Repeated integration by parts gives the Gamma integralThe logarithm of its integrand is $n\log t-t$, whose unique maximum is at $t=n$. Write $t=n+\sqrt{n}s$. For fixed $s$ and large $n$, a Taylor expansion givesThe Big O notation here is for a fixed $s$, or uniformly for $s$ in a fixed bounded interval. It does not claim a uniform approximation over the entire growing integration domain. Consequently the peak, divided by its height, approaches $e^{-s^2/2}$.

The exact substitution givesReplacing the integral by the Gaussian integral ${\int }_{\mathbb{R}}{e}^{-s^2/2}ds=\sqrt{2\pi }$ produces $S_n$. Thus the estimate is peak height $n^n{e}^{-n}$ times width $\sqrt{n}$ times the area of a standard Gaussian peak. This is Laplace's method in a concrete form; controlling the integral's tails justifies the replacement. The next terms, including their products after exponentiation, produce the Stirling correction by Gaussian moments.

For every positive integer $n$, Herbert Robbins's bounds giveEquivalently, the remainder in $\log (n!)=\log S_n+r_n$ satisfies $1/(12n+1)<r_n<1/(12n)$. This both proves that $S_n$ underestimates $n!$ and squeezes $n!/S_n$ to one, establishing Stirling's formula. Robbins published the bounds in “A Remark on Stirling's Formula” in 1955. ^[ref]

For $n=10$, the relative error of $S_{10}$ lies strictly between $1-e^{-1/121}$ and $1-e^{-1/120}$, approximately $0.823\%$ and $0.830\%$. A bound specifies the accuracy of an approximation before using it in another calculation.

An immersion of a smooth $d$-manifold in ${\mathbb{R}}^{d+1}$ supplies a rank-one normal bundle $\nu$ with $TM\oplus \nu \cong {\epsilon }^{d+1}$. The Whitney product formula for Stiefel–Whitney classes therefore requires $w(\nu )=w(TM{)}^{-1}$. A real line bundle has no positive-degree classes beyond $w_1$, so every higher-degree part of the inverse total class must vanish. For ${\mathbb{RP}}^2\times {\mathbb{RP}}^2$, the stabilized tangent bundle of real projective space gives $w(TM{)}^{-1}=(1+a)(1+b)$ in ${\mathbb{F}}_2[a,b]/(a^3,b^3)$. Its nonzero degree-two part $ab$ excludes an immersion in ${\mathbb{R}}^5$ even though the normal line need not be orientable.

For a real line bundle on a closed manifold, a transverse section has a hypersurface as its zero set. Along a closed loop transverse to this hypersurface, trivialize the pullback over the cut interval. Each simple zero reverses the sign of the component function. The total sign change is exactly the bundle's transition sign around the loop, detected by its first Stiefel–Whitney class. Thus zero-set intersection parity with every loop equals the evaluation of $w_1(L)$, proving the displayed mod-two dual class. No orientability of the manifold or bundle is required.

For north-pole stereographic projection $p=(X+iY)/(1-Z)$ on the unit sphere, the antipodal pair point has coordinate $q=-(X+iY)/(1+Z)$. Since $X^2+Y^2=1-Z^2$, this gives $p\overline{q}=-1$ when both coordinates are finite. On the whole Riemann sphere, write $q=-1/\overline{p}$, pairing zero with infinity. The two axis endpoints fixed by a sphere rotation obey this relation.

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.

Let $K$ be a Markov kernel with Stationary distribution $\pi$ on one coordinate. On $m$ coordinates, update coordinate $j$ with kernel $K$ while leaving the others unchanged. The product distribution ${\pi }^{\otimes m}$ is invariant: its updated coordinate still has law $\pi$, and independent transition randomness leaves it independent of the untouched coordinates. Choosing the updated coordinate independently of the current configuration, with any fixed selection probabilities, forms a mixture of these invariant kernels and preserves the same product law.

For a two-state kernel with $K(1,1)=a$ and $K(0,1)=b$, the stationary Probability of state one solves $p=pa+(1-p)b$. If $1-a+b>0$, thenUnder the product stationary distribution, the occupied-coordinate count has binomial distribution with parameters $m,p$, mean $mp$ and variance $mp(1-p)$. In the exceptional case $a=1,b=0$, the kernel is the identity and every $p\in [0,1]$ is stationary; the displayed fraction is undefined and must not be used.

State-dependent coordinate selection can invalidate the conclusion. On two bits, let each single-coordinate kernel refresh to a fair bit, but select coordinate one whenever its current value is zero and coordinate two otherwise. Starting with independent fair bits, the Probability that the first bit equals one after a step is $1/2+(1/2)(1/2)=3/4$, so the uniform product law is not invariant. Invariance of each coordinate-update kernel alone is insufficient when the mixture weights depend on the state.

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 $N$.