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At fixed position a two-integral distribution depends on $v_R,v_z$ only through $v_R^2+v_z^2$. Swapping these integration variables makes their second moments equal. Their odd moments and mixed second moments vanish by reflection symmetry. For an even function of $L_z$ the azimuthal velocity is also centered, so all mixed entries of the velocity ellipsoid vanish.

The singular axisymmetric logarithmic gravitational potential admits this stationary galactic distribution function withThe zero of specific orbital energy is the printed logarithmic convention. Products of Gaussian integrals normalize its two spatial density terms. For $1/2\le q^2\le 1$ the two terms are separately nonnegative; mildly prolate models require a bound on their sum.

A sufficiently regular four-partite three-uniform hypergraph, supported on sufficiently regular pair cells of positive density and with positive relative triple densities, has a positive fourth-power number of three-uniform tetrahedra. On complete pair supports, telescoping the four edge functions shows that an error of at most $\xi$ in each three-dimensional box norm gives a counting error at most $4\xi$. The relative version additionally requires control of the pair-support densities.

For a plane gravitational wave in linearized gravity with $k_{\mu }=(-\omega ,0,0,\omega )$, Lorenz gauge in linearized gravity gives $H_{3\nu }=-H_{0\nu }$. Under $\delta H_{\mu \nu }=i(k_{\mu }{X}_{\nu }+k_{\nu }{X}_{\mu }-{\eta }_{\mu \nu }{k}^{\rho }{X}_{\rho })$, choose $X_i=H_{0i}/(i\omega )$ for $i=1,2$, $X_3-X_0=i{H}_{00}/\omega$, and $X_0+X_3=(H_{11}+H_{22})/(2i\omega )$. This residual gauge symmetry of linearized gravity removes all time and longitudinal entries and the transverse trace. The remaining amplitudes $H_{+},H_{\times }$ are the two gravitational wave polarizations. The formula uses the plus-sign convention for $\delta h=2{\partial }_{(\mu }{\xi }_{\nu )}$; reversing the definition of $\xi$ reverses the gauge change.

Reflection across a cut splits Laplacian eigenfunctions into odd and even parts. The odd restriction has a Dirichlet boundary condition on the cut; the even restriction has a Neumann boundary condition there. The displayed matrix intertwines the two-tile swap with $\mathrm{diag}(-1,1)$. A connected uniform-Dirichlet rectangle of width $2a$ is therefore isospectral to the disjoint union of width-$a$ rectangles with all-Dirichlet and three-Dirichlet/one-Neumann conditions. This illustrates why changing boundary conditions can conceal connectedness in the spectrum.

A narrow dipolar flux-tube area $A(z)\propto z^3$ gives $A\rho v=\dot{M}$ for inward speed $v>0$. The isothermal magnetic Bernoulli integral in $\Phi =-GM/z$ then gives $(v-c_s^2/v)v\prime =3c_s^2/z-GM/z^2$. A smooth transonic branch crosses at $z_s=GM/(3c_s^2)$ with $v_s=c_s$, ${\rho }_s=e^{5/2}{\rho }_{\infty }$ and $\dot{M}={\rho }_s{c}_{s}A(z_s)$. The crossing must lie in the domain; subsonic branches need not attain it.

For positive $u$, integrate $t-f$ from zero to $u$; for negative $u$, reverse the integral from $u$ to zero. This proves the identity without a positivity restriction on the data. For $u,f\in L^2$, the absolute integral is bounded by $u^2/2+|fu|$, so Fubini's theorem applies. The zero-function baseline removes the divergence from the negative-height tail.

The scalar ROF denoising minimizer has perimeter-plus-forcing minimizing superlevel sets at almost every height. Conversely any finite-energy function with that property minimizes the full energy. The signed layer-cake identity for quadratic fidelity and coarea formula for BV functions prove sufficiency. To prove necessity, construct nested minimizing sets using comparison of perimeter minimizers with ordered forcing, reconstruct a function from rational levels, clip to establish energy bounds, and use uniqueness from strict convexity of the squared fidelity.

The squared-error term $\frac{1}{2}\Vert u-f{\Vert }_2^2$ is a strictly convex cost measuring agreement with data.

If $u$ minimizes scalar total variation denoising with $f\in BV\cap L^2$, set $w=T_{M}u$, $r=u-w$ and $g=f-r$. The scalar total variation splitting under clipping gives $\mathrm{TV}(u)=\mathrm{TV}(w)+\mathrm{TV}(r)$. Compare $u$ with $v+r$ and use $\mathrm{TV}(v+r)\le \mathrm{TV}(v)+\mathrm{TV}(r)$; cancelling the tail shows that $w$ minimizes the same model for $g$. It is bounded while $g$ may be unbounded, and $w-g=u-f$ exactly. This reduction concerns the actual minimizer, rather than convergence of minimizers for truncated data.

