An arrow $f:A\to B$ induces $Pf:PB\to PA$ by inverse image of predicates. Transposition of predicates on $A\times B$ gives $\mathcal{E}(A,PB)\cong \mathcal{E}(B,PA)$ and the adjunction $P^{\mathrm{op}}⊣P$. The induced monad on the topos is double power-object formation; it is distinct from the covariant powerset monad on sets.

For a real positive linear functional on the space of continuous functions on a compact space, $-\Vert f{\Vert }_{\infty }1\le f\le \Vert f{\Vert }_{\infty }1$ gives $|\varphi (f)|\le \varphi (1)\Vert f{\Vert }_{\infty }$. Evaluation at $1$ gives equality of the operator norm and $\varphi (1)$. Thus positivity alone already implies continuity in the supremum norm.

A real positive linear functional on a vector subspace of $C(K)$ containing $1$ extends positively to all of $C(K)$. Its norm is $\varphi (1)$ by order bounds. If this is positive, apply the Hahn-Banach theorem to obtain an extension of the same norm, normalize it to value one at $1$, and use the unital contraction positivity criterion. If $\varphi (1)=0$, the functional is zero and its zero extension suffices.

The symmetric steady porous thermal plume has $w=W{\mathrm{sech}}^2(x/b)$, $T-T_0=w/\beta$ and $\psi =-Wb\tanh (x/b)$. Its exact normalizations are $W{b}^2=6\kappa z$ and $(4/3)W^{2}b=Q$, so $b=(48{\kappa }^2{z}^2/Q{)}^{1/3}$ and $W=(3Q^2/(32\kappa z){)}^{1/3}$. The streamfunction supplies lateral entrainment required by continuity.

In a porous medium equation with exponent $m>1$, the effective diffusion coefficient vanishes at zero density. A nonnegative compactly supported source profile can therefore retain a finite front for every finite positive time. For example, the radial cubic-diffusion source profile has support radius proportional to $t^{1/6}$. This notion concerns a moving diffusion front; it does not require the constant characteristic cone used for a wave equation.

For any set of lengths $A$, choose at length $n$ a constant-zero Boolean circuit when $n\notin A$, and an AND of all inputs when $n\in A$. This uses $O(n+1)$ gates. If $A$ is undecidable, so is $U_A$, yet it belongs to P/poly. Since all NP languages are decidable by certificate enumeration, this proves P/poly is not equal to NP; it does not resolve containment in the other direction.

For unit-rate Poisson process arrivals $Z_i$, $X_p$ is finite almost surely when $p>1$. There are finitely many positive arrivals before time one; the expected contribution afterwards is ${\int }_1^{\infty }{z}^{-p}dz<\infty$, by the Poisson expectation identity proved on simple functions and extended by monotone convergence. Merging independent processes of rates $1$ and $\rho$ produces rate $1+\rho$; rescaling their arrivals by $1+\rho$ restores unit rate. Hence the sum of their inverse-power contributions has law $(1+\rho {)}^p{X}_p$. In particular two independent unit-rate inverse-square sums add to a variable with law $4X_2$.

A pointwise periodic self-map has a uniform positive period exactly when its finite cycle lengths are bounded. A uniform period is divisible by every length; conversely, bounded lengths divide the least common multiple of $1,\dots ,M$ for some bound $M$. Arbitrarily many bounded-length cycles are allowed. Every such map on a finite set has a uniform period. On any infinite set containing a countably infinite subset, cycles of unbounded finite lengths and fixed points elsewhere give a counterexample. Thus, with the usual axiom of choice assumption, the sets on which every pointwise periodic self-map has a uniform period are exactly finite sets.

A morphism $e:A\to B$ in a pointed category is pseudo-epic when, for every $t:B\to Z$, the equation $te=0$ forces $t=0$. If a categorical cokernel exists, this is equivalent to that cokernel having a zero object as codomain. Indeed all zero composites then factor uniquely through zero; conversely the cokernel itself must be zero, and being both zero and epic forces its codomain to have identity zero, hence be a zero object. Every epimorphism is pseudo-epic. In a preadditive category, subtraction shows the converse: $ue=ve$ implies $(u-v)e=0$, hence $u=v$. In a general pointed category no subtraction is available.

For a nearly radial Kepler orbit, $q\simeq h^2/(2GM)$, so modest changes in specific angular momentum can strongly change pericentre. A planetary torque acting for a reference orbital time gives $\delta h\sim (m_p/M)\sqrt{GM{a}_p}$. Coherent kicks change $q$ by order itself after $N\sim (M/m_p)\sqrt{q/a_p}$ orbital-scale encounters, whereas uncorrelated kicks require an order-$N^2$ random-walk count. Both estimates are conditional on encounter geometry: a close encounter with a smaller impact parameter can give a much larger kick.

