For two visible, approximately massless leptons, both measured four-momenta enterThis is a reconstructed invariant mass, including longitudinal information, rather than the lower bound obtained from transverse mass. As an original example, two opposite-azimuth leptons with $p_{T+}=p_{T-}=45\mathrm{GeV}$ and equal rapidities have $m_{\ell \ell }=90\mathrm{GeV}$. Changing their rapidity difference to one raises it to about $101.486\mathrm{GeV}$, even though their transverse momenta are unchanged.

The ideal transverse mass bound concerns the actual event's parent invariant mass, not a perfectly sharp universal number. A finite decay width spreads parent invariant masses; particle jet recoil changes daughter directions; invariant-mass resolution, momentum resolution and imperfect reconstruction can smear an observed endpoint. Recoil alone does not invalidate the ideal inequality when both daughter transverse momenta are correct. Additional invisible particles, radiated photons and a massless-daughter approximation can invalidate the simple two-body reconstruction. Likewise, a Z boson pair-mass peak requires control of identification, detection efficiency, instrumental response and backgrounds. A peak position and an endpoint are different summaries of selected event populations.

The 1983 UA2 experiment Z boson paper reports eight candidate electron pairs, including radiative decays, at $\sqrt{s}=550\mathrm{GeV}$. Four events entered its invariant mass determination. The published result was $91.9\pm 1.3\pm 1.4\mathrm{GeV}/c^2$, the second uncertainty systematic. These are that paper's reported numbers, not inputs fitted to the illustration below.^[ref]

The 1983 UA1 experiment paper used proton–antiproton collisions at $\sqrt{s}=540\mathrm{GeV}$. A high-transverse momentum electron candidate required a track matched to an electromagnetic particle shower, with little energy behind it in the hadronic particle calorimeter. Test beams checked electron and pion responses; ordinary events and particle jet samples checked the imbalance resolution. Two searches, beginning with isolated electrons and with missing transverse energy, converged on the same candidates. Conversion pairs, hadrons and heavy-flavor decays were examined as backgrounds.^[ref]

A sun valve automatically restricts a gas light's supply in daylight and allows it again as illumination falls. In Gustaf Dalén's design, a blackened rod and three polished rods respond differently to sunlight. Their relative thermal expansion operates the gas valve through a membrane; a pilot flame permits relighting. The museum dates this design to 1907.^[ref]

The useful comparison is between two thermal responses. Both kinds of rod expand when warmed; a dark surface can absorb more incident radiant power, so its temperature rises more. Comparing their extensions can reject a common ambient-temperature change while responding to illumination. The mechanism converts light into heat, heat into a difference of length, and that difference into a change of gas flow. Differential thermal model of a sun valve makes those assumptions quantitative.

By Yen-Chi Chen, explicitly named on the lecture page. Authored notes in UW STAT 535: Statistical Machine Learning, Autumn 2026, hosted on the University of Washington faculty website. No separate lecture publication or revision date is established.

Online text: faculty.washington.edu/yenchic/25A_stat535/Lec12_bootstrap.html

Cited passages: The Bootstrap, resampling procedure; High level idea of how bootstrap works, conditional empirical sampling law; Bootstrap and Statistical Functionals, conditional variance and consistency; Failure of the bootstrap, uniform-endpoint counterexample.