Use local east, north, and up unit vectors, with $x$ eastward, $y$ northward, and positive orientation $\psi$ anticlockwise as viewed from above. Let $\phi$ be signed geographical latitude, positive north, and $\Omega >0$ the magnitude of Earth's angular velocity. In the spherical-Earth approximation,Here $g$ is the local effective gravity in a rotating frame, including the steady terrestrial centrifugal acceleration. For small horizontal displacements compared with $\ell$, the horizontal restoring acceleration from tension is $-{\omega }_0^2(x,y)$. Vertical displacement is second order in the swing amplitude, so vertical velocity can be omitted at this order. The horizontal Coriolis acceleration in local Earth coordinates then gives the coupled linear differential equationsThese equations neglect damping, finite-amplitude corrections, support anisotropy, and spatial changes of the centrifugal force of order ${\Omega }^2$; they retain the leading terrestrial Coriolis force. A related derivation uses south/east axes instead.^[ref]

Introduce the complex number $z=x+iy$. Multiplying the second equation by $i$ and adding givesSet $z=e^{-i\alpha t}u$. Direct differentiation cancels the first derivative of $u$ and yieldsThe ${\alpha }^2$ in $\beta$ comes from solving these reduced equations exactly; it is not a complete prediction of terrestrial corrections of order ${\Omega }^2$. Thus the horizontal trajectory is an ellipse in coordinates rotating at angular velocity $-\alpha$ relative to the building. When $|\alpha |\ll {\omega }_0$, each rapid swing samples an almost fixed long axis; its slow precession isThe sidereal day $T_{\mathrm{sid}}=2\pi /\Omega$ is a rotation relative to distant directions. $T_{\mathrm{p}}$ denotes a full $2\pi$ turn of a continuously followed orientation; an unlabelled swing line repeats after $T_{\mathrm{p}}/2$. At the equator there is no finite precession period in this model. The Coriolis parameter is $f=2\alpha$, so the plane's rate is $-f/2$, not $-f$.

The limiting cases check the sign and scale. At the north pole $\alpha =\Omega$: the swing direction turns clockwise once per sidereal day. At the equator $\alpha =0$: both horizontal equations reduce to independent simple harmonic motions and a line launch stays on its original line. At the south pole $\alpha =-\Omega$: the direction turns anticlockwise once per sidereal day. Vanishing horizontal precession at the equator does not mean that every component of Coriolis acceleration vanishes there.

For a stipulated $\Omega =7.292115\times 10^{-5}{\mathrm{s}}^{-1}$, the sidereal day is approximately $23.9345\mathrm{h}$. At $45^{\circ }$ north the predicted rate is $-10.6356^{\circ }/\mathrm{h}$ and $T_{\mathrm{p}}\simeq 33.8485\mathrm{h}$; at $45^{\circ }$ south the rate changes sign. At the illustrative geographical latitude $34^{\circ }$ south, the prediction is $+8.4109^{\circ }/\mathrm{h}$. These calculations are model predictions, not new observations of the UNSW apparatus.

A swing plane fixed among the stars is generally only a polar picture. A north–south swing at the equator turns with Earth in an inertial frame, while having no precession against the building.^[ref]

The University of New South Wales documents an undriven, visitor-started Foucault pendulum. A fixed vertical wire and wall line define its initial north–south swing plane. The documented comparison one hour later shows reduced amplitude and about nine degrees of anticlockwise precession. The apparatus swings for several hours without an electromagnetic drive.^[ref]

To turn that demonstration into a quantitative experiment, the following is a proposed measurement protocol. Establish a rigid support with no preferred horizontal swing direction, measure the wire length to the bob's centre, and mark two perpendicular horizontal axes fixed to the building. Start with a small displacement along the sighting plane, then release with as little transverse velocity as possible. Record the actual launch method; burning a fine restraining thread is one possible proposed release, not a claim about the documented UNSW launch. Keep people clear of the bob and avoid touching the suspension during the observation.

Record the horizontal position from above with a stationary, calibrated camera, or sight the bob against the wire and wall line at recorded times. Fit the long axis of each short group of swings to estimate its orientation $\psi$; use the extrema on both sides rather than following the bob's rapidly reversing velocity. A swing line has orientation modulo $\pi$, so unwrap successive small direction changes continuously. Rotate no camera or reference between measurements. Record the fast oscillation period separately, and compare the slope of $\psi (t)$ with the Latitude law of Foucault precession. Repeat launches along different directions and monitor ellipses, pivot motion, and air currents before attributing the slow change to Earth's rotation.

The condition required of a final program state when a program finishes under its precondition. The summation loop verification requires returned value $s$ to satisfy $2s=n(n+1)$ and preserves the input $n$. Relating the answer to the original input matters: allowing the program to replace $n$ by zero would change the intended contract.

