FRET spectral overlap quantifies overlap between the FRET donor's emission and the FRET acceptor's absorption as functions of wavelength $\lambda$. With normalized donor emission $f_D$ and acceptor molar absorption coefficient ${\epsilon }_A$, one convention isMatching two emission colors is insufficient: the receiving absorption matters.^[ref]

The generalized Leibniz rule extends composition of differential operators to all integer $n$. Define $\left(\genfrac{}{}{0ex}{}{n}{r}\right)=n(n-1)\cdots (n-r+1)/r!$. For nonnegative $n$ the sum stops at $r=n$. For $n=-s<0$, its coefficients are $(-1{)}^r(\genfrac{}{}{0ex}{}{s+r-1}{r})$; in particular ${\partial }^{-1}f=f{\partial }^{-1}-f^{\prime }{\partial }^{-2}+f^{\prime \prime }{\partial }^{-3}-\cdots$. Applying $\partial$ on the left cancels consecutive derivative terms and recovers multiplication by $f$, which explains the inverse expansion.

The formal residue is the coefficient of ${\partial }^{-1}$ after coefficients have been moved to the left by the formal pseudodifferential composition rule. It is not an analytic residue at a point. For scalar coefficients, the residue of a commutator is a total derivative; integrating it with vanishing boundary terms produces the cyclic Adler trace.

For scalar coefficients, leading symbols commute. If $P=a{\partial }^p+\cdots$ and $Q=b{\partial }^q+\cdots$, their product has leading coefficient $ab$, and their commutator has coefficient $pa{b}^{\prime }-qb{a}^{\prime }$ at order $p+q-1$. Thus product order is at most $p+q$, with equality when $ab$ is not identically zero; commutator order is at most $p+q-1$, with equality when its displayed coefficient is not identically zero. Constant-coefficient operators commute. Nonzero smooth functions with disjoint supports show why product orders need not add over $C^{\infty }(\mathbb{R})$.

Treat $k$ as a formal spectral parameter near infinity. If $P={\sum }_{j\ge 0}{g}_j{\partial }^{\alpha -j}$, the formal pseudodifferential composition rule gives $Pw=k^{\alpha +\beta }{e}^{kx}{\sum }_{s\ge 0}{k}^{-s}{\sum }_{i+j+r=s}{g}_j\left(\genfrac{}{}{0ex}{}{\alpha -j}{r}\right)w_i^{(r)}$. Each coefficient contains finitely many terms. This describes a formal module action, not a claim that the infinite series converges at finite $k$.

For a cardinal number $\kappa$, ${\mathrm{MA}}_{\kappa }$ says that for every forcing with the countable chain condition for forcing and every family of at most $\kappa$ dense subsets, some filter in an ordered set meets them all. Full Martin axiom asserts this for all $\kappa <2^{{\aleph }_0}$. The strict upper bound is part of the definition.

This damped harmonic oscillator receives a constant force only while moving toward increasing $x$. For $0<k<\omega$, a positive maximum $X$ evolves without forcing to a minimum $-rX$, then with forcing to the next maximum $r^{2}X+(1+r)a/{\omega }^2$, where $r=e^{-k\pi /\sqrt{{\omega }^2-k^2}}$. Its unique periodic maximum is $a/[{\omega }^2(1-r)]$ and its period is $2\pi /\sqrt{{\omega }^2-k^2}$. Position and velocity are continuous at switches, but acceleration has jumps. The slope $r^2<1$ of this turning-point return map shows attraction of nearby turning-point amplitudes.

For an unregulated exponential basal resource and nearest-neighbor Lotka-Volterra equations, the weighted interaction matrix is skew-symmetric matrix. An even chain has an invertible tridiagonal interaction matrix; its positive equilibrium, when feasible, fixes all long-term means. An odd chain has a null direction and requires a special balance between resource growth and mortality to permit bounded full coexistence. For a three-level chain, $X_1^{b_2}{X}_3^{a_1}=C\exp [(b_{2}r-a_1{d}_3)t]$. A positive even-chain equilibrium gives a conserved coercive relative-entropy-type function, proving every species remains bounded above and away from zero.

