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For north-pole stereographic projection $z=(X+iY)/(1-Z)$ on the unit sphere, substitute $|z{|}^2=(1+Z)/(1-Z)$ into a planar circle equation. The displayed plane equation follows, and its intersection with the sphere is a circle. The plane contains the sphere's centre exactly when $r^2=1+|c{|}^2$, giving a great circle.

For two circular Kepler orbits with mutual inclination $I_m$, independently average the stellar quadrupole disturbing function over both mean anomalies. A circular ring with unit normal $\mathbf{k}$ has $⟨{\hat{r}}_i{\hat{r}}_j⟩=({\delta }_{ij}-k_i{k}_j)/2$. Contracting the two ring tensors gives $⟨{\cos }^2\psi ⟩=[1+(\mathbf{k}\cdot {\mathbf{k}}_p{)}^2]/4$, proving the displayed exact angular factor. For small orbital inclinations relative to a common plane, $I_m^2=I^2+I_p^2-2I{I}_p\cos ({\Omega }_p-\Omega )+O(I^4)$. Thus the secular angular factor is $1/4-3(I^2+I_p^2)/8+3I{I}_p\cos ({\Omega }_p-\Omega )/4$. This potential generates the slow stellar contribution to satellite nodal precession.

Choose a smooth radial bump $f$ supported inside a shell where the stellar buoyancy frequency has $N^2<0$. The displayed fluid displacement has $\nabla \cdot \mathbf{\xi }=(\rho g/\gamma p){\xi }_r$, hence its Eulerian pressure perturbation vanishes identically. Its buoyancy contribution to the stellar displacement energy identity is strictly negative and self-gravity is nonpositive. The variational principle therefore proves convective instability with no unsupported assumption that incompressibility alone removes pressure work. Large spherical-harmonic degree also makes perturbed self-gravity small.

For $W\sim N_m(\eta ,I)$, expand the squared estimation error and apply Gaussian integration by parts to each cross term. When $g$ satisfies the integrability and boundary conditions needed for that identity, this gives the displayed formula. For $g(w)=-aw/\Vert w{\Vert }^2$ and $m>2$, the origin singularity is integrable and the inner boundary term is $O({\epsilon }^{m-2})$, so the same identity applies by excision. Its divergence is $-a(m-2)/\Vert w{\Vert }^2$, giving risk $m+[a^2-2a(m-2)]\mathbb{E}\Vert W{\Vert }^{-2}$.

Under the hypotheses of steady planar ideal-MHD field-line invariants, the specific enthalpy relation and the velocity projection of momentum give $\mathbf{B}\cdot \nabla Q=0$. The horizontal total energy flux, including the Poynting vector, is $k{\mathbf{B}}_{h}Q$. The magnetic correction accounts for work exchanged between the fluid and the field.

The two axial invariants in steady planar ideal-MHD field-line invariants give the displayed identity. When $V-U\ne 0$, a finite smooth flow cannot reach horizontal Alfvén speed. When $V=U$, an everywhere Alfvénic branch is possible even with $B_z\ne 0$: constant $\rho$ and $\mathbf{u}=\mathbf{B}/\sqrt{{\mu }_0\rho }$ give an example. The nondegenerate exclusion must not be applied to this branch.

For a sinusoidal small-amplitude bed, constant current $U>0$ and buoyancy frequency $N$, put $m=\sqrt{N^2/U^2-k^2}>0$. The linear vertical velocity amplitude is $W(z)=ikU{\eta }_0\sin [m(H-z)]/\sin (mH)$. The upward-wave coefficient is $A_{+}=-kU{\eta }_0{e}^{-imH}/[2\sin (mH)]$. At $mH=n\pi$, the prescribed nonzero bed motion and homogeneous rigid-lid condition are incompatible with a bounded stationary inviscid solution. With angle $\theta$ between intrinsic group velocity and vertical, $\tan \theta =m/k$, giving $kH\tan \theta =n\pi$.

Every stationary subset of a successor cardinal ${\lambda }^{+}$ can be partitioned into ${\lambda }^{+}$ disjoint stationary subsets. For each $\beta \in [\lambda ,{\lambda }^{+})$ choose a bijection $e_{\beta }:\lambda \to \beta$. For each $\xi <{\lambda }^{+}$, the function $\beta \mapsto e_{\beta }^{-1}(\xi )$ is regressive on the stationary tail with $\beta >\xi$. The Fodor lemma gives a stationary fiber with one value $i_{\xi }<\lambda$. Some value occurs for ${\lambda }^{+}$ many indices, whose fibers are pairwise disjoint. Assign any leftover part of $S$ to one fiber.

Suppose each coordinate has stationary distribution $\pi$ for the same Markov kernel. Updating one coordinate using independent new randomness while leaving the others fixed preserves the product distribution ${\pi }^{\otimes m}$. For a two-state update with $P(1\to 1)=a$ and $P(0\to 1)=b$, its stationary occupied probability is $p=b/(1-a+b)$. Independent occupied-state indicators then have total mean $mp$ and variance $mp(1-p)$. The conclusion requires the update selection and transition randomness not to introduce dependence on the other coordinates' states.