For scalar total variation denoising with $f\in BV(\Omega )\cap L^2(\Omega )$ on a bounded Lipschitz domain in arbitrary dimension, the inequality holds with common oriented BV traces on a hypersurface. It says that every surviving jump has the same direction as its data jump and no larger amplitude. Reversing the normal changes both differences and preserves the inequality.

Use residual-preserving clipping of an ROF minimizer and the jump-amplitude inequality for a bounded ROF minimizer. At each finite jump, choose an integer clipping level above both output traces. The clipped function has those same traces, and its residual-adjusted data have the same traces as $f$. Taking a countable union over the integer clipping levels proves the result. This gives the no-new-jumps property of total variation denoising under the full $L^2\cap BV$ assumptions, without asserting regularity of unbounded-forcing perimeter minimizers.

This finite-image convex optimization model combines robust L1 norm fidelity with discrete isotropic total variation and pixel-range constraints. An exact second-order cone program uses two orthant fidelity slacks per pixel, two box slacks per pixel and a three-dimensional Lorentz cone for each gradient magnitude. The product canonical barrier has parameter $6n$ for $n$ pixels.

For an oriented rank-$r$ vector bundle, let $U\in H^r(D(E),S(E);\mathbb{Z})$ be its Thom class. Forgetting relative supports sends $U$ to ${\pi }^{\ast }e(E)$, where $e(E)$ is the Euler class. Compatibility of relative and mixed cup products givesFor odd $r$, graded commutativity of the cup product makes $2U^2=0$. The Thom isomorphism theorem is injective, so $2e(E)=0$. This explains why an odd-rank Euler class can have only two-torsion.

For equal solid and liquid mass densities, $d\mu =-sdT+dp/\rho$ and $s_l-s_s=L/T_m$ imply the displayed relation from equality of chemical potentials. $T_m$ must be absolute temperature. The phase pressures differ; this should not be confused with differentiating a coexistence curve along a common pressure.

For group homomorphisms $f,g:A\to B$, this Picard group class is symmetric and additive in both arguments. The Theorem of the Cube says precisely that the third additive difference of $h\mapsto [h^{\ast }L]$ vanishes, proving additivity of the cross-effect. The Poincaré line bundle expresses it as $(f,{\varphi }_{L}g{)}^{\ast }{P}_B$, so it depends only on the Néron-Severi group class of $L$. Its associated homomorphism is $\hat{f}{\varphi }_{L}g+\hat{g}{\varphi }_{L}f$.

For fixed forbidden induced patterns, a regularity and Ramsey refinement uses internal clique/independent types and sparse, almost-complete, or intermediate cross pairs. A clique of intermediate pairs realizes every bounded induced pattern consistent with those internal types, using both edge and nonedge regularity. This is the induced embedding consequence used in hereditary graph enumeration theorem; the exceptional-pair budget can be arbitrarily small.

With depth transport $(-{\psi }_y,{\psi }_x)$ and a depth-mean bottom-drag closure, the wind-driven equation is $\beta {\psi }_x=S-r{\nabla }^2\psi +\nu {\nabla }^4\psi$, where $S$ is wind-stress curl divided by density and $r=\gamma /({\rho }_{0}H)$ is the drag rate. Its zonal boundary modes solve $\nu {\lambda }^3-r\lambda -\beta =0$. The Munk boundary layer and Stommel boundary layer are different limits of this combined equation.

For a canonical chiral superfield, an affine translation $\Phi =\Psi +s$ preserves the kinetic action up to a vanishing full-superspace term. It removes the linear superpotential term exactly when $s$ is a critical point. For $W=\alpha +\kappa \Phi +m{\Phi }^2/2+g{\Phi }^3/6$, the shifted quadratic coefficient is $M=m+gs$ with $M^2=m^2-2g\kappa$. A real shift requires a real root; the complex field permits a complex shift as well.

Adding a constant to a global superpotential does not change the component action because the chiral superspace integral of that constant vanishes. Neither the F-term scalar potential nor the fermion mass matrix depends on it. This statement does not extend to supergravity, where the superpotential itself enters the gravitational potential.

Choose distinct odd primes and take products of the nonisomorphic Gassmann equivalent regular subgroups, selecting one of the two groups at each prime. Product conjugacy-class counts give $2^n$ Gassmann equivalent subgroups of the same finite product of symmetric groups. Their fundamental groups are distinguished by whether the unique Sylow subgroup at each chosen prime is abelian. The closed-manifold realization of a finitely presented fundamental group and Sunada theorem therefore produce $2^n$ pairwise nonhomeomorphic, and hence nonisometric, isospectral closed four-manifolds. Gassmann equivalence alone, without a topology distinction, does not imply nonhomeomorphism.