For speed $c(y)=1/y$ in $y>0$, Fermat principle gives the elapsed-time integrand $y\sqrt{1+(y^{\prime }{)}^2}$. Its symmetric stationary light rays through $(\pm d,H)$ are the displayed catenaries. Set $t=d/a$. The endpoint equation is $\cosh t/t=H/d$. This function decreases and then increases, with unique minimum where $t\tanh t=1$. Thus above the minimum there are two stationary rays, at the minimum one, and below it none. The travel time on either ray is $T=ad+(a^2/2)\sinh (2d/a)$, and its minimum height is $a$.

For $d=1$ and $H=\cosh 1$, one solution is $a=1$, with $T=1+\sinh 2/2$. A second has $a\simeq 0.7021026952$, with $T\simeq 2.8224491447$. Hence these endpoints do not uniquely specify the ray. Stationarity is distinct from global minimization: paths descending almost to $y=0$ have limiting travel time $H^2$, below either stationary value in this example.

Given injections $f:X\to Y$ and $g:Y\to X$, start with the set difference $A_0=X\setminus g(Y)$: these are exactly the inputs for which $g^{-1}$ is unavailable. Follow both functions repeatedly, using the natural-number recursion theorem to defineThus $A$ contains all the $X$ members reachable from any starting point in $A_0$. The notation $f(A_k)$ and $g(f(A_k))$ denotes images of a function. If $A_0$ is the empty set, then every $A_k$ is empty and $g^{-1}$ alone gives the desired bijection.

Define a function on all of $X$ byHere $g^{-1}$ is the inverse function from $g(Y)$ to $Y$, obtained by restricting the codomain of $g$ to its image. It is well defined because $g$ is injective. Every $x\notin A$ belongs to $g(Y)$, since $A_0\subseteq A$, so the second rule is always available where it is used.

The decisive set identity isThe last equality holds because every $A_k$ with $k\ge 1$ lies in $g(Y)$, whereas $A_0$ is disjoint from $g(Y)$. Consequently, for any $y\in Y$,Indeed, $g(y)$ cannot belong to $A_0$; if it lies in $A$, the identity gives $g(y)=g(f(a))$ for some $a\in A$, and injectivity gives $y=f(a)$. The converse follows directly by applying $g$.

Now $f$ restricts to a bijection from $A$ onto $f(A)$. The displayed equivalence says that $g$ restricts to a bijection from $Y\setminus f(A)$ onto $X\setminus A$, so its inverse function bijects the remaining inputs with the remaining outputs. The two input sets partition $X$ and the two output sets partition $Y$. Their combined function $h$ is therefore both injective and surjective.

No choice function is needed: $A$ is fixed by the given injections, and each required $g^{-1}(x)$ is unique. We neither choose a representative from each chain nor enumerate $X$. This is a proof in ordinary classical ZF, even when no well-ordering of the sets exists there.^[ref]

Take $X=\{0,1,2,\dots \}$, the nonnegative integers, and $Y=\{1,2,3,\dots \}$, the positive integers. Choose the injectionsHere $g$ is an inclusion map, while $f$ reaches only the positive even integers. In particular, $f$ is not a bijection: it misses every positive odd integer. Simply using the inverse function of $g$ on its image misses input zero.

The remedy is to follow the alternating injections starting at the missing input zero:The subscripts distinguish an input in $X$ from an output in $Y$, even when their numerical values agree. The resulting subset of $X$ is $A=\{0,2,6,14,30,\dots \}$. Its sequence of members satisfies $a_0=0$ and $a_{k+1}=2a_k+2$, giving $a_k=2^{k+1}-2$ by mathematical induction. Define the function

The image $h(A)=\{2,6,14,30,\dots \}=A\setminus \{0\}$ covers all the positive members of $A$ exactly once. The other inputs cover $Y\setminus A$ exactly once by staying fixed. These output sets are disjoint and their union is $Y$, proving that $h$ is a bijection. In particular, $h(0)=2$, $h(2)=6$, $h(6)=14$, while $h(1)=1$ and $h(4)=4$.

