For ideals $I,J$ in a commutative ring, the ideal quotient is $(I:J)=\{r\in R:rJ\subseteq I\}$. Closure under subtraction and multiplication by arbitrary ring elements proves it is an ideal. If $I\subseteq J=(a)$, then $I=a(I:J)$: write $x=ar\in I$ and observe that $rJ=xR\subseteq I$. Consequently, if also $(I:J)=(b)$, then $I=(ab)$ is a principal ideal.

A non-null hypersurface projection annihilates the unit normal vector and fixes tangent vectors to the hypersurface. It obeys $P^2=P$. Apply it to every tensor index to remove normal components, with the metric used to lower or raise the index pair for covariant slots. For a timelike unit normal in mostly-plus signature it is the spatial projection tensor. A null normal cannot be normalized this way, and a null hypersurface requires additional choices for a corresponding projection.

The codegree of a vertex set $\sigma$ in a hypergraph is the number of hyperedges containing every vertex in $\sigma$. For a singleton this is the hypergraph vertex degree; for two vertices it measures how often those vertices occur together. Subset codegrees of all sizes control the overlap relevant to the hypergraph container theorem.

For a uniform hypergraph with positive average hypergraph vertex degree $d$, the degree measure is $\mu (S)=(nd{)}^{-1}{\sum }_{u\in S}{d}_H(u)$. It is a probability measure and equals $|S|/n$ in a regular hypergraph. Every hypergraph independent set has degree measure at most $1-1/r$, even when its fraction of all vertices is close to one.

A container family is a collection of vertex subsets such that every hypergraph independent set lies in at least one member. A useful family has few members and each member is small in hypergraph degree measure or induces few hyperedges. Container fingerprints encode the members and permit bounds on the number of independent sets.

In the Poincaré half-plane model, a hyperbolic circle of radius $R$ centred at $u+iv$ is the Euclidean circle $(x-u{)}^2+(y-v\cosh R{)}^2=v^2{\sinh }^{2}R$. Its Euclidean centre is shifted upward from its hyperbolic centre, and its Euclidean radius is $v\sinh R$. The lower height $v{e}^{-R}$ is positive. The formula follows by mapping the centre to zero in the Poincare disc model and using radial distance $2\mathrm{artanh}\rho$.

With a consistent force convention, the mobility matrix maps applied generalized forces to rigid-body velocities; it is the inverse of the hydrodynamic resistance matrix. For a sphere with force on the fluid $\mathbf{F}$, its isolated mobility is $1/(6\pi \mu a)$. Coupling between bodies represents hydrodynamic interactions.

For well-separated spheres moving along their line of centres, the leading mobility matrix has diagonal entries $m_i=(6\pi \mu a_i{)}^{-1}$ and off-diagonal entries $h=(4\pi \mu \ell {)}^{-1}$. This is the axial Stokeslet interaction. Finite-size and repeated-reflection corrections enter at higher order.

Two spheres whose prescribed translations have a phase lag can exert a nonzero mean force on the fluid while each returns to its starting position. Their changing separation correlates the interaction strength with the other sphere's velocity. This pump is externally driven, so it is distinct from force-free swimming governed by the scallop theorem.

For a canonical inflaton under the slow-roll approximation, $\dot{\varphi }/H\simeq -M_{\mathrm{Pl}}^2{V}^{\prime }/V$ and $ϵ\simeq {ϵ}_V=(M_{\mathrm{Pl}}^2/2)(V^{\prime }/V{)}^2$. Differentiating $\log {ϵ}_V$ along this trajectory givesHere $\eta =d\log ϵ/d\log a$, not the alternate convention $-\ddot{\varphi }/(H\dot{\varphi })$. The logarithmic definition requires $ϵ>0$; exact de Sitter expansion is treated as a limit.

The physical Hubble radius is $c/H$, the distance at which the instantaneous Hubble law recession speed equals $c$. Its comoving counterpart is $c/(aH)$. Neither radius is generally the same as a particle horizon or a causal boundary: those depend on integrals of the past or future expansion history.

For conformal time $d\tau =dt/a$, the conformal Hubble parameter is $\mathcal{H}=a^{\prime }/a=aH=\dot{a}$, where primes and dots denote conformal and cosmic time derivatives respectively. Its inverse, in units $c=1$, is the comoving Hubble radius. It differs from the ordinary Hubble parameter by one factor of the scale factor.

For the normal flow with zero shift vector, $a_{\mathrm{loc}}=(\det h{)}^{1/6}$ obeys $H_{\mathrm{loc}}=N^{-1}{\partial }_t\log a_{\mathrm{loc}}=-K/3$ in the negative extrinsic curvature of a spatial hypersurface convention. The expansion scalar is $\theta =-K=3H_{\mathrm{loc}}$.

For full column rank $A\in {\mathbb{R}}^{m\times n}$, a full QR factorization gives $Q^{T}A=(R_1,0{)}^T$ with invertible upper triangular $R_1$. The equation $Ax=b$ is consistent exactly when the final $m-n$ entries of $Q^{T}b$ vanish. If consistent, back substitution gives its unique solution. This separates consistency from uniqueness in overdetermined systems.

For a plume of full width $w(z)$, its horizontally averaged concentration is $\overline{C}(z,t)=c(z,t)/w(z)$, where $c$ is horizontally integrated plume concentration. In a line plume with $w=\mu z$, division by $z$ generally changes the location of maximum concentration. A gradient closure for $c$ differs from one for $\overline{C}$: $w{\partial }_z\overline{C}=c_z-c/z$, so integrated diffusive flux $-Kw{\partial }_z\overline{C}$ equals $-K(c_z-c/z)$.

The complex Hopf fibration on the unit sphere of ${\mathbb{C}}^2$ sends $(z_0,z_1)$ to its complex line in ${\mathbb{CP}}^1\cong S^2$. Its fibres are circles. With compatible orientations, its Hopf invariant is one and it generates ${\pi }_3(S^2)\cong \mathbb{Z}$.

For $\varphi :S^{2q-1}\to S^q$, $q>1$, its mapping cone has integral cohomology generators $a$ in degree $q$ and $b$ in degree $2q$. The integer defined by $a⌣a=H(\varphi )b$, with fixed cell orientations, is the Hopf invariant. It measures how an attaching map changes multiplication without changing the additive groups. The complex Hopf fibration has invariant one because its mapping cone is ${\mathbb{CP}}^2$; a constant attaching map has invariant zero.

If $X$ is an $(n-1)$-connected based CW complex, $n\ge 2$, suspension induces an isomorphism ${\pi }_k(X)\to {\pi }_{k+1}(\Sigma X)$ for $k\le 2n-2$ and a surjection for $k=2n-1$. For a sphere, $\Sigma S^n\cong S^{n+1}$, giving the stable range ${\pi }_k(S^n)\cong {\pi }_{k+1}(S^{n+1})$.

Moving the basepoint around a loop acts on a higher homotopy group. Since ${\pi }_n(X)$ is an abelian group for $n\ge 2$, this action extends linearly to its integral group ring, making ${\pi }_n(X)$ a module. For a connected CW complex, its universal cover gives an equivalent description by deck transformations, together with change of basepoint upstairs. Left and right conventions are converted by inversion of the loop. The Hurewicz theorem respects this action whenever it identifies a higher homotopy group with the corresponding homology of the cover.

Based relative maps of a disk into $X$ with its boundary mapped into $A$, modulo homotopies preserving these conditions. Replacing a map by its mapping cylinder gives relative groups for a general map. They fit into the long exact sequence of relative homotopy groups and detect connectivity of the map.