For a source at comoving distance ${\chi }_s$ in a spatially flat universe, define deflection as apparent minus unlensed source direction. The transverse Weyl lensing potential gradient along the ray receives the distance weight $({\chi }_s-\chi )/{\chi }_s$. Integrating the transverse photon acceleration twice and dividing the source displacement by ${\chi }_s$ gives the formula. Reversing the definition of the deflection reverses its sign. Its angular lensing potential is $2{\int }_0^{{\chi }_s}({\chi }_s-\chi )\Psi /({\chi }_s\chi )d\chi$.

At finitely many finite places of a number field, choose square-class parameters for the local unramified quadratic extensions. Local square classes are open, including at dyadic places by the strong form of Hensel lemma. Weak approximation for number fields gives a global parameter in all the prescribed local classes. Its quadratic extension is unramified of residue degree two at those places, so their extended prime ideals remain prime. If no finite place is prescribed, an auxiliary local nonsquare condition ensures the global extension is nontrivial.

To maximize ${\sum }_i\log ({\alpha }_i+x_i)$ for positive baselines, nonnegative allocations and a positive budget $B$, the KKT conditions equalize the shifted values on allocated coordinates. The unique water level solves ${\sum }_i(\tau -{\alpha }_i{)}_{+}=B$. Inactive coordinates have baselines at least the water level. Strict convexity of the negative objective ensures uniqueness, and sorting the baselines gives an efficient water-filling algorithm.

A mixture of two orthogonal pure qubit states has Von Neumann entropy equal to the binary entropy of its weights. Its entropy difference from the first pure state is exactly $h_2(u)$ and its trace distance from that state is $u$. The elementary bound $-(1-u)\ln (1-u)\le u$ gives $h_2(u)\le u{\log }_2(e/u)$; entropy also never exceeds one bit. For small $u$, the leading term is $u{\log }_2(1/u)$.

For time-dependent integral kernels on a Riemannian manifold, composition also integrates over the intermediate point: $(A\ast B)(t,x,y)={\int }_0^t{\int }_{M}A(t-s,x,z)B(s,z,y)d{V}_{z}ds$. It is associative under the usual absolute-integrability hypotheses. The integration simplex gives factorial bounds for iterated convolutions of bounded kernels.

A polynomial-time many-one reduction represents each variable by an adjacent literal pair and each clause by a triangle in a graph. Connect each occurrence vertex to its corresponding literal vertex. A vertex cover of size $n+2m$ must choose one literal vertex per variable and two vertices per clause, and its omitted clause vertex forces a true literal. Conversely a satisfying Boolean valuation supplies such a cover. This establishes NP-hardness of the vertex-cover decision problem from 3-SAT.

The normal velocity is the scalar component of a velocity in a specified unit-normal direction. Its sign depends on the orientation of the normal. At a material interface, no penetration equates the fluid and interface normal velocities; it does not require their tangential components to agree. Surface acoustic impedance uses a fixed inward normal-velocity convention to determine the sign of energy absorbed by a boundary.

Two additive valuations are equivalent when they have the same valuation ring. Their ordered value-group images then identify compatibly with the valuations; for nontrivial real-valued rank-one valuations this is multiplication by a positive constant.

For $K=k_1+k_2+k_3$, $s_2=k_1{k}_2+k_1{k}_3+k_2{k}_3$ and $s_3=k_1{k}_2{k}_3$, the convergent in-in formalism integral is $I={\int }_{-\infty (1-i0)}^{0^{-}}{\tau }^{-2}{\prod }_i(1-i{c}_s{k}_i\tau )e^{i{c}_{s}K\tau }d\tau$. Its real endpoint divergence drops out of the expectation value, but the finite imaginary endpoint term $-c_{s}K$ must be kept. The remaining terms contribute $c_s{s}_2/K+c_s{s}_3/K^2$. Reversing the exponential requires conjugating both the contour and the outside prefactor.

For an increasing differentiable strictly concave utility function satisfying the Inada conditions, the inverse marginal utility maps each positive shadow price to the unique positive payoff with that marginal utility. For normalized CRRA utility, $I(y)=y^{-1/R}$.

An affine utility on probability measures functional satisfies $U_0(p\lambda +(1-p)\mu )=p{U}_0(\lambda )+(1-p)U_0(\mu )$ on mixtures of probability measures. Strict preferences are represented by comparing its real values, and indifference means equal values. This is cardinal utility on lotteries rather than merely an ordinal ordering of deterministic outcomes.

Six is the predicted boundary for ordinary short-range percolation to have mean-field percolation exponents, with logarithmic corrections expected at the boundary. If the Fourier connectivity has leading small-momentum behavior $|k{|}^{-2}$, the triangle integral scales as ${\int }_0^{\epsilon }{r}^{d-7}dr$, finite exactly for $d>6$. This is a scaling argument; rigorous nearest-neighbor results require their own dimensional hypotheses.

An inviscid uniform tube plug driven by entrance excess pressure $KV$ obeys $\rho L\dot{u}=KV$ and $\dot{V}=-\pi a^{2}u$. From rest, the first zero of $V$ occurs after a quarter harmonic period, independent of its initial volume. The maximum speed is proportional to that initial volume. Entrance losses, a reservoir kinetic head or independently prescribed initial flow change this ideal model.

In the Cowling approximation, a mode with phase $e^{i\omega t+im\varphi }$ has rotating-frame frequency $\sigma =\omega +m\Omega$. The horizontal momentum matrix has determinant $4{\Omega }^2-{\sigma }^2$. Eliminating velocities yields a pressure-potential equation with vertical coefficient $1-4{\Omega }^2/{\sigma }^2$ and density-gradient term $2m\Omega {\rho }_{R}W/(R\sigma )$. Frequencies zero and plus or minus $2\Omega$ need the original system rather than this inversion.

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 $0<s<1$.