For $N$ equal scattering sites on one polymer, define $g(\mathbf{k})=N^{-1}{\sum }_{m,n}⟨e^{i\mathbf{k}\cdot ({\mathbf{R}}_n-{\mathbf{R}}_m)}⟩$. It has $g(0)=N$; its normalized form factor is $g/N$. Under a rotationally invariant joint distribution, each separation vector has a uniform direction at fixed magnitude $r$. The angular average is $\frac{1}{2}{\int }_{-1}^1{e}^{ikr\mu }d\mu =\sin (kr)/(kr)$, with value one at $r=0$. Thus $g$ depends only on $k=|\mathbf{k}|$, and $g(k)=N^{-1}{\sum }_{m,n}⟨\sin (k|{\mathbf{R}}_n-{\mathbf{R}}_m|)/(k|{\mathbf{R}}_n-{\mathbf{R}}_m|)⟩$. The diagonal self terms are essential in a finite chain and give the large-$k$ limit one for a discrete Gaussian chain.

A university laboratory places an aluminium block on a Teflon-lined wooden board. For its inclined plane stage, the pulley and block string are removed; a rope and hook connect the board to a crank. The block is placed gently near the sheet's middle without prior sliding. Smooth cranking raises the board; a $0^{\circ }$–$45^{\circ }$ protractor reads the smallest angle at which sliding begins. Four readings are taken without added load, then four each with $200$, $400$, and $800\mathrm{g}$ added symmetrically at two screws. Removing the string isolates gravity-driven onset; gentle placement and smooth raising avoid premature disturbance.^[ref]

Infer ${\mu }_s=\tan {\theta }_c$ under the assumptions of static friction on an inclined plane. Repetition tests onset scatter; added loads test coefficient constancy. No measured onset values are supplied here.

For a replication, additionally check the protractor zero with a level board, retain the same contact face and surface region, and observe whether tipping or rolling precedes sliding. These are proposed additional controls. A finite angle resolution and the disturbance needed to detect first motion limit the estimate; a slow ramp does not by itself establish perfectly rigid, dry, or direction-independent contact. At the model's ${\mu }_s=0.40$ threshold, differentiating the tangent function gives $d{\mu }_s/d\theta =1+{\mu }_s^2=1.16$, with $\theta$ in radians. A hypothetical $0.5^{\circ }$ measurement error would therefore correspond to approximately $0.0101$ in the inferred coefficient at first order. This sensitivity calculation is not an uncertainty reported by the laboratory.

At an interior consumption and share-holding optimum, the Lagrange multiplier on the discrete dividend budget equation is ${\Lambda }_t={\beta }^t{U}^{\prime }(C_t)$. Varying the next holding by a variable measurable at date $t$ gives the displayed Euler equation. Thus the positive normalized process ${\zeta }_t={\Lambda }_t/{\Lambda }_0$ is a state-price density for traded dividend-inclusive gains. In an incomplete market, different investors can have different such densities; equality of their marginal-utility densities is an additional risk-sharing condition, not a consequence of trading one asset alone.

An open cluster is a relatively low-mass group of stars formed together, typically in a galactic disk. Two-body relaxation, stellar evaporation, and external tidal disturbances disperse many open clusters on much shorter times than the lifetimes of their low-mass member stars.

The thermodynamic activity is the dimensionless quantity defined by ${\mu }_i={\mu }_i^{\circ }+RT\ln a_i$, where $R$ is the molar gas constant. For an ideal dilute solution with a specified concentration reference, $a_i=c_i/c^{\circ }$; departures from ideality are represented by an activity coefficient multiplying this ratio.

The derived costalk at $x$ is $i_x^{!}E$, where $i_x:\{x\}\hookrightarrow X$. It computes local cohomology supported at $x$. On an oriented $N$-manifold the constant sheaf has costalk $\mathbb{Q}[-N]$, whose cohomology is in degree $N$.

A symmetric stable distribution has this characteristic function, with $c>0$ and location $\mu$. At location zero it is a strictly stable distribution: weighted sums of independent copies have scale factor $({\sum }_j|a_j{|}^{\alpha }{)}^{1/\alpha }$. The index $\alpha$ need not permit a finite mean or variance. The scale in this convention is $c^{1/\alpha }$, so the coefficient of $|t{|}^{\alpha }$ and the scale parameter should not be confused.

Equating an inertial eddy turnover time to the inverse buoyancy frequency gives the scale separating buoyancy-affected motions from a smaller approximately isotropic cascade. It is distinct from the viscous Kolmogorov microscales. In very stable local surface-layer scaling it is proportional to the Obukhov length, up to closure constants.

For cellular Omega-spectra $D,E$, natural degree-preserving additive operations $D^{\ast }\to E^{\ast }$ commuting with suspension are compatible families $[D_j,E_j{]}_{\ast }$ by the Yoneda lemma. Since $D\simeq {\mathrm{hocolim}}_j{\Sigma }^{-j}{\Sigma }^{\infty }{D}_j$, the Milnor exact sequence for maps of spectra realizes every such family. Its kernel consists precisely of hyperphantom maps of spectra. Thus operations determine stable maps only modulo that kernel.

The weak stable bundle adjoins the flow direction to the strong stable bundle. It is tangent to the weak stable foliation. For a geodesic flow on a hyperbolic surface, the canonical frame gives $E^{ws}=\mathrm{span}\{X,H-V\}$.

For real base-flow velocity and positive Reynolds number, multiply the homogeneous Squire equation by the conjugate mode and integrate with zero endpoint values. Taking real parts gives the displayed negative temporal growth rate. This proves decay of homogeneous Squire modes; the coupled shear-flow system can still be unstable or show transient growth because wall-normal velocity can force the Squire variable.