Spatial translation symmetry in a flat FRW metric conserves the covariant canonical momentum $q_i=m{a}^{2}d{x}^i/ds$ of a freely falling massive particle. An orthonormal spatial frame gives $p_{\mathrm{phys},i}=q_i/a$, so physical momentum redshifts as $a^{-1}$ even while the particle changes from relativistic to nonrelativistic motion. The local energy is $E=\sqrt{m^2+q^2/a^2}$ and the coordinate velocity is $d\mathbf{x}/dt=\mathbf{q}/[a\sqrt{q^2+m^2{a}^2}]$.

Suppose a collisionless particle has momentum $p=T_r\propto a^{-1}$ during pure radiation domination. Since $a\propto t^{1/2}$, integrating $\chi ={\int }_0^{t}v(t^{\prime })d{t}^{\prime }/a(t^{\prime })$ gives the displayed comoving distance. The physical distance at observation is $a\chi =H^{-1}\mathrm{arsinh}(y)/y$, $y=m/T_r$. The printed expression without $a^{-1}$ is consequently a physical distance, or a comoving distance in coordinates normalized to $a=1$ at observation. Its ratio tends to one in the relativistic limit and to $\log (2y)/y$ in the nonrelativistic limit. The accumulated scale differs from an instantaneous velocity-over-expansion-rate estimate.

Evaluate the generators as coordinate projections on ${\mathbb{R}}^n$. The lattice-ordered group they generate consists of continuous positively homogeneous piecewise linear functions with finitely many integer-linear pieces. Equivalently its elements are finite joins of finite meets of integer linear forms. The universal property follows because the real line detects every identity of Abelian lattice-ordered groups: if an identity fails, pass to a prime ordered quotient and select the finitely many active linear pieces. The resulting rational linear inequalities have a real solution by the alternative underlying Farkas' lemma, giving a real witness to failure. This is a statement about integer coefficients and positive homogeneity, not arbitrary real coefficients or affine constant terms.

Restricting the topological structure sheaf of a formal completion of a scheme to an open subscheme commutes with its inverse limit: both sheaves are ${\underleftarrow{\lim }}_n({\mathcal{O}}_U/(\mathcal{I}{|}_U{)}^{n+1})$ on $|U\cap Y|$. Equivalently, the identical infinitesimal neighbourhoods have identical restricted structure sheaves. On formal affine charts this uses completed localization of an adic ring.

// Target: toric-geometry.bigb

For a finite morphism that is flat, its direct-image algebra is finite free at every local stalk. On an irreducible scheme, every point specializes from the same generic point. Localizing each stalk further at that generic point shows that all these free ranks equal the generic rank. Hence all fibre algebras have the same dimension, even without assuming finite presentation in advance.

The Euclidean transformations $a(x,y,z)=(x+1,y,z)$, $b(x,y,z)=(x,y+1,z)$ and $s(x,y,z)=(-y,x,z+1/4)$ generate a torsion-free cocompact group with $sa{s}^{-1}=b$, $sb{s}^{-1}=a^{-1}$ and $s^4$ equal to unit translation along $z$. Every nontranslation has nonzero axial displacement modulo integers, hence no fixed point. Its maximal translation subgroup is ${\mathbb{Z}}^3$ and its holonomy is $C_4$.

These are precisely the faithful linear holonomy groups of closed orientable Euclidean three-manifolds. Cyclic cases arise from planar lattice rotations combined with axial screw translations. The noncyclic case is generated by two perpendicular half-turn screws. The list refers to the maximal translation lattice, not to arbitrary finite quotients of a fundamental group.

The theorem classifies compact flat manifolds through their crystallographic covering groups.

For a compact flat Riemannian manifold ${\mathbb{R}}^n/\Gamma$, the translations in $\Gamma$ form a normal full-rank Euclidean lattice of finite index. The quotient is its finite linear holonomy. Affine conjugacy of the torsion-free crystallographic groups classifies these manifolds up to affine diffeomorphism. An abelian $\Gamma$ must consist entirely of translations: commuting with every lattice translation forces each linear part to fix a spanning set of ${\mathbb{R}}^n$.

On a compact flat Riemannian manifold, the integrated Bochner identity for Killing vector fields makes every Killing vector field parallel. Parallel vector fields are exactly the fixed vectors of the linear holonomy. Thus the six orientable closed flat three-manifold types have identity-component isometry dimensions three for the torus, one for each cyclic nontrivial holonomy, and zero for the Hantzsche-Wendt manifold.

Embed a sheaf of abelian groups $\mathcal{F}$ into the sheaf $C(\mathcal{F})(U)={\prod }_{P\in U}{\mathcal{F}}_P$ by all its germs. Restrictions are projections, so $C(\mathcal{F})$ is flasque. Repeat on the successive sheaf cokernels to obtain a canonical flasque resolution. Its complex of global sections computes sheaf cohomology. When $\mathcal{F}$ is itself flasque, the flasque-kernel section-lifting lemma makes every successive quotient flasque and makes the global-section complex exact in positive degrees, proving $H^q(X,\mathcal{F})=0$ for $q>0$ without assuming that vanishing in the construction.

For a steady forced Fourier mode with $|\mathbf{q}|=k$, coupled viscous and magnetic response gives $\hat{\mathbf{u}}=[\eta k^2\hat{\mathbf{f}}+i(\mathbf{B}\cdot \mathbf{q})\hat{\mathbf{g}}/({\mu }_0\rho )]/D$ and $\hat{\mathbf{b}}=[\nu k^2\hat{\mathbf{g}}+i(\mathbf{B}\cdot \mathbf{q})\hat{\mathbf{f}}]/D$. Their mean-field electromotive force contains both forcing helicities and a possible weighted cross-forcing correlation. Jointly isotropic forcing removes the latter; its vanishing at zero mean field alone is insufficient for anisotropic forcing.

