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For a monic degree-$n$ polynomial $g$ over a commutative ring $R$, let $A=R[X]/(g)$ and let $\lambda$ take the coefficient of $X^{n-1}$ in the reduced representative. The pairing $(a,b)\mapsto \lambda (ab)$ is unimodular on $1,X,\dots ,X^{n-1}$: reversing one basis order gives a triangular matrix with diagonal ones. When $A$ embeds in a separable field extension, Lagrange interpolation identifies $\lambda (a)$ with $\mathrm{Tr}(a/g^{\prime }(x))$. This yields the derivative formula for the inverse different.

A Hermitian operator satisfies $\Vert X{\Vert }_1={\max }_{-I\le T\le I}\mathrm{Tr}(XT)$. Each diagonal entry of $T$ in an eigenbasis of $X$ lies between minus one and one, giving the upper bound ${\sum }_i|{\lambda }_i|$. The sign operator of $X$ attains it. The substitution $T=2E-I$ converts this formula to optimization over binary POVM effects and proves the Holevo–Helstrom theorem.

A smooth nondegenerate component of the fixed-point set of an isometry is a totally geodesic submanifold. If a geodesic starts tangent to the fixed component, its image under the isometry has identical position and tangent. Uniqueness of the geodesic equation makes the two geodesics coincide throughout their common domain, so the original geodesic stays in the fixed set.

Second-order total generalized variation combines a first derivative with an auxiliary vector field and its symmetric derivative. A standard continuous form isHere $Ew$ denotes the symmetric distributional derivative and the norms are total variations of the corresponding measures. Discrete variants replace these operators by linear maps and use sums of row norms. Unlike first-order total variation, this regularizer accommodates piecewise-affine behavior. A TGV divergence splitting makes its constrained divergence dual amenable to explicit proximal operators.

The order sum $A+B$ is the disjoint union of two total orders, preserving each order and placing every element of $A$ before every element of $B$. For example $\mathbb{N}+\mathbb{Z}$ has a least element but also has elements with infinitely many predecessors.

An order isomorphism is a bijection preserving and reflecting the order relation. It transfers well-foundedness and order type between a coded order and its ordinal interpretation.

Every nonempty bounded-above subset of a total order has a least upper bound. In the real line this is the supremum property used to extend maps defined on an order-dense subset.

A subset of a total order meeting every nonempty open interval between two distinct points. A countable order-dense subset witnesses separability in the order topology of a dense order.

Every pairwise disjoint collection of nonempty open intervals is countable. This is one of the defining properties of a Suslin line.

The topology generated by open intervals and open initial and final rays in a total order. For a dense order without endpoints, a subset is dense in this topology exactly when it meets every nonempty open interval.

Since $T^r$ is a product of circles, the Künneth theorem identifies its cohomology ring with the exterior algebra on the coordinate degree-one classes. Inversion negates each integral degree-one class, so it acts as $(-1{)}^q$ in degree $q$. Even in characteristic two, each degree-one class has square zero.

A group is locally finite when every subgroup that is a finitely generated group is finite. It is a torsion group, since the cyclic subgroup generated by one element must be finite. A restricted direct sum of groups with finite factors is locally finite. A locally finite group can be infinite, while a locally finite finitely generated group is finite by definition.

A closed target $d$-form $\omega$ pulls back under a spacetime field $\varphi$ to a closed form $\alpha ={\varphi }^{\ast }\omega$. Its dual defines an identically conserved topological current. For a compact oriented spatial $d$-manifold and an equally dimensional oriented target with $\int \omega =1$, the spatial integral is the topological degree. Conservation follows without the field equations; changing boundary flux can change the charge.

In a star-dominated Keplerian disk with $H\simeq c_s/\Omega$ and characteristic disk mass $M_D\sim \pi r^2\Sigma$, the Toomre stability criterion becomes $Q\sim (H/r)(M_{\ast }/M_D)$. Thus instability can occur at a disk-to-star mass ratio comparable to the small disk aspect ratio. The numerical coefficient depends on whether $M_D$ means a local annular estimate or integrated enclosed mass.

For a razor-thin fluid disk with sound speed or effective random speed $c$, surface density $\Sigma$ and radial epicyclic frequency $\kappa$, $Q=c\kappa /(\pi G\Sigma )$ compares pressure and rotational support with self-gravity. The local axisymmetric fluid Toomre stability criterion is $Q\ge 1$. For a Keplerian shearing sheet, $\kappa =\Omega$. Collisionless stellar disks use a different numerical normalization; gas drag can permit secular growth even when this fluid parameter exceeds unity.

For a smooth centre with density ${\rho }_c$ and pressure $p_c$, the Tolman–Oppenheimer–Volkoff equation gives $m(r)=4\pi {\rho }_c{r}^3/3+O(r^5)$ and $p(r)=p_c-2\pi ({\rho }_c+p_c)({\rho }_c/3+p_c)r^2+O(r^4)$. A fixed barotropic stellar equation of state leaves one physical central parameter; the additive lapse constant fixes time normalization.

The ratio of enclosed gravitational mass to areal radius in units $G=c=1$. In a spherical star the local value is $m(r)/r$, and the surface value is $M/R$. Bounds on this ratio constrain relativistic stellar models; it is unrelated to topological compactness.

A timelike vector field assigns a timelike vector continuously or smoothly to every event in its domain. A globally defined such field gives a time orientation, but it need not be a Killing vector field or a timelike geodesic vector field.

The matrix in time reversal of a Dirac field depends on the convention for the spin-flip phases and on the sign of the spinor identity used when relabeling the mode expansion of a Dirac field. If ${ϵ}_s=(-1{)}^{1/2-s}$ and ${ϵ}_s{u}^{-s\ast }(p_T)=-{\gamma }^{5}C{u}^s(p)$, the relabelled coefficient is ${ϵ}_{-s}{u}^{-s\ast }(p_T)={\gamma }^{5}C{u}^s(p)$, giving $B={\gamma }^{5}C$. A common phase changes $B$ but leaves its gamma matrix conjugation property and the induced fermion bilinear transformation unchanged. The antiunitary operator acts on both numerical coefficients and creation or annihilation operators.

A time-ordered product sorts operator insertions from latest time on the left to earliest time on the right. Reordering odd fermionic operators contributes the sign of their permutation. For bosonic $A,B$, $T[A(t)B(t^{\prime })]=\theta (t-t^{\prime })A(t)B(t^{\prime })+\theta (t^{\prime }-t)B(t^{\prime })A(t)$. Vacuum expectations of free time-ordered products are evaluated by the Wick theorem.