For completeness, its inverse function isThe familiar bijection $n\mapsto n+1$ also works, but this example shows something more specific: the two chosen injections can repair the missing zero by moving only a sparse sequence of inputs. The repair never has to shift the entire set.

BABAR used asymmetric electron-Positron collisions near the Upsilon meson $4S$ resonance to prepare neutral B meson pairs. It reconstructed one decay into charmonium plus a kaon, and used flavour tagging on the other B meson with charged leptons or kaons. Silicon tracking reconstructed decay vertices for the decay-time difference. Self-tagging decays calibrated mistags and time resolution; charged decays provided a control. The tagged time distributions established CP violation, limited by backgrounds, mistags and vertex alignment.^[ref]

The Lorentz boost converts vertex separation to $\Delta t\simeq \Delta z/({\beta }_{\mathrm{boost}}{\gamma }_{\mathrm{boost}}c)$ at leading order. Unequal beam energies thus make a time difference measurable as a distance.^[ref]

For a gas of fixed composition in hydrostatic equilibrium, with $z$ upward and downward gravitational acceleration of magnitude $g$, an adiabatic process gives the displaced element density change $\delta {\rho }_{\mathrm{element}}=\rho {\gamma }^{-1}(d\ln p/dz)\xi$. Comparing with the ambient change $(d\rho /dz)\xi$ yieldsThe ideal gas law $p=\rho \mathcal{R}T$ and $dp/dz=-\rho g$ give the title expression, where $c_p=\gamma \mathcal{R}/(\gamma -1)$ is the constant-pressure specific heat capacity. A temperature gradient less negative than the adiabatic lapse rate gives positive $N^2$; a steeper decline gives convective instability.

A colloidal grain in a liquid is large enough to see with a microscope, yet small enough for thermal motion to compete with its tendency to settle. In the Perrin sedimentation experiment, the equilibrium number density falls with height: a visible miniature atmosphere. Counting grains at different heights measures an energy scale; tracking their Brownian motion measures a transport rate. Connecting both to the same Avogadro constant makes the experiment a quantitative test of kinetic theory. The Wiener measure describes an idealized path law; these measurements instead test a physical model and its material parameters.

Jean Baptiste Perrin's 1909 memoir traces his sedimentation experiments to May 1908. It reports relative counts $100$, $47$, $22.6$, $12$ at heights $5$, $35$, $65$, $95\mu \mathrm{m}$, and compares radius estimates from Stokes settling velocity, counted grain mass, and rows of touching grains.^[ref]

The 1926 lecture describes resin precipitated from alcohol into water, washed and fractionated by centrifugation to obtain nearly monodisperse gamboge or mastic grains. Diameter and density were measured separately. A sealed slide-and-cover-glass chamber prevented evaporation; a vertical microscope resolved roughly $2\mu \mathrm{m}$ layers through a roughly $0.1\mathrm{mm}$ depth. Refocusing and photographing compared layer populations after equilibrium; changing temperature reversibly changed the distribution.^[ref]

In Les Atomes (1913), Jean Baptiste Perrin restricts the eyepiece view to fewer than five or six grains and alternates observations between heights. Focus displacement requires a refractive index correction to recover actual layer separation. Horizontal positions were marked with a camera lucida, usually every $30\mathrm{s}$, and distances calibrated with a micrometer; the polygon joins sampled positions rather than recording the intervening trajectory.^[ref][ref]

A photon-recoil lag correlation asks how often a photon record occurs at a specified time displacement from an electron record. Let $r_e,r_{\gamma }$ be observed separate-channel rates, $S$ the rate of detected genuine pairs before a timing cut, and $f(\ell )$ their normalized timing probability density function. For an original dilute, stationary model with observation time $T$, neglecting end effects and detector dead time, the expected pair density isThis counts all pairs of records, rather than at most one detector trigger per electron. It uses independent unrelated detections for the flat term and an added same-event contribution. A normal distribution for timing spread is a chosen approximation, not a claim about the historical instrument. Intrinsic emission delay, counter response and readout uncertainty can all affect the observed probability density function.

