The normal restoring-force coefficient of an AFM cantilever, $k_c=F/{\delta }_c$, with units $\mathrm{N}{\mathrm{m}}^{-1}$. A quantitative force measurement requires both this stiffness and the optical deflection sensitivity.

Unwanted sticking that prevents intended relative motion. In surface micromachining, a receding liquid meniscus can pull a compliant released beam onto the substrate; adhesion can then keep it there. Avoiding a liquid-vapor drying interface addresses one route to stiction, but does not remove contact adhesion during use.

Coupling orbital angular momentum $L>0$ to spin one-half produces $J=L\pm 1/2$. The dimensions add as $(2L+2)+2L=2(2L+1)$. A change between coupled and product bases changes labels, not the dimension of the Hilbert space.

For $N$ identical fermions restricted to one angular-momentum multiplet $j$, with all other one-particle labels fixed, the Pauli exclusion principle permits distinct angular momentum projections $j,j-1,\dots ,j-N+1$. Their sum is $Nj-N(N-1)/2$, provided $N\le 2j+1$. The resulting Slater determinant is killed by the total raising angular momentum ladder operator because raising any occupied state either exceeds $j$ or duplicates an occupied state. It is therefore a highest-weight vector of precisely this total angular momentum. The formula is invariant under replacing particles by holes in the full shell.

Exchanging the factors in the Clebsch-Gordan decomposition of two equal spins gives the phase $(-1{)}^{2s-j}$. This follows from the exchange symmetry of the Clebsch-Gordan coefficients. The total exchange parity includes the spatial state and any other internal factors: the spin parity alone does not impose the full symmetrization postulate. For two half-integer spins, even $j$ is spin-antisymmetric and odd $j$ spin-symmetric.

The AP2 adaptor complex is a heterotetrameric adaptor that links selected plasma-membrane cargo to clathrin during endocytosis. Recognition of membrane lipids and cargo signals helps concentrate the machinery at appropriate locations. Internalization can reduce surface receptor abundance or return reusable components to an intracellular pool. AP2 acts at a different major location from the AP1 adaptor complex, despite the related coat-adaptor design. ^[ref]

The AP1 adaptor complex is a heterotetrameric cargo-selection complex associated with clathrin-mediated traffic involving the Golgi apparatus and endosomes. It links membrane cargo and coat assembly at selected sites. Its role differs from that of the AP2 adaptor complex, which acts prominently at the plasma cell membrane. Adaptor identity helps determine which cargo is incorporated and where budding occurs. ^[ref]

A membrane pump coupling ATP hydrolysis to sodium extrusion and potassium uptake. Its gradients support electrical excitability and secondary transport; membrane location matters for epithelial directionality.

Transport of a substrate against its electrochemical tendency using downhill movement of another solute, rather than direct ATP hydrolysis by that transporter.

A membrane transport system that uses an energy source to move protons against their electrochemical gradient.

Formation of a stable assembly from which an actin filament can grow. Lowering the concentration of assembly-competent actin monomer can affect actin nucleation much more strongly than addition to an existing end.

Net addition of subunits at a actin filament end. A simple end model gives $v=k_{+}[G]-k_{-}$, with the sign changing at the relevant actin critical concentration.

The free assembly-competent actin monomer concentration at which net addition at a specified actin filament end is zero in a specified model: $C_c=k_{-}/k_{+}$. Different ends and nucleotide conditions can have different values.

A fluorescence readout of actin assembly using polymerization-dependent signal from a labelled actin fraction. Label effects, direct fluorescence changes, quenching and background must be controlled.^[ref]

Radiating material close to a compact object probes relativistic orbital motion, gravitational redshift and light propagation. A thin disk's inferred inner edge can be compared with the predicted innermost stable circular orbit, while iron-line transfer and thermal-continuum fitting probe complementary aspects of the same geometry. Agreement tests a combined spacetime-and-disk model. A black-hole spin measurement made after imposing a Kerr metric is not alone an independent verification of that metric tensor; geometry tests must vary gravitational predictions and astrophysical nuisance parameters together.

For an absolutely continuous $f$ with $f(0)=0$, this causal Abel integral equation has inversionUnder Laplace-transform growth hypotheses, the convolution theorem for Laplace transforms converts the original kernel into $\Gamma (1-\alpha )s^{\alpha -1}$. Since $\mathcal{L}[f^{\prime }]=s\mathcal{L}[f]$, the transformed inverse is $s^{-\alpha }\mathcal{L}[f^{\prime }]/\Gamma (1-\alpha )$, giving the formula. It also holds locally in the corresponding integrable convolution class: convolving the two power kernels produces the constant $B(\alpha ,1-\alpha )=\Gamma (\alpha )\Gamma (1-\alpha )$ by the beta-gamma identity, so substitution recovers ${\int }_0^t{f}^{\prime }=f(t)$. The same convolution identity proves uniqueness in that class. For example $f(t)=t$ gives $\Phi =t^{\alpha }/[\Gamma (\alpha +1)\Gamma (1-\alpha )]$.

Suppose $h$ is continuously differentiable on $[0,L]$. For $0<x\le L$, the improper integral $g(x)={\int }_0^{x}h(t)(x-t{)}^{-1/2}dt$ exists, but substituting directly into the ordinary Leibniz integral rule produces an undefined endpoint term. First use integration by parts to obtainThe new endpoint term at $t=x$ vanishes, and the differentiated kernel is locally integrable. Differentiating gives the displayed formula. One rigorous justification rewrites the second term as ${\int }_0^x[{\int }_0^s{h}^{\prime }(t)(s-t{)}^{-1/2}dt]ds$ by Fubini's theorem; the inner integral is continuous for $s>0$, so the fundamental theorem of calculus applies. Moreover $g(x)=2h(0)\sqrt{x}+O(x^{3/2})$: if $h(0)\ne 0$, the right difference quotient at zero diverges rather than defining a finite derivative.