A local ratio of stabilizing buoyancy to shear across an interface of thickness $\delta$, using reduced gravity jump $g^{\prime }$ and characteristic velocity $u$. The dimensionless value is ${\mathrm{Ri}}_i=g^{\prime }\delta /u^2$. It is distinct from a whole-layer bulk Richardson number based on the total mixed depth. Local entrainment laws use this ratio when the transition thickness and velocity characterize the turbulence-producing region.

For a finite irreducible Markov chain satisfying detailed balance with $\pi$, its transition matrix is a self-adjoint operator for $⟨f,g{⟩}_{\pi }={\sum }_x\pi (x)f(x)g(x)$. For a real orthonormal eigenbasis $(f_j)$ with eigenvalues $({\lambda }_j)$,Indeed, apply the spectral decomposition of $P^t$ to the function $1_{\{y\}}/\pi (y)$, whose coefficient against $f_j$ is $f_j(y)$. For $t=0$, every eigenvalue multiplier is interpreted as one.

Let $K$ be a reversible Markov kernel with stationary Probability measure $\pi$, and let the initial Probability measure $\nu$ have Radon-Nikodym derivative $f=d\nu /d\pi$. After $t$ steps, Detailed balance givesfor $\pi$-almost every $x$. To prove the identity, integrate a bounded measurable test function $g$ against the evolved measure: $\int f{K}^{t}gd\pi =\int g{K}^{t}fd\pi$, because reversibility makes $K^t$ self-adjoint in the integral pairing.

To express the result with ordinary probability density functions, fix a reference measure $m$, write $\pi (dx)=h(x)m(dx)/Z$, and let $r=d\nu /dm$. On the support where $h>0$, $f=Zr/h$, so the evolved density isThis formula uses the initial density $r$, not the Probability measure $\nu$ evaluated at a point. The unknown normalizing constant $Z$ cancels. Reverse simulation from $x$ can estimate this expression without evaluating the Transition density, provided the displayed integrand can be evaluated.

For example, the symmetric two-state kernel with holding Probability $3/4$ has Stationary distribution $(1/2,1/2)$. Starting at state zero gives $f=(2,0)$, then $Kf=(3/2,1/2)$; multiplying by the stationary masses recovers the one-step distribution $(3/4,1/4)$.

For four points $x_1,x_2,y_1,y_2$ in a metric space,Both steps follow from the triangle inequality. The first also holds with an absolute value around the difference, by the reverse triangle inequality. This transports a difference of distances measured from one point into a sum measured from another.

A return level plot graphs $T$ horizontally and $z_T$ vertically, often using a logarithmic scale for $T$. To read it, choose a return period, move up to the curve, and read the corresponding return level. Conversely, a threshold's horizontal coordinate gives the reciprocal of its annual exceedance probability. The usual plot of fitted quantiles against return periods, and optional pointwise confidence intervals, are documented in Mathieu Ribatet, POT package: return level plot.

The near-straight shape has a mathematical explanation: since $-\log (1-1/T)\sim 1/T$ as $T\to \infty$, the Gumbel distribution gives $z_T=\mu +b\log T+o(1)$. Multiplying a large return period by ten therefore increases the return level by approximately $b\log 10$, about $4.605$ units for this toy curve. A curve indexed by rarity puts several thresholds on one common scale while retaining their measurement units.

For a fitted curve, a pointwise confidence interval concerns uncertainty in the estimated quantile at a specified $T$; it is distinct from a prediction interval for a future annual sample maximum, and pointwise intervals need not form a simultaneous confidence band. A far-right curve value can also be an extrapolation beyond the observation record: extending the line adds no observations to support the chosen tail model. The diagram above uses fixed toy parameters, so it shows neither estimation uncertainty nor an observational record. Its interpretation throughout assumes the stated annual block and stationarity.

For an eccentric outer body and circular inner perturber, the exterior resonant argument is $\varphi =(p+q){\lambda }_{\mathrm{out}}-p{\lambda }_{\mathrm{in}}-q\varpi$. At equal mean longitudes $\Lambda$, conjunction phases areThe synodic period and coprime $p,q$ determine the visitation order, with advance $2\pi p/q$. For actual true alignment, write $x=f$ and $y=M(f)$ for the eccentric body's true anomaly and mean anomaly. Then $\varphi =(p+q)y-px$ instead of $qx$. The evenly spaced directions are a leading-small-eccentricity approximation.

For a closed linear operator $A$, the resolvent is the bounded inverse $(zI-A{)}^{-1}$ when $z$ lies in the resolvent set of an operator. This definition includes unbounded $A$. It specializes to the resolvent of an element when $A$ is an element of a Banach algebra.

Collapsing the boundary identifies the top relative homology of a disk pair with the top reduced homology of its quotient sphere. Together with the boundary connecting homomorphism, this shows that the induced quotient map and the boundary map have the same mapping degree, using the corresponding orientations.

