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The additively indecomposable ordinals are exactly the powers ${\omega }^{\delta }$.

A Von Neumann ordinal $x$ is finite exactly when $x$ and every member of $x$ are either zero or successor ordinals. Every finite ordinal has that property. Conversely, an infinite ordinal is either $\omega$, which is a nonzero limit, or is greater than $\omega$ and contains that nonzero limit ordinal. The test can be expressed by a bounded first-order formula: ordinality uses transitivity and membership comparability, and a nonempty ordinal is a successor exactly when it has a greatest member. This defines the class of natural numbers without quantifying over arbitrary infinite inductive sets.

A finite ordinal is the order type of a finite well-order; in the von Neumann representation it is one of $0,1,2,\dots$. They are exactly the ordinals $x$ such that $x$ and each member of $x$ are either zero or successor ordinals. If an ordinal is infinite, it either is the nonzero limit $\omega$ or contains $\omega$ as a member, so it fails this test. Thus the finite ordinals give the usual representation of the natural numbers.

For independent identically distributed observations with continuous cumulative distribution function $F$, fix the maximum $Y_n$ of the first $n\ge 1$ observations and let $\tau$ be the first subsequent observation exceeding it. The probability integral transform gives $U=F(Y_n)$ density $n{u}^{n-1}$ on $(0,1)$. Conditional on $U=u$, $\tau$ has a geometric distribution of parameter $1-u$. ThereforeThe masses sum to one, so the waiting time is finite almost surely, but the tail-sum formula for expectation gives $E\tau =\infty$. These conclusions require only continuity of $F$, not a density or a strictly increasing distribution function.

For $n$ independent and identically distributed rate-$\lambda$ exponential random variables, let $M_0=0$ and $M_1<\cdots <M_n$ be their order statistics. The gaps $D_i=M_i-M_{i-1}$ are independent and have rates $(n-i+1)\lambda$. After the first clock rings, the memoryless property makes the surviving clocks fresh independent exponentials, independently of the elapsed minimum and the winning label. Induction gives the result. Equivalently, the ordered joint probability density is $n!{\lambda }^n\exp [-\lambda {\sum }_i{m}_i]$ on $0<m_1<\cdots <m_n$; changing to gaps gives a product of exponential densities because ${\sum }_i{m}_i={\sum }_i(n-i+1)D_i$.

For two independent random variables with the uniform distribution on $[0,1]$, put $U=\min (X,Y)$ and $V=\max (X,Y)$. Then $P(U>u)=(1-u{)}^2$, so $\mathbb{E}U=1/3$. Since $U+V=X+Y$ and $UV=XY$, $\mathbb{E}V=2/3$ and $\mathbb{E}(UV)=1/4$. Thus their covariance is $1/36$, although the original variables are independent. Rescaling the interval to $[a,b]$ multiplies the covariance by $(b-a{)}^2$.

For independent and identically distributed random variables, suppose normalized sample maxima have limiting distribution function $G$. At a threshold $a_{n}x+b_n$, where $x$ is a continuity point of $G$ and $G(x)>0$, the exceedance count has limiting Poisson distribution with parameter $-\log G(x)$. The $r$th largest observation lies below the threshold exactly when at most $r-1$ observations strictly exceed it, including in samples with ties. Summing the first $r$ limiting Poisson probabilities gives the displayed limit for every fixed positive integer $r$. The equivalence follows from $F(a_{n}x+b_n{)}^n\to G(x)$ and $-\log (1-p)\sim p$ as $p\downarrow 0$.

The rank of an observation records where that observation lies among the order statistics of a statistical sample. When all values are distinct, $r_i=1+\#\{j:x_j<x_i\}$ and $x_i=x_{(r_i)}$. For $(0.7,0.1,0.9,0.4,0.2)$, the ranks of observations in original order are $(4,1,5,3,2)$: the first observation is fourth smallest, while the fourth observation is third smallest.

Ties require a convention. A tied block occupying sorted positions $a,\dots ,b$ can assign each member the average rank of an observation $(a+b)/2$, the minimum position $a$, or the maximum position $b$; an explicit rule can also break ties into distinct positions. R's official documentation describes these choices and uses average ranks by default. ^[ref].

For example, $(0.1,0.4,0.4,0.9)$ has order statistics $(0.1,0.4,0.4,0.9)$. The two $0.4$ observations occupy positions $2$ and $3$; average ranks of observations are $(1,2.5,2.5,4)$, whereas minimum-position ranks are $(1,2,2,4)$. A fractional average rank is a summary of tied positions, rather than an index of one order statistic. Without omissions, distinct ranks and average ranks sum to $n(n+1)/2$, since averaging each tied block preserves its sum; minimum-position ranks need not.

Wilcoxon signed-rank tests rank the absolute nonzero paired differences, while Wilcoxon rank sum tests rank the pooled observations from two samples; these are different rankings.

