Identical linear springs of stiffness $k$ have equivalent stiffness $2k$ in parallel and $k/2$ in series. Parallel springs have equal displacement and add forces; series springs carry equal force and add extensions. A mass $m$ attached to an equivalent massless spring has angular frequency $\sqrt{k_{\mathrm{eff}}/m}$.

A zero-rest-length spring has quadratic potential energy in its actual length $s$, with no subtracted natural length. It is a useful idealization, not an ordinary unstretched spring of arbitrary length. For an end-to-end spring on a two-link pendulum, changing the natural length changes the quadratic bending stiffness about the straight configuration.

The speed of one side of newly formed crust relative to the ridge axis in a symmetric spreading model. It is half the full relative opening speed; the width formed during a polarity interval is this speed times the interval duration.

Seafloor spreading creates oceanic crust at a spreading ridge and carries it away with the adjoining plates. With symmetric steady spreading, an age interval maps to distance by the half-spreading rate, not the full relative opening speed. A time sequence of field polarities can therefore become paired marine magnetic stripes.

For independent moments with magnetic energies $\mp \mu B$, $Z=(2\cosh x{)}^N$ and $S=N{k}_B[\ln (2\cosh x)-x\tanh x]$, where $x=\mu B/(k_{B}T)$. This counts the magnetic orientations only, excluding translational and lattice entropy.

For a spin-one Hamiltonian $H=A{S}_z^2+B(S_x^2-S_y^2)$, Cartesian basis energies are ${\hslash }^2(A-B)$, ${\hslash }^2(A+B)$ and zero along the $x,y,z$ basis labels. The $S_x=\hslash$ state combines the latter two basis vectors equally. Their relative phase produces the displayed spin-reversal probability. Complete reversal occurs at odd multiples of $\pi /(|A+B|\hslash )$ if $A+B\ne 0$; when it vanishes those two components remain degenerate. Individual matrix signs depend on basis conventions, while the commutators and physical probability must remain consistent.

A model using exponentially correlated magnetic fluctuations. For homonuclear dipolar relaxation, $T_1^{-1}=C[{\tau }_c/(1+{\omega }_0^2{\tau }_c^2)+4{\tau }_c/(1+4{\omega }_0^2{\tau }_c^2)]$. Its rate maximum occurs near ${\omega }_0{\tau }_c=0.616$; a single-frequency Lorentzian instead peaks at one.

For $s_1=s_2=3/2$, angular momentum addition gives one copy of $j=0,1,2,3$. The identity $2{\mathbf{S}}_1\cdot {\mathbf{S}}_2={\mathbf{J}}^2-{\mathbf{S}}_1^2-{\mathbf{S}}_2^2$ gives the four eigenvalues in the title, with state degeneracy $1,3,5,7$. These exhaust the $4\times 4$ spin space. If the particles are identical fermions, an appropriate spatial exchange parity accommodates each sector; only a specified spatial restriction can remove sectors.

With polar angle $0\le \theta \le \pi$ and $0<|ah|\le 1$, the spherical-geodesic first integral gives ${\theta }_{\min }=\arcsin |ah|$ and ${\theta }_{\max }=\pi -\arcsin |ah|$. The maximum absolute latitude is the different angle $\arccos |ah|$. At $|ah|=1$ the great circle is the equator; at $h=0$ it is a meridian reaching both poles.

Differentiating the spherical-geodesic first integral givesThe same equation follows from the affine geodesic equation after eliminating the arc-length derivatives. The squared first-derivative equation also fixes the normalization constant; a second-order solution must retain that first integral.

Broadening shared by otherwise identical emitters, often described by a common optical coherence time $T_2$. Exponential field-coherence decay produces a Lorentzian line shape with full angular-frequency width $2/T_2$. Population decay, dephasing and collisions can all contribute, with their time constants distinguished.

An ensemble-wide distribution of resonance frequencies, such as velocity-dependent Doppler shifts or spatially varying static-field shifts. The observed profile averages the individual responses. A velocity class need not have the full width of the ensemble's thermal Doppler broadening.

Finite interaction time with a spatially confined optical beam gives a finite spectral response width. Its detailed shape depends on the spatial beam profile, particle trajectory and interaction envelope; a duration alone does not imply a universal Lorentzian profile.

Let $L=w^{-1}[-(d/dx)(pd/dx)+q]$ be the regular self-adjoint operator of a Sturm-Liouville problem, with eigenfunctions orthonormal for the weighted inner product $\int w\overline{u}v$. For $\mu$ outside its spectrum, the displayed kernel satisfies $(L_x-\mu )G_{\mu }=\delta (x-\xi )$ with an ordinary, unweighted Dirac delta. Indeed weighted completeness gives $w(\xi ){\sum }_n{y}_n(x)y_n(\xi {)}^{\ast }=\delta (x-\xi )$. Thus $u(x)=\int G_{\mu }(x,\xi )f(\xi )d\xi$ solves $(L-\mu )u=f$. A kernel integrated against $w(\xi )d\xi$ instead omits the prefactor. For real coefficients, the weighted reciprocity relation is $G_{\mu }(x,\xi )/w(\xi )=G_{\mu }(\xi ,x)/w(x)$ when the spectral parameter is real and outside the spectrum. A zero eigenvalue prevents inversion at $\mu =0$ without a forcing compatibility condition or a projected inverse.

For the fundamental representation of SU(N), take traceless Hermitian generators with $\mathrm{Tr}(T^a{T}^b)={\delta }^{ab}/2$. Along with the identity matrix, they form an orthogonal basis of all complex matrices. Expanding a matrix in this basis and subtracting its identity component proves the displayed completeness relation. It separates exchanged-index and unchanged-index contributions to color amplitudes.

For a monochromatic uniformly bright incoherent disk of angular diameter $\delta$, the mutual coherence at baseline $b$ is proportional to $2J_1(\pi \delta b/\lambda )/(\pi \delta b/\lambda )$. The first zero is $b\simeq 1.22\lambda /\delta$. A nonuniform disk or finite spectral band changes the detailed curve.

For a distant spatially incoherent quasimonochromatic source in the small-angle regime, the equal-delay mutual coherence across projected baseline $\mathbf{b}$ is the Fourier transform of its angular intensity. With matched collector response, the normalized degree is $g(\mathbf{b})=\int I(\mathbf{\theta })e^{-i2\pi \mathbf{b}\cdot \mathbf{\theta }/\lambda }{d}^2\theta /\int I(\mathbf{\theta })d^2\theta$. Equal collector intensities and matched polarizations make fringe visibility $|g|$. Finite bandwidth and unmatched optical paths introduce additional temporal-coherence averaging.

A photovoltaic device using a perovskite-structure absorber. Selective contacts extract carriers; recombination and contact selectivity determine the attainable photovoltage.

A rear doping gradient that reduces minority-carrier access to a recombination-active contact. In a p-type silicon cell, a more heavily doped $p^{+}$ rear region provides a common example.

Maximum power point tracking adjusts the electrical load presented to a source so that it operates near its maximum power point. In a photovoltaic system, a controlled power converter can change the panel's operating voltage while delivering energy to a battery or another load.

One elementary strategy perturbs the operating voltage, measures the resulting power, and continues in the direction that increases it. If illumination changes during the measurement, the power change may reflect the weather rather than the chosen perturbation, so the control must distinguish these effects over an appropriate time scale. Partial shading can also produce several local maxima in an array's power curve. Tracking changes the operating condition; it does not raise the power available on that instantaneous curve.