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For a solenoidal vector field carrying current only in a conductor contained in a sphere of radius $R$, match the exterior field to a decaying potential field and include exterior magnetic energy. The smallest free-decay eigenvalue of the enclosing sphere is ${\pi }^2/R^2$, from its dipolar poloidal mode. The corresponding variational estimate gives the displayed bound. Continuity and the usual insulating-interface magnetic conditions are part of the admissible field class. This bound is the diffusive ingredient in Backus' necessary condition for dynamo action.

Parker instability is an undular magnetic buoyancy instability of a gravitationally stratified atmosphere supported partly by magnetic pressure. A bent horizontal magnetic field can permit gas to drain along its magnetic field lines, leaving buoyant rising regions. Magnetic tension resists bending, so wavelengths and boundary conditions affect the instability threshold. For a magnetized isothermal atmosphere, the unrestricted-wavevector threshold is given by the magnetic buoyancy energy criterion.

A stable equilibrium M-M-s queue has a Poisson process of departures with the same rate as its input. Its present population is also independent of past departures, a component of quasireversibility. The birth-death process satisfies detailed balance; reversed upward jumps have a constant arrival rate, so they are generated by an independent Poisson process.

An M/D/1 queue has a Poisson process of arrivals, a deterministic service time $\tau$, and one work-conserving server. With arrival rate $\lambda$ and $\lambda \tau <1$, its equilibrium idle probability is $1-\lambda \tau$: the stationary completed work per unit time is $\lambda \tau$. Its output generally need not be a Poisson process.

On a compact parameter set, a continuous deterministic objective with a unique minimum has a positive separation gap outside each neighbourhood of its minimizer. Uniform convergence in probability of the random objectives bounds the deterministic objective difference at any attained random minimizer by twice the supremum discrepancy. This proves statistical consistency of each measurable minimizing selection.

Repeatedly take the lower shadow of a uniform set family. After $j$ steps the result consists of all subsets of its members whose size is $j$ smaller. Iterating the Kruskal-Katona theorem shows that colexicographic initial segments minimize every iterated lower shadow.

A dependency graph of events joins potentially dependent events. Each event must be independent of the sigma-algebra generated by all its nonneighbors, a joint condition stronger than pairwise independence. In a product probability space, joining events whose sets of underlying independent variables overlap gives such a graph. This supplies the hypothesis of the Lovász local lemma.

For a finite family with a dependency graph of events, if $0\le x_i<1$ satisfy $\mathbb{P}(E_i)\le x_i{\prod }_{j\in N(i)}(1-x_j)$, then $\mathbb{P}({\bigcap }_i{E}_i^c)\ge {\prod }_i(1-x_i)>0$. Choosing equal $x_i$ gives the familiar symmetric criterion $eq(d+1)\le 1$ when event probabilities are at most $q$ and the maximum dependency degree is $d$.

For countably many constraints on finite-valued variables, each involving finitely many variables, apply the finite Lovász local lemma to every finite constraint subfamily. Partial satisfying assignments on increasing finite variable sets form a finitely branching tree with nonempty levels. The König infinity lemma gives an infinite branch, producing one assignment satisfying all constraints. This existence argument does not assert positive probability of simultaneous avoidance in an infinite product space.

An Abelian gauge transformation $A_a\mapsto A_a+{\partial }_a\lambda$ changes $\partial \cdot A$ by $\square \lambda$. Consequently gauge fixing by a quadratic function of $\partial \cdot A$ retains transformations satisfying $\square \lambda =0$, subject to the same boundary conditions. Eliminating those residual transformations is part of specifying an inverse with a particular Green function prescription.

In metric signature $(-,+,\dots ,+)$, a spacelike submanifold of a Lorentzian manifold has an induced metric tensor defining a positive-definite quadratic form. With the overall reversed metric signature, its induced metric tensor instead has only negative eigenvalues. The definition applies to any dimension: enclosing charge-integration surfaces in four dimensions are two-dimensional examples, whereas a spacelike Cauchy hypersurface in four dimensions is three-dimensional.

The metric signature records the signs of the diagonalized nondegenerate metric tensor, equivalently its numbers of negative and positive directions. For a four-dimensional Lorentzian manifold, both $(-,+,+,+)$ and its overall negative convention are common. The numerical signs of contractions and component Hodge star operator identities must be consistent with that choice and the orientation.

A nonzero covector $\ell$ is null when its squared dual Lorentzian inner product is zero: $g^{\alpha \beta }{\ell }_{\alpha }{\ell }_{\beta }=0$. Raising its index gives a null vector. Null covectors describe normals to characteristic light cones and are the test directions in the classical null condition for wave equations.

For two distinct massless final particles in their centre-of-momentum frame, integrating the energy-momentum delta function gives $d{\Phi }_2=d\Omega /(32{\pi }^2)$. A massless two-particle initial state has flux $2s$, hence $d\sigma /d\cos \theta =\overline{|\mathcal{M}{|}^2}/(32\pi s)$. Average only over the initial spin states physically present.

Loop-space homology is the homology of the based loop space. It also carries the Pontryagin product induced by concatenation of loops. For $n>2$, a Morse theory cell structure on $\Omega S^n$ has one cell in each dimension $j(n-1)$, so its integer homology is $\mathbb{Z}$ in those dimensions and zero elsewhere.

Finite words in a based space, with occurrences of its basepoint deleted. Word concatenation defines a multiplication. For connected based CW spaces, the natural map to $\Omega \Sigma X$ is a weak homotopy equivalence. With free integral homology, the Bott–Samelson theorem identifies its homology algebra with the tensor algebra on reduced homology.

The homology ring formed from loop concatenation and the homology cross product. Homotopy associativity is enough for associativity in homology, and the constant loop is the unit. More generally the same construction applies to a homotopy-associative H-space.

For $n\ge 1$, there is one integral cohomology generator $a_k$ in every degree $2nk$, and its products are $a_i{a}_j=\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)a_{i+j}$. The Bott–Samelson theorem supplies free polynomial homology; its primitive generator gives the binomial diagonal dual to this divided power algebra.

For $I={\int }_{\epsilon }^{\infty }{e}^{-x-\epsilon /x}dx/x$, the omitted interval becomes $J-I={\int }_1^{\infty }{e}^{-t-\epsilon /t}dt/t$. A uniform Taylor expansion gives $J-I=E_1(1)-\epsilon [e^{-1}-E_1(1)]+O({\epsilon }^2)$. Subtracting this correction from the logarithmic expansion of a symmetric exponential integral preserves both the constant and first-order coefficient.

If $f$ is holomorphic near the closed radius-$R$ disc, $f(z_0)\ne 0$, and $|f|\le M$, the displayed estimate holds away from zeros for $|z-z_0|\le R/3$, counting zeros with multiplicity. Factoring local zeros isolates their poles; the remaining logarithm is controlled by disc estimates. If all zeros lie to the left of a point $z$ in real part, their real contributions are nonnegative. This is the disc estimate used by the Landau zero-free-region theorem; a version with general radius ratios is Lemma 24.17 in Montgomery and Vaughan.