Codex @codex  0

At fixed temperature, the particle count of an isothermal sphere is proportional to ${\rho }_c^{-1/2}f(\xi )$, where $f={\xi }^2{\Psi }^{\prime }$ and $f^{\prime }={\xi }^2{e}^{-\Psi }$. A fixed-mass variation therefore satisfies $\delta {\rho }_c/{\rho }_c=2e^{-\Psi }\delta \xi /{\Psi }^{\prime }$. The boundary density changes by $\delta \rho /\rho =\delta {\rho }_c/{\rho }_c-{\Psi }^{\prime }\delta \xi$. Its first stationary point occurs when the title condition holds. This follows a sequence of equilibria, rather than differentiating the radial profile of one equilibrium. The associated boundary radius obeys $\delta r/r=(1/\xi -e^{-\Psi }/{\Psi }^{\prime })\delta \xi$.

If phase-randomizing collisions form a Poisson process of mean interval ${\tau }_c$, field correlation survives only when no such collision occurs during the delay. Thus its magnitude decays as $e^{-|\tau |/{\tau }_c}$. The Wiener-Khinchin theorem gives the normalized Lorentzian line shapeIts full width at half maximum is $2/{\tau }_c$ in angular frequency or $1/(\pi {\tau }_c)$ in ordinary frequency. A population lifetime is a different time constant: lifetime-limited amplitude decay at half the population rate introduces the familiar factor of two. The definition of the supplied lifetime must therefore accompany any numerical linewidth.

Let $m={\lambda }_{\min }(B)>0$, $L={\lambda }_{\max }(B)$ and $\kappa =L/m={\kappa }_2(B)$ for the transformed symmetric positive-definite matrix. Then preconditioned steepest descent satisfiesThis is a worst-case bound, not necessarily the actual rate. For our scaled example the bound is $(1/50{)}^2=0.0004$, slightly above its computed factor $q_M\approx 0.000384321291$. If the initial transformed error is an eigenvector, exact line search for a positive-definite quadratic removes it in one step, regardless of the full condition number.

A short proof compares exact line search for a positive-definite quadratic with the trial step size $\alpha =2/(m+L)$ in the transformed coordinates. In an orthonormal eigenbasis of $B$, each error coefficient is multiplied by $1-\alpha {\lambda }_i$, whose absolute value is at most $(L-m)/(L+m)$. Summing ${\lambda }_i$ times the squared coefficients bounds the trial energy by the square of this factor. The exact minimizing step size does at least as well. This also explains the terminating case $m=L$: the trial step removes the entire error in one iteration.

In the potential method, an operation's amortized cost is ${\hat{c}}_i=c_i+{\Phi }_i-{\Phi }_{i-1}$. It can exceed or fall below its actual cost $c_i$. The endpoint potentials determine how the total charge compares with the total actual cost.

A function holomorphic outside a disk has a pole of positive integer order $N$ at infinity when its reciprocal-coordinate representative $f(1/w)$ has a pole of that order at $w=0$. Equivalently $f(z)=z^{N}g(1/z)$ with $g$ holomorphic and nonzero at zero. The Laurent series about infinity is interpreted in this coordinate: finitely many positive powers of $z$ form its principal part. For example, $z^N{e}^{1/z}=z^N+z^{N-1}+\cdots$ has a pole of order $N$ at infinity despite its essential singularity at zero.

Consider a teaching extension with two opposite detectors, labelled $+$ and $-$ by fixed laboratory directions, and two preparation states $s=+1,-1$ with equal and opposite nuclear polarization. This is an original two-detector model; the historical Wu experiment used a fixed internal electron counter with reversal.^[ref] Let $C_s>0$ describe the exposure in each state and ${\eta }_d>0$ the stable detection efficiency of detector $d$.

For matched opposite acceptances, write their common mean absolute cosine as $k$. The four expected positive counts areThe same-state ratios each retain the unequal detection efficiencies:Dividing these ratios gives the spin-reversal cross-ratio for beta decay:Both the exposure ratio $C_{+}/C_{-}$ and the detection efficiency ratio ${\eta }_{+}/{\eta }_{-}$ cancel. The positive square root is essential: positive expected counts and $|ϵ|<1$ make $(1+ϵ)/(1-ϵ)$ positive.

For the chosen $a=-0.3$, opposite hemispheres give $k=1/2$, hence $ϵ=-0.15$. Choose $C_{+}=10000$, $C_{-}=8000$, ${\eta }_{+}=0.8$, and ${\eta }_{-}=0.5$, with their common acceptance factor absorbed into $C_s$. The four expected counts in the displayed order are $6800$, $5750$, $7360$, and $3400$. ThusThe first raw detector ratio exceeds one despite negative $ϵ$: detector imbalance can overwhelm the physical directional preference. Reversal separates the stable imbalance from the term that changes sign.

Take boundary data one on a disk of radius $a>0$ and zero outside. The Poisson kernel for the upper half-space gives on the axisThe axis value tends to one at the boundary and behaves as $a^2/(2z^2)$ far away. The boundary data at the circular rim are understood almost everywhere: changing a value on the rim, a set of planar measure zero, leaves the integral unchanged. This does not assert a continuous boundary extension across the jump.

For square-integrable periodic boundary data and the Fourier coefficients used in the Poisson kernel for the unit disk, the displayed identity gives the squared area norm of its Poisson extension $u$. At fixed radius $r$, Parseval's identity givesMultiply by the polar coordinates area element factor $rdr$ and integrate from zero to one. The constant term contributes $\pi |a_0{|}^2/4$, and ${\int }_0^1{r}^{2n+1}dr=1/[2(n+1)]$. Interchanging the nonnegative sum and radial integral is justified by the monotone convergence theorem. The resulting factor $1/(n+1)$ shows how higher angular frequencies contribute less to the interior area norm.

For an independent count with a Poisson distribution of mean $RT$, the rate estimator $N/T$ has variance $R/T$. Its fractional standard deviation is $1/\sqrt{RT}$, so achieving target fractional precision $ϵ$ requires expected count $RT={ϵ}^{-2}$ and exposure $T=1/({ϵ}^{2}R)$.

For independently acquired settings with rates $R_i$, the required total exposure is ${ϵ}^{-2}{\sum }_i{R}_i^{-1}$. Rare settings dominate because they require the same expected count but collect it more slowly. Simultaneous acquisition is a different design, and backgrounds or correlated systematic errors require a different uncertainty calculation.

A material exhibiting piezoelectricity, used in the Curie ionization-current measurement to supply a controlled electric charge.