For each particle the moment includes its signed charge and inverse mass. Writing both moments with one reference magneton requires effective signed magnetic factors. A shared symbol must not hide the sign change or imply equal magnitudes; the reference magneton is distinct from the SI vacuum permeability.

A generic instrument for force and torque uses a magnetically responsive bead in a controlled magnetic field, with optical tracking. Translational force requires a field gradient. Permanent or anisotropic moments can feel orienting torque; an isotropic induced moment parallel to the field has no such torque. This definition concerns the physical instrument, without biological applications.

A magnetic dipole of fixed magnitude $m_B$, free to rotate about an axis with moment of inertia $J$, has potential energy $-m_{B}B\cos \theta$ in a uniform magnetic field. Near stable alignment, $J\ddot{\theta }=-m_{B}B\theta$, so its small-angle period is $2\pi \sqrt{J/(m_{B}B)}$. This neglects damping, bearing friction and any other restoring torque.

A uniform-radius magnetic arc of length $L$ and radius $R$, two linear tapers of length $s$ to radius $r$, and an unfringed vacuum gap of length $h$ have the displayed gap field. Here ${\mu }_r=\mu /{\mu }_0$ and $B_g={\mu }_0{H}_g$. Tapering concentrates flux, but at fixed winding current the gap-dominated limit is $H_g\simeq NI/h$ regardless of pole area. Saturation and leakage prevent unrestricted field enhancement.

A square resistive loop of side $a$, moving normally across a sharp field edge at speed $v$, has induced current magnitude $Bav/R$ and drag magnitude $B^2{a}^{2}v/R$ while partially overlapping the field. Completely enclosed in a uniform static field it has no changing flux and, with neglected self-inductance, no current.

The instantaneous-current approximation uses $I=\mathcal{E}/R$ and requires the electrical relaxation time $L/R$ to be negligible relative to the relevant mechanical changes. Otherwise circuit self-inductance changes the braking transient.

A square wire loop partly in a uniform field has linear drag $B^2{a}^{2}v/R$, giving velocity-relaxation time ${\tau }_t=mR/(B^2{a}^2)$. For uniform wire of mass density ${\rho }_m$ and resistivity ${\rho }_e$, ${\tau }_t=16{\rho }_m{\rho }_e/B^2$, independent of wire dimensions.

A thin square loop rotating about a central in-plane axis has $\mathcal{I}=m{a}^2/6$. With ohmic current and decay slow compared with a rotation, the averaged torque is $-B^2{a}^4\omega /(2R)$. Its angular-speed decay time is $mR/(3B^2{a}^2)$; its kinetic-energy decay time is half that value.

A lineated pattern associated with differently magnetized crust formed at a spreading ridge during changes of field polarity. The age-distance relation is set by spreading, while the measured anomaly is a spatially filtered field of the magnetized material.

The tensor connecting spin to magnetic moment in an effective spin Hamiltonian. Its principal values specify direction-dependent Zeeman coupling.

A displacement of a ferromagnetic hysteresis loop's field centre caused by coupling to a prepared, sufficiently pinned magnetic environment, commonly an antiferromagnet. Field cooling through the ordering/blocking range can establish a unidirectional interfacial spin preference. With switching fields $H_{-},H_{+}$, the shift is $H_E=(H_{+}+H_{-})/2$, distinct from coercive half-width $(H_{+}-H_{-})/2$. Reversing the cooling field, testing a full major loop and comparing uncoupled/zero-field-cooled controls help identify genuine bias.

A directional magnetic preference that distinguishes a Magnetization direction from its reverse, unlike a purely uniaxial easy-axis energy. A pinned interface can supply such a preference and produce exchange bias. In a simple model its interfacial energy term is proportional to $-J_{\mathrm{int}}\cos \theta$.

A magnetic energy preference for an axis, with the two opposite directions equivalent in a purely uniaxial model. An easy-axis energy can be written $K{\sin }^2\theta$ for $K>0$. A unidirectional term distinguishes the two directions and can instead shift a hysteresis.

The anti-Hermitian generators of local qubit rotations become real antisymmetric matrices in the magic basis. Their exponentials preserve $b^{\mathsf{T}}b$, and hence its modulus and pure-state concurrence. A common phase changes that bilinear quantity by a phase but not its modulus.

A prefix of a word that is itself a Lyndon word. The longest Lyndon prefix is useful when ranking necklaces and substrings of a De Bruijn sequence.

Let true parallax $p$ have prior density proportional to $p^{-4}$ on the valid finite population range, and let an observation supply a constant likelihood on $0<p_{-}\le p\le p_{+}$. The normalized posterior is $3p^{-4}/(p_{-}^{-3}-p_{+}^{-3})$. Integrating gives cumulative probability $(p_{-}^{-3}-p^{-3})/(p_{-}^{-3}-p_{+}^{-3})$, hence the displayed median. A fixed half-width equal to half the observed center gives $p_{-}=p_0/2$, $p_{+}=3p_0/2$ and $p_{\mathrm{med}}=p_0(27/112{)}^{1/3}$. Parameter-dependent error widths require their own normalized likelihood and support.

In an appropriate spinful one-dimensional interacting conductor, low-energy density and spin disturbances form distinct collective modes, generally with different velocities. A bare injected electron can excite both modes. Boundary conditions and attached leads still matter for electrical conductance; a bulk interaction parameter alone need not set the two-terminal value of a clean contacted wire.