For a radial conservative force plus the magnetic-type force $\dot{\mathbf{r}}\times (\mu \mathbf{r}/r^3)$, the torque equals $\mu d\hat{\mathbf{r}}/dt$. Subtracting $\mu \hat{\mathbf{r}}$ from ordinary angular momentum therefore produces a constant vector. Its dot product with $\hat{\mathbf{r}}$ is $-\mu$, so motion lies on a cone about that vector. At $\mu =0$, ordinary planar central force motion is recovered. This is the classical radial magnetic monopole angular-momentum correction.

Consider particles of the same constant mass $m$, interacting through equal-and-opposite central internal forces, with external forces $F_i=-k{\dot{r}}_i$ for constant $k$. About a fixed point $a$, internal pair torques cancel and the external torque is $G_a=-k{\sum }_i(r_i-a)\times {\dot{r}}_i=-(k/m)L_a$. Hence ${\dot{L}}_a=-(k/m)L_a$ and the displayed exponential decay follows. The same result holds for angular momentum about the centre of mass, because the moving-origin term $-\dot{R}\times P=-\dot{R}\times M\dot{R}$ vanishes. Unequal masses do not generally permit factoring out a single $k/m$.

A charged particle in a uniform field $\mathbf{B}=B{\mathbf{e}}_z$ and an axially symmetric potential conserves the displayed canonical angular momentum, with $\rho$ the distance from the field axis and $q$ its charge. It follows directly by differentiating the expression and using the Lorentz force. For charge $q=-e$, the equivalent invariant is $m(\dot{\mathbf{r}}\times \mathbf{r})\cdot \mathbf{B}+e|\mathbf{r}\times \mathbf{B}{|}^2/2$. A radial restoring force has no axial torque.

A gyroscope is a rotating instrument whose rapidly spinning, nearly symmetric rotor provides a direction reference through its angular momentum. In an inertial frame, negligible external torque approximately preserves that direction by conservation of angular momentum. A transverse torque turns the angular momentum instead: the surprising sideways motion of a supported spinning wheel is precession. An ideal freely falling test gyroscope in general relativity has its spin undergo parallel transport along its timelike geodesic. Comparing that direction with distant reference axes measures Lense-Thirring precession; the direction need not remain fixed in arbitrary rotating coordinates.

A gyroscope direction reference is useful precisely because its direction can persist while its housing turns. A spinning-wheel precession experiment deliberately supplies gravitational torque to reveal how that persistence fails. The gyroscope vector geometry explains the direction, and slow steady gyroscope precession gives a numerical prediction with a check on its assumptions.

For nonzero total momentum $\mathbf{P}$, translating the origin by $\mathbf{b}$ changes angular momentum to $\mathbf{L}-\mathbf{b}\times \mathbf{P}$. The displayed line of origins makes this vector parallel to $\mathbf{P}$, equal to $(\mathbf{L}\cdot \mathbf{P})\mathbf{P}/|\mathbf{P}{|}^2$. The invariant $\mathbf{L}\cdot \mathbf{P}$ cannot be removed by translation; if it is zero, angular momentum vanishes on this line.

Move the fixed origin from $O$ to $O^{\prime }=O+\mathbf{a}$, keeping the axes parallel and the inertial frame's velocity measurements unchanged. The new positions are ${\mathbf{r}}_i^{\prime }={\mathbf{r}}_i-\mathbf{a}$, so distributivity of the cross product givesHere $\mathbf{P}$ is total linear momentum. For the angular momentum example with $\mathbf{a}=(1,0,0)\mathrm{m}$, the new position is $(1,0,0)\mathrm{m}$ and $L_{O^{\prime },z}=6-3=3\mathrm{kg}{\mathrm{m}}^2/s$. If total momentum is zero, angular momentum is independent of this origin shift. A moving origin additionally contributes $-\dot{\mathbf{a}}\times \mathbf{P}$ when this expression is differentiated while retaining the original momenta; see angular momentum about the centre of mass for an important case where that term vanishes.

