A rigid pendulum tilted from the upward vertical and driven at fixed azimuthal angular velocity $\Omega$ has effective potential $\mu {\theta }^2/2+mga\cos \theta -m{a}^2{\Omega }^2{\sin }^2\theta /2$. With $\mu =2mga$, its upright equilibrium loses linear stability when ${\Omega }^2$ passes $g/a$. At equality the leading restoring torque is cubic and the potential has a positive quartic minimum. This gives a stable quartic oscillator with zero infinitesimal harmonic frequency, rather than a freely drifting neutral state.

The momentum density of a top-hat wavefunction decays as an oscillatory $1/p^2$ envelope. Its momentum second moment therefore diverges, although the probability density itself is normalizable. The central diffraction-lobe width is not the standard deviation. Even the first moment requires care: the symmetric principal value is zero, but the absolute first moment diverges logarithmically.

A time complexity bounded by $O(1)$ independently of input size, under a stated computational cost model. Fixed-size machine arithmetic may be constant time, whereas arbitrary-precision arithmetic depends on operand length.

A time complexity bounded by $O(n)$ for input size $n$. A single pass with constant work per input entry has this bound.

A time complexity bounded by $O(n^2)$ for input size $n$. Examining every ordered pair of entries with constant work per pair has this bound.

For a dispersion relation $E(k)=E_0+t\cos (ka)$, $t<0$, occupied states at zero temperature form the interval $|k|<k_F$ in the first Brillouin zone. Its occupied fraction is $f=2k_F/(2\pi /a)=k_{F}a/\pi$. A spin-degenerate band therefore holds $2f$ electrons per site. If the band is independent of the other two wavevector coordinates, its Fermi surface consists of two planes. For $t>0$ the occupied region is instead near the zone edges, bounded by $|k|=\pi (1-f)/a$.

In an orthogonal one-orbital nearest-neighbor tight-binding model, the hopping integral is $t=⟨{\varphi }_j|H|{\varphi }_{j+1}⟩$ and the dispersion is the displayed cosine. Using the same orbital-phase convention at every site, equal s-orbital coefficients usually make a bonding state at the zone center, so $t<0$ and $k=0$ is a band minimum. For $p_x$ orbitals along the chain, with every positive lobe pointing right, equal site coefficients bring opposite-sign lobes together between atoms. This is antibonding; alternating coefficients are bonding, giving the usual $t_{\sigma }>0$ and a band maximum at zero. Changing conventions requires transforming the Bloch basis consistently. The physical statement concerns the curvature and orbital character of a specified Bloch state, not an isolated hopping sign divorced from its basis.

Opposite parity of same-centre s orbitals and axial p orbitals makes their left and right nearest-neighbour hopping integrals opposite in sign. Their Bloch wave sum is consequently odd in wavevector and vanishes at the zone centre and boundary. Away from those points, orbital hybridization repels coupled levels and can produce an avoided crossing.

For a chain along $x$, translated $p_z$ orbitals form a side-on bonding combination at $k=0$, whereas translated $p_x$ orbitals form an end-on bonding combination at $k=\pi /a$. With $E=ϵ+2t\cos ka$ and identically translated orbitals, these correspond to $t_{\pi }<0$ and $t_{\sigma }>0$. Changing the sign convention of the hopping Hamiltonian changes the parameter signs, not the physical band extrema.

A nonlinear optics process producing a field at the sum of two input angular frequencies. The driven electric polarization contains a term proportional to the product of the input complex amplitudes. With undepleted plane-wave inputs, integration along a crystal gives an output amplitude proportional to $L{e}^{i\Delta kL/2}\mathrm{sinc}(\Delta kL/2)$; the sinc function therefore determines the ideal length and phase mismatch dependence. A nonzero initial output field also interferes with this generated field.

Difference-frequency generation produces a wave at the positive difference ${\omega }_i={\omega }_p-{\omega }_s$ of two incident angular frequencies, through a second-order nonlinear optical susceptibility. The product of two oscillating electric fields contains both sum and difference frequencies, and the corresponding polarization density drives the generated wave.

For plane waves in a uniform medium, coherent buildup is favored by ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$, the relevant phase matching condition. Two shorter-wavelength inputs can thereby generate longer-wavelength infrared radiation. Collecting the output after removing both incident frequencies distinguishes frequency generation from scattered pump light.^[ref]

An optical parametric amplifier transfers energy from a pump beam into a signal and an idler through nonlinear optics. For the common three-wave mixing process, their angular frequencies obey ${\omega }_p={\omega }_s+{\omega }_i$. A strong pump and a weaker signal enter a nonlinear medium; the signal grows and an idler is generated.

Efficient growth requires phase matching so that contributions generated along the medium reinforce. Changing crystal orientation or temperature can tune the matched frequencies. Measuring the signal with and without the pump tests amplification, while observing the corresponding idler checks the frequency-conversion mechanism. The pump supplies the extra energy; the process does not require a stored population inversion.^[ref]

Conservation relations for resonant, lossless wave mixing. For sum-frequency generation with ${\omega }_3={\omega }_1+{\omega }_2$, the intensities obey $I_1^{\prime }/{\omega }_1=I_2^{\prime }/{\omega }_2=-I_3^{\prime }/{\omega }_3$. Thus $I_1/{\omega }_1+I_3/{\omega }_3$ and $I_2/{\omega }_2+I_3/{\omega }_3$ are constant. Their difference gives a third, dependent invariant. Adding the intensity derivatives yields conservation of energy.

The exchange interpretation divides each intensity by its photon energy $\hslash {\omega }_j$. One conversion removes a photon from each lower-frequency mode and adds one to the upper mode. For classical coupled-wave equations, differentiating the squared complex amplitudes gives the same common exchange rate multiplied by $({\omega }_1,{\omega }_2,-{\omega }_3)$. Loss or externally replenished pumps require corresponding source terms.

If three quarks are hypothetically spin-zero bosons, their total state is symmetric under particle exchange. In an $L=0$ symmetric spatial state, the unchanged antisymmetric three-quark colour singlet therefore requires an antisymmetric quark flavour state. With three flavours its space is ${\bigwedge }^3{\mathbb{C}}^3$, of dimension one. The sole flavour state contains one quark of each flavour and has total angular momentum zero.

In a three-level laser whose lower lasing level is the ground state, fast relaxation from a pumped higher level feeds a long-lived upper lasing level. With equal lasing-level degeneracies, negligible pump-level population and pump rate R per ground-state emitter, the no-light stationary rate equation gives $N_2/N_1=R{\tau }_{21}$. Positive population inversion therefore requires more than half the active emitters to leave the ground state. This is a transparency threshold; optical resonator loss imposes an additional positive-gain requirement. Degenerate sublevels require population comparisons weighted by their statistical weights and transition strengths.