A Gaussian scale mixture is a random variable $X=RZ$, where $Z$ has a standard normal distribution and is independent of a positive random scale $R$. Its probability density function is the average of the conditional normal densities. Random scales can produce larger kurtosis than a single normal law.

For a minimally coupled scalar field with a Levi-Civita connection, metric compatibility and the symmetric Hessian of a scalar field give the displayed off-shell identity. It is a special case of the diffeomorphism Noether identity for a scalar field, and gives stress-energy conservation when the scalar equation is satisfied. No gravitational field equation is needed.

In $k[u,v]$, the ideal $I=(u^2,uv)$ has radical of an ideal $(u)$, a prime ideal, and hence a unique minimal prime over an ideal. Nevertheless $uv\in I$, $u\notin I$, and $v^j\notin I$ for all $j\ge 1$, so $I$ is not a primary ideal. The same example works after adjoining inverses of $1+u$ and $1+v$; thus it also gives an example in a two-variable Laurent polynomial ring.

Every proper ideal of a unital commutative ring is contained in a minimal prime over that ideal. A maximal ideal supplies a prime containing it, and an intersection of a decreasing chain of primes is still prime. Zorn's lemma, with reverse inclusion, therefore supplies a minimal member. No Noetherian assumption is needed.

The derivative of the local phase in $\Psi \mapsto e^{-iq\alpha }\Psi$ cancels the added gradient in the electromagnetic potential. Thus $D_{\mu }^{\prime }{\Psi }^{\prime }=e^{-iq\alpha }{D}_{\mu }\Psi$, and the charged Dirac equation maps solutions to solutions. The same cancellation makes its action gauge invariant.

Two coaxial opposite-handed helices linked by an internal motor share a translation speed but counterrotate. In an additive local-drag model, a length ratio $n$ multiplies the second helix's diagonal resistances by $n$ and reverses its coupling. With ${\Omega }_1-{\Omega }_2=\omega$, force and torque balance give $U=-2nBD\omega /[AD(1+n{)}^2-B^2(1-n{)}^2]$.

A subset of the radius-$R$ Euclidean ball admits an $\epsilon$-metric net of size at most $(1+2R/\epsilon {)}^n$. Take a maximal separated set: its disjoint radius-$\epsilon /2$ Euclidean balls lie in the radius-$(R+\epsilon /2)$ ball. Comparing Lebesgue measures gives the estimate. In particular, the unit sphere admits a net of radius one with at most $3^n$ points.

For a symmetric matrix $H$ and an $\eta$-metric net $U$ of the unit sphere, with $0<\eta <1/2$, approximate a maximizing unit vector $x$ by $u\in U$. Expanding $x^{\top }Hx-u^{\top }Hu=(x-u{)}^{\top }Hx+u^{\top }H(x-u)$ bounds its absolute value by $2\eta \Vert H{\Vert }_{\mathrm{op}}$. Rearrangement proves the bound. The volumetric bound for Euclidean metric nets limits the number of needed directions, allowing a union bound to control a random matrix.

Suppose the unit ball $B$ of a $k$-dimensional normed vector space has metric covering number $N(r,B)\le (A/r{)}^k$ at $r=\delta /2$, where $A,\delta >0$. Cover $B$ by at most $(2A/\delta {)}^k$ balls of radius $\delta /2$. Discard every ball missing the unit sphere, and select one point of the unit sphere from each remaining ball. These selected points form a radius-$\delta$ metric net $N_{\delta }$ of the unit sphere, withIndeed, a unit sphere point and the selected point from the same covering ball each lie within $\delta /2$ of its center, so the triangle inequality bounds their distance by $\delta$. The selection ensures that all centers of the new metric net have unit norm; the original covering centers need not. For Euclidean balls, the volumetric bound for Euclidean metric nets supplies the initial metric covering number estimate.

An $\epsilon$-net of a metric space is a set $G$ such that every point lies within distance $\epsilon$ of a point of $G$. A Cartesian grid of coordinate spacing $\eta$ is a finite $\sqrt{d}\eta$-net of a bounded cube in the Euclidean norm. A Lipschitz bound transfers control on the grid to control over the whole cube.

For a two-mode tensorial squirmer flow of radius $a$ and a passive equal sphere held at $X=Re$, with $a/R\ll 1$, let $s=e\cdot Be$. Faxén's first law gives holding force $F_h=-9\pi \mu a^{3}se/R^2+O(\mu a^4|A|/R^3+\mu a^5\Vert B\Vert /R^4)$. Its reflected Stokeslet changes swimming velocity by $-9a^{3}se/(4R^3)$. Since this leading force is radial, its vorticity at the swimmer vanishes. The next holding force, generated by the potential dipole, yields rotation $a^{4}A\times X/(4R^6)$. Terms from the passive sphere's finite radius, reflected stresslet and rotlet are needed at higher order. These formulas assume fixed slip and no externally applied swimmer force or torque.

