Two deep layers of incompressible flow with upper mass density ${\rho }_1$, lower mass density ${\rho }_2$, and tangential velocities $U_1,U_2$ have this dispersion relation when surface tension is absent. Decaying velocity potentials and the linearized kinematic boundary condition determine their amplitudes; the dynamic boundary condition for an inviscid interface gives the displayed relation. For $k>0$ the squared departure of the phase velocity $c$ from the density-weighted mean velocity is $g({\rho }_2-{\rho }_1)/[k({\rho }_1+{\rho }_2)]-{\rho }_1{\rho }_2(U_1-U_2{)}^2/({\rho }_1+{\rho }_2{)}^2$. A negative value gives Kelvin-Helmholtz instability despite stable density stratification.

The intrinsic planetary luminosity $L_{\mathrm{int}}$ is the power supplied by interior cooling, contraction, nuclear reactions if present, or internal heating, excluding reflected or passively reradiated stellar light. For fixed mass and no accretion or nuclear source, global conservation of energy gives $L_{\mathrm{int}}=Q_{\mathrm{deep}}-d(U+\Omega )/dt$, where $Q_{\mathrm{deep}}$ is any additional power deposited in the interior, such as tidal heating or Joule heating. For $Q_{\mathrm{deep}}=0$ and a gas-supported quasi-static sphere of fixed specific-heat ratio $\gamma$ and negligible surface pressure, the stellar virial theorem gives $\Omega =-3(\gamma -1)U$, henceFor a dynamically stable gas-supported model take $\gamma >4/3$. For $\gamma =5/3$ this is half the gravitational release. Degenerate giants require a different internal energy reservoir; the half-factor is not universal.

For $S={\sum }_i{a}_i{ϵ}_i$ and $F=\Vert S\Vert$, the even-function spectral gap on a hypercube gives $2\mathrm{Var}(F)\le -\mathbb{E}[FLF]$. A norming functional supplied by the Hahn-Banach theorem and convexity of the norm gives $LF\ge -F$, because $LS=-S$. Hence the Dirichlet form is at most $\mathbb{E}{F}^2$, proving the inequality. Two equal scalar coefficients attain equality, so the constant is sharp.

For a permutation-symmetric density in the Kac master equation, integrate over all but the first velocity. Collisions among integrated coordinates cancel. The $N-1$ pairs containing the first coordinate give equal terms, with coefficient $N(N-1)/(2\pi (\genfrac{}{}{0ex}{}{N}{2}))=1/\pi$. The resulting exact evolution depends on the two-coordinate marginal distribution; it is not a closed equation for the first marginal without additional asymptotic independence.

For the Kac collision operator, expanding the squared Hilbert space norm gives $⟨F,(I-Q)F⟩=(4\pi (\genfrac{}{}{0ex}{}{N}{2}){)}^{-1}{\sum }_{i<j}{\int }_0^{2\pi }{\int }_{{\mathbb{R}}^N}|F(R_{ij,\theta }v)-F(v){|}^{2}dvd\theta$. The nonnegative form measures failure of invariance under the pair rotations. It vanishes precisely on the radial kernel of the Kac collision operator.

For inner radius $a$, mean gap $\Delta \ll a$, fixed eccentricity parameter $|\alpha |<1$, gap $h=\Delta (1+\alpha \sin \theta )$, and inner angular velocity $\Omega$, the constant circumferential flux isIt follows by integrating the lubrication theory relation $p_s=6\mu \Omega a/h^2-12\mu q/h^3$ over a full period. The leading holding force is horizontal:Integration by parts computes this force without a closed-form pressure. The fluid couples about the separate cylinder axes obey $C_i+C_o=\alpha \Delta F_x$; moments about a common axis balance exactly at this order.

Let the inner-centre displacement from a fixed outer centre have magnitude $r=\Delta e$, with $0\le e<1$. The leading full-film journal bearing resistance to translation in the displacement direction and perpendicular to it isFor $h=\Delta (1-e\cos \theta )$, the Reynolds lubrication equation gives radial squeezing flux $a{v}_r\sin \theta$ and tangential squeezing flux $a{v}_t[-\cos \theta +3e/(2+e^2)]$. The additive constant enforces periodic pressure. Integrating the pressure traction gives these coefficients; at $e=0$ they coincide, as required by rotational symmetry.

A freely translating inner axle with prescribed positive angular velocity $\Omega$ and downward load $F>0$ has full-film journal bearing force balance equationswhere $R_r,R_t$ are the squeeze resistance of an eccentric journal bearing. Rotation supplies tangential fluid force $R_t\Omega r/2$. The equilibrium lies horizontally to the right and satisfies $F=R_t\Omega r/2$. Linearization there has imaginary eigenvalues: the prescribed rotating axle continuously supplies energy, so viscous dissipation does not imply settling. For an initially concentric axle, putting $e=r/\Delta$ and $K=6\pi \mu \Omega a^3/{\Delta }^2$ gives the closed orbitThis follows by eliminating time and integrating $x_e+3ex/[2(1-e^2)]=\Omega {\Delta }^2{R}_{0}e/[2F(1-e^2{)}^{3/2}]$. It applies to the leading inertia-free full-film model; adding cavitation, constraints or finite-length leakage changes the prediction.

A full-film journal bearing model assumes that liquid fills the entire annular gap and that pressure is single-valued around a complete circumference. No cavitation boundary is imposed. The full-film eccentric journal-bearing rotation and full-film journal-bearing whirl formulas use this assumption; their predicted motion depends on keeping it.