For an iid sample of size $n\ge 2$ from a continuous cumulative distribution function, the probability integral transform makes the transformed sample uniform. Integrating the joint density of its minimum and maximum gives range density $n(n-1)r^{n-2}(1-r)$ on $0<r<1$. Its distribution function is $n{r}^{n-1}-(n-1)r^n$. The result describes population content between the extreme order statistics, independently of the underlying continuous distribution.

Let commuting elements $a,b$ have finite coprime orders $m,n$. Their cyclic subgroups have trivial intersection by Lagrange's theorem. If $(ab{)}^k=1$, then $a^k=b^{-k}$ belongs to this intersection, so $m\mid k$ and $n\mid k$. Thus $mn\mid k$. Conversely, $(ab{)}^{mn}=1$ because the elements commute. This proves the displayed order formula.

Over a cone point of order $m$, a cycle of length $\ell$ in the monodromy on sheets has ramification degree $\ell$ and residual cone angle $2\pi \ell /m$. The lifted inherited hyperbolic metric is smooth precisely when every such cycle has length $m$. Equivalently the subgroup defining the cover meets no conjugate of a nonidentity power of a cone generator.

Pair the $2s$ edges of an oriented hyperbolic polygon by length-preserving identifications reversing their boundary directions. Label the pairs by generating elements $A_1,\dots ,A_s$ of a finite group $T$. Glue the corresponding edge of a copy labelled $g$ to its mate in the copy labelled $g{A}_k$. Left multiplication of labels defines the $T$ action. A vertex cycle of a paired polygon has angle sum ${\theta }_j$ and accumulated gluing word $C_j\in T$ of order $q_j$; above it lie $|T|/q_j$ vertices, each of cone angle $q_j{\theta }_j$. The action is free away from vertices, and the vertex stabilizers are conjugates of $⟨C_j⟩$. If the base cone has integer order $m_j=2\pi /{\theta }_j$ and $q_j\mid m_j$, this is an orbifold covering, with lifted cone order $m_j/q_j$. The lift is smooth exactly when $q_j=m_j$.

An $n$-sheeted surface cover of an orbifold of underlying genus $g_0$ with $r$ cone points has the displayed ordinary Euler characteristic, where $c_i$ is the number of cycles of the $i$th cone monodromy permutation. The ramification deficit over that point is $n-c_i$, so the formula follows by counting lifted vertices, edges and faces, or by the Riemann-Hurwitz formula. If every cone cycle has its full prescribed length, this equals $n{\chi }_{\mathrm{orb}}$.

Phase-contrast microscopy converts specimen-dependent phase differences into visible light intensity differences. It can reveal structures in living, unstained biological cells, allowing their changes to be followed without first making an absorbing preparation.^[ref] The key question is how an optical microscope can read a phase when an ordinary detector records light intensity.

Consider a transparent specimen with complex amplitude transmission $t(x,y)=e^{i\varphi (x,y)}$. In ideal in-focus coherent imaging with all relevant spatial frequencies transmitted, its image has $I=|t{|}^2=1$: varying $\varphi$ alone produces no brightness detail. This is a particular model, not a claim that all transparent specimens are invisible. Diffraction, defocus, absorption and a finite optical pupil can already change a real image.

A phase-contrast microscope images an illuminating annulus onto a phase plate ring near the objective's rear focal plane. The aligned ring attenuates and shifts direct light relative to most diffracted light; their interference changes image brightness.^[ref]

The method is beautiful because the bright direct beam becomes its own reference: a small specimen-dependent complex amplitude can interfere with it and change the image at first order. Merely magnifying an invisible phase pattern would leave $|t{|}^2$ unchanged.

In a simple two-mirror oscillator, laser threshold is the small-signal optical gain at which a resonant optical normal mode exactly replaces its round-trip intensity loss. Assume constant intensity coefficients $g$ and $\alpha$ throughout a one-way distance $L$, and mirror intensity reflectivities $0<R_1,R_2\le 1$. One round trip has multiplierTwo traversals give the distance $2L$; multiplying the two mirror factors completes the bookkeeping. Setting $M=1$ and taking a logarithm givesThe reflectivities here are intensity fractions, not field-amplitude reflectivities. A shorter gain region or nonuniform loss requires replacing the path exponent by the appropriate path integral. The balance also presumes a permitted resonant optical normal mode. ^[ref].

For $L=0.50\mathrm{m}$, $R_1=1$, $R_2=0.90$ and $\alpha =0.10{\mathrm{m}}^{-1}$, the threshold is $g_{\mathrm{th}}\approx 0.2053605{\mathrm{m}}^{-1}$. At $g=0.15{\mathrm{m}}^{-1}$, $M\approx 0.946144$: a small circulating signal decays even though the material has positive optical gain. At $g=0.30{\mathrm{m}}^{-1}$, $M\approx 1.099262$: the signal initially grows. Raising passive loss or lowering mirror reflectivity raises laser threshold.

