Let $K(\Delta )={\max }_t\#\{r:\Vert {\theta }_r-t\Vert \le \Delta /2\}$. Without a separation assumption,For $0<\Delta \le 1$, each point belongs to at most $K(\Delta )$ integration arcs in the Sobolev–Gallagher inequality. For $\Delta \ge 1$, $K(\Delta )$ is the total number of points, and the Cauchy-Schwarz inequality bound $|S{|}^2\le N\sum |a_n{|}^2$ suffices.

Since the differential of the Exponential map of a Lie group at zero is the identity, the inverse function theorem makes it a diffeomorphism on a sufficiently small neighborhood of zero. Its local inverse is a logarithm near the group identity. It need not extend to a global logarithm, even for the real general linear group.

A convex normal neighbourhood is an open set whose points have unique smoothly varying joining geodesics, with the joining geodesic contained in the set. The corresponding star-shaped restriction of each exponential map is a diffeomorphism onto the neighbourhood. Hence its differential is invertible and no such joining geodesic admits a nonzero Jacobi field vanishing at both endpoints. Every point of a Riemannian manifold has arbitrarily small such neighbourhoods. This is stronger than merely requiring the existence of a minimizing geodesic in the set.

For the Riemannian exponential map ${\exp }_p(v)={\gamma }_v(1)$, uniqueness of the geodesic equation gives ${\exp }_p(tv)={\gamma }_v(t)$. Differentiation at $t=0$ yields $(d{\exp }_p{)}_{0}v=v$. Smooth ODE dependence and the inverse function theorem therefore make ${\exp }_p$ a diffeomorphism on a neighborhood of zero, producing geodesic normal coordinates. This is a local assertion and requires no geodesic completeness.

For simple random walk on a finite connected loopless unweighted graph, start at $a\ne z$ and stop on first hitting $z$. Ifthen $v_{az}(z)=0$ and $L{v}_{az}={\delta }_a-{\delta }_z$ for the graph Laplacian $L$. The incoming-visit balance equation is $g_{az}(u)=1_{\{u=a\}}+{\sum }_{w\sim u}{g}_{az}(w)/\mathrm{deg}(w)$ for $u\ne z$; at $z$ the Laplacian value is $-1$ because all its coordinates sum to zero. Thus $v_{az}$ is the voltage for unit current from $a$ to $z$, and $v_{az}(a)=R_{\mathrm{eff}}(a,z)$. In particular, the expected number of directed transitions $u\to w$ before $T_z$ is $v_{az}(u)$.

For simple random walk on two disjoint complete graphs of size $n\ge 2$ joined through one new vertex $w$, let $h_1$ be the expected hitting time of $w$ from either vertex attached to $w$, and $h_o$ the expected hitting time from any other clique vertex. First-step analysis gives $h_1=1+(n-1)h_o/n$ and $h_o=h_1+n-1$, so $h_1=n^2-n+1$ and $h_o=n^2$. The expected hitting time from $w$ is zero. In discrete time the random walk is an aperiodic Markov chain only for $n\ge 3$.

For a common unconstrained Markov kernel on the edges of a rooted tree, let $N_{ab}$ be the conditional expectation of the number of transitions $a\to b$, given observed leaves and the old Markov kernel. The expectation-maximization algorithm maximizes ${\sum }_{a,b}{N}_{ab}\log K_{ab}$ over row probability distributions, giving $K_{ab}^{\mathrm{new}}=N_{ab}/{\sum }_c{N}_{ac}$. A row with zero total count is unrestricted by this objective. Belief propagation computes the counts exactly on a tree.

The exobase marks the transition between a collisional atmosphere and its exosphere. For a neutral, approximately hydrostatic ideal gas, take mean free path $\ell \simeq 1/(n{\sigma }_c)$ and atmospheric scale height $H=k_{B}T/(mg)$. The criterion $\ell \sim H$ givesAn order-one collision convention can change this coefficient. The pressure is not universal: the collision cross-section, composition, local gravity, and ionization must be specified. A strongly escaping, non-hydrostatic atmosphere requires an appropriate density-gradient length instead of a hydrostatic $H$.

An exoplanet passes in front of its host star as seen by an observer and blocks part of the stellar flux. Exoplanet transit photometry infers geometry and radii from the resulting light curve. A true alignment involves the true anomaly, not generally the mean anomaly.

