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A vector space whose coordinates are the components of a particle's momentum; a distribution in momentum space describes momenta rather than positions.

For an integer $n\ge 1$, add residue classes by adding their representatives and taking the class modulo $n$. This is well-defined because replacing either representative by a multiple-of-$n$ translate changes the sum by a multiple of $n$. The associativity, identity class $[0{]}_n$ and inverse $[-a{]}_n$ are inherited from integer addition. The result is the additive cyclic group $\mathbb{Z}/n\mathbb{Z}$.

Respiratory control can be investigated outside an intact biological cell by supplying isolated mitochondria with substrates and ADP. This makes the boundary between oxygen consumption and ATP formation experimentally accessible. The historical mitochondrial oxygen electrode assay describes the 1955 experiment; the following authored protocol adds modern preparation and inhibitor controls.

In Functional Assessment of Isolated Mitochondria In Vitro (2009), Ian R. Lanza and K. Sreekumaran Nair describe human skeletal-muscle biopsy mincing, cold protease treatment, homogenization and differential centrifugation. A low-speed spin removes coarse material; subsequent spins collect and wash mitochondria. A stirred, closed $2\mathrm{mL}$ Oxygraph-2k chamber at $37{}^{\circ }\mathrm{C}$ uses a calibrated polarographic oxygen electrode. Its signal gives dissolved oxygen concentration; the negative time derivative gives consumption. Instrument corrections include electrode consumption and oxygen leakage.^[ref]

Their additions include glutamate plus malate, an ADP pulse, then saturating ADP. Added cytochrome c tests outer-membrane damage: stimulation suggests loss of endogenous cytochrome c. Later oligomycin inhibits ATP synthase; FCCP produces respiratory uncoupling; antimycin A inhibits respiratory complex III to estimate residual background. Figure 6 includes intervening succinate dehydrogenase and respiratory complex I interventions; the modeled traces below simplify that sequence. Isolation can damage or select mitochondria, and saturating assay conditions differ from those in living biological tissue.^[ref]

The control logic matters more than a single steep slope. If ADP accelerates consumption and a finite pulse is followed by slowing, demand is affecting respiratory flux. If oligomycin slows consumption, but subsequent FCCP accelerates it, the electron transport chain can still turn over while ATP synthase remains inhibited. That contrast separates restricted coupled turnover from a completely inactive chain. Failure of FCCP to restore a rate is less specific: substrate limitation, damage, low oxygen or excessive uncoupler can also restrict respiration.

Neither the cytochrome c test alone nor a large respiratory control ratio establishes every aspect of mitochondrial integrity. Likewise, oxygen consumption is not a direct ATP assay: respiratory uncoupling provides a concrete counterexample. The preparation, additions and normalization denominator determine what comparison the experiment can support.

The outer mitochondrial membrane separates the surrounding cytosol from the intermembrane space. Channels permit passage of many small molecules, while the inner mitochondrial membrane supplies the main selective barrier between them and the mitochondrial matrix.^[ref]

Small-solute access does not imply unrestricted access for proteins. The cytochrome c addition in the isolated mitochondria respiration experiment tests a consequence of damage to this boundary. Organelle protein import describes purposeful delivery across mitochondrial boundaries.

In a mitochondrion, the intermembrane space lies between the outer mitochondrial membrane and inner mitochondrial membrane, extending into the interiors of cristae. It includes the side of the inner mitochondrial membrane from which protons return through ATP synthase to the mitochondrial matrix.

A useful sidedness check is to follow a single arrow: outward pumping goes from green to yellow in the diagram, while coupled return goes from yellow to green. These are opposing movements of protons, not a circuit in which electrons return through ATP synthase. The mitochondrial respiratory chain and chemiosmosis articles distinguish the two ledgers.

The mitochondrial matrix is the compartment enclosed by the inner mitochondrial membrane, containing mitochondrial DNA, ribosomes and many enzymes. Most citric acid cycle steps occur here; succinate dehydrogenase is membrane-associated. The citric acid cycle supplies NADH for the mitochondrial respiratory chain.^[ref]

Distinguish the location of a chemical reaction from the location of its detection. An external oxygen electrode can report consumption by an intact preparation without entering the mitochondrial matrix. That signal does not locate every intermediate, establish protein targeting or directly count newly formed ATP.

The inner mitochondrial membrane separates mitochondrial matrix from intermembrane space. Its selective permeability permits distinct chemical conditions, while embedded mitochondrial respiratory chain proteins and ATP synthase support oxidative phosphorylation. Cristae increase its surface area.^[ref]

A selective boundary and a productive enzyme have different jobs. Even active respiratory enzymes cannot sustain a useful electrochemical gradient if protons return freely across the boundary. Conversely, an intact boundary without an electron donor does not supply the energy for continued pumping. The isolated mitochondria respiration experiment separates some of these dependencies by additions to the same preparation.

For a two-body W boson decay into a charged lepton $\ell$ and a neutrino, neglect both daughter masses and use natural units $c=1$. Let $\Delta \varphi$ be the angle between the daughter transverse momenta. Identifying the missing transverse momentum with the neutrino's givesThis observable uses no $p_{z\nu }$. The 1983 UA1 experiment paper already used transverse mass as a lower bound on parent mass, before making additional decay-model assumptions for a mass estimate.^[ref]

The bound follows directly from the energy–momentum relation. For each massless daughter write its four-momentum, using rapidity $y$ and azimuth $\varphi$, asUsing the Minkowski inner product with positive time sign, $p^2=E^2-|\mathbf{p}{|}^2$, the two daughter four-momenta give the true parent invariant mass $M$:Since $\cosh u\ge 1$, $m_T\le M$, with equality at equal daughter rapidities when both transverse momenta are nonzero. Thus the unmeasured longitudinal information can enlarge the true invariant mass, but cannot lower it below this ideal transverse mass.

For an original example, stipulate $p_{T\ell }=p_{T\nu }=40\mathrm{GeV}$ and $\Delta \varphi =\pi$. Then $m_T=80\mathrm{GeV}$. Equal rapidities give $M=80\mathrm{GeV}$; choosing $y_{\ell }-y_{\nu }=1$ with exactly the same transverse readout gives $M=80\cosh (1/2)\simeq 90.210\mathrm{GeV}$. One measured transverse configuration therefore permits different true parent masses.