For $M>1$, the fixed-gain recurrence predicts startup amplification, not infinite actual output. Growing light intensity depletes the available population inversion, reducing optical gain through saturation. In the simplest steady oscillator, the saturated round-trip gain balances loss and pumping supplies the outgoing power and dissipated energy. The small-signal model locates the onset; it does not determine the final output power. ^[ref].

Consider one selected polarization of waves, represented by complex fields $E_1(t)$ and $E_2(t)$ at two sampling positions. Suppose their second moments are finite and stationary: $J_j=⟨|E_j(t){|}^2⟩$ and the following uncentered wave-field correlation do not depend on the absolute time $t$. DefineThe star denotes the complex conjugate; brackets denote an ensemble average. A long detector average estimates this quantity only if it adequately samples the fluctuations. The superscript $1$ identifies a field correlation: it does not mean the autocorrelation of the measured light intensity. The one-position version is $g^{(1)}(\tau )=⟨E^{\ast }(t)E(t+\tau )⟩/⟨|E{|}^2⟩$, so $g^{(1)}(0)=1$ for every nonzero field covered by this definition; this alone says nothing about its correlation at nonzero delay. ^[ref].

The Cauchy-Schwarz inequality gives $0\le |g_{12}^{(1)}(\tau )|\le 1$. At a specified delay, magnitude one means $E_2(t+\tau )=a{E}_1(t)$ in mean square for a fixed nonzero complex $a$: the samples have a reproducible relation in both wave amplitude and wave phase. Magnitude zero means the averaged field cross term vanishes. It need not mean statistical independence. A deterministic phase difference changes the argument of $g^{(1)}$, not its complex modulus; a distorted but fixed wavefront can therefore remain fully coherent. ^[ref].

To see the observable, recombine the samples in the same detected spatial mode with matched wave polarizations. With a common positive conversion factor $C$, let $I_j=C{J}_j$ be the single-arm light intensities reaching that detector. Adding $E_1(t)+e^{i\varphi }{E}_2(t+\tau )$ givesThis follows simply by expanding the squared complex modulus and averaging, then taking the maximum and minimum as $\varphi$ spans a full cycle. The extra wave phase scan must leave $I_1,I_2$ and the magnitude of the correlation effectively unchanged. Thus interferometric visibility equals $|g^{(1)}|$ only for equal nonzero light intensities and ideal mode matching. For $I_2/I_1=1/4$, perfectly coherent beams have interferometric visibility $0.8$; a measured value $0.4$ would imply $|g^{(1)}|=0.5$ under these assumptions. Orthogonal linear polarizations, background light, unresolved interference fringes or vibrations can reduce measured contrast further. One absent arm gives zero contrast and makes the normalized two-sample correlation undefined.

At the boundary of an Oppenheimer-Snyder cloud, its areal radius is $R=a\sin {\chi }_0$. Matching to a Schwarzschild metric gives $M=4\pi \rho R^3/3=a_{\max }{\sin }^3{\chi }_0/2$. Thus ${\sin }^2{\chi }_0=2M/R_0$ and $a_{\max }=\sqrt{R_0^3/(2M)}$. The boundary is a radial timelike geodesic with specific Killing energy $\sqrt{1-2M/R_0}=\cos {\chi }_0$. Its radial equation is ${\dot{R}}^2=2M/R-2M/R_0$. The induced metric and the angular extrinsic curvature agree on both sides, so no surface layer is required.

A boundary clock in Oppenheimer-Snyder collapse reads $\tau =a_{\max }(\eta +\sin \eta )/2$. Its event horizon crossing occurs at $R=2M$, giving the displayed angle, and the final curvature singularity occurs at $\eta =\pi$. The remaining proper time is $a_{\max }(\pi -{\eta }_H-\sin {\eta }_H)/2$. For $R_0\gg 2M$ this approaches $4M/3$, but it is not exactly that value for finite release radius.

Radial outgoing null geodesics in the interior of Oppenheimer-Snyder collapse obey $d\chi /d\eta =1$. Trace the ray crossing the boundary at $({\eta }_H,{\chi }_0)$ backward: it starts at the centre at $\eta ={\eta }_H-{\chi }_0$. This ray is the event horizon; it can exist inside the cloud before the surface crosses $R=2M$. An event horizon is determined by global escape to future null infinity, rather than by a local stopwatch signal.

The comoving dust ball $0\le \chi \le {\chi }_0<\pi /2$ in the Oppenheimer-Snyder model is an Oppenheimer-Snyder cloud. Its pressureless matter has uniform density on each constant-proper time slice and reaches a simultaneous final singularity in the ideal homogeneous model.