For equal spheres of radius $a$ separated by $R\gg a$, one held fixed and the other forced with $\mathbf{F}=6\pi \mu a\mathbf{V}$, put $\zeta =6\pi \mu a$, $\mathbf{n}=\mathbf{X}/R$ and $\mathsf{A}=3a(I+\mathbf{nn})/(4R)$. The method of reflections for Stokes flow gives $\mathbf{U}=\mathbf{V}-{\mathsf{A}}^2\mathbf{V}+O(V{a}^4/R^4)$, since the fixed sphere creates the reflected Stokeslet with force $-6\pi \mu a\mathsf{A}\mathbf{V}$. The leading hydrodynamic mobility matrix correction therefore reduces longitudinal motion more strongly than transverse motion.

A first-order structure $A$ embeds into a model of $T$ exactly when $A\models T_{\forall }$. For the nontrivial direction apply the compactness theorem to $T\cup \mathrm{Diag}(A)$. If a finite diagram conjunction were inconsistent with $T$, its universal negation after replacing the new constants by variables would belong to $T_{\forall }$ and contradict its truth in $A$.

In a flat universe containing pressureless matter and smooth radiation in cosmology, put $y=a/a_{\mathrm{eq}}$ and $C=H_0^2{\Omega }_{m,0}^2/{\Omega }_{r,0}$. The Friedmann equation gives ${\mathcal{H}}^2=C(1+y)/y^2$, so $y^{\prime 2}=C(1+y)$ and $y^{\prime \prime }=C/2$. The subhorizon cosmological Poisson equation gives ${\delta }^{\prime \prime }+\mathcal{H}{\delta }^{\prime }=3C\delta /(2y)$. The chain rule produces the displayed equation. Smoothing radiation neglects its rapid acoustic forcing after horizon entry; it is not an exact treatment of all radiation perturbations.

The midplane galaxy rotation curve fixes the radial derivative of the Newtonian gravitational potential, not every exterior boundary condition. Adding $a|z|$ leaves that radial derivative unchanged but adds a uniform sheet of surface density of a disk $a/(2\pi G)$. The pure scale-free Mestel disk excludes such extra sources.

For an initial pion-nucleon state and outgoing meson octet-baryon octet pair, conservation of baryon number and strangeness allows $\pi N$, ${\eta }_{8}N$, $K\Lambda$, and $K\Sigma$, subject to electric charge and production thresholds. Total isospin is $1/2$ or $3/2$; the ${\eta }_{8}N$ and $K\Lambda$ channels contain only $1/2$.

The progression version of the Mertens second theorem gives ${\sum }_{\begin{array}{c}p\le y\\ p\equiv 1(\mathrm{mod}4)\end{array}}1/p=\frac{1}{2}\log \log y+c+O(1/\log y)$. Subtracting from the all-prime sum shows that the forbidden primes $p\equiv 3(\mathrm{mod}4)$ contribute half the logarithmic density. This is the source of the square root in a half-dimensional interval sieve.

For a torsion-free effective modular group with genuine, untwisted modular cusp periods, a meromorphic weight-$k$ form has integer local orders. The tensor differential $f^{12}(d\tau {)}^{6k}$ has interior order $12\mathrm{ord}f$ and modular cusp order $12\mathrm{ord}f-6k$. Its canonical degree and the genus formula for a modular curve yield the displayed valence formula, with $d$ the projective index. Regular modular cusp periods are important in odd weights.

For a torsion-free modular curve with regular modular cusps, let $f$ be a nonzero meromorphic weight-$k$ form and $C$ the reduced sum of modular cusp points. The local-order divisor of $f$, minus $C$, imposes exactly holomorphy in the interior and vanishing at every modular cusp on the product $f\phi$. Dividing any cusp form by $f$ gives the reverse identification with a Riemann-Roch space. The regular-cusp valence formula on a torsion-free modular curve and Riemann-Roch theorem compute the dimension when the resulting divisor has degree greater than the canonical degree.

For a merged pair held at the original velocity and reduced gravity, pure plume balance requires the original individual radius, not the larger area-preserving radius. The required pure area is half the combined area. Shrinking to that area at unchanged velocity and reduced gravity would halve volume, momentum and buoyancy fluxes, contradicting conservation of the merged fluxes. A conservative merger must change its properties or begin away from pure balance.