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Coma makes an off-axis optical point source appear asymmetric, often with a compact head and a flaring tail like a comet. Different parts of the optical pupil send its light rays to different transverse image positions. A sharp central star image therefore does not guarantee sharp images across an optical telescope's field of view. In an idealized, aligned system with rotational symmetry, coma vanishes on the optical axis; misalignment can introduce coma even at the nominal field centre.^[ref]

Coma differs from spherical aberration, whose dependence on optical pupil radius can blur an on-axis point without selecting a transverse direction. Astigmatism gives different optical foci in two perpendicular sections of an off-axis bundle; a through-focus sequence can change its elongated image's orientation. Diffraction persists when these optical aberrations vanish. A real asymmetric point spread function may mix several effects, so its appearance alone is not a unique diagnosis of coma.

The classical primary description belongs to the nineteenth-century Seidel aberration classification of five primary monochromatic image errors.^[ref] It remains useful in modern design: OpticStudio's Seidel Coefficients by ANSYS, Inc. documentation lists coma as $W_{131}$ and tabulates each surface's contribution as well as the system sum (Discussion). Cancellation of contributions can reduce coma without making every surface individually free of it. The same documentation limits this calculation to specified systems with axially symmetric surfaces; higher-order terms and broken rotational symmetry require more general ray tracing.

Spherical aberration makes rays at different pupil radii focus at different axial positions. Its primary pupil dependence is quartic; the balanced Zernike spherical mode also contains quadratic and constant terms to remove defocus and piston.

Phasing adjusts relative piston and tilt so adjacent segments form a continuous optical wavefront. Near normal reflection a surface piston changes optical path length by twice that displacement.

For locally bed-tangent drag force and an upslope angle $a$, a spherical-grain force balance gives ${\tau }_{\mathrm{th}}(a)={\tau }_{\mathrm{th},0}(\cos a+\sin a/{\mu }_s)$. The small-slope term is $\delta {\tau }_{\mathrm{th}}=({\tau }_{\mathrm{th},0}/{\mu }_s){\eta }_x$. A horizontal drag force instead changes the normal contact force and gives a different slope correction.

The scalar second mass moment is the trace of the second mass moment tensor. The moment of inertia about a specified axis instead integrates squared distance from that axis. For an isotropic distribution each axial moment of inertia is $2I/3$; the sum over three orthogonal axes is $2I$.

With the upper-half-plane branch, ${\theta }_t\in (0,\pi )$ before the Loewner swallowing time. Write $Z_t=X_t+i{Y}_t$. The Itô formula and the Chordal Loewner equation giveAt parameter four its drift vanishes. At other parameters the drift describes the competition between the deterministic conformal evolution and the Brownian driver.

Using $V=k_0^2(n^2-1)$, the scalar Lippmann-Schwinger equation is $\psi ={\psi }_i+{\int }_D{G}_{k_0}V\psi$. The Born approximation replaces the total field inside the integral by the known incident field:For a unit plane wave, its far-field pattern is $(4\pi {)}^{-1}{\int }_{D}V({\mathbf{r}}^{\prime })e^{i{k}_0({\hat{\mathbf{r}}}_0-\hat{\mathbf{r}})\cdot {\mathbf{r}}^{\prime }}d{\mathbf{r}}^{\prime }$. Thus each incident direction samples a shifted sphere in the Fourier transform of the potential.

A pressure scale height is the local distance over which pressure would change by a factor $e$ if its logarithmic gradient were constant. In hydrostatic equilibrium, $H_P=P/(\rho g)$; for an ideal gas it is $k_{B}T/(\mu m_{u}g)$. A small logarithmic pressure interval has depth approximately $H_P\Delta \log P$.

In vacuum analysis, an exact flat direction of a scalar potential is a nonconstant continuous family of degenerate stationary vacua. The scalar potential stays constant along this family and its gradient vanishes there. An ordinary level set does not suffice: $V(x,y)=x^2+y^2$ is constant on every circle, but its vacuum at the origin has no flat direction. A vanishing eigenvalue of the quadratic Hessian matrix also does not suffice: $V(x)=x^4$ has zero quadratic mass at zero but no family of nearby vacua. Exact flat directions can be lifted by corrections absent from the classical scalar potential.

This $C^1$ flux has $|f^{\prime }|\le 1$ globally and agrees with the Inviscid Burgers equation flux on $[-1,1]$. It allows Burgers examples to satisfy a global bounded-speed hypothesis. For Riemann data $-1$ to the left and $1$ to the right, both the stationary expansion discontinuity and the fan $u=x/t$ on $|x|<t$, with exterior values $\pm 1$, are bounded weak solutions. Only the fan is an entropy solution.