In an Einstein-de Sitter universe, $H(z)=H_0(1+z{)}^{3/2}$ and the radial comoving distance is $2H_0^{-1}[1-(1+z{)}^{-1/2}]$. A physical Hubble radius $H(z{)}^{-1}$ has comoving size $(1+z)/H(z)$. Dividing by the radial distance gives the displayed small angle, in radians. A Hubble diameter has twice this angle; the matter-era particle-horizon radius also equals twice the Hubble radius. These distinctions matter when comparing approximate CMB angular scales.

For a parameter of interest $\psi$ and nuisance parameter $\lambda$, a partial ancillary for $\psi$ has a marginal law independent of $\psi$, possibly depending on $\lambda$. This differs from a fully ancillary statistic, whose law is independent of both. For independent Poisson counts with means $\lambda \psi$ and $\lambda (1-\psi )$, their total is Poisson with mean $\lambda$, while the first count conditional on the total is binomial with success probability $\psi$. Conditioning on the total eliminates $\lambda$. A partial ancillary does not automatically eliminate a nuisance parameter; the conditional law must actually be free of that parameter.

A maximal ancillary statistic has no strictly finer ancillary partition of the sample space, but different maximal ancillary partitions can exist. Let $(U,V)$ take values in $\{0,1{\}}^2$, with probabilities $\theta /2$ on each equal pair and $(1-\theta )/2$ on each unequal pair, for $0<\theta <1$. Both $U$ and $V$ have the Bernoulli distribution with success probability $1/2$ for every $\theta$. Their joint law depends on $\theta$, so combining them is not ancillary. Each partition is maximal: splitting one of its two-point cells singles out a point whose probability depends on $\theta$. There is no common ancillary refinement, and conditioning on one ancillary choice need not specify a unique conditioning principle for all models.

For independent observations uniform on $(\theta -1/2,\theta +1/2)$, let $U$ and $V$ be the sample minimum and maximum, $R=V-U$, and $C=(U+V)/2$. For $n\ge 2$ their joint probability density function in $(c,r)$ is $n(n-1)r^{n-2}$ on $0<r<1$, $|c-\theta |<(1-r)/2$. Integrating over $c$ gives $f_R(r)=n(n-1)r^{n-2}(1-r)$, independent of $\theta$, so $R$ is an ancillary statistic. Moreover $C\mid R=r$ is uniform on $(\theta -(1-r)/2,\theta +(1-r)/2)$. Consequentlyis an exact conditional $(1-\alpha )$ confidence interval for $\theta$. The realized range determines the appropriate precision, although the range is not itself evidence about the value of $\theta$.

Three Fourier wavevectors sum to zero and form a hexagonal star. Translations give each amplitude the phase of its own wavevector; cyclic rotations permute them, and a half-turn conjugates them. These symmetries permit the resonant quadratic term and force its coefficients to be real. A spatial reflection would interchange the two cross-couplings and require $b=c$, but chiral patterns need not have that extra symmetry. The equations describe rolls, hexagons, phase locking and oscillatory changes in dominant orientation.

Independent sign changes of three amplitudes and a cyclic coordinate permutation force each cubic component to be its own amplitude times a cyclic permutation of one quadratic polynomial. One convenient parametrization is $\dot{x}=x(\mu -aX-c{y}^2+e{z}^2)$ with cyclic companions and $X=x^2+y^2+z^2$. The axial and diagonal isotropy types give primary branches; two-coordinate branches can emerge in secondary pitchfork bifurcations when cross-couplings change sign.

The Newell–Whitehead–Segel equation is an amplitude equation for slowly modulated stationary rolls in an isotropic pattern-forming medium. With the displayed sign convention, a roll $A=R{e}^{iqX}$ has $R^2=\mu -q^2$. A transverse phase modulation has linear growth rate $q{k}^2-k^4/4$, so long-wave transverse instability occurs for $q>0$. Reversing the carrier-wave convention reverses the sign attached to its detuning $q$; one must derive the stability sign from the operator actually used.

