Electromagnetic minimal coupling replaces the free kinetic momentum in an action or Hamiltonian by a combination involving the electromagnetic potential. For $L=-m\sqrt{{\dot{x}}^a{\dot{x}}_a}-q{A}_a{\dot{x}}^a$, the canonical covector is $P_a=-m{\dot{x}}_a/\sqrt{{\dot{x}}^2}-q{A}_a$, so $P_a+q{A}_a$ recovers the potential-independent covector. Under $A_a\mapsto A_a+{\partial }_a\chi$, the action changes by an endpoint term. The fixed-endpoint equations are the relativistic Lorentz force equation and are gauge invariant.

In relative survival, the background hazard is an individual's expected mortality hazard in a suitable reference population. Its dependence on age, sex and calendar time can be obtained from population life tables. In a heterogeneous cohort, estimating a common excess hazard requires averaging these known background hazards over the current risk set, rather than over the original cohort.

An excess hazard measures mortality above the background hazard. When it is common across individuals, each observed survivor function factors into its individual background survivor function and a common relative survivor function. A nonnegative excess hazard gives a nonincreasing relative survivor function; a negative excess hazard can instead reflect better survival than expected in the reference population.

Under the additive hazard function model $h^{(i)}=h_B^{(i)}+h_E$, the survivor function factorization gives $F_E=F^{(i)}/F_B^{(i)}$, independently of $i$. If the excess hazard is nonnegative, this has a survival-probability interpretation for a hypothetical process with just that hazard. If excess hazards are negative, relative survival can exceed one and is a comparison ratio rather than an ordinary survivor function.

An index state is a chosen orthonormal basis vector in the indexing subsystem of a relative state decomposition. It labels the corresponding partial inner product; it need not be an eigenvector of the subsystem's reduced density matrix.

For the half-normal distribution, the density ratio to $\mathrm{Exp}(\lambda )$ is maximized at $y=\lambda$. Completing the square gives the envelope constant above and acceptance probability $e^{-(y-\lambda {)}^2/2}$. Its logarithmic derivative is $\lambda -1/\lambda$, so the optimal rate is $\lambda =1$. Give an accepted nonnegative value an independent fair sign to obtain the standard normal distribution. The expected number of proposals is $M$, while the probability of rejecting one proposal is $1-1/M$; these are different quantities.

For a nonnegative integrable function $h$ with integral $Z$, its ratio-of-uniforms region has area $Z/2$. A uniform point in the region transforms under $v=ux$ with Jacobian determinant $u$; integrating $2u/Z$ over $0<u\le \sqrt{h(x)}$ gives density $h(x)/Z$. A rectangle containing the region gives a practical rejection sampling algorithm using only uniform random variables, with acceptance $Z/(2\mathrm{area}(B))$.

Write $a=\nu /2$. Against an exponential distribution proposal of mean $\nu$, the chi-squared distribution density has envelope constant $M=a^a{e}^{1-a}/\Gamma (a)$. The density ratio divided by $M$ is $\exp ((a-1)(\log (y/\nu )-y/\nu +1))$. Thus generate $Y=-\nu \log U$ and accept when $\log V\le (a-1)(\log (Y/\nu )-Y/\nu +1)$ for independent uniform distributions. The inequality $\log t\le t-1$ validates the acceptance probability. The accepted probability density function is the target because rejection sampling weights the proposal by its target-to-envelope ratio.

Accepting a fresh uniform deck permutation with a probability depending only on whether its endpoint ranks match preserves uniform individual rank marginals by symmetry. It changes their joint law. For four cards of each of thirteen ranks and rejection probability $p$ on a mismatch, the accepted matching probability is $1/(17-16p)$. The two accepted ranks are independent precisely at $p=1/4$; equality of marginals alone does not imply independence.

With proper prior $\pi (\theta )$ independent of the neutral Kingman's coalescent times, propose $(\theta ,T)$ from their joint prior and accept with probability $e^{-\theta L/2}(\theta L/2{)}^k/k!$. The accepted density is proportional to the prior times the observed-count likelihood, hence is the posterior given $S=k$. The acceptance rate is the prior predictive probability $P(S=k)$. For $k>0$, divide the acceptance function by its maximum $M_k=e^{-k}{k}^k/k!$, achieved at $\theta L/2=k$, to improve the rate to $P(S=k)/M_k$ without altering the accepted distribution. For $k=0$, $M_0=1$.

For $a,b>1$, the maximum of $r$ on $[0,1]$ occurs at $(a-1)/(a+b-2)$. If one parameter is one, the maximum is attained at the corresponding endpoint; if both are one, $r$ is constant. Let $M=\sup r$. Accept an independent uniform distribution proposal $X$ when another uniform draw is at most $r(X)/M$. The accepted probability density function is the beta distribution, and acceptance probability is $B(a,b)/M$.