Using the unweighted neutrino Boltzmann hierarchy convention and ${\Pi }_{ij}=-(4{\bar{\rho }}_{\nu }/3){\Pi }_{\nu }({\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3)$, the scalar scalar anisotropic stress is ${\Pi }_{\nu }=-3{\Theta }_2/5$. Only the quadrupole contributes, since the trace-free tensor $e_i{e}_j-{\delta }_{ij}/3$ has angular degree two. Different stress and multipole sign conventions change this displayed coefficient.

For two fluids with dynamic viscosities ${\mu }_1<{\mu }_2$, uniform permeability of a porous medium $k$, porosity $\varphi$, and imposed Darcy velocity $U$, the Saffman–Taylor instability of a planar interface has growth rate $\sigma =\alpha U({\mu }_2-{\mu }_1)/[\varphi ({\mu }_2+{\mu }_1)]$ without capillarity. With constant interfacial tension $\gamma$ and the stabilizing pressure-jump convention, a perturbation of transverse wavenumber $\alpha >0$ instead hasThe fastest-growing mode satisfies ${\alpha }_m^2=({\mu }_2-{\mu }_1)U/(3k\gamma )$.

Let ${\psi }_R={\psi }_i{e}^{{\varphi }_1}$ with unit-modulus incidence and ${\varphi }_1={\int }_{D}K(\mathbf{r},{\mathbf{r}}^{\prime })V({\mathbf{r}}^{\prime })d{\mathbf{r}}^{\prime }$. For a centered real Gaussian random field $V$, the exponential moment of a Gaussian linear functional givesHere $C_V$ is the autocorrelation function of a random field. Both the covariance $\mathbb{E}|{\varphi }_1{|}^2$ and the generally nonzero term $\mathrm{Re}\mathbb{E}{\varphi }_1^2$ enter the exponent; dropping the latter is not justified for an arbitrary finite real random medium.

Let $I=[n]$ for odd $n\ge 3$, embed it in the cyclic group $G=\mathbb{Z}/(2n+1)\mathbb{Z}$, and let $A\subseteq I$ have density of a finite subset $\alpha$. If $A$ has no nonconstant three-term arithmetic progression and $n\ge 4{\alpha }^{-2}$, its balanced indicator function $f=1_A-\alpha 1_I$ satisfiesIndeed, the normalized trilinear arithmetic progression count has the Fourier analysis on a finite abelian group formula $\Lambda (h_1,h_2,h_3)={\sum }_r{\hat{h}}_1(r){\hat{h}}_2(-2r){\hat{h}}_3(r)$. Its values on $1_A$ and $\alpha 1_I$ are $\alpha n/|G{|}^2$ and ${\alpha }^3(n^2+1)/(2|G{|}^2)$. Telescoping their difference into three terms containing $f$, the Parseval identity and Cauchy-Schwarz inequality bound its magnitude by $3\max |\hat{f}|\alpha n/|G|$. The difference is at least ${\alpha }^3{n}^2/(4|G{|}^2)$, giving the claim. The zero Fourier coefficient vanishes by the definition of $\alpha$.

If $R_2=R_L\simeq Ca(M_2/M{)}^{1/3}$, Kepler's third law givesThus a Roche lobe-filling star's period measures its inverse square-root mean mass density, largely independently of the companion mass in this approximation. A local mass-radius relation $R_2\propto M_2^{\zeta }$ gives $P\propto M_2^{(3\zeta -1)/2}$.

The kernel of a ring homomorphism $\phi :R\to S$ is the ideal $\{r\in R:\phi (r)=0\}$. The ring homomorphism is injective exactly when this ideal is zero.

A multiplicative subset $S$ of a ring is a right Ore set if $1\in S$, $0\notin S$ and it satisfies the right Ore condition. In a noncommutative domain, this condition lets nonzero denominators be used in Ore localization.

A proper right ideal $L$ is maximal if no proper right ideal strictly contains it. Equivalently, $A/L$ is a simple module. The Jacobson radical is the intersection of all maximal right ideals.

At pericentre, mean longitude equals longitude of periapsis, so resonant argument $\varphi =(p+q)\lambda -p{\lambda }_p-q\varpi$ becomes $\varphi =p(\varpi -{\lambda }_p)$. The pericentre angle in the planet's rotating frame is $\varphi /p$ modulo $2\pi /p$. For coprime $p,q$, there are $p$ distinct pericentre directions before the trajectory closes. Resonant-argument libration restricts these directions and supplies resonance protection.

For resonant trapping of dust in an exterior $(p+q):p$ mean-motion resonance, let Poynting–Robertson drag have parameter $A$. Holding semi-major axis $a$ fixed and retaining secular rates through second order in orbital eccentricity givesFor $5p>q$, squared orbital eccentricity approaches this equilibrium exponentially in the truncated model. If the predicted value is not small, higher-order terms are needed for quantitative dynamics. Finite planetary torque also imposes a minimum eccentricity for balance, which alone does not guarantee capture during migration.