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For $dX=\mu dt+\sigma dB$ on $[0,T]$, take $\theta =(\nu -\mu )/\sigma$ and use density $\mathcal{E}(\int \theta dB{)}_T$. If $\mu ,\nu$ are bounded and $\sigma$ is continuous and bounded below by a positive constant, the exponential has bounded bracket and defines an equivalent measure. Under it, $\tilde{B}=B-\int \theta dt$ is Brownian and $dX=\nu dt+\sigma d\tilde{B}$. Set $\theta =0$ after $T$ to define the density on the entire original sigma-algebra and the new Brownian motion on the whole time axis.

A family of definable arrows covers a formula when the theory proves that their images jointly exhaust that formula. Such covers encode disjunction and existential quantification. The resulting site presents the classifying topos of the geometric theory. Adding geometric axioms adds covering sieves, yielding the duality between geometric quotients and subtoposes.

The metric defines this globally smooth Hamiltonian on the cotangent bundle. With $\omega =d{x}^i\wedge d{p}_i$ and ${\iota }_{X_H}\omega =dH$, its Hamiltonian vector field projects to affinely parametrized geodesics. The momentum is $p_i=g_{ij}{\dot{x}}^j$, and its conserved value is half the squared speed.

Suppose these sets exhaust a manifold, and its Riemannian metric is Euclidean beyond some radius in the end coordinates. A finite-time geodesic has fixed speed and cannot increase its exterior radius faster than that speed. It remains in one compact core enlargement, with velocity in a compact disk bundle. The geodesic equation therefore continues beyond every finite parameter endpoint. The compactness hypothesis is substantive: the open exterior of a closed Euclidean ball, with empty core, is incomplete at its inner boundary despite having a Euclidean metric at infinity.

Pointwise domination of Riemannian metrics gives domination of their distances. A Cauchy sequence for the larger distance is Cauchy for the complete smaller one. Smooth positive metrics induce the same manifold topology, so its limit is also a limit in the larger distance. The Hopf-Rinow theorem turns metric completeness into geodesic completeness. Mere positivity of the second metric, without domination, is insufficient.

For endomorphisms of a finite-dimensional complex vector space, suppose repeated commutation with $T$ annihilates $S$. Write the blocks of $S$ using the generalized eigenspaces of $T$. On $\mathrm{Hom}(V_{\lambda },V_{\mu })$, the operator $\mathrm{ad}T$ is $(\mu -\lambda )I$ plus a nilpotent endomorphism. If $\lambda \ne \mu$, it is invertible, so $(\mathrm{ad}T{)}^{N}S=0$ forces that off-diagonal block of $S$ to vanish. Thus $S$ preserves each generalized eigenspace. This allows a nilpotent Lie algebra representation to be split by generalized eigenvalues even when its acting operators do not commute.

A subject-specific random intercept and random slope may have a joint bivariate normal distribution with arbitrary positive-definite covariance matrix $D$. With independent normal measurement errors of variance ${\sigma }^2$, the marginal covariance between a person's readings at $s,t$ is $D_{00}+(s+t)D_{01}+st{D}_{11}$, plus ${\sigma }^2$ on the diagonal. Different subjects remain independent. Allowing $D_{01}$ distinguishes this model from an independent random-intercept and random-slope model.

The two-stage Gauss--Legendre Runge-Kutta method uses nodes $c_{1,2}=1/2\mp \sqrt{3}/6$, weights $b_1=b_2=1/2$ and coefficients $a_{11}=a_{22}=1/4$, $a_{12}=1/4-\sqrt{3}/6$, $a_{21}=1/4+\sqrt{3}/6$. The Butcher order conditions give order four. Its stability function is the displayed rational function, which is A-stable. The algebraic stability matrix is zero, so positive weights also give nonlinear contractivity for dissipative vector fields.

For the convention $s\bar{c}=-b$, the action $S_{\mathrm{YM}}-s\Psi$ contains $-b\partial \cdot A+\xi b^2/2-\bar{c}\partial \cdot Dc$ after integration by parts. Integrating the Nakanishi-Lautrup field gives $-(\partial \cdot A{)}^2/(2\xi )$. The Grassmann Gaussian integral represents the Faddeev-Popov determinant.

With a mostly-plus metric signature, $e^{iS}$, and gauge-fixing term $-(\partial \cdot A{)}^2/2$, the free adjoint gauge-field kernel is $-p^2{\eta }_{\mu \nu }{\delta }_{ab}$. Inverting it gives the displayed Feynman propagator. This convention uses the same denominator and sign as a massless real scalar propagator.

The neutral pion has positive charge conjugation eigenvalue and its spin-one isospin half-turn has eigenvalue minus one. Hence the G parity eigenvalue is minus one. Since G parity commutes with isospin, both charged members have the same value. Charged-state phase changes alter individual charge-conjugation and rotation formulas together, leaving this eigenvalue unchanged.

A futures contract delivery quote above the current spot price for the maturity considered; a rising quoted curve across maturities is also described as contango. This compares delivery prices rather than the current zero value of a newly settled contract.

A futures contract delivery quote below current spot for the maturity considered, or a falling futures curve across maturities. This spot comparison should be distinguished from comparisons with the future physical-measure expected spot.

Convergence in the displayed sense combines strong function convergence with weak-star convergence of derivative vector measures against $C_0$ tests. Strong $L^1$ convergence and a uniform variation bound imply the derivative convergence by integration by parts and uniform test-function approximation. Conversely the Uniform boundedness principle bounds the derivative measures of a convergent sequence.

For a bounded Lipschitz domain, functions can be extended with the displayed bound and common compact support. Reflect across local Lipschitz boundary graphs, combine the reflected functions with a partition of unity, and multiply by a fixed cutoff function. Lipschitz coordinate changes control derivative measures; the product rule bounds cutoff function terms by the $L^1$ norm. Zero extension is also possible with an estimate for BV traces on a hypersurface, but its boundary jump must be included in the variation.

For a smooth function, integrate its directional derivative along each segment of length $|h|$ and then integrate in space. This gives the displayed bound with $\int |\nabla w|$. Smooth BV approximation and sequential lower semicontinuity give the same inequality for general functions in the BV space. Averaging it over a mollifier proves the BV mollification error estimate.

The functional $\phi \mapsto -\int u\mathrm{div}\phi$ is bounded in the uniform norm exactly when $u$ belongs to the BV space. Extend it from compactly supported smooth tests to $C_0$, then apply the Riesz-Markov-Kakutani representation theorem to obtain the unique finite vector Radon measure $Du$. Its dual norm is the defining variation supremum. This representation should be distinguished from the deeper rectifiability results used in the decomposition of a BV derivative.

For a scalar function in the BV space, derivative variation is the integral of the relative perimeter measures of its superlevel sets. The formula holds on every Borel subset $A$ of the domain, not just for total mass. It converts the continuous ROF denoising problem into linked minimization problems for finite-perimeter sets.

An approximate mean limit of a function is a value satisfying the displayed condition. Functions in the BV space have such finite limits outside an approximate discontinuity set up to negligible sets in the appropriate measure. Their approximate jump points instead have two distinct half-ball limits.

Both functions represent the same $L^1$ equivalence class. Fubini's theorem shows that the interval-mass function has derivative $Du$, so its difference from $u$ is constant. Continuity of a finite measure gives left continuity for $u_l$ and right continuity for $u_r$. Their opposite one-sided limits differ by the measure atom $Du(\{t\})$. There are only countably many nonzero measure atoms, since every set of measure atoms above a fixed magnitude threshold is finite.