For target density proportional to $x^{\alpha -1}{e}^{-\beta x}$ on $x>a>0$, use the Pareto distribution proposal $g(x)=b{a}^b{x}^{-b-1}$ with $b>0$. Its density ratio is proportional to $x^{\alpha +b}{e}^{-\beta x}$, maximized at $(\alpha +b)/\beta$ if that point exceeds $a$. The tight envelope yields the highest acceptance probability for this fixed proposal. An inverse draw is $a{U}^{-1/b}$ with $U$ uniformly distributed on $(0,1)$.

If $f\ge 0$, $C=\int f\in (0,\infty )$ and $f(x)\le Mg(x)$ for a proposal probability density function $g$, accept a proposal $Y$ when an independent uniform distribution value is at most $f(Y)/(Mg(Y))$. Integrating the joint acceptance probability density function gives $p_{\mathrm{acc}}=C/M$ and conditional accepted probability density function $f/C$. For $n$ independent proposals the accepted count has binomial distribution with parameters $n,C/M$.

For $0<|Q|<M$, a Reissner-Nordstrom spacetime has an outer event horizon and an inner Cauchy horizon. Between them $r$ is timelike; inside the inner horizon $r$ is spacelike and the $r=0$ curvature singularity is timelike. The exact analytic extension contains repeated exterior blocks beyond the Cauchy horizons. Small perturbations produce severe inner-horizon blueshift and mass inflation, so that exact continuation is not a robust prediction for collapse. At $|Q|=M$ the horizons merge into a degenerate Killing horizon; for $|Q|>M$ the spherical electrovacuum solution instead has a naked singularity.

For $M>0$ and $M\ge |Q|$, put $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The stationary Killing vector field normalized at infinity generates the outer Killing horizon. In an ingoing null chart its covector is $dr$ on that horizon; the Killing equation then gives ${\nabla }_{k}k=\frac{1}{2}{f}^{\prime }(r_{+})k$, where $f=(r-r_{+})(r-r_{-})/r^2$. Thus the positive nonextremal surface gravity is $(r_{+}-r_{-})/(2r_{+}^2)$. It tends to zero when $M=|Q|$, where the horizon becomes a degenerate Killing horizon. The past horizon branch has the opposite signed nonaffinity for the same time-translation field.

For $M=|Q|>0$ and the exterior areal radius $r>M$, integrate $d\rho /\rho =dr/(r-M)$ and choose $\rho =r-M$. The Reissner-Nordstrom metric becomes the displayed form. Constant-time proper distance to $\rho =0$ diverges, but these exterior coordinates do not cover the horizon; ingoing null coordinates are regular there. Infinite spatial throat length therefore does not establish absence of an event horizon.

A neutral radial timelike geodesic of an extremal black hole has conserved specific energy $E=(1-M/r{)}^2\dot{t}$, giving the displayed radial equation. An infalling geodesic reaches $r=M$ with finite nonzero $\dot{r}=-E$ and finite proper time. In ingoing coordinates, outgoing radial null curves have $dr/dv=(1-M/r{)}^2/2$; those starting inside approach $r=M$ only at infinite $v$ and cannot return to the same exterior future null infinity. This distinguishes causal access from length on a static slice, and supplies an operational event horizon test even without strictly trapped round spheres.

For $f=(r-r_{+})(r-r_{-})/r^2$, near-horizon Euclidean geometry has polar form $d{\rho }^2+[f^{\prime }(r_{+})/2{]}^2{\rho }^{2}d{\tau }^2$. Smoothness gives inverse temperature $4\pi /f^{\prime }(r_{+})$ and the displayed formula. This assumes a nondegenerate Killing horizon and time normalized at infinity. Its nonextremal limit is zero at $M=|Q|$.

For full hydrogen ionization after instantaneous reionization, $n_{e0}=3{\Omega }_b{H}_0^2/(8\pi G{m}_p)$ and $H(z)=H_0(1+z)\sqrt{1+{\Omega }_{m}z}$. Integrating Thomson scattering givesFor ${\Omega }_m=1$, $\kappa ={\Omega }_b{C}_H[(1+z_{\mathrm{re}}{)}^{3/2}-1]$. At fixed baryon density and redshift, lowering the matter density reduces the expansion rate and increases the optical depth.

// Target: cosmology.bigb

For independent exponential claims of mean $\mu$ and a zero-based geometric count of parameter $p$, excess of loss reinsurance at per-claim reinsurance retention $M$ has expected recovery $(1-p)\mu e^{-M/\mu }/p$. The zero-based geometric sum of exponential variables has tail $(1-p)e^{-px/\mu }$, so aggregate stop loss reinsurance at reinsurance retention $\tilde{M}$ has expected recovery $(1-p)\mu e^{-p\tilde{M}/\mu }/p$. Equating them gives the displayed reinsurance retention relation. This comparison concerns claim payouts, before any reinsurance premium is charged.