For background shear $\Omega y\hat{\mathbf{x}}$ and uniform field $B_0\hat{\mathbf{x}}$, first-order smoothing gives the magnetic shearing-wave amplitude equationIf magnetic relaxation is fast compared with shear, $c_y,c_z\simeq i{k}_x{B}_0(v_y,v_z)/(\eta K^2)$ after free transients decay. Thus the doubled Fourier-mode electromotive quantity $\mathcal{E}=\mathrm{Re}(v_y{c}_z^{\ast }-v_z{c}_y^{\ast })$ is $-B_{0}H/(\eta K^2)$, where $H=i{k}_x(v_y{v}_z^{\ast }-v_z{v}_y^{\ast })$. The spatially averaged mean-field electromotive force of real waves has an extra factor $1/2$. Using the helicity invariant of an inviscid shearing wave, and assuming the approximation holds uniformly in time with $\Omega k_x\ne 0$, givesIndeed $K^2=A+{\Omega }^2{k}_x^2{t}^2$, $A=k_x^2+k_z^2$, and substitution $z=|\Omega k_x|t/\sqrt{A}$ uses ${\int }_{-\infty }^{\infty }(1+z^2{)}^{-2}dz=\pi /2$.

For a locally uniform mean magnetic field, first-order smoothing solves ${\partial }_t\mathbf{b}-\eta \Delta \mathbf{b}=(\overline{\mathbf{B}}\cdot \nabla )\mathbf{u}$. Inserting its causal diffusion propagator into the mean-field electromotive force and taking the trace of the isotropic response tensor gives the displayed relation to kinetic helicity. When diffusion during a correlation time is negligible, it reduces to $\alpha =-{\tau }_H⟨\mathbf{u}\cdot \nabla \times \mathbf{u}⟩/3$. At small magnetic Reynolds number, a quasistatic monochromatic velocity instead gives $\alpha =-⟨\mathbf{u}\cdot \nabla \times \mathbf{u}⟩/(3\eta k_0^2)$. The effective correlation time and diffusion weighting should not be silently discarded.

Three equal-amplitude transverse waves along orthogonal axes, with the same circular polarization, each satisfy $\nabla \times {\mathbf{u}}_j=\sigma k{\mathbf{u}}_j$, $\sigma =\pm 1$. Distinct spatial phases have zero cross averages. The periodic first-order smoothing response to a uniform test field satisfies $({\partial }_t-\eta \Delta ){\mathbf{b}}_j=(\overline{\mathbf{B}}\cdot \nabla ){\mathbf{u}}_j$. Writing $a=\eta k^2$, the response is ${\mathbf{b}}_j=k{\overline{B}}_j[a{\partial }_{\varphi }{\mathbf{u}}_j-\Omega {\mathbf{u}}_j]/(a^2+{\Omega }^2)$. Since ${\mathbf{u}}_j\times {\partial }_{\varphi }{\mathbf{u}}_j=-\sigma U^2{\mathbf{e}}_j$, summing yields the displayed isotropic alpha effect. It includes the resistive and temporal weighting absent from an unqualified instantaneous helicity rule.

The symplectic canonical class is the negative First Chern class of the tangent bundle for a compatible almost complex structure. The contractibility of compatible almost complex structures makes it independent of that choice. It is the class appearing in the symplectic adjunction formula.

Introduce a variable $z_I$ for every nonempty $I\subseteq [k]$ and the homogeneous equations $z_I-{\sum }_{i\in I}{z}_{\{i\}}=0$ for $|I|\ge 2$. In their matrix $A$, partition the columns by the displayed blocks. For each $j$, the vector with coordinate $1$ exactly when $j\in I$ lies in the kernel of a linear map of $A$. Its nonzero coordinates occur in $B_j$ and earlier blocks only; every coordinate in $B_j$ is one. Hence the first block's column sum is zero and every subsequent block sum is in the linear span of earlier columns. The columns condition holds, so Rado's theorem forces a positive monochromatic solution, and all nonempty finite sums of its singleton coordinates have the same colour.

A finite deterministic Turing machine can be represented by a finite semigroup presentation whose generators are tape symbols, state symbols, a boundary marker $h$, and an accept symbol $q$. Defining equalities replace the local scanned symbol and state according to machine instructions, with separate boundary cases for implicit blanks. Only terminal-state equalities erase tape letters and then replace $h{q}_{H}h$ by $q$. A derivation from an initial description to $q$ must encounter a terminal state before its first erasure. Simulation equalities preserve halting membership in either direction: for a deterministic transition $C\to D$, $C$ halts if and only if $D$ halts. Thus reverse use of defining equalities cannot create a false halting certificate. Boundary and erasure rules require explicit checking for every chosen encoding.

Suppose $K$ is an open normal subgroup of a pro-p group $G$ and has a finite pro-p presentation. Choose lifts of generators of the finite quotient $G/K$. Add finitely many relators giving their conjugation and inverse conjugation on the generators of $K$, and replace each quotient relator by its value in $K$. The resulting finite finite pro-p presentation maps onto $G$. Its closed subgroup generated by the kernel generators is isomorphic to $K$, and its quotient by that subgroup is exactly $G/K$, so the map is also injective.

The pullback of $X\to Z\leftarrow Y$ is a universal commutative square with an object mapping to $X$ and $Y$. In the Category of sets, it is the set of pairs with equal images in $Z$.

The equalizer of parallel arrows $f,g:A\rightrightarrows B$ is a universal arrow $e:E\to A$ satisfying $fe=ge$.