The stretch method rescales the light curve's time axis by $s(1+z)$ and corrects peak apparent magnitude by $\alpha (s-1)$, fitting $\alpha$ jointly with cosmology.^[ref]

Here is a deliberately stipulated local model that exposes the signs. Let $t_{\mathrm{obs}}$ be observer time in days, $t_{\max }$ the observed peak time and $T(u)$ a dimensionless template with $T(0)=1$. In a single photometric passband, writeHere $u$ is rest-frame time in days; the template's argument carries those units. For illustration, assign the intrinsic relation $M_B(s)=M_{B,0}-\alpha (s-1)$ with $\alpha >0$: broader events are intrinsically brighter. Define the sign of the K correction by $m_{\mathrm{obs}}=M_B(s)+\mu +A_B+K$, where $A_B$ is interstellar extinction in magnitudes. ThenRemoving attenuation makes the magnitude smaller; standardizing a broad, intrinsically bright event makes it larger. The fitted cosmological time dilation must be removed before comparing intrinsic durations. This schematic factorization explains the bookkeeping; an actual multicolour fit must also model changing intrinsic colours and correlated uncertainties.

For two equally standardized sources, cancellation of their common absolute magnitude gives an especially clean result:A stipulated residual of $+0.20$ mag therefore means a distance ratio $10^{0.04}=1.09648$ and a radiative flux ratio $10^{-0.08}=0.83176$ relative to the comparison prediction, provided luminosity and attenuation have been correctly standardized. This is a conversion of a hypothetical residual, not a measured survey value.

In his 1909 lecture, Karl Ferdinand Braun recalled two experimental stations on Strasbourg forts in summer 1902. Known capacitances and calculated inductances set the transmitter tuning; the receiver searched for its two coupled oscillations. Its receiving coil fed a second coil in a tuned indicator circuit. Separating these coils increased both the sharpness of resonance and excitation intensity, until further separation reduced the response. The varied coil spacing was the control; transmitter coupling need not be loose for this receiver effect. For intensity measurements, Braun described an air-thermometer readout in the related resonant measuring apparatus.^[ref]

The comparison tests receiver coupling, not an unrestricted law that looser coupling always helps. The paragraph does not identify the particular indicator used in the coil-spacing test or give a response curve from which to reconstruct its optimum. The plotted curves below are an original linear model with stated losses, rather than invented historical measurements.

A triangular system is a linear system whose coefficient matrix is a triangular matrix. Nonzero diagonal entries give a unique solution.

Electron shelving distinguishes an internal quantum state outside a cycling optical transition from one that scatters many photons. In the mercury-ion example in David J. Wineland's lecture, $S\leftrightarrow P$ excitation produces $194\mathrm{nm}$ fluorescence; transfer to a metastable atomic level $D$ suppresses it. A photomultiplier records bright/dark intervals. Coherent $282\mathrm{nm}$ interrogation precedes detection to avoid perturbation by the detection beam.^[ref]

“Dark” means weak scattering under a particular detection setting. The motional ground state is dark to an ideal red motional sideband for a different reason: there is no lower number state. It can still fluoresce strongly on a cycling detection transition. Shelving's population readout also need not preserve a quantum superposition's quantum coherence; scattered photons carry information about the internal state, and their recoil can disturb motion.

Resolved sideband cooling removes motional quanta by selectively driving a red motional sideband and then resetting the internal quantum state through spontaneous emission or optical pumping. Spectral resolution requires the effective cooling-transition width smaller than the motional angular frequency in the same units. A narrow Raman transition can meet that requirement even when the optical transition used for Doppler cooling is broad.

The reset is essential: the coherent red motional sideband merely exchanges internal and motional excitation reversibly. To see the distinction, an ideal pulse sends $|g,1⟩$ to $-i|e,0⟩$; the same interaction can send it back. An ideal dissipative reset instead takes $|e,0⟩$ to $|g,0⟩$, with information and energy carried into the environment. Repeating this cycle accumulates population near the motional ground state. Recoil during the reset can add motion, and off-resonant excitation can disturb $|g,0⟩$; neither is included in the ideal pulse below.

