If the entire integrand path is independent of the integrating Brownian motion, conditioning on that path gives a deterministic square-integrable integrand. The Itô isometry and Gaussian approximation by simple integrands show that its stochastic integral is conditionally centered normal with the stated variance. A strictly positive finite conditional variance implies that the unconditional distribution has no atoms.

Globally Lipschitz drift and diffusion coefficients have linear growth and give a nonexplosive strong stochastic solution for a prescribed initial state and Brownian motion. The solution satisfies pathwise uniqueness. Local versions apply after stopping inside compact regions.

A long disturbance trapped on a depth step is sometimes called a double Kelvin wave. With $H_{+}>H_{-}$, $f>0$ and $c_{\pm }=\sqrt{g{H}_{\pm }}$, cross-step geostrophic balance and the along-step acceleration yield the transport equation shown for a background height jump ${\eta }_0\mathrm{sgn}(x)$. Its zero-initial-amplitude solution is $A={\eta }_0[\mathrm{sgn}(x)-\mathrm{sgn}(x-(c_{+}-c_{-})t)]$. The adjusted region diverts cross-step volume flux along the step; the propagating front still carries flux. This outer approximation does not resolve fast adjustment at the fronts.

A transiting body hides different velocity regions of a rotating stellar disc and causes an anomalous measured Doppler shift. Its time dependence constrains the sky-projected spin-orbit angle, which is not generally the full stellar obliquity.

Integrating galactic stellar mass against a $x^{-7/4}{e}^{-\sqrt{x}}$ mass function requires ${\int }_0^{\infty }{x}^{-3/4}{e}^{-\sqrt{x}}dx$. Substitution $u=\sqrt{x}$ turns it into $2\Gamma (1/2)=2\sqrt{\pi }$. Thus the stellar density is $2\sqrt{\pi }$ times characteristic number density times characteristic galactic stellar mass, in the same volume convention as the mass function.

Consider a spherical uniform-density stellar model of mass density $\rho$, radius $R$, negligible surface pressure and a fixed-composition ideal gas with adiabatic index $\gamma >1$. Define ${\omega }_d^2=4\pi G\rho /3$, so the inward gravitational acceleration has magnitude $g(r)={\omega }_d^{2}r$ and $P(r)=\rho {\omega }_d^2(R^2-r^2)/2$.

For an adiabatically displaced parcel in pressure balance with its surroundings, the squared buoyancy frequency isA constant background mass density does not mean an incompressible parcel: its pressure and temperature change during the displacement. The negative $N^2$ means a rising parcel is lighter than its surroundings and continues to rise. Equivalently, the ideal-gas specific entropy decreases outward.

Thus this thermally stratified equilibrium has stellar convective instability even if a separate test of homologous adiabatic stellar stability finds restoring radial compression for $\gamma >4/3$. Mechanical equilibrium, radial compression stability and buoyant stability answer different questions. The singular near-surface expression also reflects the zero-pressure idealized boundary and should not be extrapolated to a realistic photosphere unchanged.

A narrow surface density of a disk source approaching the outer edge from inside, with $S=S_0\delta (r-r_2)$ normalized to have full weight, gives $\dot{M}=2\pi r_2{S}_0$ and the displayed steady interior profile. This follows by replacing the inner source integral by $r_2{S}_0$ and integrating $u^{-1/2}$. The flux jumps from $\dot{M}$ just inside to zero outside the injecting edge. Thus a zero exterior flux is consistent, whereas a continuous zero interior boundary trace is not. A half-weight convention for a symmetric endpoint Dirac delta function represents a different total injected mass; it must not be silently mixed with the full-weight boundary source.

A zero of the angular drift in a cold single-grain-size dust tail produces a formally divergent number density per unit angle. The divergence is integrable when the exact local angle-to-age map is retained. A spread of launch velocities or radiation-pressure coefficients removes the singularity.

For a localized terrain source in uniform flow, the stationary phase method applied to $kx+m(k)z$ gives phase $\Phi =(N/U)\sqrt{x^2+z^2}$ on the downstream branch. Permanent phase lines are circular arcs, whereas an individual monochromatic component has straight phase lines. Reflected arcs use an image source; transmitted phase is continuous across the velocity interface.

