For a finite positive measure on Euclidean space, cover a compact subset of the maximal superlevel set by finitely many high-density balls. Wiener covering lemma selects disjoint balls whose triples cover it. Their total measure mass is at most the mass of the original measure. Inner regularity of Lebesgue measure yields the displayed bound. Applied to approximation errors, it proves the Lebesgue differentiation theorem.

The fraction of pore volume occupied by a fluid phase. For a completely fluid-filled two-phase pore space the saturations sum to one. Residual saturation refers to fluid remaining immobile under the modeled displacement; it can impose zero relative mobility at the endpoint.

The fraction of total advective Darcy flux carried by one phase when capillary and gravity corrections are omitted. For two phases it is ${\lambda }_w/({\lambda }_w+{\lambda }_n)$. This quantity lies between zero and one for nonnegative phase mobilities. Its nonlinear dependence on fluid saturation determines characteristic speeds and shocks in the Buckley-Leverett equation.

The one-dimensional mass conservation equation for saturation when total Darcy flux is constant and capillary pressure and gravity are neglected. It is a scalar conservation law with dimensional characteristic speed $(Q/\varphi )F^{\prime }(s)$. Shock speeds obey a Rankine-Hugoniot condition; physically admissible solutions satisfy an entropy selection. Capillary diffusion regularizes the saturation transition.

Injecting water to displace oil through a porous reservoir. Spatial permeability contrasts cause preferential flow and early water breakthrough, while unswept slower regions retain oil. The recovery also depends on fractional flow, capillary pressure and phase mixing; a passive streamline model alone does not capture all two-phase effects.

When a nonwetting phase recedes from pores, surface tension and pore geometry can retain a residual saturation $s$. If $h$ is the mobile invaded thickness and $M$ its maximum past value, stored phase volume per plan area is $\varphi [(1-s)h+sM]$. Newly invading pores have $M=h$; during recession $M$ remains fixed. With mobile Darcy velocity $u$, conservation gives advance speed $u/\varphi$ and recession speed $u/[\varphi (1-s)]$. The latter is faster for $0<s<1$.

Two equal point masses can move around the same fixed circle of radius $a$ in diametrically opposite positions. Their stationary centre of mass is the circle's center. A separation $2a$ gives gravitational force $G{m}^2/(4a^2)$ on each mass, and Newton's second law equates it to $ma{\omega }^2$. The common angular speed is therefore fixed by the displayed relation. Opposite tangential velocities produce the same sense of angular rotation.

Build an infinite oracle as an increasing union of finite binary strings. With the diagonal halting set as an oracle, decide whether some finite extension makes a specified Turing functional halt with a binary output at a fresh input. In a yes case, preserve that finite computation and commit the opposing set's bit to the opposite value. In a no case, no subsequent infinite extension can yield a characteristic-function value there. Length growth makes the resulting sets computable in the halting oracle.

For $f(t,x,v)=f_0(x-ta(v),v)$, apply the area formula to the velocity map and bound the initial data by their velocity essential supremum. This gives the displayed mixed Lebesgue norm estimate whenever the weighted inverse multiplicity is essentially bounded. If $a$ is injective and $|\det Da|\ge \alpha >0$, the constant is at most $1/\alpha$. In dimension one a derivative bounded away from zero gives a global diffeomorphism; in higher dimensions determinant bounds alone do not.

The differential operators $Z_j$ commute with $D={\partial }_t+a(v)\cdot {\nabla }_x$. Indeed $[D,{\partial }_{v_j}]=-{\sum }_i({\partial }_{v_j}{a}_i){\partial }_{x_i}$, canceled by differentiating the explicit time factor in $Z_j$. Hence $Z_{j}f$ satisfies the same transport equation as $f$, and obeys the same L2 norm conservation when boundary fluxes vanish. For the standard Jacobian matrix $(Da{)}_{ij}={\partial }_{v_j}{a}_i$, the vector of these derivatives is ${\nabla }_{v}f+t(Da{)}^T{\nabla }_{x}f$.