For an edge-on orbit crossing the centre of a uniformly bright stellar disk at distance $r\gg R_{\star }$, projected speed is $h/r$ and chord time is $2R_{\star }r/h$. A steady cross-sectional-area current $J_{\sigma }=2fh$ therefore places area $4f{R}_{\star }r$ in front of the star, givingThis requires a geometrically narrow, optically thin wire with negligible overlap. A noncentral chord, limb darkening, finite angular extent and large optical depth change the estimate; it cannot predict dimming greater than one.

In leading mean-conjunction geometry a common observer phase $x$ for two resonant arguments requires $qx={\varphi }_1$ and $nx={\varphi }_2$ modulo $2\pi$. With $d=\gcd (q,n)$, the exact modular criterion isNecessity follows by subtracting the two congruences after multiplication; sufficiency follows from the integer linear combinations of $q$ and $n$, whose set is $d\mathbb{Z}$. For exterior centres $\pi$ and interior centres $0,\pi ,0$ at orders one to three, only $q=n=2$ satisfies it. This is phase compatibility, not a guarantee of an observed exoplanet transit: timing, observer orientation, resonant-argument libration widths and true-versus-mean longitude must also be considered.

A departure of transit arrival times from a constant-period ephemeris can reveal gravitational interactions with other bodies. Dynamical fits may constrain an exoplanet mass independently of transit depth, providing a mass to combine with radius for exoplanet interior models.

If $T$ has algebraically prime models and every inclusion between models of $T$ is a simple closure, then $T$ has quantifier elimination. For a common base $A$, embed its algebraically prime extension into both models. Simple closure transfers a witness into that extension; the second embedding transfers it to the other model.

For reservoir kinetics ${\mathcal{R}}_t=-({\gamma }_R+R_{R}n)\mathcal{R}+P$, adiabatic elimination gives $\mathcal{R}=P/({\gamma }_R+R_{R}n)$. Expanding for $R_{R}n/{\gamma }_R\ll 1$ and inserting into the condensate equation gives a complex Ginzburg–Landau equation withFast reservoir response and weak depletion are separate assumptions. Without the number density expansion the gain is saturable, not cubic. The coefficients depend on position when the pump does.

For the cubic complex Ginzburg–Landau equation with $\alpha ,\beta >0$, $\Psi =\sqrt{\alpha /\beta }{e}^{-i\mu t}$ is spatially uniform, with $\mu =g\alpha /\beta -s$. In the reservoir reduction of a polariton condensate, its pump threshold is $P_{\mathrm{th}}={\gamma }_C{\gamma }_R/R_R$ and its number density is $({\gamma }_R/R_R)(1-P_{\mathrm{th}}/P)$. The full two-field homogeneous model instead gives $(P-P_{\mathrm{th}})/{\gamma }_C$. These agree near threshold only to the expansion's accuracy. Existence of the homogeneous solution does not imply stability for arbitrary interaction coefficients.

In a compact connected orientable three-manifold with no spherical boundary components, every embedded knot is homotopic to a knot with an excellent three-manifold as exterior. In particular this can preserve the generator class in $S^1\times S^2$, or the winding number of a satellite pattern one class in a solid torus. Homotopy here is weaker than isotopy.

On a nonempty compact $2n$-dimensional symplectic manifold without boundary, with $n\ge 1$, the class $[\omega {]}^n$ in top-degree de Rham cohomology is nonzero, since its integral in the symplectic orientation is positive. In particular $\omega$ cannot be an exact differential form: if $\omega =d\alpha$, then ${\omega }^n=d(\alpha \wedge {\omega }^{n-1})$ and the Generalized Stokes theorem would make that integral zero. Thus vanishing second de Rham cohomology obstructs a closed positive-dimensional symplectic manifold.

Under independent exponential death and censoring, a rate fit using only observed-death times estimates $s=\lambda +\gamma$. Multiplying by the observed death fraction consistently estimates $\lambda$. This plug-in correction generally differs in a finite sample from the full maximum-likelihood estimate, which uses observed exposure from both deaths and censored subjects.

An option whose payoff depends on an average of the underlying stock prices over time. The displayed arithmetic-average call differs from a geometric-average call, which uses the geometric mean. A convex payoff and a nondecreasing positive numéraire give an upper bound by the average costs of same-strike European call options at the averaging dates.

A maturity-$T$ claim is attainable if a predictable self-financing strategy with fixed initial capital has terminal wealth equal to the claim payoff almost surely. In discrete time its wealth is $X_t=x+{\sum }_{s=1}^t{H}_s\cdot (P_s-P_{s-1})$. The strategy replicates the claim; attainability is a statement about exact pathwise replication, not only equality of expectations.