For a Lie algebra ideal $I$, the quotient vector space $\mathfrak{g}/I$ has Lie bracket $[x+I,y+I]=[x,y]+I$. The ideal condition makes this independent of representatives. Its Lie algebra ideals correspond to the ideals of $\mathfrak{g}$ containing $I$, by inverse image under the quotient Lie algebra homomorphism.

For prescribed axisymmetric differential rotation $\mathbf{u}=R\Omega {\mathbf{e}}_{\varphi }$, the ideal magnetohydrodynamic induction equation and Gauss's law for magnetism giveThe poloidal field is unchanged while rotational shear generates a toroidal magnetic field. For time-independent $\Omega$, the toroidal component grows linearly until neglected dynamical feedback becomes relevant.

In curvature $-1$, a regular $n$-gon of circumradius $R>0$ has interior angle $\alpha (R)=2\arctan (\cot (\pi /n)/\cosh R)$, by the right-triangle angle form of the hyperbolic law of cosines. Thus its hyperbolic polygon area is $(n-2)\pi -n\alpha (R)$, continuously and strictly increasing from $0$ to $(n-2)\pi$. Every area in that open interval is therefore attained at exactly one circumradius.

For a one-dimensional first-order system $v_t+A(v,x,t)v_x=s$, hyperbolicity means that the coefficient matrix has real eigenvalues and a complete set of eigenvectors. Distinct real eigenvalues give strict hyperbolicity and separate characteristic curves. Along a curve of speed $\lambda$, a left eigenvector $\ell A=\lambda \ell$ gives the compatibility equation $\ell dv/dt=\ell s$.

If $g$ has infinite order in a hyperbolic group, the map $n\mapsto g^n$ is a quasi-isometric embedding of the integers. Equivalently $|g^n|\ge c|n|-C$ for some $c>0$. This is the homogeneous quasigeodesic theorem: local-to-global control of metric geodesics in a hyperbolic Cayley graph gives a quasigeodesic orbit for a suitable conjugate of a positive power; undoing conjugation and passing between consecutive powers changes only uniform constants. It implies $\tau (g)>0$. The theorem and its hypotheses are given in Löh, Theorem 6.5.9. It is not true for arbitrary finitely generated groups, as the Baumslag-Solitar group relation $ab{a}^{-1}=b^3$ shows.

Using the idealized CNO cycle law $ϵ\propto \rho T^{16}$ and constant electron-scattering opacity, homology relations for radiative stars giveThese retain ideal-gas pressure support; significant radiation pressure or convection changes the model.

With proton–proton chain energy generation $ϵ\propto \rho T^4$ and Kramers' opacity law $\kappa \propto \rho T^{-7/2}$, homology relations for radiative stars giveThe Hertzsprung-Russell diagram relation has $d\log L/d\log T_{\mathrm{eff}}=284/69$, with effective temperature increasing toward the left in the usual plotting convention.

A laminar column can contract through nearly hydrostatic equilibrium when its cooling time greatly exceeds its vertical dynamical timescale of a disk. For cooling $A{T}^{1+\beta }$ and fixed column mass, a similarity solution has $\rho \propto {\eta }^{-1}$, $T\propto {\eta }^2$, $P\propto \eta$, and vertical velocity $u_z=\dot{\eta }z/\eta$. The coefficient $C$ also depends on the initial mass density normalization.

Choose a cycle $\Gamma$ surrounding $\sigma (x)$ and a cycle $\Delta$ surrounding $f(\sigma (x))$, with $f(\Gamma )$ inside the region of winding number one for $\Delta$. The algebra-homomorphism property givesSubstitute this into the contour formula for $g(f(x))$, interchange integrals, and apply the Cauchy integral formula to the inner $\zeta$ integral. This proves composition as an equality of algebra elements, not merely an equality on characters, which need not separate a nonsemisimple algebra.

The displayed series converges absolutely in a Banach algebra. It is the holomorphic functional calculus value of the analytic branch $z\mapsto \log (1-z)$ on the unit disk. The composition rule for holomorphic functional calculus and $\exp (\log (1-z))=1-z$ give $\exp (\log (1-x))=1-x$. Thus every sufficiently small norm perturbation of the identity is an exponential.

Near a non-characteristic plane $x=0$, use the small cap ${\omega }_{\epsilon }=\{x>0,x+y^2<\epsilon \}$. Its curved face is non-characteristic by continuity of the principal symbol. A polynomial coordinate change flattens that face. Uniform real analytic coefficient bounds and the uniform Cauchy radius for polynomial forcing solve $P^{\ast }v=g$ throughout a common cap, with zero boundary jet on the curved face. The complete boundary-jet condition for formal adjoints then gives ${\int }_{{\omega }_{\epsilon }}ug=0$ for every polynomial $g$. Uniform approximation of $\bar{u}$ gives $\int |u{|}^2=0$. Repeating on the other side of the plane proves two-sided local uniqueness.