A postcondition alone gives no promise of algorithm termination. It also need not specify a unique output: $s\ge 0$ allows many answers, while the exact triangular number equation determines one integer answer for each permitted $n$. Stronger postconditions express more obligations; proving only a weaker consequence cannot justify the stronger requirement.

The condition required of a starting program state for a stated correctness guarantee. In the summation loop verification, $n\in \mathbb{Z}$ and $n\ge 0$ admit $n=0$ and $n=4$, but exclude $n=-1$. The accompanying model assumes exact integer arithmetic and terminating elementary operations. The initial values of $i,s$ need no restriction because initialization overwrites them before use.

A precondition is a logical assumption, not automatically an executable check. Writing require in a procedure states its contract; an implementation that checks or rejects invalid inputs needs that behavior specified too. Weakening a precondition admits more starts and therefore demands more proof: dropping $n\ge 0$ here makes $n=-2$ a terminating counterexample to $2s=n(n+1)$, since initialization returns zero while $n(n+1)=2$.

A forest containing one minimum spanning tree of each connected component of a graph of a finite undirected weighted graph, including isolated vertices. For disconnected input, Kruskal's algorithm returns a minimum spanning forest.

For a lightly damped forced harmonic oscillator with known force amplitude $F_0$, measure the static displacement $x_s$, the resonance angular frequency ${\omega }_r$ and the peak displacement $x_p$. The response is $|x_0|=F_0/\sqrt{(k-m{\omega }^2{)}^2+b^2{\omega }^2}$. To leading order at large quality factor,The exact displacement resonance is ${\omega }_r^2=k/m-b^2/(2m^2)$, whereas the undamped natural angular frequency is ${\omega }_0=\sqrt{k/m}$. A lightly damped half-power bandwidth obeys $\Delta \omega \simeq b/m$ and $Q\simeq {\omega }_0/\Delta \omega$. Half power corresponds to displacement $x_p/\sqrt{2}$, not $x_p/2$. Graph readings should be reported as estimates rather than extra significant digits from the model inversion.

For a supported mass $m$, total spring constant $k$, and viscous coefficient $b$, a harmonic applied force has response $|x|=F_0/\sqrt{(k-m{\omega }^2{)}^2+b^2{\omega }^2}$. Far above the natural angular frequency, light damping gives $|x|\simeq F_0/(m{\omega }^2)$. For a prescribed displacement bound $x_{\max }$, this estimates $m\gtrsim F_0/({\omega }^2{x}_{\max })$, but the finite spring term can make the estimated equality slightly violate the bound. If $F_0/x_{\max }>b\omega$, the high-mass exact threshold isThis model describes a specified external force. Motion transmitted through a moving support instead has a base-displacement forcing term and a different dependence on damping.

For overdamped Brownian motion at uniform temperature in a potential energy $U$, constant diffusion coefficient $D$ gives the displayed Fokker-Planck equation, with $\beta =(k_{B}T{)}^{-1}$. The probability current is $\mathbf{J}=-D(\nabla p+\beta p\nabla U)$. Zero current recovers the Boltzmann distribution; absorbing boundaries instead permit transport and mean first-passage times. This diffusion equation should be distinguished from the similarly named coagulation equation.

A screen whose fluorescence makes the spatial distribution of incident electrons or other radiation visible.

A reduction in fluorescence caused by a process competing with photon emission, for example a collision that removes excitation energy without emitting a photon. Measuring less light because fewer excitation photons reach the sample does not by itself establish fluorescence quenching.

For composition-independent $\chi$, the spinodal of Flory–Huggins solution theory is $2\chi =1/(N\varphi )+1/(1-\varphi )$. At its thermodynamic critical point, a second condition $-1/(N{\varphi }^2)+1/(1-\varphi {)}^2=0$ gives $1-{\varphi }_c=\sqrt{N}{\varphi }_c$, proving the displayed values. Hence ${\varphi }_c\sim N^{-1/2}$ and ${\chi }_c-1/2\sim N^{-1/2}$. Under $\chi =C/T$, the theta temperature is $\Theta =2C$ and $T_c=\Theta /(1+N^{-1/2}{)}^2$. Thus $T_c$ approaches a constant while $\Theta -T_c$ scales as $N^{-1/2}$. Converting ${\chi }_c$ to a temperature always requires a specified or inferred $\chi (T)$ law.

The Flory interaction parameter measures the excess free-energy cost of unlike contacts in Flory–Huggins solution theory. For a simple lattice of coordination number $z_c$ and contact energies ${ϵ}_{ps}$, ${ϵ}_{pp}$ and ${ϵ}_{ss}$,Real effective parameters may also contain entropic contributions and composition dependence. With site volume $v_0$ and composition-independent $\chi$, the effective two-segment excluded volume is $v_{\mathrm{eff}}=v_0(1-2\chi )$. Positive, zero and negative $v_{\mathrm{eff}}$ correspond respectively to a good solvent, the theta temperature condition and a poor solvent. In particular, $\chi =1/2$ means cancellation of leading repulsive and attractive contributions; it does not mean that every molecular interaction vanishes.