For a smooth map of disk pairs whose graph of a function is transverse to the zero slice, orient the graph by its domain. With the tangent space of the zero slice ordered before that of the graph, the local smooth intersection number at a zero is $\mathrm{sgn}\det DF$. Summing these signs gives the relative mapping degree and hence the boundary mapping degree. Reversing the order of these two $m$-dimensional tangent spaces changes the sign by $(-1{)}^m$.

Inclusion, the map sending an absolute disk class to a relative one, and restriction to the disk boundary give this exact sequence. Its low-dimensional end is interpreted with pointed sets and the usual fundamental-group action. For a mapping-cylinder pair it measures the failure of a map to induce homotopy isomorphisms.

Let $U_1,\dots ,U_k$ be an open cover of $Y$, and suppose ${\alpha }_i\in H^{d_i}(Y;R)$ restricts to zero on $U_i$. The pair long exact sequence lifts each ${\alpha }_i$ to $H^{d_i}(Y,U_i;R)$. Their relative cup product lies in $H^{\sum d_i}(Y,{\bigcup }_i{U}_i;R)=H^{\sum d_i}(Y,Y;R)=0$, so ${\alpha }_1⌣\cdots ⌣{\alpha }_k=0$. This proves characteristic-class products vanish from local trivializations without cancelling possible zero divisors in the coefficient ring.

An error bound $A\delta /\alpha +Bc\alpha$, with positive $A,B,c$, is minimized at ${\alpha }_{\ast }=\sqrt{A\delta /(Bc)}$, with value $2\sqrt{ABc\delta }$. Differentiate the scalar bound, or apply the arithmetic-geometric mean inequality. This is an a priori regularization parameter choice for a linear regularization with inverse-parameter noise amplification and first-order approximation bias; the parameter must also lie in the method's allowed interval.

For fixed $r$ and positive density threshold $\lambda$, sufficiently regular pairwise dense bipartite graphs between $r$ disjoint equal-sized vertex classes contain at least $\delta n^r$ transversal cliques, for some $\delta >0$. A greedy embedding maintains common neighbourhoods and discards the small set of vertices with atypical degrees into them. The constants depend only on $r,\lambda$.

An open set satisfies an exterior cone condition at $z$ if a solid cone with vertex $z$ lies outside the set in some neighbourhood of $z$. The stronger global version requires the closures of the cone and the set to meet only at $z$. In two dimensions, a power barrier for an exterior cone makes such a point a regular boundary point for the Dirichlet problem. Local cone conditions suffice for local boundary regularity; a global cone is an additional geometric assumption.

An irreducible Markov chain is positive recurrent when its mean return times are finite. For a nonexplosive Markov chain on a countable state space, this is equivalent to the existence of a stationary distribution on its irreducible class. For continuous time, a return cycle includes departure and return, rather than counting immediate residence as a return.

On a closed oriented Riemannian manifold of dimension four, the Hodge star operator preserves harmonic two-forms and splits them into orthogonal spaces ${\mathcal{H}}^{+}$ and ${\mathcal{H}}^{-}$ with eigenvalues $+1$ and $-1$. The real de Rham intersection form in dimension four equals the positive $L^2$ inner product on ${\mathcal{H}}^{+}$ and its negative on ${\mathcal{H}}^{-}$. It is nondegenerate, since $Q(h,\ast h)=\Vert h{\Vert }_2^2$. Its signature pair is $(\dim {\mathcal{H}}^{+},\dim {\mathcal{H}}^{-})$.

Every ordered field has an ordered algebraic real closure. Combined with the Artin-Schreier ordering criterion, every formally real field embeds into a real closed field. This is one embedding direction in the model companion relation between their theories.

In the faithful matrices $g=\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right)$ with $a>0$, the Maurer-Cartan form has entries $da/a$ and $db/a$. Thus the left-invariant differential forms ${\sigma }^D=da/a$, ${\sigma }^T=db/a$ obey $d{\sigma }^D=0$ and $d{\sigma }^T=-{\sigma }^D\wedge {\sigma }^T$. The Lie algebra generators satisfy $[D,T]=T$.

The vertical field of a disk has kernel $Gz/(|\mathbf{R}-{\mathbf{R}}^{\prime }{|}^2+z^2{)}^{3/2}>0$. The triangle inequality therefore bounds a signed-surface density field by the field of its absolute surface density. If that surface density is everywhere smaller than a nonnegative comparison surface density, their vertical fields satisfy the displayed strict comparison, provided the integrals exist and the strict bound holds on positive measure. This is useful when comparing gravity-equivalent magnetic surface density with actual disk mass.

A local dispersion relation $\omega =\Omega (\mathbf{k};\mathbf{X},T)$ yieldsThe spatial and temporal derivatives on the right hold the wavevector fixed. Translation symmetries conserve the corresponding wavevector components, while time independence conserves frequency.