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For a positive modulus $n$, a reduced residue system contains exactly one representative of each residue class coprime to $n$. Its size is the Euler totient function $\phi (n)$. Multiplication by any unit modulo n permutes these classes, which proves the Fermat-Euler theorem after multiplying and cancelling their unit product.

For a radial function in three dimensions, $(\Delta -1)y=g(x)$ becomes $y^{\prime \prime }+2y^{\prime }/x-y=g(x)$. Setting $u=xy$ gives $u^{\prime \prime }-u=xg(x)$, reducing it to a constant-coefficient linear ordinary differential equation. If $g=e^{-3x}/x^2$, a decaying particular solution obtained by variation of parameters isThe small-argument expansion of the exponential integral givesThe undetermined decaying homogeneous coefficient $\alpha$ is fixed by a boundary condition or by a matched asymptotic expansion.

The differential equation defining modified Bessel functions of order $\nu$. For real fixed $\nu$ and positive $z$, its independent standard solutions are $I_{\nu }(z)$ and $K_{\nu }(z)$.

The solution of the Modified Bessel differential equation with power series $I_{\nu }(z)={\sum }_{j\ge 0}(z/2{)}^{2j+\nu }/[j!\Gamma (j+\nu +1)]$, interpreted through the reciprocal gamma function at its zeros. For fixed real order it grows exponentially along the positive real axis.

For fixed real $\nu$ as $z\to +\infty$, the Modified Bessel function of the first kind and Modified Bessel function of the second kind have asymptotic expansionsThe expansion and its error bounds establish actual growing and decaying solutions, beyond formal matching of a differential equation. These formulas are documented in NIST DLMF, equations 10.40.1–2.

For a free field, its positive-frequency part consists of the annihilation terms multiplying $e^{-i\omega t}$ with $\omega >0$; its negative-frequency part consists of the creation terms. The sign refers to positive energy, even though the exponential has a minus sign. Imposing the Lorenz gauge only through $(\partial \cdot A{)}^{(+)}|\mathrm{phys}⟩=0$ is the subsidiary condition in Gupta-Bleuler quantization, not an operator identity for the whole divergence.

An arcsecond is $1/3600$ of a degree, equivalently $\pi /648000$ radians. There are approximately $206265$ arcseconds in a radian.

A minimal subsystem is a nonempty closed forward-invariant subset on which the restricted dynamical system is a minimal dynamical system. Every continuous map on a nonempty compact Hausdorff space has one: intersections of chains of nonempty closed invariant sets remain nonempty by compactness, so the Zorn lemma gives a minimal member.

A point is minimal when its forward orbit closure is a minimal dynamical system. This does not require the point to be a fixed point. In a finite-alphabet full shift, minimal points are exactly the uniformly recurrent sequences.

A cobounded group action by isometries on a nonempty geodesic metric space admits an orbit map that is a quasi-isometry for a possibly infinite generating set of a group. If $Gx$ is $R$-dense, take $S=\{g\ne 1:d(x,gx)\le 2R+1\}$. Subdivide a metric geodesic into segments of length at most one and approximate subdivision vertices by orbit points, keeping the endpoints exact. The resulting group increments generate and giveThe first estimate includes zero displacement: a nontrivial stabilizer element has length one. No finiteness assertion about $S$ survives without properness.

For the Levi-Civita connection, the difference of two connections is a tensor. Varying the coordinate expression for the Ricci tensor then gives the displayed covariant identity. It is the starting point for the metric variation of scalar curvature.

For a continuous path $\gamma :[a,b]\to X$ in a metric space, take the supremum over finite partitions $a=t_0<\cdots <t_n=b$ of ${\sum }_{i=1}^{n}d(\gamma (t_i),\gamma (t_{i-1}))$. This definition allows $+\infty$ and needs no derivative. A metric geodesic parametrized on $[a,b]$ has length $b-a$. In a Riemannian manifold, it agrees with the usual arc length integral for a smooth curve.

A network meta-analysis jointly compares several treatments using a graph of direct randomized controlled trials and indirect treatment comparisons. If effects are measured relative to a reference treatment, consistency means ${\delta }_{ij}={\delta }_{i0}-{\delta }_{j0}$. A connected graph identifies all relative effects under that model. Transitivity in network meta-analysis is the substantive comparability assumption supporting these relations; inconsistency can be assessed when the network has loops.

For independent study estimates $y_i$ with within-study variances $v_i$, a usual approximate model is $y_i\mid {\delta }_i\sim N({\delta }_i,v_i)$ and ${\delta }_i\sim N(\mu ,{\tau }^2)$. Its marginal distribution is $N(\mu ,v_i+{\tau }^2)$, so for fixed heterogeneity ${\tau }^2$ the inverse-variance weighted mean uses weights $(v_i+{\tau }^2{)}^{-1}$. Here $\mu$ describes the mean effect across comparable studies; ${\tau }^2$ represents between-study variation. Estimation and uncertainty for heterogeneity are essential, especially with few studies.

Refit a meta-analysis after deleting each study, comparing fitted effects, uncertainty and heterogeneity to the full fit. For fixed weights $w_i>0$, total weight $W$ and weighted effect $\hat{\mu }$, direct subtraction of the two weighted means gives $\hat{\mu }-{\hat{\mu }}_{(-i)}=w_i(y_i-\hat{\mu })/(W-w_i)$. Re-estimating heterogeneity additionally measures influence through the weights. This is a sensitivity diagnostic, not an automatic rule for removing discordant studies.

Suppose $f$ decays faster than every power at infinity and, near zero, $f(y)={\sum }_{j=1}^r{c}_j{y}^{{\sigma }_j}+O(y^{{\sigma }_{r+1}})$ for every $r$, with ${\sigma }_j$ strictly increasing and tending to infinity. Subtracting finitely many terms in the integral over $(0,1)$ givesThe last integral is holomorphic on $\mathrm{Re}s>-{\sigma }_{r+1}$. These expressions continue the Mellin transform meromorphically to the whole plane, with simple poles precisely at $-{\sigma }_j$ for nonzero $c_j$.

For $C(k)=\mathbb{E}(e^{\xi }-e^k{)}^{+}$ and $\mathrm{Re}z>1$ in the exponential moment domain, the moment-generating function satisfies $\mathbb{E}{e}^{z\xi }={\int }_{\mathbb{R}}C(k)z(z-1)e^{(z-1)k}dk$.

A countable measurable partition is a countable family of pairwise disjoint measurable sets whose union has full measure. Its members are its atoms; null atoms can be discarded in statements made almost everywhere.

A random measure assigns a measure ${\mu }_{\omega }$ to each outcome so that $\omega \mapsto {\mu }_{\omega }(A)$ is a random variable for every measurable $A$. A finite-variation process gives pathwise signed random measures through Lebesgue–Stieltjes integration.

A measurable set $N$ is a null set for a measure $\mu$ when $\mu (N)=0$. A property holds almost everywhere when its exceptions lie in a null set. With a complete measure, every subset of a null set is measurable and is itself a null set.