If a local martingale has an integrable running absolute supremum on each fixed finite horizon, localization and conditional dominated convergence make it a true martingale. This upgrades polynomial Itô local martingales when maximal moments and bracket moments control all terms.

The polynomial $X^4-6X^2⟨X⟩+3⟨X{⟩}^2$ has zero Itô drift. Finite fourth maximal and second bracket moments make it a true martingale. If $\mathbb{E}{X}_t^2=t$ and $\mathrm{Cov}(X_t^2,⟨X{⟩}_t)=0$, then $\mathbb{E}{X}_t^4=3t^2-3\mathrm{Var}(⟨X{⟩}_t)$. Equality at every time forces a deterministic clock and hence Brownian motion by the Lévy characterization of Brownian motion.

A continuous local martingale converges to a finite limit on the event that its terminal bracket is finite. Stopping at each integer bracket level gives an L2-bounded martingale and its almost sure limit. On the finite-bracket event, some such stopped process coincides with the original process forever. This supplies terminal values for a finite DDS lifetime.

For $f(E,L)=f_E(E)/L$, with $L=rv\sin \eta$ and $E=\psi -v^2/2$, velocity integration gives $r\rho =2{\pi }^2{\int }_0^{\psi }{f}_E(E)dE$. Thus $f_E(E)=(2{\pi }^2{)}^{-1}d(r\rho )/d\psi$ at $\psi =E$. The factor $1/L$ is locally integrable under the velocity-volume measure. Its angular measure is uniform in $\eta$; averages of ${\cos }^2\eta$ and ${\sin }^2\eta$ are both $1/2$, so the total tangential second moment equals the radial second moment and the velocity-anisotropy parameter is $\beta =1/2$. At the largest allowed radius $\psi (r_E)=E$, the derivative becomes $[\rho +r(d\rho /dr)]/(d\psi /dr)$, not $[\rho +d\rho /d\psi ]/(d\psi /dr)$. Positivity requires $d(r\rho )/d\psi \ge 0$.

In a spherical galactic distribution function with $F=L^{-1}{f}_E(E)$, write $L=r{v}_T$ and $d^{3}v=d{v}_r{v}_{T}d{v}_{T}d\psi$. The velocity Jacobian cancels the apparent singularity at $v_T=0$. Integration of Gaussian energy components then reduces to one full-line and one half-line Gaussian integral. The angular average makes each tangential velocity dispersion half the radial one, so the velocity-anisotropy parameter is $1/2$.

Putting $\lambda =l^2\nu$ into the sideband cubic and retaining $l^4$ givesFor $r=1-q^2>0$, $\mu /\sigma >(1-3q^2)/(1-q^2)$ makes the constant coefficient negative. There is a positive real $\nu$, so sufficiently small nonzero $l$ grows. On a finite period $L$, however, only $l=2\pi j/L$ are allowed. For $q=0$, $\mu =2$, $\sigma =1$, $L=1$, every nonzero allowed mode has negative growth rates despite this long-wave inequality. Thus an arbitrarily-small-wave-number conclusion cannot be asserted without the domain-size qualification.

A potential for a conservative vector field $X$ is a function with scalar values $\Phi$ whose gradient is $X$. Potentials differ by a constant on each connected component. The fundamental theorem for line integrals gives ${\int }_{\gamma }X\cdot d\mathbf{x}=\Phi (\gamma (1))-\Phi (\gamma (0))$. In mechanics a conservative force is conventionally written $-\nabla V$, so its potential energy is $V=-\Phi$. This spatial potential differs from the scalar potential $V(\varphi )$ that denotes the derivative-free energy density of a scalar field.

If a periodic vector field has a periodic potential of a conservative vector field, the line integral between points separated by a period vector must vanish, since it is the potential difference. A periodic gradient field need not satisfy this: the constant field $(1,2)$ on the plane has potential $x+2y$, but its integrals over the two unit translations are $1$ and $2$. On the periodic quotient these translation integrals are nonzero periods, obstructing a single-valued potential there.

Negligible spin and a circular orbit give $J=M_1{M}_2\sqrt{Ga/M}$. Fixed $M$ and $J$ imply $a(M_1{M}_2{)}^2$ is constant. Kepler's third law then gives the period-product invariant $P(M_1{M}_2{)}^3$.

