Two minimal left covering nets for the same identity neighbourhood $V$ can be matched bijectively so that paired translates intersect. Join $f\in F$ to $f^{\prime }\in F^{\prime }$ when $fV\cap f^{\prime }V\ne \varnothing$. For $S\subseteq F$, its neighbours $N(S)$ cover every point covered by $S$, so $(F\setminus S)\cup N(S)$ still covers the group. Minimality forces $|N(S)|\ge |S|$. Hall's marriage theorem supplies the matching, and an intersection $fv=f^{\prime }{v}^{\prime }$ gives $f^{-1}{f}^{\prime }=v{v}^{\prime -1}\in V{V}^{-1}$. Averaging a continuous function over matched nets therefore gives values differing by at most its uniform oscillation on right displacements in $V{V}^{-1}$. This is the finite combinatorial step in constructing Haar measure on a compact group.

Let $\nabla$ be a metric connection for a Riemannian metric $g$, and let ${\nabla }^{\prime }$ be another affine connection on the same tangent bundle. The difference of affine connections is a tensor $A=\nabla -{\nabla }^{\prime }$. Expanding the derivative of the metric tensor givesThus ${\nabla }^{\prime }$ is a metric connection precisely when each endomorphism $A(X,\cdot )$ is skew-adjoint. This imposes no antisymmetry in $X,Y$; that separate condition characterizes agreement of parametrized geodesics. If both conditions hold, $g(A(X,Y),Z)$ is alternating in all three arguments.

The ambient Levi-Civita connection induces this metric connection on the normal bundle of an embedded submanifold. Its tangential counterpart appears in the Weingarten formula. Projecting the derivative preserves the connection on a vector bundle Leibniz rule and the fiber inner product.

Nevanlinna theory studies the growth and value distribution of meromorphic functions through their integrated pole counts and logarithmic proximity to target values. The Nevanlinna characteristic combines both quantities, while the Nevanlinna first main theorem compares it with the corresponding counts and proximity for any fixed target.

A meromorphic function on the plane with $|f(z)|\le C|z{|}^n$ outside a disc has no poles there and only finitely many poles inside. Subtract their finite principal parts to obtain an entire function. For $n\ge 0$, Cauchy estimates make every derivative of order above $n$ vanish, so the remainder is a polynomial. For $n<0$ the remainder tends to zero at infinity and is zero by Liouville theorem. Adding back the principal parts gives a rational function in either case.

Mittag-Leffler's theorem prescribes the principal parts of a meromorphic function. Given an open subset $D$ of the complex plane, a subset $A$ having no accumulation point in $D$, and a finite sum of negative powers $P_a(z)={\sum }_{j=1}^{m_a}{c}_{a,j}(z-a{)}^{-j}$ for each $a\in A$, there is a meromorphic function on $D$ with exactly these principal parts and no other poles. The difference of any two such meromorphic functions is a holomorphic function on $D$, because their principal parts cancel at every prescribed pole.

Poles are isolated, including in the coordinate $w=1/z$ around infinity. Compactness of the Riemann sphere therefore leaves only finitely many poles. Subtract the finite principal parts at finite poles and the polynomial principal part at infinity. The remainder is entire with a finite limit at infinity, hence bounded and constant by Liouville's theorem. The original function is a sum of a polynomial and finitely many partial fractions, so is rational.

A meromorphic function $f$ has an algebraic addition theorem if some nonzero polynomial $P\in \mathbb{C}[U,V,W]$ satisfies $P(f(z),f(w),f(z+w))=0$ wherever the three values are finite. For a polynomial $f$ of degree $d>0$, take roots $z_i$ of $f(z)-U$ and $w_j$ of $f(w)-V$; the product ${\prod }_{i,j}(W-f(z_i+w_j))$ has polynomial coefficients in $U,V$ by the Fundamental theorem of symmetric polynomials, is monic in $W$, and supplies an addition relation. An elliptic function also has such a relation, by finite algebraic extension of its function field and the Weierstrass elliptic function addition formula.

For a meromorphic function viewed as a holomorphic map to the unit Riemann sphere, this is the length density of the pulled-back spherical metric. Its square is the pulled-back area density. At a pole use the coordinate $1/f$, which gives the same density. Some conventions omit the factor two and use a sphere of area $\pi$; all area normalizations must then change accordingly.

This is the supremum of reliable classical bits per use of a memoryless quantum channel when sender and receiver share unlimited prior quantum entanglement. If $|{\psi }_{\rho }{⟩}_{RA}$ purifies the input, let ${\omega }_{RB}=({\mathrm{id}}_R\otimes \mathcal{N})(|{\psi }_{\rho }⟩⟨{\psi }_{\rho }|)$. The single-letter capacity formula is ${\max }_{\rho }[S(\rho )+S(\mathcal{N}(\rho ))-S({\omega }_{RB})]$, the maximum quantum mutual information between reference and output. The preshared resource is independent of the message.

A quantum erasure channel transmits an input state unchanged with probability $1-p$ and replaces it by an orthogonal, recognizable erasure flag with probability $p$. Its output space has one more dimension than the input. This is a quantum channel for $0\le p\le 1$ and preserves all input coherences on the nonerased branch.

