The nonnegative Dirichlet Laplacian is the self-adjoint realization of $-\mathrm{div}\mathrm{grad}$ with zero boundary values. On a bounded domain, its weak form is the Dirichlet energy on $H_0^1$. For polygonal domains the weak operator domain is preferable to a requirement of smoothness at corners.

The nonnegative Neumann Laplacian is the self-adjoint realization of $-\mathrm{div}\mathrm{grad}$ with vanishing outward normal derivative. Its weak quadratic form is the Dirichlet energy on $H^1$. On a connected compact domain the constants give a one-dimensional zero eigenvalue space.

For a geodesic $\gamma$ with initial velocity $v$, $(f\circ \gamma {)}^{\prime \prime }(0)=\mathrm{Hess}f(v,v)$, because the covariant acceleration vanishes. Summing over an orthonormal basis of the tangent space gives $\Delta f(p)=-{\sum }_i(f\circ {\gamma }_i{)}^{\prime \prime }(0)$ for the nonnegative Laplace-Beltrami operator.

The positive convention for the Laplace-Beltrami operator is ${\Delta }_{+}f=\delta df$, where $\delta$ is the codifferential. It is the restriction of the Hodge Laplacian to functions. In coordinates,On a compact boundaryless Riemannian manifold, or for compactly supported functions, $⟨f,{\Delta }_{+}f⟩=\Vert df{\Vert }^2\ge 0$. The opposite $\mathrm{div}\mathrm{grad}$ convention is also standard; formulas involving the sign must specify which is used.

For a number field $K$, a finite set $S$ and $m\ge 2$, let $K(S,m)$ consist of classes $[a]\in K^{\ast }/K^{\ast m}$ whose valuations outside $S$ are divisible by $m$. It fits into $0\to {\mathcal{O}}_{K,S}^{\times }/({\mathcal{O}}_{K,S}^{\times }{)}^m\to K(S,m)\to \mathrm{Cl}({\mathcal{O}}_{K,S})[m]\to 0$. The S-unit group is finitely generated and the ideal class group is finite, so $K(S,m)$ is finite. When $S$ contains primes dividing $m$, this controls unramified multiplicative Kummer classes outside $S$.

For nonnegative densities of possibly different total masses, this divergence includes the mass correction $-x+y$ as well as the logarithmic term. It is the Bregman distance of the maximum-entropy regularization functional at a positive reference $y$. Convexity of $s\log s-s$ proves nonnegativity. For equal masses the correction integrates to zero, recovering the Kullback-Leibler divergence; for probability density functions both masses are one. Use the usual extended-value conventions at zero reference density.

For probability distributions $P,Q$, relative entropy satisfies $D_{\mathrm{KL}}(P\Vert Q)\ge 0$, with equality exactly when $P=Q$. Thus truthful prediction uniquely minimizes expected log-loss whenever the comparison is well defined, explaining the strictly proper scoring rule property of the logarithmic scoring rule.

For a short proof with probability density functions $f,g$ and $g>0$, set $r=f/g$. Their integrals both equal one, givingThe strictly convex function $r\log r-r+1$ is minimized only at $r=1$, so equality means $f=g$ almost everywhere. Its value at zero is one by $0\log 0=0$. Where $g=0<f$, relative entropy is infinite; where both vanish, the contribution is zero. These conventions extend the argument beyond positive reference probability density functions.

For sufficiently regular positive probability densities, this compares their spatial logarithmic derivatives. It is the dissipation functional for relative entropy under suitable diffusion equations. For standard Gaussian $g=\gamma$, it becomes $\int |\nabla \log f+v{|}^{2}f$, and in one dimension equals $\int |f^{\prime }{|}^2/f+\int v^{2}f-2$ at unit mass.

For a measurable event $A$, applying Jensen inequality to $u\log u$ separately on $A$ and its complement shows that the Kullback-Leibler divergence between two probability measures is at least the divergence between the two event indicators. This reduces a general probability measure comparison to a binary one and is the key step in a proof of Pinsker's inequality.

Every maximal radial timelike geodesic in positive-mass Kruskal spacetime is incomplete in at least one direction. Its equations are ${\dot{r}}^2=E^2-1+2M/r$ and $\ddot{r}=-M/r^2$. A finite turning point is a maximum, and each trajectory reaches $r=0$ at one or both ends. The remaining proper time is finite because $d\tau \sim \sqrt{r/(2M)}dr$. Complete circular timelike geodesics exist, but they have nonzero angular momentum.

A strictly increasing finite sequence of prime ideals. Its length is the number of strict inclusions, so the displayed chain has length $s$. The Krull dimension of a ring is the supremum of such lengths; the height of a prime ideal restricts to chains ending at that prime.

If a compact convex set is the closed convex hull of a subset $S$, every extreme point lies in the closure of $S$. A neighborhood avoiding $S$ around a putative missing extreme point gives finitely many closed convex caps covering $S$ and excluding that point. The convex hull of a finite union of compact convex sets is compact: group the terms by their set and use the simplex parametrization. The point must lie in that hull, contrary to extremality. The closures are in the given Hausdorff locally convex topology.

Let $K$ be a compact Hausdorff space. A test subspace $H\subset C(K,\mathbb{R})$ containing a strictly positive function is sufficient for uniform convergence of positive linear operators on continuous functions if, for each $x\in K$, the only positive linear functional agreeing with point evaluation on $H$ is that evaluation itself. By the Riesz-Markov-Kakutani representation theorem, this is uniqueness of the positive measure with the prescribed moments. Boundedness from the positive test function and the Banach-Alaoglu theorem prove the assertion by compactness. When $1\in H$, nonnegative tests with a unique zero at each prescribed $x$ give a convenient sufficient condition.

Komar angular momentum is the Komar charge associated with an axial Killing vector field in four-dimensional general relativity. In geometrized units its normalization is $(16\pi {)}^{-1}{\int }_S\star dm$, where $m$ is the metric-dual differential one-form. With zero cosmological constant, vacuum Einstein field equations make $d\star dm=0$, so homologous surrounding surfaces in the vacuum region give the same charge.

A description machine is a partial computable map from finite binary programs to finite binary outputs. It is optimal if, for every other description machine $V$, there is a constant $c_V$ such that $C_U(\sigma )\le C_V(\sigma )+c_V$. Encoding an interpreter index in a fixed prefix gives the constant-overhead descriptions used in incompressibility arguments.

For a fixed optimal description machine $U$, the plain Kolmogorov complexity of a binary string is the minimum length of a finite program producing it. The admissible programs need not satisfy a prefix restriction. There are at most $2^n-1$ descriptions shorter than $n$, which is the counting fact behind the existence of incompressible strings.

For integers $n\ge 2k\ge 2$, the chromatic number of the Kneser graph is $n-2k+2$. The upper bound colours by the least element among the first $n-2k+1$ elements, with one final colour for all other sets. The Gale hemisphere lemma and the Lusternik-Schnirelmann-Borsuk theorem give the matching lower bound.

This binary integer program selects each item at most once. Exact optimization is NP-hard; the fractional knapsack problem gives an upper bound and supports a half-approximation algorithm for knapsack.

This linear programming relaxation allows a fraction of each item. With positive weights, sort decreasing profit-to-weight ratios and fill capacity in that order, using at most one fractional item. An exchange argument proves optimality. Negative-profit items are omitted and positive-profit zero-weight items are included for free.

The quadratic knapsack problem includes pairwise interactions between selected items. For symmetric $p_{ij}$, a sum over all ordered pairs counts each unordered interaction twice; an alternative convention sums only over $i<j$. The objective convention must be kept consistent when bounding or fixing variables.