Weak equal-mass small-angle gravitational encounters give a single-star transverse kick $2Gm/(bu)$ and a relative-velocity kick twice as large. With encounter rate $2\pi bdbnu$, the single-star squared-velocity diffusion rate is the displayed $D_v$, while the relative rate is $4D_v$. A chosen reference squared velocity divided by either rate defines a relaxation estimate; specifying which rate and reference are used is essential to a numerical prefactor. If $u=\sqrt{2}\sigma$, the convention $3{\sigma }^2/(4D_v)$ gives $3{\sigma }^3/(16\pi \sqrt{2}n{G}^2{m}^2\ln \Lambda )$. The single-star convention $3{\sigma }^2/D_v$ is four times larger. Velocity-distribution averaging further changes numerical coefficients, while the scaling and Coulomb logarithm in stellar dynamics persist. The factor $n$ must be number density, not a dimensionless population count.

Half the peak-to-peak line-of-sight orbital velocity range of a star in a single-companion Kepler orbit. It equals $K=2\pi a_{\star }\sin i/[P\sqrt{1-e^2}]$, where $a_{\star }$ is the stellar barycentric semi-major axis, $P$ the orbital period, $i$ the sky-plane orbital inclination, and $e$ the orbital eccentricity. The Doppler spectroscopy formula follows from Kepler's third law for the relative orbit, $a^3=G(M_{\star }+M_p)P^2/(4{\pi }^2)$.

In a well-mixed closed-box model of galactic chemical evolution, gas consumed into locked stellar mass supplies a natural cumulative mass coordinate $\delta$. The mean stellar metallicity is ${\overline{Z}}_{\ast }(\delta )={\delta }^{-1}{\int }_0^{\delta }{Z}_{\mathrm{birth}}(s)ds$, giving the displayed derivative relation. If the birth abundance increases linearly, $Z_{\mathrm{birth}}=b\delta$, its cumulative mean is half the current maximum. More generally the maximum of any nonnegative abundance distribution cannot lie below its mean. A fixed initial mass function and the same survival selection are required to identify this mass-weighted distribution with a star-count distribution.

For indicator test functions $g$, the equation has a solution with $\Vert \Delta f{\Vert }_{\infty }\le 1$. One proof uses an immigration--death process with immigration rate $\mu$ and per-particle death rate one. Its potential $u(k)=-{\int }_0^{\infty }(\mathbb{E}g(Z_t^k)-\mathbb{E}g(Z))dt$ satisfies its generator equation, and $f(k)=u(k)-u(k-1)$. Coupling two additional initial particles bounds the second difference of $u$ by ${\int }_0^{\infty }2e^{-2t}dt=1$.

For indicators $I_a$ with means $p_a$, let $B_a$ contain $a$ and suppose $I_a$ is independent of the entire family outside $B_a$. Put $\mu ={\sum }_a{p}_a$, $b_1={\sum }_a{\sum }_{b\in B_a}{p}_a{p}_b$ and $b_2={\sum }_a{\sum }_{b\in B_a\setminus \{a\}}\mathbb{E}(I_a{I}_b)$. The Poisson Stein equation and its bound $\Vert \Delta f{\Vert }_{\infty }\le 1$ give the displayed total variation distance estimate. Stronger Stein factors can improve the bound when $\mu$ is large.

Let $W={\sum }_i{I}_i$ and $\lambda ={\sum }_i{p}_i>0$, where each indicator random variable $I_i$ is independent of all indicators outside $\{i\}\cup J_i$ jointly. Write $V_i={\sum }_{j\in J_i}{I}_j$ and $W_i=W-I_i-V_i$. For a test-event solution of the Poisson Stein equation, independence givesTelescoping each difference and using $\Vert \Delta g{\Vert }_{\infty }\le \min (1,{\lambda }^{-1})$ bounds the total variation distance by that factor times ${\sum }_i{p}_i^2+{\sum }_i{\sum }_{j\in J_i}(p_i{p}_j+\mathbb{E}{I}_i{I}_j)$. Directed dependence neighborhoods are allowed. If $\lambda =0$, the count is identically zero.

With a fixed axial seed field and prescribed localized differential rotation, the winding source is time independent. Early induction gives $B_{\varphi }\sim B_0\Omega t$, while the steady balance has $B_{\varphi }\sim B_0\Omega {\ell }^2/\eta$. Their comparison gives the magnetic diffusion time, not merely the rotation period. It diverges as conductivity becomes infinite. In an unbounded domain, this is a local equilibration estimate on scale $\ell$, not finite-time exact convergence everywhere. Lorentz force feedback may invalidate the prescribed-flow approximation sooner.

A connected nonvacuum perfect fluid with equation of state $p=\rho /3$ obeys the stationary radiation density law $\rho =C/N^4$ with $C>0$. On a compact region of a regular spacetime, a smooth Killing vector field has bounded norm. Consequently $\rho$ cannot tend to zero at a finite regular free surface. A vanishing timelike norm instead makes the density diverge. A radiation-filled cavity can have a material wall with nonzero boundary pressure; this is a different boundary condition.

Conditioning on the first observation of a centered autoregressive process of order one makes the maximum-likelihood estimator of its coefficient the least-squares ratio above, when the optimum is interior. The stationary initial normal distribution contributes an additional factor $\sqrt{1-{\varphi }^2}$ and quadratic term $(1-{\varphi }^2)X_1^2$. These change the exact maximizer, but for a fixed interior stationary parameter their score is order one while the conditional score has order $T$, giving the same leading estimator.