Define the accepted lag interval by $|\ell |<w/2$: $w$ is its full width. Integrating the flat baseline gives the existing accidental coincidence expectation $T{r}_e{r}_{\gamma }w$. The additional area is $TS{\int }_{-w/2}^{w/2}f(\ell )d\ell$. Moving a window of the same full width away from the peak tests the baseline under the model's stationarity assumption. It does not remove correlated instrumental backgrounds or changing rates automatically.

For the lag model, choose $T=1000\mathrm{s}$, $r_e=2{\mathrm{s}}^{-1}$, $r_{\gamma }=0.5{\mathrm{s}}^{-1}$, $S=0.1{\mathrm{s}}^{-1}$ and timing standard deviation $\sigma =1\mathrm{ms}$. A full width $w=2\mathrm{ms}$ gives $2$ expected accidental coincidences and $100\mathrm{erf}(1/\sqrt{2})\simeq 68.27$ accepted genuine pairs, using the error function. The total is approximately $70.27$, whereas the displaced interval from $4$ to $6\mathrm{ms}$ has approximately $2.003$ expected pairs. A finite sample fluctuates around these expectations; these values are not measured counts or a significance calculation.

The right panel isolates the information lost by retaining singles totals alone. In a separate idealized model, stipulate $M=10^6$ independent $1\mathrm{ms}$ bins, with Bernoulli distribution indicators $X,Y$ for electron and photon detection in a bin. Set $p_e=\mathrm{Pr}(X=1)=0.002$ and $p_{\gamma }=\mathrm{Pr}(Y=1)=0.0005$. Independence predicts $\mathrm{Pr}(X=1,Y=1)=p_e{p}_{\gamma }=10^{-6}$, hence one expected double detection. Alternatively chooseThese nonnegative probabilities sum to one and preserve both marginal probabilities, yet predict $101$ double detections in expectation. Their covariance is $0.000101-p_e{p}_{\gamma }=0.000100$. This is an exact joint probability distribution illustrating pairing; it does not model the timing spread in the left panel or reconstruct either historical hypothesis quantitatively. Its lesson is why two singles totals cannot decide whether the observed events share a cause.

In $d$ dimensions, isotropic modes with dispersion $ϵ=A{k}^s$ have density of states proportional to ${ϵ}^{d/s-1}$. Their equilibrium number is not conserved, so the excitation chemical potential is zero. Integrating $ϵD(ϵ)/(e^{ϵ/(k_{B}T)}-1)$ and rescaling by $k_{B}T$ gives the displayed energy power and heat capacity proportional to $T^{d/s}$. In three dimensions, linear acoustic phonons give $T^4$ energy and $T^3$ heat capacity; quadratic gapless ferromagnetic magnons give $T^{5/2}$ energy and $T^{3/2}$ heat capacity. A gap changes the asymptotically low-temperature law to activated behaviour. The mode approximation applies below the relevant band cutoff and when interactions between excitations are negligible.

The scattering interpretation connects an incoming preparation to the relative frequency of outgoing directions and final atomic states. Max Born's 1926 paper treats a distant incoming electron and a chemical atom, expands the outgoing wavefunction into final atomic states, and imposes conservation of energy. Its translated correction footnote specifies the square of the coefficient; the modern complex modulus notation below is ours.^[ref]

In his 11 December 1954 lecture, Max Born recalled the difficulty of interpreting bound electrons alone. He instead described an incident electron wave generating an outgoing spherical wave, with direction-dependent squared amplitude determining relative scattering probabilities. Distinct final atomic states describe excitation with corresponding electron energy loss. This is a retrospective account of an interpretation, rather than the report of a new apparatus.^[ref]

The distinction from the modern Born rule matters. The historical question was which collision outcome occurs: a direction of motion and a final atomic state. The general projective measurement formula applies to arbitrary observables, including degenerate outcomes and mixed states; a positive operator-valued measure describes still more general detector statistics. Those formulations should not be read back as the explicit contents of the short 1926 scattering paper. Nor does the scattering interpretation itself supply a rule for the conditional post-measurement state.