For a regular infinite $\theta <\kappa$, with $\kappa$ a regular uncountable cardinal number, $S_{\theta }^{\kappa }=\{\delta <\kappa :\mathrm{cf}(\delta )=\theta \}$ is stationary. Build a strictly increasing continuous $\theta$-sequence in any club set, and take its supremum. Closure places it in that club and the cofinality of an increasing ordinal supremum gives cofinality $\theta$. In particular the two disjoint sets $S_{\omega }^{{\omega }_2}$ and $S_{{\omega }_1}^{{\omega }_2}$ show that the club filter on ${\omega }_2$ is not an ultrafilter.

A function on a set of ordinals with $f(\alpha )<\alpha$ at every nonzero point of its domain. On a stationary subset of a regular uncountable cardinal, Fodor lemma makes it constant on a stationary subset.

There are cofinal sequences $A_{\delta }\subseteq \delta$ of order type $\omega$ for the countable limit ordinals, such that every uncountable $X\subseteq {\omega }_1$ contains some $A_{\delta }$.

A sequence $D_{\alpha }\subseteq \alpha$ guesses every $X\subseteq {\omega }_1$ stationarily often: $\{\alpha :D_{\alpha }=X\cap \alpha \}$ is stationary.

A family $(A_{\alpha ,n}{)}_{\alpha <{\omega }_1,n<\omega }$ whose row at $\alpha$ partitions $(\alpha ,{\omega }_1)$ and whose fixed-column cells are pairwise disjoint. Choose injections $e_{\beta }:\beta \to \omega$ and put $A_{\alpha ,n}=\{\beta >\alpha :e_{\beta }(\alpha )=n\}$. This gives a matrix used to split stationary sets into many stationary pieces.

For an uncountable regular cardinal $\kappa$, choose cofinal sequences $c_{\alpha }:\omega \to \alpha$ for the stationary set of ordinals of cofinality $\omega$. Above any bound, some fixed coordinate exceeds the bound on a stationary subset; Fodor lemma makes that coordinate constant on a stationary subset. Thus the stationary constant fibers, over all coordinates, have unboundedly many values. Regularity makes one coordinate have $\kappa$ such values. Its fibers are disjoint stationary sets; adding all leftover ordinals to one piece partitions $\kappa$.

The pathwise identity ${\sum }_{t=1}^T(\Delta S_t{)}^2=S_T^2-S_0^2-2{\sum }_{t=1}^T{S}_{t-1}\Delta S_t$ turns a squared-increment claim into a terminal-square claim and predictable stock gains. On integer terminal support $\{0,\dots ,N\}$, $S_T^2=S_T+2{\sum }_{K=1}^N(S_T-K{)}^{+}$, so a static call strip plus a dynamic stock hedge replicates the claim.

For a bounded payoff vector $P$, the condition that $h\cdot p\le 0\le h\cdot P$ almost surely forces $h=0$ makes $F(h)=\mathbb{E}{e}^{-h\cdot P}+h\cdot p$ coercive. At a minimizer $h_{\ast }$, differentiation gives $\mathbb{E}[P{e}^{-h_{\ast }\cdot P}]=p$. The resulting positive pricing kernel is bounded above and away from zero. A direct coercivity proof uses compactness of the unit sphere and separates directions with negative-payoff probability from directions with nonnegative payoffs.

A state-price density converts terminal wealth and consumption into initial cost. For a nonnegative admissible self-financing portfolio with consumption, the deflated wealth plus cumulative deflated consumption is a supermartingale, giving $\mathbb{E}[{\zeta }_{T}X]+\mathbb{E}{\int }_0^T{\zeta }_t{c}_{t}dt\le w$. In a complete market, an integrable nonnegative terminal claim with full budget equality is replicable.

A self-financing investor with consumption satisfies $X=H\cdot P$ and $dX=H\cdot dP-cdt$. The Itô product rule and $d[Y,X]=H\cdot d[Y,P]$ give the displayed equation. For a positive local martingale deflator $Y$, the first term is a stochastic integral against the vector of deflated asset-price local martingales, not their current levels.

Two-body relaxation redistributes stellar energies and feeds stars above the escape threshold. The escaping population removes mass and generally nonnegative energy at infinity. Radius evolution and external tidal limits must be considered before inferring the change in cluster density.