For $p=k_{B}T[n+B_2(T)n^2]$, the coefficient is $\mu =N[T{B}_2^{\prime }-B_2]/[C_p(1+2B_{2}n)]$. On the stable dilute branch it is positive precisely when $d(B_2/T)/dT>0$.

Over an algebraically closed field, a linear operator $X$ has a unique decomposition into a commuting diagonalisable endomorphism $X_s$ and nilpotent endomorphism $X_n$. On its generalized eigenspace for $\lambda$, set $X_s=\lambda I$ and $X_n=X-\lambda I$. The Chinese remainder theorem makes both parts polynomials in $X$, so they preserve every $X$-invariant subspace. Uniqueness follows by restricting any other commuting decomposition to those generalized eigenspaces. Over a perfect field the semisimple part need only become diagonalizable after scalar extension.

A copula is a joint distribution function on a unit cube whose one-dimensional marginal distributions are uniform distributions. Applying each continuous marginal distribution to its random variable produces uniform marginals by the probability integral transform; their copula records the remaining dependence. Independent uniform coordinates and one common uniform coordinate give different copulas with the same marginals.

The Jeans swindle is the removal of the unperturbed uniform density's gravitational field when treating an infinite homogeneous Newtonian medium as static. Without such subtraction or an external supporting prescription, a nonzero constant density and zero background gravitational acceleration cannot satisfy Poisson's equation. Linear perturbations then obey ${\nabla }^2\delta \varphi =4\pi G\delta \rho$. This is a specified idealization, not an exact isolated infinite equilibrium.

For the screened attraction $-Gm{e}^{-\alpha r}/r$ with $\alpha \ge 0$, $({\nabla }^2-{\alpha }^2)\varphi =4\pi G\rho$. A static homogeneous barotropic medium with sound speed $v_s>0$ has ${\omega }^2=k^2[v_s^2-4\pi G{\rho }_0/(k^2+{\alpha }^2)]$. A growing mode exists iff $4\pi G{\rho }_0>v_s^2{\alpha }^2$, with $0<k<\sqrt{4\pi G{\rho }_0/v_s^2-{\alpha }^2}$. For nonzero screening the growth rate tends to zero as $k\to 0$; the homogeneous mode is not an exponentially growing density fluctuation. Finite interaction range can therefore remove the instability entirely, while all sufficiently small nonzero wavenumbers remain unstable when the threshold is exceeded.

For a Lie group with a bi-invariant Riemannian metric, vary $\gamma (t)=e^{tA}$ by conjugation: $F(s,t)=e^{sX}{e}^{tA}{e}^{-sX}$. Its Jacobi field is $J_X(t)=X{e}^{tA}-e^{tA}X$. If $e^A$ is central, all these fields vanish at both endpoints. Their space has dimension $\dim \mathfrak{g}-\dim \mathfrak{z}(A)$, since the kernel is the centralizer of an element of a Lie algebra.

For $j_{n}f(x)={\int }_{-\pi }^{\pi }f(x-t)J_n(t)dt$, evenness and unit mass reduce the error to an integral of second differences. The Jackson kernel bound and the scaling inequality for the second modulus of smoothness yield $\Vert j_{n}f-f{\Vert }_{\infty }\le C{\omega }_2(f,1/n)$. Since the degree is $2(n-1)$, a degree-$N$ estimate uses $n=\lfloor N/2\rfloor +1$. For $f\in C^2$ this gives $E_N(f)\le C^{\prime }{N}^{-2}\Vert f^{\prime \prime }{\Vert }_{\infty }$.

A nondegenerate quadratic form over a number field is isotropic if and only if it is isotropic over every completion. Over $\mathbb{Q}$, these are the real field and all P-adic numbers. It turns a rational existence problem into local tests; denominators can then be cleared for an integral homogeneous equation.

For a surface of revolution whose induced metric is $g=E(\rho )d{\rho }^2+{\rho }^{2}d{\varphi }^2$, with $\rho >0$ and $E>0$ smooth on a coordinate interval, define $\sigma (\rho )={\int }_{{\rho }_{\ast }}^{\rho }\sqrt{E(r)}dr/r$. Then ${\sigma }^{\prime }>0$ and the conformal factor ${\rho }^{-1}$ gives ${\rho }^{-2}g=d{\sigma }^2+d{\varphi }^2$. This proves the rescaled metric is locally the Euclidean metric without solving a curvature equation. Poles and places where $\rho$ is not a coordinate must be treated in other charts; the angular coordinate is also understood locally.

The isometry group of a metric space consists of its bijective isometries, with composition as the group operation. For a Riemannian metric, these are the diffeomorphisms preserving the metric tensor. Right translations of a right-invariant Riemannian metric supply a faithful subgroup isomorphic to the underlying Lie group.

For initially well-separated Kerr black holes with no orbital angular momentum, exact axisymmetry gives $J_f=J_1+J_2$. Put $X=(A_1+A_2)/(8\pi )$. The black-hole area theorem and the irreducible mass identity implyIndeed, $X\ge |J_1|+|J_2|\ge |J_f|$, and $\frac{1}{2}(Y+J_f^2/Y)$ is nondecreasing for $Y\ge X$. Opposite spin signs lower $J_f^2$ at fixed initial areas and hence raise the allowed radiated-energy bound; this need not be achievable in an actual merger.