hamiltonian when every fundamental vector field is a Hamiltonian vector field. A Hamiltonian group action additionally supplies Hamiltonians as components of an equivariant moment map. Constants in the component Hamiltonians do not affect their vector fields, but the ability to choose them with the required equivariance is an additional global condition.

An evolution equation has Hamiltonian form when a Hamiltonian operator applied to the variational derivative of a functional gives its time derivative. The induced functional bracket must satisfy the Poisson properties, including Jacobi.

Along Hamilton's equations, spatial derivatives of the Hamiltonian function cancel in its total time derivative. Thus $dH/dt={\partial }_{t}H$. An autonomous Hamiltonian function is conserved; explicit time dependence generally prevents conservation even though phase space is still preserved.

The halting set of a Turing machine $T$ is the set of inputs on which it eventually halts. For a deterministic machine with a designated terminal state, eventual halting is invariant along every transition edge, even when that edge is traversed backwards. This observation justifies passing from forward computations to symmetric equality derivations in a semigroup presentation.

In a finite bipartite graph, suppose every left vertex has positive degree and each edge joins vertices with $d(x)\ge d(y)$ from left to right. Then $|N(A)|\ge {\sum }_{x\in A}{\sum }_{y\sim x}1/d(y)\ge |A|$, so a matching covers the left side.

Let $S$ be constructible in an algebraic group, and $C=\overline{S}$. The closure is a subgroup. Choose a dense open $U\subseteq C$ contained in $S$. For every $c\in C$, the dense opens $U,cU$ intersect, giving $u=cv$ for $u,v\in S$. Hence $c=u{v}^{-1}\in S$ and $S=C$. The Chevalley constructibility theorem therefore proves closedness of homomorphism images.