The $n$th cumulant is the coefficient of $t^n/n!$ in the cumulant-generating function. The first two cumulants are the expected value and variance; the third is the third central moment.

Normalize three distinct points of the Riemann sphere by a Möbius transformation $f$ sending them to $\infty ,0,1$. A fourth point lies on their generalized circle exactly when $f$ takes a real value there: the image of that generalized circle is the extended real line. Other standard cross-ratio ordering conventions differ by real fractional-linear transformations and preserve this reality criterion.

For cross-intersecting families $\mathcal{A},\mathcal{B}$ with member sizes at most $r,s$ respectively, there is a finite set $X$ with $|X|\le r{\sum }_{j=0}^s{r}^j$ such that $A\cap B\cap X\ne \varnothing$ for every $A\in \mathcal{A},B\in \mathcal{B}$. Take a bounded-size transversal kernel ${\mathcal{A}}_0$ preserving hitting sets of size at most $s$ and set $X={\bigcup }_{A\in {\mathcal{A}}_0}A$. For each $B$, the set $B\cap X$ is a hitting set for ${\mathcal{A}}_0$, hence for all of $\mathcal{A}$. This proves the assertion even for infinite set families; if either family is empty, take $X=\varnothing$.

An echellogram is the two-dimensional detector pattern of separated diffraction orders from a cross-dispersed echelle spectrograph.

A cross-disperser is a second dispersing element, often an optical prism or diffraction grating, oriented to separate optical spectra perpendicular to their principal dispersion direction.

For an infinite subset $A\subseteq \mathbb{N}$, repeatedly selecting the least element not yet used gives a bijection $f:\mathbb{N}\to A$. The well-order ensures each minimum exists, infinitude ensures the remaining set is nonempty, and strict increase gives injectivity. If $a\in A$ were omitted forever, all selected values would be at most $a$, impossible inside the finite set $\{0,\dots ,a\}$.

If $\mathcal{C}$ has finite colimits and a monad $T$ preserves reflexive coequalizers, its Eilenberg-Moore category has finite colimits. The forgetful functor creates reflexive coequalizers preserved by $T$. The coproduct presentation for monad algebras yields binary coproducts, and $F0$ is initial. Any pair $a,b:X\rightrightarrows Y$ can then be replaced by the reflexive pair $[a,1],[b,1]:X+Y\rightrightarrows Y$, with identical coequalizers. Finite coproducts and coequalizers give all finite colimits.

For a population with randomized vertical phases, orbital inclination $I$, and $R_p\ll a_{p}I$, the effective volume probability near the planetary plane is proportional to ${\Sigma }_c/(a_{p}I)$. The collision time then scales as $I{Q}^{3/2}{M}_p^{-2/3}{a}_p^2$, at fixed stellar mass and planetary mass density. An individual inclined Kepler orbit whose orbital nodes do not intersect the planet's orbit need not collide at all.

For an irreducible Markov chain on a finite or countable state space, a positive recurrent state implies a unique stationary distribution $\pi$. If the chain is also an aperiodic Markov chain, its transition probabilities satisfy the displayed limit for every pair of states. The same limit holds from any initial probability distribution by bounded convergence. Positive recurrence cannot be omitted in the countably infinite case, and aperiodicity is required for ordinary, rather than averaged, convergence. This is the standard discrete-time convergence theorem; the finite-state statement alone does not cover unbounded renewal countdowns.

The occupation time records the duration spent in a state of a continuous-time Markov chain. Conditional on $Z(0)=r$, its expected value is ${\int }_0^t{p}_{rs}(u)du$, where $p_{rs}$ is the transition probability.

The entry count $N_s(t)$ counts jumps of a continuous-time Markov chain into state $s$ by time $t$, excluding initial occupation. It is a counting process. For a finite-state chain with homogeneous transition intensities, its mean has derivative ${\sum }_{i\ne s}{p}_{ri}(t)q_{is}$ conditional on initial state $r$, so its expected value is the sum of incoming rates times the expected occupation times of a continuous-time Markov chain. This density is an expected count rate, not necessarily a normalized probability density function.

A partition-valued Markov chain in which every unordered pair of current blocks merges at rate one. On $n$ initial singleton blocks, the holding time with $j$ blocks is exponential with rate $\left(\genfrac{}{}{0ex}{}{j}{2}\right)$. Restrictions to subsets of labels have the same law.