An internal qubit uses two distinguishable electronic or hyperfine structure quantum states, denoted $|g⟩,|e⟩$. The ion's centre-of-mass motion has a different Hilbert space. In one approximately harmonic direction, a number state $|n⟩$ has energy $E_n=\hslash \nu (n+1/2)$, where $\nu$ denotes a motional angular frequency, not ordinary frequency. The tensor product of quantum systems gives $|g,n⟩$. Cooling towards $n=0$ prepares motion; it does not automatically prepare the internal qubit. Nor does the ground state mean a stationary point particle: zero-point energy and position spread remain.

For several ions, Coulomb forces couple their displacements into collective normal modes. Each selected normal mode has its own quantum harmonic oscillator number states, while each ion retains internal qubit states. A pulse addressing one ion can change a shared mode. This supplies a route to correlations between distant internal quantum states; it also makes residual motional excitation and unwanted excitation of other normal modes relevant. A single-ion pulse calculation alone establishes no experimental multi-ion quantum gate performance.

In a matched lossless parallel-plate transmission line, a single travelling wave obeys $V/I=Z_0=(d/a)\sqrt{{\mu }_0/\epsilon }$. The electric attraction per unit length is $f_E=\epsilon a{V}_{\mathrm{rms}}^2/(2d^2)$; each plate feels the field of the other plate, excluding its self-field. Opposite sheet currents give magnetic repulsion $f_B={\mu }_0{I}_{\mathrm{rms}}^2/(2a)$. Substituting the travelling-wave impedance proves their equality, so the ideal net transverse electromagnetic force is zero at every phase. A standing wave or independent voltage and current need not have this cancellation.

Cell-free translation uses isolated ribosomes and soluble components to make polypeptide chains from messenger RNA outside an intact biological cell.

Two identical torsional loads of moment of inertia $I$ hang in series from massless fibres of lengths ${\ell }_1,{\ell }_2$ and common filament torsional stiffness $K_t$. Put $R={\ell }_2/{\ell }_1$ and ${\omega }_0^2=K_t/(I{\ell }_1)$. The elastic energy is $K_t{\varphi }_1^2/(2{\ell }_1)+K_t({\varphi }_2-{\varphi }_1{)}^2/(2{\ell }_2)$ and the kinetic energy is $I({\dot{\varphi }}_1^2+{\dot{\varphi }}_2^2)/2$. Their normal modes have the displayed squared angular frequencies. Writing $\lambda ={\omega }^2/{\omega }_0^2$, their amplitude ratio is $A_2/A_1=1/(1-R\lambda )$. The lower mode twists in phase and approaches a combined inertia $2I$ as $R\to 0$, giving ${\omega }_{-}^2\to {\omega }_0^2/2$. The upper mode approaches the original single-load oscillation as $R\to \infty$, giving ${\omega }_{+}^2\to {\omega }_0^2$.

A clock following a timelike circular Schwarzschild geodesic of radius $R_s$ advances at $\sqrt{1-3\mu /R_s}$ per unit Schwarzschild coordinate time. A static clock at radius $R_E>2\mu$ advances at $\sqrt{1-2\mu /R_E}$. Their ratio over the same coordinate-time interval gives the displayed result. For periodic signal comparisons, successive corresponding signals must also have the same propagation delay, as happens for repeated overhead passages in a fixed circular orbit observed from a static pole. To first order in $\mu /R$, the ratio is $1+\mu /R_E-3\mu /(2R_s)$.

A source on a timelike circular Schwarzschild geodesic at radius $r>3GM/c^2$ emits light to a static observer at infinity on the orbital symmetry axis. Such a ray has zero conserved axial angular momentum. The locally measured photon energy is the contraction of its four-momentum with the observer's four-velocity. Comparing the emitter with the observer at infinity therefore gives ${\nu }_{\mathrm{em}}=U^t{\nu }_{\infty }$, where $U^t=[1-3GM/(c^{2}r){]}^{-1/2}$, proving the formula.

