A one-way arm-length difference $\delta$ gives optical path difference $2\delta$ and phase difference $2k\delta$. An ideal constructive output port has intensity $I=I_{\mathrm{in}}[1+\cos (2k\delta )]/2$. The other output is complementary.

Changing both arms of a balanced Michelson interferometer by the same length changes the common phase of both returning fields. Their relative phase and detector intensity remain unchanged at every frequency. Thus balanced interference cannot select a minimum positive common arm length.

Moving one mirror by $d$ changes the round-trip optical path by $2d$. Starting at a common bright centre for two commensurate wavelengths, the next common bright centre occurs when $2d=m{\lambda }_1=n{\lambda }_2$ for the smallest positive integers. This is a common-multiple condition on optical path, not on the mirror displacement itself. Incommensurate exact wavelengths have no finite exact recurrence in the ideal model.

For isotropic Brownian motion started at zero with independent coordinate variances $2Dt$, the Gaussian distribution has density $(4\pi Dt{)}^{-3/2}{e}^{-|x{|}^2/(4Dt)}$. Integrating the radius with spherical volume element $4\pi r^{2}dr$ gives $\mathbb{E}|X_t|=4\sqrt{Dt/\pi }$. The root mean square displacement is instead $\sqrt{6Dt}$, and the mean vector is zero. None of these is the microscopic path length.

Compute a nonnegative integer power $A^n$ by repeated squaring. If $n=2k$, use $A^n=(A^k{)}^2$; if $n=2k+1$, use $A^n=A(A^k{)}^2$. This needs $O(\log n)$ matrix multiplications and permits fast evaluation of fixed-order linear recurrences via a companion matrix. It is distinct from computing the matrix exponential $e^A$.

Time-dependent deformation after an earthquake, including viscous or viscoelastic flow as distinct from fault afterslip. Joint spatial and temporal observations are needed to distinguish those mechanisms and the layers that deform.

A real symmetric one-family Dirac-Majorana mass matrix can have a negative algebraic eigenvalue. A rephasing of the corresponding Weyl field puts its physical rest mass at the absolute value. Relative Majorana phases can still affect coherent lepton-number-violating amplitudes.

A mobile spin-triplet excitation above a singlet background in a quantum magnet. At low density, an appropriate triplet branch can be described as bosonic quasiparticles with an effective mass and repulsive interaction. Applied magnetic field can lower one branch toward condensation. The term does not imply that the total excitation number is conserved on thermodynamic timescales.

A dipole's magnetic pressure scales as $B_{\ast }^2{R}_{\ast }^6/({\mu }_0{r}^6)$. Spherical free fall with $\rho =\dot{M}/(4\pi r^{2}v)$ and $v\propto (GM/r{)}^{1/2}$ has ram pressure proportional to $\dot{M}(GM{)}^{1/2}{r}^{-5/2}$. Equating them gives the seventh-root scaling. Disc geometry changes the coefficient; the estimate ceases to describe an exterior magnetosphere if the resulting radius lies inside the star.

A junction with magnetic electrodes separated by a thin insulating barrier. Spin-dependent tunnelling can make resistance depend strongly on the relative electrode magnetizations.

A large field-induced resistivity reduction in correlated magnetic materials. Double exchange links carrier mobility to spin alignment, while lattice coupling, localization and competing phases can magnify the response.

With magnetic spin equilibrium at the magnetospheric truncation radius, $T\propto R_M^{3/2}{M}^{-1/2}\propto B_{\ast }^{6/7}{R}_{\ast }^{18/7}{\dot{M}}^{-3/7}$ at fixed mass. Inserting the surface accretion Eddington rate changes the radius power to $15/7$. The relation assumes the boundary remains outside the star; very weak fields instead encounter surface-accretion and mass-shedding limits.

A tangential force integral around an instantaneous moving loop is not generally the force work along the actual particle trajectory. For an expanding circular guide, the particle has both radial and tangential velocity; the magnetic radial and tangential powers cancel exactly even when the tangential circulation is nonzero.

A trial domain pattern costs wall energy ${\gamma }_w{A}_w$ and can release stray energy $\Delta U$. For a stable uniform ground state, ${\gamma }_w{A}_w\ge \Delta U$ within that trial model. Substituting the entire initial stray energy assumes that it is nearly all removed; a numerical lower bound independent of the domain geometry does not follow from single-domain observation alone.

Let a conducting loop of area A rotate about z with normal $(\cos \omega t,\sin \omega t,0)$ in a fixed field $B\hat{\mathbf{x}}$. Neglect inductance so $I=(BA\omega /R)\sin \omega t$. Put $m_0=B{A}^2\omega /R$. Its vector magnetic moment isThe static component does not radiate. The remaining component rotates at twice the mechanical frequency and has $|\ddot{\mathbf{m}}{|}^2=4m_0^2{\omega }^4$. Thus $\overline{P}=2{\mu }_0{B}^2{A}^4{\omega }^6/(3\pi c^3{R}^2)$. Treating the same sinusoidal current as a fixed-axis moment underestimates this rotating geometry by a factor of eight. The energy source is mechanical rotation, with additional ohmic dissipation and magnetic braking.

The phase margin measures the remaining phase lag before a negative-feedback loop reaches the critical phase $-\pi$ at unity loop magnitude, using a continuous phase convention. Positive adequate margin is a useful stability and transient-response design diagnostic for standard single-crossover amplifier loops. More general unstable open-loop systems and multiple crossovers require the full stability criterion.

A linearly damped oscillator $m\ddot{x}+b\dot{x}+kx=0$ has $x=A{e}^{-\gamma t}\cos ({\omega }_{d}t+\varphi )$, with $\gamma =b/(2m)$ and ${\omega }_d^2=k/m-{\gamma }^2>0$. After one damped period $T=2\pi /{\omega }_d$, both displacement and velocity are multiplied by $e^{-\gamma T}$. Their quadratic mechanical energy is therefore multiplied exactly by $e^{-2\gamma T}$, regardless of the starting phase. In particular successive same-direction equilibrium crossings have this ratio even though potential energy is zero at each. A cosine phase maximum is not exactly a displacement turning point when damping is nonzero.

Imaging of biological events in a living organism. Repeated cell tracking can establish event ordering but requires controls for phototoxicity, imaging access and sampling.

After a bipartite spin rotation, a leading Holstein–Primakoff transformation expansion gives $H_2=-J\Delta N{S}^2+JS{\sum }_k[2\Delta n_k+\cos k(a_k{a}_{-k}+a_k^{†}{a}_{-k}^{†})]$. The bosonic Bogoliubov diagonalization gives the displayed spectrum and vacuum energy $-J\Delta N{S}^2+JS{\sum }_k[\sqrt{{\Delta }^2-{\cos }^{2}k}-\Delta ]$. The anisotropy gap is $2JS\sqrt{{\Delta }^2-1}$. At $\Delta =1$ the finite-angle transformation is singular at its zero modes, and quantum depletion of Néel order is infrared divergent in one dimension.

For $0<\Delta <1$, choose a planar Néel reference and rotate local axes to align its longitudinal spins. One phase convention gives the displayed full-zone spectrum, linear near $k=0$ with slope $JS\sqrt{2(1+\Delta )}$, and gapped at $k=\pi$ by $2JS\sqrt{2(1-\Delta )}$. Shifting a sublattice phase can interchange the momentum locations. Folding the magnetic zone separates the gapless in-plane mode from the gapped out-of-plane polarization. An easy-axis reference in this regime is unstable and does not supply this physical expansion.