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A Squire mode has zero wall-normal velocity and nonzero wall-normal vorticity, satisfying the homogeneous Squire equation. For real base velocity, positive finite Reynolds number, and zero boundary terms,for a nontrivial mode in a finite channel with no-slip walls. This follows by multiplying the homogeneous equation by $\overline{\eta }$, integrating by parts and taking the real part. Such modes are damped even when the coupled velocity-vorticity system can exhibit transient growth from non-normal modes.

For an unpolarized initial state, average the squared scattering amplitude over its equally populated spin states. With two incoming spin-one-half particles this gives a factor $1/(2\cdot 2)=1/4$. Unobserved final spins are summed rather than averaged.

For fixed mass $M$, a uniform sphere of physical radius $R$ has $W_G=-3G{M}^2/(5R)$ and $W_{\Lambda }=-\Lambda M{R}^2/10$, using the convention in which its outward acceleration is $\Lambda r/3$. The virial theorem reads $2T+W_G=2W_{\Lambda }$. Defining $x=R_f/R_{\mathrm{ta}}$ and $\eta =\Lambda /(4\pi G{\rho }_{\mathrm{ta}})$, conservation of energy gives $2\eta x^3-(2+\eta )x+1=0$. The collapsing branch has $x=1/2-\eta /8+O({\eta }^2)$. For small positive cosmological constant, it is more compact at fixed turnaround data, even though the outward force inhibits turnaround and collapse.

The elastohydrodynamic penetration length balances the transverse viscous force $c_{\perp }\omega y$ against the bending force $By/{\ell }_{\omega }^4$ in a periodically driven filament. The sperm number $L/{\ell }_{\omega }$ distinguishes nearly rigid, weakly deformed filaments from long filaments whose response is localized near the actuator.

For a self-adjoint operator $T$, a real number $\lambda$ lies in the continuous spectrum when $T-\lambda I$ is injective with dense range but is not surjective. Such spectral values need not be eigenvalues of normalizable states. The free-particle Hamiltonian on the real line has a continuous energy spectrum.

In the disjoint convention for a bounded linear operator $T$, the residual spectrum consists of $\lambda$ for which $T-\lambda I$ is injective but has nondense range. Together with the point spectrum and continuous spectrum it partitions the spectrum of a bounded operator. Some conventions also include noninjective operators with nondense range, so the convention must be stated.

A multi-object spectrograph records optical spectra of many selected objects simultaneously. The main designs are multi-slit spectrographs and fiber-fed spectrographs. The ESO instrumentation overview describes both arrangements.

A cross-dispersed echelle spectrograph uses an echelle grating for strong dispersion and a cross-disperser in the perpendicular direction to separate overlapping diffraction orders. It records many orders simultaneously in an echellogram.

Spectral resolving power is $R=\lambda /\Delta \lambda$, where $\Delta \lambda$ is the wavelength separation just resolved under the specified instrumental and source conditions.

An integral field spectrograph records a spectrum for each of many spatial elements across a two-dimensional field, producing a spectral data cube. Field reformatters include lenslet arrays, optical fibers and image slicers.

Asymptotic regularization stops the gradient flow $x^{\prime }=-A^{\ast }(Ax-f)$ at $t=1/\alpha$, starting from zero. With the singular system of a compact operator convention $A{v}_j={\sigma }_j{u}_j$, its spectral filter givesThe scalar coefficient solves $c_j^{\prime }=-{\sigma }_j^2{c}_j+{\sigma }_j⟨f,u_j⟩$ with zero initial value. Since $1-e^{-s}\le \min (s,1)$ for $s\ge 0$, the operator norm is at most ${\alpha }^{-1/2}$. On the domain of the Moore–Penrose inverse of an operator, dominated convergence theorem of the squared spectral coefficients proves $R_{\alpha }f\to A^{†}f$. The noise-bias decomposition for linear regularization then gives noisy-data convergence when $\delta /\sqrt{\alpha }\to 0$.