Let $f(\varphi )$ be the mixing free energy per lattice site in Flory–Huggins solution theory, $v_0$ the site volume, and $\varphi$ the polymer volume fraction. The solvent mixing chemical potential is $f-\varphi f^{\prime }$, so its balance against the pure-solvent pressure contribution gives $\Pi =(\varphi f^{\prime }-f)/v_0$. For composition-independent $\chi$,The first term is the ideal chain translational contribution; the second encodes the effective two-segment excluded volume. At the theta temperature condition it vanishes but the positive third-order term remains. Treating $f$ as a free-energy density without dividing by $v_0$ loses the physical dimensions of osmotic pressure.

One resolved spin branch of a two-dimensional Landau level has degeneracy $eB/h$ per area. Thus the total carrier density is $n_s=\nu eB/h$, where the filling factor $\nu$ counts all occupied spin and other internal branches, including partial filling. If consecutive unsplit orbital peaks change filling by an unresolved degeneracy $g_d$,For only spin degeneracy, $g_d=2$. Once spin splitting is resolved, adjacent individual branches differ by one rather than two; counting split subpeaks as unsplit orbital peaks gives a factor-two error.

A connected loop Feynman diagram has at least one independent internal loop momentum. With $I$ internal lines and $V$ vertices, the number of independent loops is $L=I-V+1$. A tree-level Feynman diagram has $L=0$. A multi-leg amplitude generated by a loop is not an extra fundamental interaction vertex of the original Lagrangian.

Feynman rules translate a Feynman diagram into a contribution to a scattering amplitude or correlation function in perturbation theory. They specify factors for external states, interaction vertices, internal propagators, momentum-conservation constraints and loop integrals, with statistics-dependent signs and symmetry factors. For quantum electrodynamics, a photon-fermion vertex contains the coupling $e$, while a photon propagator supplies an inverse squared four-momentum. The complete amplitude also contains currents, so propagator powers alone do not determine the energy dependence of a scattering cross-section.

At a magnetic compensation temperature, $M_A=M_B\ne 0$: an ordered ferrimagnet has zero net magnetization. Such cancellation need not occur in every material.^[ref]

For an original stipulated example, let $t=T/T_C$, with $0\le t\le 1$, and prescribe sublattice magnitudes per common volume byTake A along $+z$ and B along $-z$, retaining a fixed sublattice orientation convention. These are toy functions, not measured curves or a self-consistent mean-field approximation. Their difference factors exactly:At $t=0$, $M_z=-M_0/2$. At $t=1/3$, both magnitudes equal $2M_0/3$ and $M_z=0$: this is compensation with finite order. At $t=1/2$, $M_z=M_0/8>0$. At $t=1$, both magnitudes vanish: that endpoint represents the stipulated Curie temperature, not another compensation point. For $t>1$ stipulate both ordered magnitudes zero; do not extrapolate the polynomials into negative sublattice magnitudes.

The sign change refers to the net component relative to the fixed A direction. A freely reorienting specimen in an applied magnetic field need not display this signed branch directly. The example isolates the cancellation mechanism; it predicts neither a material's transition temperature nor its response to magnetic domains or an applied magnetic field.

For a two-dimensional energy band, a Fermi contour is the set of wavevectors at the Fermi energy. It is the two-dimensional counterpart of a Fermi surface. At a regular value it consists locally of curves; at a band extremum or a crossing it can acquire isolated points or intersecting branches. At zero temperature, each band's occupied region satisfies $E_n(\mathbf{k})<E_F$, so the contours must be labelled by band when determining occupations. The Fermi contours of an isotropic Rashba band illustrate why two contours need not belong to two different bands.

A Fermi point is a point of the zero-temperature Fermi surface in one-dimensional momentum space. For a spin-degenerate band with one occupied interval, its two boundaries are usually $-k_F$ and $+k_F$. Low-energy excitations near them form left- and right-moving branches. A perturbation at wavevector $2k_F$ can mix those branches, as in a Peierls distortion.

The Sommerfeld expansion approximates integrals weighted by the Fermi-Dirac distribution when $k_{B}T\ll \mu$ and $f$ is smooth near $\mu$. Applying it to the number integral at fixed particle number gives $\mu -E_F=-({\pi }^2/6)(k_{B}T{)}^2{g}^{\prime }(E_F)/g(E_F)+\cdots$. The sign depends on the slope of the density of states. For an unbounded parabolic band, $g(E)\propto E^{d/2-1}$, yielding opposite quadratic shifts in one and three dimensions. In two dimensions the leading algebraic shift vanishes, but the finite band-bottom cutoff still gives an exponentially small negative shift.