For the Hermitian difference $X=\rho -\sigma$, define its positive part of a Hermitian operator $Q=X_{+}$ and negative part of a Hermitian operator $R=X_{-}$. Their supports are orthogonal and $|X|=Q+R$. Equal traces give $\mathrm{Tr}Q=\mathrm{Tr}R$, so the trace distance is both $\frac{1}{2}(\mathrm{Tr}Q+\mathrm{Tr}R)$ and $\mathrm{Tr}Q$. This explains why the positive spectral projector attains the variational characterization of trace distance.

A topological space is separable when it has a countable dense subset. The topology is part of the assertion: a dual may be separable in its weak-star topology and not in its norm topology. A compact metric space is separable by taking the union of finite nets at scales tending to zero.

Three intervals joined at one endpoint form a compact connected metric space. It is second countable, Hausdorff and paracompact, but is not a topological manifold. At regular arm points a putative manifold has dimension one; deleting the junction from a small neighborhood instead gives three components, incompatible with an interval chart. This local obstruction precedes any question about a smooth structure.

Glue two copies of a compact manifold along their boundaries to form its double. A collar gives the double a smooth structure when $W$ is smooth. Folding the copies onto $W$ gives a retraction; in particular, inclusion of either copy is injective on homology. For an oriented manifold, use the opposite orientation on the second copy.

For equivalent minima with nearest-neighbour hopping magnitude $\Delta$, paths with $r$ right steps and $s$ left steps have $r-s=n-m$. Summing $(\Delta \tau /\hslash {)}^{r+s}/(r!s!)$ and applying the Fourier representation of a Kronecker delta gives the displayed modified Bessel function kernel in a normalized localized-site basis. Its Fourier exponent gives the tight-binding model energy $E(\theta )=E_{\mathrm{well}}-2\Delta \cos \theta$. Position-endpoint kernels can have an additional common local-wavefunction prefactor.

The fractional mass density decrease per temperature increase at fixed pressure. In the Boussinesq approximation, a temperature excess $\theta$ gives reduced gravity $g^{\prime }=g{\alpha }_T\theta$. It converts a heat budget into a buoyancy budget in natural ventilation.

In static thermal equilibrium, the local temperature times the stationary lapse is constant. Energy measured locally redshifts to conserved Killing energy by that same lapse; requiring a single equilibrium Boltzmann exponent gives the relation. The Killing vector field normalization fixes the constant but not the ratio of temperatures at different locations.

For pore velocity $u(y)=2Uy/H$ on $0<y<H$, with reflecting transverse boundaries, solve $D_T{\chi }^{\prime \prime }=u-U$, ${\chi }^{\prime }(0)={\chi }^{\prime }(H)=0$, $\overline{\chi }=0$. The Taylor dispersion enhancement is $D_T\overline{({\chi }^{\prime }{)}^2}=U^2{H}^2/(30D_T)$. Here $U$ is the mean advective pore velocity, not the Darcy velocity, and $D_L$ is the original longitudinal dispersion. The averaged equation applies after transverse equilibration and on suitably long longitudinal scales.

For a nonempty compact topological manifold $X^d$ without boundary, its suspension is a topological manifold without boundary exactly when $X$ is homeomorphic to $S^d$. The forward implication is a local homology from a link calculation at a suspension vertex: ${\tilde{H}}_j(X)$ is $\mathbb{Z}$ in degree $d$ and zero elsewhere. For $d\ge 2$, put a coordinate ball between two nested cone neighbourhoods. The inclusion of their punctures is a homotopy equivalence but factors through the simply connected punctured ball, forcing ${\pi }_1(X)=0$. Thus $X$ is a homotopy sphere, and the topological generalized Poincare theorem gives the result. The dimensions zero and one follow from the finite-set and circle classifications. Conversely $S(S^d)\cong S^{d+1}$. If manifolds with boundary are allowed, the interval and its disk suspension show why this formulation needs the boundary restriction.