For fixed $E,l,I$, separated motion obeys $p_{\tau }^2=[E\tau +G(\tau )-I+(\alpha -\beta )l^2/(2(\tau +\alpha ))]/[2(\tau +\alpha )(\tau +\beta )]$. Allowed intervals are where this is nonnegative, with turning points and coordinate-fold endpoints treated separately. A generic nonresonant bound orbit with $l\ne 0$ circulates azimuthally and densely samples a projected volume bounded by confocal spheroids and hyperboloids. It is not uniformly distributed in volume. Resonant or zero-angular-momentum orbits can project to lower-dimensional regions; integrability alone does not imply that every orbit fills a three-dimensional shell.

The steady upstream film of an infinite blade has $h_x=-3(h-1)/h^3$. Thus $X(h)=h^3/9+h^2/6+h/3+\log (h-1)/3$. The head is fixed by under-blade leakage balancing the incoming film. The deep pile has a cubic profile, while the actual upstream transition has an exponential tail towards $h=1$.

For a rounded blade with local parabolic lubrication gap $d=ϵ+x^2/(2\delta )$, $ϵ\ll \delta$, the Couette-Poiseuille flow in a thin gap yields a floor-driven flux $2ϵ/3$ and pressure-driven leakage proportional to ${ϵ}^{5/2}\Delta p/\sqrt{\delta }$. Positive $J$ is from the upstream pile towards the rear.

A long blade with negligible under-blade leakage diverts incoming fluid laterally. A deep-pile profile $h^3=9[x_N(y)-x]$ and integrated flux conservation give $d(\int h^{4}dx)/dy=-12y$, hence $x_N=[126({\alpha }^2-y^2){]}^{3/7}/9$. This leading approximation excludes the nose and blade-tip transition regions.

The risk of an estimator under squared Euclidean error is its expected squared distance from the true parameter. Pointwise risk domination can be strict even when the supremum risk agrees, because the supremum can be approached at parameters tending to infinity.

Given finitely many points in a right triangle with hypotenuse endpoints $a,b$, there is a path from $a$ to $b$ visiting them all with sum of squared edge lengths at most $\Vert a-b{\Vert }^2$. Repeated altitude subdivision reduces to cells containing one point. Joining the child paths preserves the squared-cost bound by the Pythagorean theorem; deleting an auxiliary right-angle vertex preserves it by the law of cosines.

The square class of $a\in K^{\ast }$ is its image in the square-class group of a field $K^{\ast }/K^{\ast 2}$. Two nonzero elements have the same square class exactly when their quotient is a square. Every element of this quotient has order dividing $2$. Over the rational numbers, a square class has a unique signed squarefree integer representative.

The valuation contributes one factor $C_2$. The unit decomposition of the 2-adic Gaussian field contributes ${\mu }_4/{\mu }_4^2\cong C_2$ and the additive quotient ${\mathbb{Z}}_2[i]/2{\mathbb{Z}}_2[i]\cong C_2^2$. Thus there are sixteen square classes and fifteen quadratic extensions over this field.

On a spin-$s$ field $f$, the spin-raising and lowering operations are $ðf=-{\sin }^s\theta ({\partial }_{\theta }+i\csc \theta {\partial }_{\phi })({\sin }^{-s}\theta f)$ and $\overline{ð}f=-{\sin }^{-s}\theta ({\partial }_{\theta }-i\csc \theta {\partial }_{\phi })({\sin }^s\theta f)$. The intermediate spin changes after each application. For an axisymmetric scalar $f(\mu )$, either operation applied twice yields $(1-{\mu }^2)f^{\prime \prime }(\mu )$.

Equal spheres of radius $r$ occupy the sites of a cubic lattice of spacing $2r$. A cell contains one sphere, so its packing fraction is $(4\pi r^3/3)/(2r{)}^3=\pi /6$. This is an ideal arranged deposit rather than a claim about all settling suspensions.

Every element of a nonzero complex unital Banach algebra has a nonempty compact spectrum of an element. If the resolvent of an element existed everywhere, composing it with any bounded linear functional would give a bounded entire function tending to zero at infinity. The Liouville theorem and point separation by the Hahn-Banach theorem would force the resolvent to be zero, contradicting the inverse equation. Nonunital algebras are treated by unitization.

An approximate eigenvalue of a bounded operator $T$ is a scalar $\lambda$ admitting unit vectors $x_n$ with $\Vert (T-\lambda I)x_n\Vert \to 0$. Every approximate eigenvalue belongs to the spectrum; for a normal operator, all spectral points are approximate eigenvalues.

Linear spectral dispersion is the derivative of the photodetector coordinate in the dispersion direction with respect to wavelength. Near the optical axis, $q=f_{\mathrm{cam}}d\beta /d\lambda$.

A spectral resolution element is a spectral interval at the adopted instrumental resolution. At spectral resolving power $R$, its width is $\Delta \lambda =\lambda /R$. The number of photodetector detector pixels sampling it is distinct from $R$.

Two Riemannian manifolds are isospectral for a specified differential operator if their eigenvalues, counted with multiplicities and with the same boundary conditions, agree. Riemannian isometries preserve the Laplace-Beltrami operator spectrum, but the converse can fail. The transplantation theorem and the Sunada theorem give systematic constructions.

For the unit $S^d$ and the nonnegative Laplace-Beltrami operator, the eigenvalues are $\ell (\ell +d-1)$, with multiplicities $\left(\genfrac{}{}{0ex}{}{\ell +d}{d}\right)-\left(\genfrac{}{}{0ex}{}{\ell +d-2}{d}\right)$ for $\ell \ge 0$. The eigenfunctions are restrictions of degree-$\ell$ harmonic polynomials. The harmonic decomposition of homogeneous polynomials and the Stone-Weierstrass theorem prove completeness.

Filter a fixed local term using a nested sequence of neighbourhood Hamiltonians $H_s$, and call the results $G_s$. The differences $G_s-G_{s-1}$ give a shell decomposition. When $H_0=h_Z$ and $H_D=H$, normalization gives $G_0=h_Z$ and telescoping gives $g_Z=h_Z+{\sum }_{s=1}^D(G_s-G_{s-1})$. Individual shell terms need not commute with the full ground projector.

The condition $[g_Z,P_0]=0$ makes a unique global ground state an eigenvector of every filtered term, but its termwise eigenvalues can exceed the corresponding minima. For example $h_{12}=|1⟩⟨1{|}_1+|0⟩⟨0{|}_2$ and $h_{23}=|1⟩⟨1{|}_3+2|1⟩⟨1{|}_2$ commute with their sum. Their sum is uniquely minimized by $|000⟩$, yet $h_{12}$ is lower on $|010⟩$. Filtering leaves these commuting terms unchanged, so it cannot force frustration freeness.

If each local term is already minimized by the global ground state, subtract its minimum $a_Z$ to obtain a positive operator. Nonnegative normalized spectral filtering of Hamiltonian terms preserves positivity and leaves the ground eigenvalue $a_Z$ unchanged. Thus the same state minimizes every filtered term. This preserves frustration freeness but does not create it for an initially frustrated decomposition.

In a telescoping local-shell decomposition, split the spectral filter integral at a time proportional to the shell distance. A Lieb-Robinson bound controls short times, while the two-sided almost-exponential spectral filter tail controls long times. Polynomial shell-weight growth then gives the displayed decay. The estimate is asymptotic at large distance; small shells retain the elementary operator-norm bound. The evenization of a nonnegative bandlimited filter can supply the two-sided tail from a one-sided existence statement.