If creation operators obey $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, set $|\mathbf{p}⟩=\sqrt{2E_{\mathbf{p}}}{a}_{\mathbf{p}}^{†}|0⟩$. Their inner product then has the displayed value. The associated completeness measure is $d^{3}p/((2\pi {)}^{3}2E_{\mathbf{p}})$, the invariant positive-energy mass shell measure. Specifying this normalization prevents hidden external energy factors in a scattering amplitude.

A one-particle state belongs to the one-excitation sector of a Fock space. In a momentum basis with $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, the wavepacket $\int d^{3}pf(\mathbf{p})a_{\mathbf{p}}^{†}|0⟩/(2\pi {)}^3$ has norm squared $\int d^{3}p|f(\mathbf{p}){|}^2/(2\pi {)}^3$. Sharp-momentum states have distributional rather than unit normalization; factors of $\sqrt{2E_{\mathbf{p}}}$ convert to relativistic normalization of a one-particle state.

A fluorescence microscope maps photon emission to an image. Excitation light passes through a fluorescence excitation filter and reflects from a wavelength-selective beam splitter into the objective optical lens. Returning fluorescence passes through that beam splitter and a fluorescence emission filter to the photodetector. Rejecting excitation makes faint emission visible.^[ref]

An ordinary image records emission-band counts at each pixel. Adding pulse-delay measurements gives a fluorescence lifetime map; scanning emission wavelength gives spectral information instead. Confocal microscopy adds spatial rejection and confocal optical sectioning, as developed in its own branch. An unlabeled specimen under the same illumination reveals intrinsic fluorescence and excitation leakage; excitation-blocked frames reveal detector background. A blank cannot perfectly predict background in a differently prepared specimen.

For the constant-rate model in fluorescence quantum yield, fluorescence lifetime is the mean residence time in the emitting excited state and the $1/e$ timescale of its exponential decay. The detected fluorescent photon rate after an ideal instantaneous pulse is $I(t)=\eta k_{\mathrm{r}}{N}_0{e}^{-t/\tau }$ for $t\ge 0$. Its amplitude and its decay timescale are separate quantities.

For a stipulated example, rates $(k_{\mathrm{r}},k_{\mathrm{nr}})=(0.05,0.20){\mathrm{ns}}^{-1}$ give $\tau =4\mathrm{ns}$ and $\Phi =0.20$. Rates $(0.20,0.05){\mathrm{ns}}^{-1}$ give the same $\tau$ but $\Phi =0.80$. Conversely, doubling $N_0$ doubles the ideal amplitude without changing $\tau$. Equal fluorescence lifetimes need not imply equal brightness or equal fluorescence quantum yields.

Different environments can supply several populations, giving $I(t)={\sum }_j{A}_j{e}^{-t/{\tau }_j}$ rather than one exponential decay. A fitted single timescale then depends on the selected model and detection band. FRET donor lifetime uses the change in a donor's decay when transfer introduces an additional loss route; the FRET branch develops that interpretation.

Photobleaching is an irreversible loss of fluorescent capacity caused by illumination. It changes the number of active fluorophores over experimental time; fluorescence lifetime concerns the delay after an individual excitation. Reversible dark states and ordinary return to the ground state are different processes.

In a stipulated population-loss model, let $T$ denote elapsed illumination time and $t$ denote delay after a pulse. If active population declines as $e^{-k_{\mathrm{b}}T}$ while surviving fluorophores retain the same decay rates, then $I(t;T)=A_0{e}^{-k_{\mathrm{b}}T}{e}^{-t/\tau }$. Brightness falls while $\tau$ stays fixed. This example does not assert that all bleaching follows exponential decay: chemical changes and selective loss from a mixture can also change the observed decay shape. The existing FRET photobleaching controls illustrate why deliberate bleaching requires careful interpretation.