The full norm is $|u{|}_{k,\alpha ;U}=|u{|}_{k;U}+[D^{k}u{]}_{\alpha ;U}$, combining the derivative supremum norm with the top-derivative Hölder seminorm. At order zero it is the sum of the supremum norm and the Hölder seminorm.

If $D$ is nonzero numerically with $D^2=0$ and $D\cdot A>0$ for ample $A$, then $D\cdot E=0$ implies $E^2\le 0$, with equality only when $[E]$ is proportional to $[D]$. To see this, use the Hodge index theorem for algebraic surfaces to split off the positive $A$ direction and diagonalize the remaining negative definite form. Orthogonality to a nonzero isotropic vector then leaves a negative semidefinite form whose radical is precisely that vector's span.

The Hochschild cohomological dimension of an associative algebra over a field is $\mathrm{Dim}(A)={\mathrm{pd}}_{A\otimes A^{\mathrm{op}}}A$. Equivalently, $H{H}^n(A,M)$ vanishes above that dimension for every bimodule $M$. A bound by one implies $H{H}^2(A,A)=0$ and hence formal rigidity from vanishing second Hochschild cohomology.

Write $f\circ g={\sum }_{i=0}^{p-1}(-1{)}^{i(q-1)}f{\circ }_{i}g$ for insertion of a degree-$q$ cochain into a degree-$p$ one. With the unsigned Hochschild cup product and the left graded Leibniz rule, take $[f,g]=(-1{)}^{(p-1)(q-1)}f\circ g-g\circ f$. It descends to Hochschild cohomology and gives a Gerstenhaber algebra. The alternative insertion bracket $f\circ g-(-1{)}^{(p-1)(q-1)}g\circ f$ differs by the displayed degree sign and obeys the corresponding right rule. Degree-one brackets are commutators of derivations in either convention.

For the polynomial ring $A=k[X_1,\dots ,X_r]$ in characteristic zero, antisymmetrization identifies $H{H}^{\ast }(A,A)$ with ${\bigwedge }_A^{\ast }{\mathrm{Der}}_k(A)$. The map on a wedge of $p$ derivations is $\frac{1}{p!}{\sum }_{\sigma \in S_p}\mathrm{sgn}(\sigma ){\prod }_j{D}_{\sigma (j)}(a_j)$. The Hochschild cup product becomes the exterior product, and the left Gerstenhaber bracket becomes the left Schouten-Nijenhuis bracket. This assertion requires a smoothness hypothesis when generalized beyond polynomial rings.

For a planar hit-and-run sampler, let $\ell (x,u)$ be the chord length and $a(x)$ the fraction of directions with positive chord length. With zero-length chords redrawn, the probability density function isfor almost every $y$ in the body. Interior points have $a(x)=1$. The two signed representations of a direction contribute the factor two in the polar-coordinate calculation. If the body has diameter $D$, a version of this density is bounded below by $1/(\pi D^2)$. This is a global Doeblin condition after normalization by the body's area.

The planar Hill equations describe small displacements about a circular Kepler orbit in a frame rotating at angular frequency $n$. With $x$ radial and $y$ along the orbit, their unforced solution isDirect differentiation gives $\ddot{x}-2n\dot{y}-3n^{2}x=0$ and $\ddot{y}+2n\dot{x}=0$.

The $A,B$ terms describe epicyclic motion with tangential amplitude twice the radial amplitude. The constant radial offset $C$ causes an accumulating along-track displacement through Keplerian shear, while $D$ is an initial along-track offset. This separation makes the solution useful for comparing nearly neighbouring orbits and for identifying which motion requires additional encounter forcing.

If a fixed graph $F$ has chromatic number $r+1$, every extremal $F$-free graph $G$ on $n$ vertices has minimum degree of a graph $(1-1/r+o(1))n$. Use a high-minimum-degree multipartite stability subgraph $H$: replacing any vertex by one adjacent to all but one class of $H$ preserves $F$-freeness. Any supposed new copy of $F$ can replace the new vertex by an unused common neighbour in the omitted class. Extremality therefore gives the lower bound on every degree of a vertex; the Erdős-Stone theorem gives the upper bound on the average degree of a vertex.

For generators normalized by $\mathrm{Tr}(t^a{t}^b)={\delta }^{ab}/2$, a constant adjoint vacuum expectation value ${\Phi }_0$ gives this matrix. A generator commuting with ${\Phi }_0$ corresponds to a massless gauge boson.