Equivalently the gravitational redshift factor $\sqrt{1-2GM/(c^{2}r)}$ multiplies the transverse Doppler effect factor $1/\gamma$ measured by a local static observer. For $R_S=2GM/c^2\ll r$,The coefficient contains both gravitational and orbital time dilation. It assumes an observer at infinity on the symmetry axis; other viewing directions generally introduce a first-order longitudinal Doppler effect.

Atmospheric cosmic rays generate muons whose mean lifetime at rest is about $2.2\mu \mathrm{s}$. For a selected speed $v$ and production-to-detection distance $h$ in the terrestrial frame, special relativity predicts survival fraction $e^{-h/(\gamma v{\tau }_0)}$, where $\gamma$ is the Lorentz factor. Without time dilation, the exponent would instead be $-h/(v{\tau }_0)$. Comparing altitude-dependent counts at controlled energies, allowing for acceptance and energy loss, tests these alternatives. In the muon rest frame the terrestrial distance is shortened to $h/\gamma$ by length contraction, so the proper flight time and survival fraction agree with the terrestrial calculation. This frame comparison does not turn the same decay measurement into two independent tests.

Accelerator-produced muons circulating in a magnetic storage ring allow a direct comparison between their measured laboratory mean lifetime and $\gamma$ times the rest lifetime. The 1977 experiment at $\gamma =29.33$ measured positive- and negative-muon lifetimes near $64.4\mu \mathrm{s}$, consistent with time dilation. This is a controlled complement to atmospheric survival measurements, with different systematic effects.^[ref]

A two-state degree of freedom represented mathematically like spin one-half, without necessarily being physical spin. The sublattice pseudospin in graphene labels amplitudes on two sites per primitive cell.

A hopping integral is an off-diagonal Hamiltonian matrix element between localized orbitals on different lattice sites. It has units of energy and controls the dispersion in a tight-binding model. Its sign and phase depend on the orbital phase convention; observable band energies do not. The sign of a real nearest-neighbour $t$ in $E(k)=E_0+2t\cos (ka)$ determines the location of the minimum in that convention.

In the orthogonal tight-binding approximation, localized basis states satisfy $⟨i|j⟩={\delta }_{ij}$ while the Hamiltonian retains nonzero hopping integrals. These are independent matrix elements: orthogonal states need not have vanishing Hamiltonian coupling. One can construct localized orthonormal orbitals from overlapping atomic orbitals; truncating the resulting Hamiltonian to nearest neighbours is a further approximation. An overlap matrix must instead be retained for raw nonorthogonal orbitals.

For one localized orbital per site in a chain, take on-site matrix elements $H_{jj}=E_0$, nearest-neighbour matrix elements $H_{j,j\pm 1}=h$, overlaps $S_{jj}=1$ and $S_{j,j\pm 1}=s$, with other terms neglected and $h,s$ real. A Bloch state reduces the generalized eigenvalue problem $Hc=ESc$ toThe overlap metric must be positive for the retained model; $|s|<1/2$ is sufficient for all $k$. Dropping it yields the cosine band of the orthogonal tight-binding approximation.

In an orthonormal basis of localized atomic orbitals, a ring of three equivalent sites with real hopping $t$ has Hamiltonian $H=(E_0-t)I+tJ$, where $J$ is the three-by-three all-ones matrix. The normalized symmetric state $(|1⟩+|2⟩+|3⟩)/\sqrt{3}$ has energy $E_0+2t$. Its two-dimensional orthogonal complement has energy $E_0-t$. Thus the symmetric state is the unique spatial ground state for $t<0$, with an energy difference $3|t|$ to the excited eigenstates. Equivalently, the three discrete Bloch states have energies $E_0+2t\cos (2\pi l/3)$, $l=0,\pm 1$. Nonorthogonal atomic orbitals instead require a generalized eigenvalue problem.