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.

For a real triplet potential $\lambda (|\varphi {|}^2-v^2{)}^2/8$ and canonical gauge coupling $g_c$, unitary gauge gives one neutral Higgs mode, two oppositely charged massive vectors and one massless Abelian vector. The scalar kinetic contribution is $\frac{1}{2}(\partial \eta {)}^2+g_c^2(v+\eta {)}^2{W}^{+}{W}^{-}$, and its potential is $\lambda v^2{\eta }^2/2+\lambda v{\eta }^3/2+\lambda {\eta }^4/8$. This potential convention is distinct from using $\lambda (|\varphi {|}^2-v^2{)}^2/4$.

Before the difference $Z$ hits zero, its diffusion coefficient has magnitude at most $KZ$ and its drift is at least $-LZ$. Itô's formula bounds the drift of $f_{ϵ}(Z)$ by $(L+K^2)f_{ϵ}(Z)$. Its stochastic integrand is bounded by $K/(4ϵ)$, making expectation legitimate on finite horizons. The Gronwall inequality and the value $f_{ϵ}(0)=1/ϵ$ then exclude finite-time contact from a strictly positive starting difference.

When $A_S$ is injective, this vector solves $A_S^{\ast }{h}_0=\mathrm{sgn}(x_S)$. Any other solution differs by a vector in $\ker A_S^{\ast }$, orthogonal to the range of $A_S$ containing $h_0$. The Pythagorean identity therefore proves that $h_0$ has the smallest Euclidean norm. It is a strict dual certificate for basis pursuit exactly when each inactive coordinate of $A^{\ast }{h}_0$ has magnitude less than one. Failure of that test does not rule out another certificate: a null space component of $A_S^{\ast }$ can alter those inactive coordinates.

For a simple symmetric random walk, the infinite mean first passage of a simple symmetric random walk gives a finite almost sure first passage $T_1$ to $1$. The stopped martingale $M_n=1-S_{n\wedge T_1}$ is nonnegative and eventually zero almost surely. Nevertheless $\mathbb{E}{M}_n=1$ for every $n$, so it has almost sure convergence but no convergence in L1. It is therefore not uniformly integrable. Before absorption the nearest-neighbour walk is at most zero, ensuring nonnegativity.

For an upward-oriented band of the circular elliptic paraboloid between radii $r_0$ and $R$, the vector field $B=(-y^3,x^3,z^3)$ has vertical curl $3(x^2+y^2)$. Its surface integral is $3\pi (R^4-r_0^4)/2$. Stokes theorem gives the same value from the counterclockwise outer circle minus the counterclockwise inner circle. Equivalently the induced inner boundary direction is clockwise. Forgetting the inner boundary or its reversed orientation changes the answer.

In one base and one phase variable, take $\Phi =x\theta$ and $a=x\eta (\theta )/|\theta |$, with an even smooth cutoff function equal to zero near zero and one for large $|\theta |$. This is a symbol of order $-1$. Its oscillatory integral is $-2x\log |x|$ plus a smooth function near zero. Thus it is singular there even though the restricted function $a(0,\cdot )$ has empty support. The joint closed conic support of an oscillatory amplitude retains the limiting base point and its stationary directions. A support-sensitive regularity theorem must use that conic neighborhood information, not merely vanishing of an amplitude at one base point.