For a fixed mass on a Keplerian circular orbit, $L_s=M_s\sqrt{G{M}_{\ast }{r}_0}$ has derivative $M_s{r}_0\Omega /2$. The torque on the satellite is minus the torque $\Gamma$ on the disk. This gives the displayed drift law. In the density-slope satellite torque model, ${\dot{r}}_0=-(8\chi /9)\beta q({\Sigma }_0{r}_0^2/M_{\ast })r_0\Omega (H/r_0{)}^{-2}$.

A plane-parallel atmosphere is geometrically accurate only if the relevant line-forming region is thin relative to the stellar radius. An isothermal estimate gives $H=k_{B}T/(\mu m_u{g}_{\mathrm{eff}})$. Radiative acceleration reduces the effective gravity and increases extension. A small photospheric $H/R$ does not justify a plane-parallel description of wind lines formed far above that photosphere.

For integer $r\ge 1$, the truncated power is zero for $x\le t$ and equals $(x-t{)}^r$ for $x>t$. It is globally $C^{r-1}$ and its $r$th derivative has a jump at $t$. Order zero is the step-function convention $1_{\{x>t\}}$. Knot divided differences of these functions produce B-splines.

A physical-strip pole of a diagonal S-matrix can come from exchange of an existing particle in the crossed channel. For external masses $m_a,m_b$, use the momentum difference, rather than the sum, to identify the exchanged mass through the displayed invariant. Crossing symmetry turns the pole at $i\alpha$ into a direct pole at $i(\pi -\alpha )$ of the particle-antiparticle channel. For $m_B=2m\cos (u/2)$, the pole of $S_{BA}$ at $iu/2$ has $t=m^2$ and exchanges the existing charge-one particle; it is not a new charge-three bound state.

For $J_h^{\mu }={\sum }_f{Q}_f{\bar{q}}_f{\gamma }^{\mu }{q}_f$ without a factor of $e$, use $i\int e^{iqx}⟨T{J}_h^{\mu }(x)J_h^{\nu }(0)⟩d^{4}x=(q^{\mu }{q}^{\nu }-q^2{g}^{\mu \nu }){\Pi }_J(q^2)$. The inclusive tensor is $W_J^{\mu \nu }=2\pi {\rho }_J(s)(q^{\mu }{q}^{\nu }-s{g}^{\mu \nu })$. A massless free-quark calculation gives ${\rho }_J=N_c{\sum }_f{Q}_f^2/(12{\pi }^2)$; strong interactions alter this function.

For $q=(q^0,\mathbf{q})$ with $\mathbf{q}\ne 0$, the Coulomb gauge propagator hasThe temporal component is instantaneous; the spatial part projects onto transverse photon polarizations. Although these components are not individually Lorentz covariant, their contraction with conserved external currents equals the covariant answer. Zero-spatial-momentum limits should be taken after combining the current contractions.

Define the inclusive photon-field tensor for hadronic final states by $W_A^{\mu \nu }=2\pi {\rho }_h(s)(-g^{\mu \nu }+q^{\mu }{q}^{\nu }/s)$ for timelike $q$ with $s=q^2$ and $q^0>0$. Here each final state has relativistic normalization and the sum includes the four-momentum delta function. This hadronic photon spectral density has mass dimension $-2$. At leading electromagnetic order, ${\rho }_h=e^2{\rho }_J/s$, with ${\rho }_J$ the dimensionless hadronic electromagnetic-current spectral density. Stating the tensor convention prevents mismatched powers of $s$ or $e$ in an inclusive cross-section.

The interaction vertex $-i\lambda$ and scalar propagator $-i/(p^2-iϵ)$ give one four-point bubble diagram for each of three momentum channels, each with Feynman-diagram symmetry factor $1/2$. The two-point tadpole diagram is zero as a scaleless integral in dimensional regularization. If full proper vertices include the free quadratic kernel, ${\hat{\tau }}_2^{(0)}=-p^2$; the interaction self-energy convention instead starts at zero.

The excess free energy per area of a solid-liquid phase boundary. Curvature changes the equilibrium temperature through the Gibbs--Thomson relation. Crystal anisotropy can make this energy and its associated surface stiffness depend on orientation.

Premelting is the persistence of a liquid layer at a surface or interface below the bulk melting temperature. Molecular disjoining pressure and phase equilibrium can stabilize the film. Its hydraulic connection to a reservoir permits frost heave when water freezes onto the base of loaded ice.