Actual experimental settings are described in the Davisson–Germer experiment and the Franck-Hertz experiment. The former scans collected electric current as an electron beam scatters from prepared nickel; changing direction, accelerating voltage and surface preparation probes electron diffraction. The latter uses acceleration through mercury vapor and a retarding collector to infer discrete energy losses from current minima, with vapor density and retarding settings as controls. Angular resolution and surface backgrounds limit the first inference; initial energy spread and collision positions limit the second. These linked experiments supply the preparation, apparatus and readout; the examples below are original calculations, not Born's measured data.

Bootstrap Methods: Another Look at the Jackknife by Bradley Efron, 1979 journal article appeared in The Annals of Statistics 7(1), pp. 1–26, in January 1979. The original article's bibliographic cover identifies the author, journal, date and pages.^[ref] Stanford's earlier technical report BIO 32, also by Bradley Efron and with the same title, is dated July 1977; that report date is distinct from the journal publication date.^[ref]

Take the original observations $(1,2,4)$, with sample mean $\overline{x}=7/3$. All $3^3=27$ ordered bootstrap samples have probability $1/27$. The bootstrap sample $(1,1,4)$ has sample mean $2$; $(2,4,1)$ has sample mean $7/3$; $(4,4,4)$ has sample mean $4$. These are illustrative possibilities, not three simulated measurements used to estimate the distribution.

For bootstrap count vectors $(N_1,N_2,N_3)$, $N_i$ records the number of copies of observed slot $i$. Their multinomial distribution isand ${\overline{X}}^{\ast }=(N_1+2N_2+4N_3)/3$. Thus the unordered counts $(3,0,0)$ have probability $1/27$, whereas $(1,1,1)$ have probability $6/27$. Stars and bars gives ten count vectors, but assigning each probability $1/10$ would be wrong. Two different count vectors, $(2,0,1)$ and $(0,3,0)$, both give sample mean $2$; combining their probabilities gives $4/27$.

The conditional bootstrap variance of a sample mean gives a compact check on the enumeration. Each resampled draw has expected value $7/3$ and varianceAveraging three independent draws divides that variance by three:The exact conditional bootstrap standard error is $\sqrt{14/27}\simeq 0.720$. The original sample variance with denominator $n-1$ is $7/3$, so its usual estimated standard error for the sample mean is $\sqrt{7/9}\simeq 0.882$. Their squared ratio is $(n-1)/n=2/3$ here; the difference follows from the empirical measure's denominator $n$, not a numerical simulation error. Neither number is the known true sampling standard error of the unknown source probability distribution.

The part of entropy associated with the number of distinct spatial or occupancy arrangements consistent with chosen macroscopic constraints. For equally weighted configurations, the Boltzmann entropy formula gives the displayed expression. Unequal probabilities require $-k_B{\sum }_i{p}_i\ln p_i$ over configurations. For $n$ indistinguishable empty sites among $M$ otherwise occupied sites, ${\Omega }_{\mathrm{conf}}=\left(\genfrac{}{}{0ex}{}{M}{n}\right)$. Vibrational or internal degrees of freedom can contribute additional entropy even at fixed site arrangement.

For spins $s_i\in \{-1,0,1\}$ with ferromagnetic nearest-neighbour exchange and no single-ion anisotropy, a homogeneous mean-field approximation givesThe self-consistency equation is $S=2\sinh h/(1+2\cosh h)$ with $h=\beta (zJS+B)$. Its small-$h$ expansion is $2h/3-h^3/9+O(h^5)$. Hence $k_B{T}_c=2zJ/3$ and, with $t=T/T_c-1$, the nonzero branches satisfy $S^2=(8/3)(-t)+O(t^2)$. The critical exponent is the mean-field value $1/2$, not an exact two-dimensional exponent. At zero source the finite-system equilibrium magnetization is zero by spin inversion symmetry; the broken branches describe the thermodynamic limit with a phase selected by a vanishing source.

