The virial temperature characterizes the thermal energy of gas supported in a gravitational potential. It is of order $\mu m_{p}GM/(k_{B}R)$, with a structural coefficient determined by the density profile. For a uniform self-gravitating gas sphere the virial theorem gives $k_{B}T=GM\mu m_p/(5R)$; a conventional halo circular-speed definition often uses $k_{B}T\simeq \mu m_p{v}_c^2/2$.

For positive masses in their centre of mass frame, ${\sum }_{j<k}{m}_j{m}_k|{\mathbf{r}}_j-{\mathbf{r}}_k{|}^2=M{\sum }_k{m}_k{r}_k^2$. Thus the scalar inertia $I=\frac{1}{2}{\sum }_k{m}_k{r}_k^2$ equals $(2M{)}^{-1}{\sum }_{j<k}{m}_j{m}_k{r}_{jk}^2$. The identity compares total spatial extent with pairwise separations.

For an isolated Newtonian point-mass system, $\ddot{I}=T+E\ge E>0$. Twice integrating gives a quadratic lower bound on $I$. The maximum-separation inertia bounds then give at-least-linear asymptotic growth of the diameter $R$ for a trajectory existing for arbitrarily large times. The bound does not assert instantaneous monotonicity or escape of every particle.

The vertical velocity is the component of a fluid velocity field along the selected upward vertical direction. In incompressible flow, impermeable horizontal boundaries impose $w=0$. Eliminating pressure often yields a closed hydrodynamic stability equation for this component.

Set-valued analysis studies relations whose output at one input is a set. Solution maps, subdifferentials and generalized normal mappings are important examples. Graph geometry provides definitions of continuity, local stability and differentiation.

Normal cones describe generalized supporting directions to sets. For a convex set, the normal condition is $⟨w,z^{\prime }-z⟩\le 0$ for every $z^{\prime }$ in the set. For nonconvex sets, the Fréchet normal cone and the limiting normal cone distinguish local regular support from limits of such supports.

Writing $f=1-2M(v)/r$, the radial Christoffel symbols are ${\Gamma }^v{}_{vv}=M/r^2$, ${\Gamma }^r{}_{vv}=fM/r^2+M^{\prime }/r$ and ${\Gamma }^r{}_{vr}=-M/r^2$. The angular couplings are ${\Gamma }^v{}_{\theta \theta }=-r$, ${\Gamma }^r{}_{\theta \theta }=-fr$, with ${\sin }^2\theta$ factors for the corresponding $\varphi \varphi$ terms, ${\Gamma }^{\theta }{}_{r\theta }={\Gamma }^{\varphi }{}_{r\varphi }=1/r$, ${\Gamma }^{\theta }{}_{\varphi \varphi }=-\sin \theta \cos \theta$ and ${\Gamma }^{\varphi }{}_{\theta \varphi }=\cot \theta$. The lower indices are symmetric. The plus sign of $M^{\prime }/r$ follows from the ingoing $+2dvdr$ cross term.

The two radial null geodesic families are $v=\text{constant}$ and $dr/dv=(1-2M(v)/r)/2$. The first has affine radius, while the second has tangent $k={\partial }_v+(f/2){\partial }_r$ obeying ${\nabla }_{k}k=(M/r^2)k$, so advanced time is generally a nonaffine parameter. An affine parameter $\lambda$ for the outgoing family satisfies $d\log (dv/d\lambda )/dv=-M/r^2$.

Every nonzero nonunit in a unique factorization domain is a finite product of irreducible elements. Such a factorization is unique up to permutation and multiplication of the factors by units. Irreducibles are prime elements, so divisibility can be checked by comparing their multiplicities. This applies to a polynomial ring over a field and is useful for intersections of localizations inside its fraction field.

A uniformly continuous real function on a closed bounded interval can be approximated uniformly by finite step functions. Given $\epsilon >0$, choose a mesh smaller than the uniform-continuity distance and set the approximant equal to a sampled value on each interval. Then the supremum norm of the approximation error is at most $\epsilon$. If the function has a Lipschitz bound $L$, a mesh of maximum length $h$ gives the explicit error bound $\Vert f-g{\Vert }_{\infty }\le Lh$.

For a continuous periodic function setIt measures smoothness through second differences without requiring a derivative. For $h>0$, factoring the translation difference gives ${\omega }_2(f,t)\le (1+t/h{)}^2{\omega }_2(f,h)$. The second-difference integral formula gives ${\omega }_2(f,h)\le h^2\Vert f^{\prime \prime }{\Vert }_{\infty }$ for twice continuously differentiable functions.

A closed superstring theory whose left and right Ramond sector projections have opposite ten-dimensional Majorana-Weyl chiralities. It has 32 real spacetime supercharges and is nonchiral. Its massless fields form type IIA supergravity; the RR gauge potentials have odd degrees, including a one-form and a three-form.

A closed superstring theory whose left and right Ramond projections have the same ten-dimensional chirality. It has 32 real supercharges and is chiral. Its low-energy fields form type IIB supergravity. Circle T-duality exchanges this theory with type IIA superstring theory.

The closed-string sector with NS sector excitations on both chiral sides. Its massless transverse tensor product is $8_v\otimes 8_v=35_v+28+1$, giving the graviton, Kalb–Ramond field and dilaton. These fields are common to the type II theories.

The closed-string sector with Ramond sector states on both chiral sides. The left and right spinor tensor product describes differential-form gauge fields. Opposite chiralities in type IIA superstring theory give the massless one-form and three-form, with transverse dimensions $8$ and $56$. Same chiralities instead give the type IIB form-field content.

A filling box model describes a turbulent plume rising through a confined fluid and spreading beneath its ceiling. Fluid entrainment transfers initially unmodified ambient fluid into the plume; the discharged fluid forms a descending filling-box first front. A maintained heat source can have negligible source volume flux while maintaining nonzero buoyancy flux. A finite injected fluid flux requires an equal exhaust if the container volume is fixed.

A buoyant plume with insufficient momentum for the corresponding pure plume balance. In the constant-entrainment top-hat model it has plume balance parameter $\Gamma >1$, unlike the momentum-excess forced plume. The equations accelerate the plume and drive the balance parameter toward one downstream. The definition depends on the chosen flux and entrainment convention; it is not merely a synonym for slow vertical speed.

An eddy diffusivity is a transport closure representing unresolved turbulence by a mean scalar flux proportional to a negative mean concentration gradient. A local estimate is $K\sim U\ell$, where $U$ and $\ell$ are turbulent speed and length scales. In a line plume with constant characteristic speed and width proportional to $z$, this gives $K\propto z$. Its coefficient is an independent closure parameter; fluid entrainment conservation alone does not determine it.

Mixing by turbulence is represented approximately by a diffusive flux down a mean concentration gradient. Its coefficient is the eddy diffusivity. A constant-coefficient model gives a $D{\nabla }^2\sigma$ term and a smoothing rate $D{k}^2$ for a Fourier mode. Effective diffusion can suppress slow gravitational concentration even when microscopic diffusion is negligible.

Product-to-sum formulas turn products of trigonometric functions into sums at combined arguments. The identity $2\sin A\sin B=\cos (A-B)-\cos (A+B)$ is especially useful for verifying orthogonality of sine Fourier series modes on an interval.