For an equilibrium chain with strictly positive stationary probabilities, reversed jump rates are $q_{ij}^{\ast }={\pi }_j{q}_{ji}/{\pi }_i$. Under detailed balance they equal the original rates, making the stationary path law reversible. Holding times and jump directions reverse together; this is more than symmetry of one-time distributions.

A function is discontinuous at a point in its domain of a function when it fails the epsilon-delta definition of continuity there. Two basic failures are an incorrect assigned value and unequal one-sided limits. They demand different repairs.

For a removable discontinuity, considerThe limit at zero is $0$, but $r(0)=1$. Replacing that one value by $0$ gives the continuous square function. The sequence $x_n=1/n$ makes the mismatch visible: $r(x_n)=1/n^2\to 0\ne r(0)$. A removable discontinuity can also describe an expression with a missing value, in which case the question is whether a continuous extension to the missing input exists.

For a jump discontinuity, take $j(x)=0$ for $x<0$ and $j(x)=1$ for $x\ge 0$. The left and right one-sided limits at zero are $0$ and $1$. No assignment to $j(0)$ can equal both, so changing only the point value cannot make the function continuous there. Contrast both examples with the corner of $|x|$: that function is continuous at zero even though its derivative is absent.

Inside the constructible universe, every hereditarily countable set appears in a countable level of the constructible hierarchy. Take a countable elementary substructure of a sufficiently large $L_{\eta }$ containing its transitive closure pointwise. The condensation lemma for the constructible universe identifies the collapse with $L_{\beta }$ for countable $\beta$, and the collapse fixes the original set. Conversely, a countable-indexed level is countable inside the constructible universe.

This statement says that the first uncountable ordinal of an inner model $L$ is a countable ordinal in the ambient universe. It can hold together with the Generalized continuum hypothesis: start with constructibility and use a finite-function collapse to countable size on ${\omega }_1^L$. The GCH preservation by a finite-function collapse calculation gives the relative consistency result already from the consistency of ZFC.

For radius $a$ and tangential velocity ${\mathbf{u}}_s={\mathbf{I}}_s\mathbf{Ax}$ with symmetric traceless $\mathbf{A}$, the surface divergence is $-3\mathbf{n}\cdot \mathbf{An}$. At small surface Péclet number $a^2|\mathbf{A}|/D_s$, neglecting advection of the concentration perturbation givesThe result follows because this quadratic is a degree-two spherical harmonic with surface Laplacian $-6/a^2$ times itself and zero spherical mean.

An affine connection on a smooth manifold is a connection on a vector bundle for its tangent bundle. It satisfies ${\nabla }_{fX}Y=f{\nabla }_{X}Y$ and ${\nabla }_X(fY)=X(f)Y+f{\nabla }_{X}Y$. Its curvature isThe Leibniz rule makes curvature tensorial in all three slots. An affine connection is also called a linear connection on the manifold; a Levi-Civita connection is the distinguished torsion-free metric-compatible example.

For a metric vector bundle connection over a Riemannian manifold, the rough Laplacian is the formally nonnegative connection operatorwhere $e_i$ is a local orthonormal frame. The correction term makes the expression independent of the orthonormal frame. On a closed manifold, integration by parts gives $⟨{\nabla }^{\ast }\nabla s,s{⟩}_{L^2}=\Vert \nabla s{\Vert }_{L^2}^2$.

For odd $r\ge 1$ and $W_g={\#}_{i=1}^g(S^r\times S^r)$, the integral cohomology has one copy of $\mathbb{Z}$ in degrees zero and $2r$, ${\mathbb{Z}}^{2g}$ in degree $r$, and zero elsewhere. Choose the top orientation class $\omega$ and degree-$r$ classes ${\alpha }_i,{\beta }_i$ from the two factors of summand $i$. Their nonzero positive-degree basis products areAll ${\alpha }_i{\alpha }_j$ and ${\beta }_i{\beta }_j$ vanish, and $\omega$ annihilates positive-degree classes. The Künneth theorem computes each sphere product; the Mayer–Vietoris sequence computes the connected sum groups, and its degree-one pinch maps determine the products. Thus its middle-degree Poincare duality pairing is a symplectic vector space after changing coefficients to a field of characteristic different from two. The case $r=1$ is the usual cohomology ring of a closed oriented surface.