A complex bundle connection induces a connection on its conjugate vector bundle by the displayed rule on real tangent fields. The conjugate scalar action gives the correct Leibniz rule. In conjugate frames the matrix is the complex conjugate of the original connection matrix, with evaluation of one-forms on real tangent vectors understood.

On the square $[-1,1{]}^2$, write normal traces in ${\varphi }_n(s)=\sin [n\pi (s+1)/2]$. Set $p_n=n\pi /2$, ${\omega }_n=\sqrt{k^2+p_n^2}$, $r_n=({\omega }_n+p_n)/k$. Paired spectral samples $\{r_n,r_n^{-1}\}$, $\{-r_n,-r_n^{-1}\}$, $\{i{r}_n,i/r_n\}$, $\{-i{r}_n,-i/r_n\}$ give adjoint tests $e^{\mp {\omega }_{n}y}{\varphi }_n(x)$ and $e^{\mp {\omega }_{n}x}{\varphi }_n(y)$. Each test vanishes on the adjacent sides, removing their unknown normal derivatives.

For an even permutation $g$, its symmetric group class splits into two equal alternating group classes exactly when its symmetric-group centralizer contains no odd permutation. If an odd commuting element exists, an odd conjugator can be multiplied by it to become even; otherwise the index-two subgroup has two conjugation orbits. In cycle notation the splitting types are precisely distinct odd cycle lengths, counting fixed points as length-one cycles: rotating odd cycles is even, whereas an even cycle or the exchange of two equal odd cycles supplies an odd centralizer element.

For a conformal bijection $\varphi$ and planar Brownian motion $B$, this increasing clock measures the common quadratic variation of the two coordinates of $\varphi (B)$. The transformed Brownian path is $\varphi (B_{A^{-1}(t)})$. The inverse clock runs from image time to original time.

The map $z\mapsto z^{\pi /\alpha }$ takes a wedge of opening $\alpha$ to a half-plane and the circular boundary of radius $r$ to radius $r^{\pi /\alpha }$. Conformal invariance of planar Brownian motion preserves which boundary part is reached first, although it changes the clock. Use the branch defined by the wedge argument; the origin is a polar point for planar Brownian motion, so localization handles the vertex.

For a nondegenerate multivariate normal distribution, the conditional independence graph is read from zeros in its precision matrix. Conditional independence of two components given all others is equivalent to a zero cross term in their conditional Gaussian quadratic density, hence to ${\Omega }_{jk}=0$. It does not generally correspond to a zero in the covariance matrix.

Fit the conditional law of each variable given all others and use its predictor support to estimate its neighbours. In a Gaussian graphical model, this is linear regression with coefficients given by conditional regression coefficients from a Gaussian precision matrix. The Nodewise Lasso supplies sparse fits; the OR rule for nodewise graph selection and AND rule for nodewise graph selection combine possibly disagreeing directional supports into an undirected graph.

For $F(y)=-e^{-ay}$ below hurdle $w_0$ and $F(y)=-e^{(b-a)w_0-by}$ above it, with $b>a>0$, an interior common tangent touches at $\ell =w_0-[b/a-1-\log (b/a)]/(b-a)$ and $h=w_0+[\log (b/a)-1+a/b]/(b-a)$. The slope is $a{e}^{-a\ell }$ and $h-\ell =1/a-1/b$. The formula requires $\ell \ge 0$; a smaller hurdle gives a different endpoint-at-zero concavification.

With zero interest and zero risk premium, a manager whose current wealth lies between common-tangent contacts $\ell ,h$ can improve expected utility maximization through a fair lottery paying those two values. The probability of $h$ is $(w_0-\ell )/(h-\ell )$. In a Brownian filtration, replicating a bounded terminal lottery gives a nonnegative wealth martingale throughout.

The dual of the complex tautological line bundle. On the complex projective line the induced frames satisfy $e_1=w{e}_0$ on the overlap of its standard affine charts.

A complex manifold is Stein if its global holomorphic functions separate points, give local coordinates, and make it holomorphically convex: the holomorphic hull of every compact set is compact. Closed complex submanifolds of affine complex space are examples, including products of copies of $\mathbb{C}$ and ${\mathbb{C}}^{\ast }$. Cartan theorem B makes these spaces useful for acyclic covers in sheaf cohomology.