The maximum power point is the operating voltage $V_{\mathrm{mp}}$ and output electric current $I_{\mathrm{mp}}$ at which a source delivers its largest electrical power, $P_{\max }=V_{\mathrm{mp}}{I}_{\mathrm{mp}}$. It is found along the device's current–voltage curve under specified illumination and temperature.

For a smooth solar-cell output curve written with delivered current positive, an interior maximum satisfies $d(VI)/dV=I+VdI/dV=0$. The short-circuit current and open-circuit voltage are useful endpoints but neither delivers power, because one factor is zero at each endpoint. Changing illumination or temperature moves the maximum power point, motivating maximum power point tracking in the connected power converter.

The short-circuit current is the terminal electric current of a device when its terminal voltage is zero. For a solar cell, it is usually denoted $I_{\mathrm{sc}}$ and is obtained from the illuminated current–voltage curve where the two terminals are connected through a sufficiently low-resistance measuring path.

A nonzero short-circuit current establishes photogenerated charge collection, but the delivered terminal power $P=VI$ is zero at an ideal short circuit. Useful electrical output instead requires a finite operating voltage, often near the maximum power point. The short-circuit current depends on illumination, active area and collection efficiency; short-circuit current density divides it by the specified area, allowing devices of different sizes to be compared.

A photovoltaic device combining absorbers with different bandgaps to convert different spectral regions. A monolithic two-terminal stack has a common series current and therefore requires photocurrent matching; independently contacted junctions relax that constraint.

An organic solar cell converts light into electrical output using organic semiconductors, such as conjugated molecular or polymeric absorbers. Light absorption commonly produces a bound exciton, so useful charge generation requires more than simply creating an excited molecule.

In a donor–acceptor design, an exciton reaches an interface where charge transfer can separate its electron and electron hole. Selective contacts collect the separated charge carriers before they undergo recombination. A bulk heterojunction distributes donor–acceptor interfaces throughout the absorber, shortening the distance an exciton must travel. Its morphology must also supply connected transport pathways to the contacts. Absorption, exciton diffusion length, separation and charge collection therefore constrain performance together.

Given three mutually externally tangent circles, the inner Soddy circle lies in their curvilinear triangular gap and is tangent to all three. The Descartes' circle theorem determines its radius from the three given radii.

If $[L_a,L_b]={ϵ}_{abc}{L}_c$, $[L_a,K_b]={ϵ}_{abc}{K}_c$ and $[K_a,K_b]={ϵ}_{abc}{L}_c$, then $L_a^{\pm }=(L_a\pm K_a)/2$ generate commuting copies of $\mathfrak{su}(2)$. This Lie-algebra decomposition does not remove the central quotient in the corresponding global group.

A smectic A phase has layered positional order with the average molecular orientation along the layer normal. Molecular positions within each layer remain liquid-like. Its orientational axis is described by a nematic director.

A smectic C phase has layered positional order with the nematic director tilted away from the layer normal. The tilt supplies additional orientational information beyond the density modulation of a smectic phase.

For uniaxial stress $\sigma$, projection onto a slip-plane normal and a slip direction gives the displayed shear stress. The angles $\varphi$ and $\lambda$ are measured from the tensile axis to that normal and direction respectively.

Dislocation glide in a hexagonal crystal on its basal $(0001)$ plane, ordinarily along a close-packed basal direction in the $⟨11\overline{2}0⟩$ family. Direction indices must be converted using the actual hexagonal metric, including the ratio $c/a$.

Plastic flow involving more than one active slip system. Interactions between dislocations on intersecting systems can give substantially faster work hardening than easy single slip.

Near the first addition threshold of a repulsive single-orbital defect, the empty and singly occupied grand weights are 1 and $2e^{\beta (\mu -ϵ)}$. Equal weights occur at $\mu =ϵ-k_{B}T\ln 2$. Near the second threshold, the single/double aggregate weights are in ratio $2:e^{\beta (\mu -ϵ-U)}$, giving half-step position $ϵ+U+k_{B}T\ln 2$. These are entropic finite-temperature shifts, which vanish in energy units as $T\to 0$. They must still be retained when evaluating statistics exactly at an unshifted degeneracy point.

With $z=e^{\beta (\mu -ϵ)}$, $q=e^{-\beta U}$ and $\Xi =1+2z+q{z}^2$, direct four-state averaging givesFor $\beta U\gg 1$ it has two separated peaks of height $1/4$ at the spin-degeneracy-shifted charge steps. At the lower bare crossing $\mu =ϵ$, three microstates share equal limiting weight, giving $\bar{n}=2/3$ and variance $2/9$. The upper bare crossing similarly gives mean $4/3$ and variance $2/9$. Between crossings occupation is pinned near one and fluctuations are suppressed. The identity is grand-canonical; fixing total particle number would change the fluctuation question.

For a repulsive single orbital at temperatures small compared with U, chemical potentials strictly between the first and second addition energies select a singly occupied defect. Its mean occupation is nearly one while its charge-number variance is exponentially small away from the thermally broadened step regions. The remaining spin degeneracy does not itself produce charge-number fluctuations.

Stochastic optical reconstruction microscopy reconstructs an image from the estimated positions of individual fluorescent labels that are observed in sparse subsets over many frames. The experiment repeatedly switches labels between detectable and dark states, fits isolated point spread functions, and combines the localizations.

This separates emitters in time instead of requiring all neighboring labels to be spatially resolved in one exposure. Localization precision depends on detected photons and background; drift, incomplete labeling and simultaneous overlapping emitters constrain final optical resolution. In the original demonstration, reversibly switched dye labels enabled repeated sparse localization.^[ref]

Uncertainty in an estimated emitter position. It can be much smaller than the width of its point spread function, but it does not by itself guarantee that densely packed emitters or an entire labelled structure are resolved.

A surface black in the thermal infrared beneath a shortwave-transparent, isothermal grey layer has atmospheric balance ${ϵ}_a\sigma T_s^4=2{ϵ}_a\sigma T_a^4$. For nonzero layer emissivity, $T_a^4=T_s^4/2$. Direct surface transmission plus upward layer emission then gives $F_{\mathrm{out}}=(1-{ϵ}_a/2)\sigma T_s^4$, proving the title. This model only permits $1/2\le {ϵ}_{\mathrm{eff}}\le 1$ for $0\le {ϵ}_a\le 1$. A real vertical opacity/temperature profile is not uniquely determined by this one fitted parameter.

Adaptive Simpson's method compares one Simpson's rule panel $S$ on $[a,b]$ with two panels totaling $S_2$ after bisection. In the smooth asymptotic regime, the leading error drops by a factor of sixteen, so $(S_2-S)/15$ estimates $I-S_2$, and $S_2+(S_2-S)/15$ is the corrected estimate.

If the discrepancy exceeds the local tolerance, subdivide again and split the error budget between the children. Endpoint and midpoint values can be reused. The factor fifteen follows by eliminating the leading fourth-order error; it is not a rigorous error bound without quantitative assumptions. A narrow peak missed by all sampled nodes can make both estimates agree while both are wrong, as finite samples do not certify quadrature accuracy explains.

The bacterial SRP receptor GTPase that partners with Ffh in membrane targeting.

Reciprocal regulation of the receptor and SRP GTPases couples recognition to transfer of the nascent chain toward a membrane translocation site. This illustrates a targeting cycle in which simple binding is only one stage of productive delivery. ^[ref]

The eukaryotic receptor subunit whose GTPase partners with the SRP54 GTPase during targeting.