Fluorescence quantum yield counts emitted fluorescent photons per absorbed excitation photon. It differs from quantum efficiency, which describes the photodetector.^[ref]

For an original constant-rate model, let $k_{\mathrm{r}}$ be the radiative decay rate and $k_{\mathrm{nr}}$ the sum of competing rates without fluorescent photon emission. An initial population $N_0$ in the emitting excited state obeys the rate equationIntegrating $k_{\mathrm{r}}N(t)$ over all time givesHere every absorbed excitation is assumed to reach this emitting population; additional losses before that stage require another factor. Under steady weak excitation, absorbed photon rate $R_{\mathrm{abs}}$ and total detection fraction $\eta$ give $C=\eta \Phi R_{\mathrm{abs}}$. The count rate therefore mixes excitation, yield, and detection, whereas $\tau$ specifies a decay timescale.

A fluorescence emission spectrum resolves emitted photons by wavelength at fixed excitation settings. Subtract a matched sample-container and solvent background, and correct the wavelength-dependent throughput and photodetector sensitivity when estimating the emitted distribution. A camera image through one fluorescence emission filter instead measures a weighted band integral, not a whole fluorescence emission spectrum. A fluorescence lifetime resolves delay after excitation; its horizontal axis is time, not wavelength.

A fluorescence excitation spectrum scans excitation wavelength while detecting fluorescence in a fixed emission band. A fluorescence emission spectrum does the converse. The inspected spectrofluorometer example uses an excitation monochromator scan with the emission monochromator fixed, and reverses their roles for emission.^[ref]

An operational model explains why this need not reproduce absorption. For one dilute, unsaturated emitting species, write $a(\lambda )$ for absorbed fraction, $R_{\mathrm{in}}(\lambda )$ for incident photon rate, $\Phi (\lambda )$ for fluorescence quantum yield, and ${\eta }_{\mathrm{band}}(\lambda )$ for the fraction of emitted photons counted in the selected band. The background-subtracted count rate isOnly after correcting incident photon rate, and with constant yield and emission-band collection, is this proportional to absorbed fraction. A nonfluorescent absorber contributes to absorption but not this readout. Thick samples, multiple emitting species, and reabsorption of fluorescence complicate the model.

The Stokes shift compares the lowest absorption-band maximum with the fluorescence emission spectrum maximum. Relaxation often makes $\Delta \lambda >0$. This comparison differs from using an arbitrary excitation wavelength. A spectral-density maximum can move when changing from per unit wavelength to per unit spectroscopic wavenumber; specify the spectral convention.^[ref]

An original illustrative energy budget makes the distinction concrete. Excitation at $450\mathrm{nm}$ and emission at $520\mathrm{nm}$ correspond to about $2.76$ and $2.38\mathrm{eV}$ per photon, using $E=hc/\lambda$ with Planck constant $h$ and speed of light $c$ in vacuum. The difference, about $0.37\mathrm{eV}$, must go elsewhere in the complete energy balance. These chosen wavelengths are not measured band maxima for a particular fluorophore.

Anti-Stokes emission has shorter wavelength than excitation. Additional energy, for example from a thermally populated level of molecular vibration or multiple excitation photons, makes it possible. The one-photon, initially relaxed drawing does not cover these cases.^[ref]

A Jablonski diagram arranges molecular energy eigenstates vertically and draws arrows for changes between them. In the usual closed-shell molecular example, $S_0$ is the electronic ground state, $S_1$ and $S_2$ are electronic excited states with singlet state spin, and $T_1$ has triplet state spin. Each electronic state also has levels of molecular vibration. The diagram records possible processes, not molecular paths in space.^[ref]

Photon absorption can populate an upper level of molecular vibration in $S_1$. Relaxation removes excess energy without photon emission; fluorescence commonly starts from relaxed $S_1$.^[ref] The drawing also includes relaxation from a higher electronic excited state and a competing spin-changing route to a triplet state.

A fluorophore is a fluorescent molecule or fluorescent part of a molecule. Its usefulness as a label depends on how efficiently it absorbs the chosen excitation light, its fluorescence quantum yield, and its resistance to photobleaching. A large fluorescence quantum yield alone does not guarantee a bright image if few excitation photons are absorbed or collected.

The stigma is the carpel's pollen-receiving region. Compatible pollen may form a pollen tube that grows toward an ovule; arrival alone does not establish fertilization.^[ref]

A useful diagnostic question is what was actually observed: pollen on a stigma, a growing pollen tube, or a new plant embryo? Each observation tests a different stage. A pollen tube germination assay isolates the middle process for direct viewing.