A weak perturbing relative acceleration $\mathbf{f}$ changes specific orbital energy at rate $\dot{\epsilon }=\mathbf{v}\cdot \mathbf{f}$. Since $\epsilon =-\mu /(2a)$ for a Kepler orbit,For a circular Kepler orbit under $\mathbf{f}=-\kappa \mathbf{v}+\mathbf{g}$ with constant $\mathbf{g}$, the constant-force term averages to zero over an unperturbed orbit. Thus $⟨\dot{a}⟩=-2\kappa a$. This is a leading-order orbital average; a strong constant force can distort the orbit instead.

The specific angular momentum of an elliptic Kepler orbit is $h=\sqrt{\mu a(1-e^2)}$. At periapsis, $r=a(1-e)$, soThe mean motion understates instantaneous angular speed near periapsis. A particle in an exterior mean-motion resonance can briefly overtake the planet in angle despite completing full orbits more slowly.

Every real orthogonal transformation of odd dimension with determinant one has eigenvalue $+1$. Nonreal eigenvalues come in conjugate pairs with product one, while real eigenvalues are $+1$ or $-1$. There are an odd number of real eigenvalues and an even number of negative ones, leaving at least one positive eigenvalue. The corresponding eigenvector is fixed.

The SO(3) group consists of real three-by-three orthogonal matrices with determinant one. It acts by rotation in three dimensions. Its group manifold is ${\mathbb{RP}}^3$, since the Adjoint double cover from SU(2) to SO(3) identifies antipodal points of the three-sphere.

A surjective linear isometry $U:C(K)\to C(L)$ between compact Hausdorff function spaces has the form $(Uf)(s)=u(s)f(h(s))$, where $h:L\to K$ is a homeomorphism and $u$ is continuous with modulus one. Thus the Banach space structure of $C(K)$ determines $K$ up to homeomorphism.

Weak convergence to zero gives pointwise convergence by the continuous evaluation functionals and uniform boundedness in the supremum norm by the weakly bounded set criterion. The dominated convergence theorem for Lebesgue measure then proves the displayed implication. The finite measure of the interval and uniform norm bound are the relevant hypotheses.

For a normed vector space $X$, take $K=B_{X^{\ast }}$ with the weak-star topology. Banach-Alaoglu theorem makes it compact Hausdorff, and each displayed evaluation function is continuous. Its supremum norm is $\Vert x\Vert$ by the Hahn-Banach theorem, giving a linear isometric embedding into $C(K)$.

On a bounded smooth domain, the Sobolev trace theorem can be strengthened toFor smooth $u$, extend the outward unit normal to a smooth vector field $X$ near the closure of $U$ with $X\cdot \nu =1$ on its boundary. The divergence theorem gives ${\int }_{\partial U}{u}^2={\int }_U(\mathrm{div}X)u^2+2uX\cdot Du$, and the Cauchy-Schwarz inequality gives the estimate. Density extends it to $H^1(U)$. A bound containing only the product $\Vert u{\Vert }_2\Vert Du{\Vert }_2$ is false for nonzero constant functions. Equivalently, for every $\epsilon >0$, the Young inequality gives $\Vert Tu{\Vert }_2^2\le \epsilon \Vert Du{\Vert }_2^2+C_{\epsilon }\Vert u{\Vert }_2^2$.

For a cube $Q$ of side length $r$ in ${\mathbb{R}}^n$, $p>n$, and a continuously differentiable function $v$, its average $v_Q=|Q{|}^{-1}{\int }_{Q}v$ satisfiesAveraging the fundamental theorem of calculus along a line segment from $y$ to $z\in Q$ and changing variables gives $|v(y)-v_Q|\le C_n{\int }_Q|Dv(w)||w-y{|}^{1-n}dw$. The Holder inequality applies because $(n-1)p^{\prime }<n$, and the kernel norm is $O(r^{1-n/p})$. Approximation extends the estimate to the continuous representative of a Sobolev space function. On ${\mathbb{R}}^n$ it yields a Hölder continuous function of exponent $1-n/p$ representing every $W^{1,p}$ class.