For $p>0$ and a nonnegative integer $n$, integration by parts gives, when $n>0$,The endpoint term $[s(1-s^p{)}^n{]}_0^1$ vanishes. ThereforeRepeated application yields $J_n(p)={\prod }_{j=1}^{n}pj/(pj+1)$. The integration by substitution $w=s^p$ also identifies $J_n(p)=B(1/p,n+1)/p$ with an Euler beta function. A related weighted integral follows immediately from the same substitution:

For positive real $p,q$, substitute $x={\sin }^2\theta$ in the defining Euler beta function integral. The increasing substitution maps $[0,\pi /2]$ onto $[0,1]$ and has $dx=2\sin \theta \cos \theta d\theta$. Combining this factor with $x^{p-1}(1-x{)}^{q-1}$ gives the displayed identity. Positivity of both parameters ensures convergence at the two endpoints, even when the integrand is unbounded there.

In $B(n,n)={\int }_0^1[x(1-x){]}^{n-1}dx$, put $v=2x-1$. Then $x(1-x)=(1-v^2)/4$, and evenness givesThe further substitution $u=v^2$ turns this into $2^{1-2n}B(1/2,n)$. The symmetry of the Euler beta function yields the title identity. This proof works for every positive real $n$, without requiring $n$ to be an integer or using the Gamma function.

In 2006, Zoran Hadzibabic, Peter Krüger, Marc Cheneau, Baptiste Battelier and Jean Dalibard split a cooled rubidium-87 cloud into two independent planar gases using an optical lattice. After release, the expanding sheets overlapped; absorption imaging recorded their matter-wave interference. Smooth fringe waviness revealed thermal phase fluctuations, while sharp dislocations indicated isolated vortices. The initial evaporative-cooling endpoint controlled the preparation; central fringe contrast served as a degeneracy proxy because direct thermometry in the lattice was difficult.^[ref]

Loss of extended coherence coincided with increased vortex signatures. The optical trap and finite cloud produced a crossover; tightly bound pairs were unresolved, and short-wavelength modes could contaminate high-temperature counts. These were observations of the vortex-unbinding mechanism, not a direct measurement of phase stiffness or a sharp homogeneous universal stiffness jump. The authors left the sample's superfluidity open.^[ref]

Consider an original counting example: a square two-dimensional XY model of side $L$, a microscopic core radius $a$, and a fixed phase stiffness ${\rho }_s$. This deliberately neglects screening by other vortices. Away from a unit phase vortex, $\theta =\arg (x+iy)$ and $|\nabla \theta |=1/r$. Integrating the Goldstone-mode effective free energy over an annulus givesBoundary shape changes finite terms. There are order $(L/a{)}^2$ core positions, so the positional entropy is $S_{\mathrm{v}}=2k_B\log (L/a)+O(k_B)$, where $k_B$ is the Boltzmann constant. With $K={\rho }_s/(k_{B}T)$, subtracting $T{S}_{\mathrm{v}}$ gives the displayed free energy.

For $L/a=1000$ and $E_c/(k_{B}T)=2$, the dimensionless free energy cost is about $5.546$ at $K=0.8$ and $-0.965$ at $K=0.5$. The same logarithm appears in the cost and the positional reward: increasing the system size strengthens whichever has the larger coefficient. The corresponding one-defect Boltzmann factors, about $0.00390$ and $2.62$, are relative statistical weights, not normalized probabilities. The latter signals failure of a dilute single-defect estimate.

A single net phase vortex requires compatible boundaries: on a periodic sample use neutral pairs, as required by vortex charge neutrality. This argument identifies a possible threshold, but predicting the universal stiffness jump requires the renormalized phase stiffness, after smaller pairs have responded. It does not justify setting the transition temperature equal to a bare lattice coupling divided by a universal constant.

The neutral axis is the line in the cross-section of an elastic beam where longitudinal bending strain vanishes. For a homogeneous beam in pure bending, zero resultant longitudinal force gives ${\int }_{A}zdA=0$, so the neutral axis passes through the centroid. Fibres on its opposite sides are stretched and compressed. The second moment of area about this axis controls the relation between bending moment and curvature.