Two Ornstein-Uhlenbeck processes with the same driving Brownian motion have the displayed difference. If $Y_0$ is standard normal and independent of future Brownian increments, replacing it by $-Y_0$ produces another stationary process with the same covariance and second moment, but a different pathwise coupling. Equal initial values are necessary for indistinguishability of stochastic processes.

Put $m=(4-\phi ){\zeta }^2/\phi$. The derivative of the stationary branch of five-mode magnetoconvection at $a^2=0$ is $r_2=1+q+q(2-\phi )/m$. For fixed positive $q$ and small $\zeta$, it changes from a forward to a backward pitchfork bifurcation direction at the displayed value. Expanding the exact numerator $q\phi (2-\phi )+(4-\phi ){\zeta }^2(1+q)$ gives this result. The expansion in amplitude requires $a^2\ll m$; branch direction alone does not determine stability against all other modes.

On the linear span of $1,x$, define $g(a1+bx)=a+b\lambda$. Affine spectral mapping gives $|a+b\lambda |\le \Vert a1+bx\Vert$, making this well-defined with norm one and $g(1)=1$. A norm-preserving Hahn-Banach theorem extension to the whole C-star algebra remains unital; the norm characterization of a positive functional on a C-star algebra makes it a state on a C-star algebra attaining the spectral value.

Write $x=h+ik$ with both terms Hermitian. At least one is nonzero; its norm equals its spectral radius, so it has a nonzero spectral value. Spectral values attained by C-star states gives a state taking that nonzero value on the chosen term. Both state values on $h,k$ are real, so the corresponding complex value on $x$ cannot vanish. This also detects nonzero quasinilpotent elements, which cannot be separated by looking only at their own spectrum.

On a smooth toric variety, the blowup along the orbit closure in a toric variety of a cone generated by part of a lattice basis $v_1,\dots ,v_k$ is the star subdivision at their sum. Locally the center has ideal $(x_1,\dots ,x_k)$; its blowup charts adjoin $x_j/x_i$, producing precisely these subdivided cones. This extends the toric blowup at a torus-fixed point to invariant centers of any codimension.

For independently driven nonnegative squared Bessel processes $X,Y$, put $Z=X+Y$. Where $Z>0$, use Brownian weights $\sqrt{X/Z}$ and $\sqrt{Y/Z}$. Where $Z=0$, fill the weights with $(1,0)$. Their squared norm is always one and their cross-variation is zero, so their stochastic integral is Brownian motion by the Lévy characterization of Brownian motion. Multiplying by $2\sqrt{Z}$ recovers the sum's martingale part even at zero, and the drifts add to $\alpha +\beta$.

For bounded predictable $\sigma$ on $[0,T]$, the stochastic exponential $Z_s=\exp (\frac{1}{2}{\int }_0^s{\sigma }_{u}d{W}_u-\frac{1}{8}{\int }_0^s{\sigma }_u^{2}du)$ is a positive martingale by the Novikov condition. Under $dQ/dP=Z_T$, the Bayes formula for conditional expectation givesIndeed, the square-root stock ratio is $(Z_T/Z_t)\exp (-\frac{1}{8}{\int }_t^T{\sigma }_u^{2}du)$. This exact identity gives the bounds on square-root stock implied volatility by bounding integrated variance pathwise. The Girsanov theorem gives $W^Q=W-\frac{1}{2}\int \sigma du$. The measure is an auxiliary pricing identity for this payoff, not a claim that it is an equivalent martingale measure for the original stock.

For Positive elements of a C-star algebra $x\le y$, inversion reverses the order of strictly positive elements, so $(t1+x{)}^{-1}\ge (t1+y{)}^{-1}$ for $t>0$. Subtracting their square-root resolvent integral in a C-star algebra formulas givesThe integrand is positive, and the positive cone is closed, so the norm limit of these integrals is positive. No commutativity of $x$ and $y$ is required. In particular, $b^2\ge a^2$ for positive $a,b$ implies $b\ge a$, even though squaring itself is not generally order preserving.

After $y={\epsilon }^{2}Y$ and $x=\epsilon \xi$ or a translated version, the layer equation is $Y^{\prime \prime }-\sqrt{Y}+k=0$. Multiplying by $Y^{\prime }$ and matching $Y=k^2$, $Y^{\prime }=0$ fixes the energy constant to $k^3/3$ and gives the displayed factorization. The increasing branch below $k^2$ approaches it at positive infinity; the increasing branch above it leaves it at negative infinity and grows like ${\xi }^4/144$. This supplies both interfaces of the square-root reaction problem with nested boundary layers.

For solutions with independent Brownian drivers in a common filtration, combine the drivers with weights $\sqrt{Z/(Z+Z^{\prime })}$ and $\sqrt{Z^{\prime }/(Z+Z^{\prime })}$ where the sum is positive. At zero use one original driver. The resulting continuous local martingale has bracket $t$ and is Brownian motion by the Lévy characterization of Brownian motion. Multiplying its differential by $\sqrt{Z+Z^{\prime }}$ gives exactly the sum of the original stochastic differentials.

Replace the square-root coefficient near zero by smooth globally Lipschitz cutoffs that vanish below half the cutoff level. With a common Brownian driver, local pathwise uniqueness makes the solutions agree until the larger cutoff level. The exit times increase as the levels decrease. The compatible integrands $1_{\{s\le T_n\}}\sqrt{Z_s}$ have increasing squares and expected time-integrals at most $zt$ by the nonnegative local martingale bound. Their limit is square-integrable on every finite horizon. The Itô isometry gives a continuous global limiting stochastic integral; at a finite limiting exit time its value is zero, and it remains absorbed afterward.