Two counterpropagating modes transform as $(A,B)\mapsto (e^{ik\delta }A,e^{-ik\delta }B)$ under spatial translation. Forcing at twice their natural frequency breaks continuous temporal phase symmetry but preserves simultaneous sign reversal. Hence $\overline{B}$ transforms like $A$ and $\overline{A}$ like $B$, allowing linear parametric coupling. An equal-amplitude phase-locked standing wave has $|A{|}^2=|B{|}^2=z>0$ with $|\beta +\gamma {|}^2{z}^2-2\mu \mathrm{Re}(\beta +\gamma )z+{\mu }^2-{\nu }^2=0$. A nonnegative discriminant is only an algebraic compatibility condition: a physical nonzero wave also requires a positive root.

A forcing at the carrier wavenumber breaks continuous spatial translation symmetry and adds a constant to its complex amplitude equation. In a frame following a moving forcing pattern, its phase drift gives $i\Lambda A$. A nonzero forcing coefficient is normalized to one by scaling amplitude by its cube root and time by its two-thirds power. A steady intensity $s=|A{|}^2$ satisfies $s[(\mu -s{)}^2+{\Lambda }^2]=1$, and its phase is uniquely determined by the steady complex equation.

Assume $\sigma >0$. With periodic boundary conditions, the spatial mean of $B$ is a conserved quantity. For zero mean write $B=H_X$ with periodic zero-mean $H$. The functional $V=⟨\mu A^2/2+\alpha A^3/3-A^4/4-A_X^2/2-A^2{H}_X/2-H_X^2/(4\delta )⟩$ generates $A_T=\delta V/\delta A$, $H_T=2\delta \sigma \delta V/\delta H$. Thus $V_T=⟨A_T^2+H_T^2/(2\delta \sigma )⟩\ge 0$. Completing the square in $H_X$ bounds $V$ above by a coercive quartic polynomial when $0<\delta <1$, ruling out nonconstant recurrent motion on regular precompact trajectories.

The partial differential equation $T_t=-\mu \Delta T-{\Delta }^{2}T+\nabla \cdot (|\nabla T{|}^2\nabla T)$ on a $2\pi$-periodic square conserves the spatial mean. On a fixed-mean subspace its first instability is at $\mu =1$ and has four real critical amplitudes. With $\mu =1+{\epsilon }^2\nu$ and $T=\epsilon (A{e}^{ix}+B{e}^{iy}+\text{complex conjugate})+\cdots$, projection onto the critical Fourier modes gives the displayed equation and its $A,B$ interchange. For $\nu >0$, squares have $|A{|}^2=|B{|}^2=\nu /5$, amplitude eigenvalues $-2\nu ,-2\nu /5$, and neutral translation phases. Rolls have a positive transverse growth rate $\nu /3$.

A frog is an amphibian of the tailless adult group. Many species develop from aquatic tadpoles, while others have different reproductive modes. The adult’s jumping body plan and the larva’s feeding apparatus belong to distinct developmental stages.

In vacuum magnetostatics, the Ampère-Maxwell equation reduces to $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$. Applying Stokes theorem gives ${\oint }_C\mathbf{B}\cdot d\mathbf{\ell }={\mu }_0{I}_{\mathrm{enc}}$, where $I_{\mathrm{enc}}$ is the electric current crossing an oriented spanning surface. Time-dependent electric fields require the additional displacement current term in the Ampère-Maxwell equation.

Cysteine is an amino acid whose amino acid side chain contains sulfur, represented by Cys or C for an incorporated amino acid residue. Two cysteine positions can be joined through a protein disulfide bond. See IUPAC Blue Book, P-103.1: cysteine's structure and symbols and Cooper: cysteine crosslinks in proteins. The C in a sequence means cysteine, whereas the C-end label in the toy diagram means C-terminus.