In ideal nanotube zone folding, a single-walled carbon nanotube with indices $(n,m)$ satisfying $n-m\not\equiv 0(\mathrm{mod}3)$. Circumferential quantization misses the Dirac valleys in graphene, opening a band gap.

The graphene translation ${\mathbf{C}}_h=n{\mathbf{a}}_1+m{\mathbf{a}}_2$ that becomes a tube circumference when its endpoints are identified. It determines wrapping indices and diameter $|{\mathbf{C}}_h|/\pi$; it is perpendicular to the tube axis.

An approximation that samples the graphene bands on wave vectors satisfying $\mathbf{k}\cdot {\mathbf{C}}_h=2\pi \ell$, where ${\mathbf{C}}_h$ is the carbon nanotube chiral vector. It produces one-dimensional subbands and the ideal metallicity criterion $n-m\equiv 0(\mathrm{mod}3)$; curvature and interactions can modify its predictions.

An optical excitation between nanotube electronic subbands, often labelled $E_{ii}$ for prominent axial-polarization transitions. One-dimensional Van Hove singularities enhance the joint density of states, but selection rules, oscillator strengths and strong excitons interactions determine actual absorption peaks. Dielectric environment, bundling, doping and defects can change those peaks.

An enriched population selected by wrapping indices or electronic type. Sorting can narrow optical transition distributions or remove metallic leakage paths from semiconductor networks. Purity, yield, residual sorting agents and the processed-device environment remain separate performance variables.

A thin lens placed immediately after an aperture can reproduce its free-propagated pattern from distance L on a nearer screen d if $1/f=1/d-1/L$. In the paraxial Fresnel integral, the lens changes the aperture's quadratic propagation phase from $k{x}^2/(2d)$ to $k{x}^2(1/d-1/f)/2=k{x}^2/(2L)$. The cross term $-kxy/d$ then matches free propagation at the old-screen coordinate $Y=Ly/d$. Intensities have the same shape under transverse scale d/L, with an overall brightness factor in a lossless lens of sufficient aperture. For a narrow uniformly illuminated slit, the far-field first-zero estimate on the new screen is $d\lambda /a$.

For mutually incoherent wavelengths, intensities add rather than amplitudes: $I(\theta )=\int S(\lambda ){\mathrm{sinc}}^2[\pi a\sin \theta /\lambda ]d\lambda$, where $\mathrm{sinc}q=\sin q/q$ and S includes source strength and detector response. All wavelengths share the central maximum but their minima occur at different angles. A smooth white spectrum therefore has a white centre, wavelength-dependent edge colours, no common first dark minimum and increasingly overlapped, low-contrast outer fringes. A particular hue needs the actual spectrum and colour response.

A system's natural frequency describes a free oscillation in the absence of external driving. Distinguish the undamped angular frequency ${\omega }_0$ from a damped oscillation frequency and from the driving frequency that maximizes a particular measured response.

A uniform harmonic chain couples neighbouring displacements by quadratic spring energies. With periodic boundaries and spacing $a$, a Fourier transform gives ${\omega }_k=2\sqrt{\kappa /m_0}|\sin (ka/2)|$. Quantization makes each nonzero normal mode a phonon. The zero-frequency uniform displacement is a free centre-of-mass coordinate unless pinned, not an ordinary oscillator.^[ref]

In a uniform U-tube containing incompressible liquid of density $\rho$, cross section $A$ and total moving length $l$, let one surface fall by $y$ while the other rises by $y$. The height difference is $2y$, and potential energy relative to equilibrium is $U=\rho gA{y}^2$. The whole moving mass is $\rho Al$, so $K=\rho Al{\dot{y}}^2/2$. Newton's second law or energy conservation gives $\ddot{y}+(2g/l)y=0$ and ${\omega }_0=\sqrt{2g/l}$. This assumes small excursions that keep the free surfaces in the uniform vertical limbs.

Subtracting an exactly known mean sky level leaves source-plus-sky variance of a Poisson distribution $S+B$. If the sky per source pixel is estimated from $n_b$ equal-area, equally exposed independent sky pixels, its additional variance is $B/n_b$, so $R=S/\sqrt{S+B+B/n_b}$ before read and dark terms. The variance of removed background photons does not disappear with their mean.

The conserved eukaryotic SRP subunit containing a signal-sequence-binding region and a GTPase domain; bacterial Ffh is its homolog.

The bacterial homolog of the SRP54 protein, containing a GTPase involved in cotranslational membrane targeting.

Its signal-recognition and GTPase functions connect a nascent hydrophobic segment with the bacterial SRP receptor FtsY. The pathway directs selected translating ribosomes toward the membrane before the complete protein has been synthesized. ^[ref]

The time derivative of engineering shear strain. A Newtonian simple-shear stress is $\eta \dot{\gamma }$. This scalar is twice the off-diagonal symmetric rate-of-strain-tensor component.

The ladder identity $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$ proves the matrix element in the title. In time-dependent perturbation theory, a linear force excites only $n=1$ directly from $n=0$. Parity alone rules out even excited states; the ladder identity also rules out odd $n>1$. Higher-order evolution can reach other levels, so the driven oscillator is not an exact two-level system.

A sinusoidal trap displacement with amplitude $\epsilon$ and duration $T$ produces a resonant first-order excitation amplitude proportional to $\epsilon \omega T/a_0$, where $a_0$ is the oscillator length. Keeping its square much smaller than one is the long-time condition for weak excitation. The small-displacement condition $\epsilon /a_0\ll 1$ alone does not ensure uniform validity as $T$ grows. Exact linear driving remains solvable as a coherent state after this first-order approximation fails.

For series resistance $R$, inductance $L$ and capacitance $C$, with output measured across the inductor, the $e^{i\omega t}$ transfer function is $H_L=i\omega L/[R+i(\omega L-1/(\omega C))]$. At current resonance ${\omega }_0=1/\sqrt{LC}$ its modulus is ${\omega }_{0}L/R=Q$. Large and oppositely phased inductor and capacitor voltages nearly cancel in the loop sum. This is voltage magnification in a passive circuit, not power gain. The maximum of $|H_L|$ need not occur exactly at the current-resonance frequency.

A Lambert series has the form ${\sum }_{n\ge 1}{a}_n{q}^n/(1-q^n)$. Expanding each denominator as a geometric series shows that its coefficient of $q^m$ is ${\sum }_{d\mid m}{a}_d$.

For a positive integer $a_0$, define $a_{k+1}=a_k/2$ when $a_k$ is an even number, and $a_{k+1}=3a_k+1$ when $a_k$ is an odd number. Starting at $6$ gives $6,3,10,5,16,8,4,2,1$. Computing many trajectories benefits from memoization, since trajectories that reach the same value have identical continuations.

An integer sequence is a sequence whose terms are integers. It can be specified by a formula, a recurrence relation, a counting interpretation or a finite computational rule. Examples include $a_n=n^2$, the Fibonacci numbers and the prime numbers.

A finite prefix does not determine a unique infinite sequence. If a polynomial matches $a_0,\dots ,a_N$, adding $c{\prod }_{j=0}^N(x-j)$ leaves those values unchanged and usually changes later ones. Identifying a sequence therefore requires its definition, indexing convention and domain, rather than a few suggestive values alone.