In a reflection-symmetric mean-field dynamo model, dipole parity can be represented by an even poloidal-potential amplitude and an odd toroidal-field amplitude; quadrupole parity reverses these assignments. The names describe the reflection classes appropriate to the model, not arbitrary electric multipoles. A complex amplitude $P=A+iB$ allows a phase rotation by $i$ to interchange the two classes when the unperturbed equations have that phase symmetry.

For $k(t)=k_0\sqrt{1+(t/T{)}^2}$, factor resistive damping out of the shearing-mode equations by $R=e^{-D}G$, $S=e^{-D}H$, $D=\eta \int k^{2}dt$. They imply $G^{\prime \prime }={\alpha }^2{k}^{2}G$, so a growing WKB approximation has $G\propto k^{-1/2}{e}^{|\alpha |\int kdt}$ and $H\simeq i\mathrm{sgn}(\alpha )kG$. Its magnetic energy is proportional to $k\exp [2|\alpha |\int kdt-2\eta \int k^{2}dt]$. At long times the positive quadratic exponent is overtaken by negative cubic resistive damping. A well-separated growing interval exists for sufficiently small positive magnetic diffusivity, with $\eta k_0/|\alpha |\ll 1$ at fixed nonzero $|\alpha |k_{0}T$, and the displayed leading peak time. This is transient amplification, not an asymptotically self-sustaining dynamo.

For a nonempty $A\subseteq \{0,1{\}}^n$ of relative size at least $\epsilon \in (0,1]$, its Hamming distance function $F(x)=d(x,A)$ has the bounded differences property with all constants one. If $\mu =\mathbb{E}F$, the concentration bound at $F=0$ gives $\epsilon \le 2e^{-2{\mu }^2/n}$, so $\mu \le t/2$ for $t=\sqrt{2n\log (2/\epsilon )}$. Then $\mathrm{Pr}(F>t)\le 2e^{-2(t-\mu {)}^2/n}\le \epsilon$, proving that the radius-$t$ neighbourhood contains at least $(1-\epsilon )2^n$ cube points.

Vary the antisymmetric auxiliary electromagnetic field tensor and the electromagnetic four-potential independently. The first variation sets $F_{ab}={\nabla }_a{A}_b-{\nabla }_b{A}_a$ algebraically, and variation of the potential gives ${\nabla }_b{F}^{ab}=0$. Substitution of the algebraic equation reduces the first-order action to the usual negative quadratic Maxwell action. Metric variation gives the same stress tensor after eliminating the auxiliary field, because the chain-rule term multiplying its variational equation then vanishes.

For $n$ independent uniform points in a rectangle, condition on one point with normalized coordinates $(x,y)$. It is a maximal external point if all others avoid the upper-right rectangle of area $(1-x)(1-y)$, giving conditional probability $[1-(1-x)(1-y){]}^{n-1}$. Integrating gives $H_n/n$, where $H_n$ is a harmonic number. Linearity of expectation over the $n$ indicator random variables gives $\mathbb{E}{K}_n=H_n$. The same answer holds for independent continuous coordinate distributions, by transforming each coordinate with its distribution function.

The row rank of a matrix is the dimension of the vector space spanned by its rows. Elementary row operations preserve this row space; multiplication on the right by an invertible matrix preserves all linear relations among the rows.

To see why row rank equals column rank, write a reduced row echelon form as $R=EA$, where $E$ is a product of elementary matrices and is an invertible matrix. Reversibility of the row operations gives equal row spaces for $R$ and $A$. Also $Rz=0$ if and only if $Az=0$, so the columns have exactly the same linear relations. If $R$ has $r$ nonzero rows, its pivot columns are the first $r$ standard basis vectors of ${\mathbb{F}}^m$ and span its column space. Its nonzero rows are linearly independent, since their matrix entries in the pivot positions form an identity matrix. Both dimensions therefore equal $r$, and both counts transfer back to $A$.

The column rank of a matrix is the dimension of its image of a linear map, equivalently the dimension of the linear span of its columns. Multiplication on the left by an invertible matrix preserves all column linear relations, and elementary column operations preserve their linear span. In row echelon form the pivot columns form a basis of the column space. Its dimension equals the number of nonzero rows, and hence the row rank.

For composable linear maps, $\mathrm{im}(AB)=A(\mathrm{im}B)$ lies in $\mathrm{im}A$ and has dimension at most that of $\mathrm{im}B$. This proves the bound on the rank of a matrix product. When $A$ is $m\times p$, $B$ is $p\times n$, and $m,n\ge p$, the bound $p$ is attained by $A=\left(\genfrac{}{}{0ex}{}{I_p}{0}\right)$ and $B=(I_{p}0)$.

For matrices $A\in {\mathbb{F}}^{m\times n}$ and $B\in {\mathbb{F}}^{n\times p}$, their matrix product is the matrix $AB\in {\mathbb{F}}^{m\times p}$ with entries $(AB{)}_{ij}={\sum }_{k=1}^n{A}_{ik}{B}_{kj}$. The operation producing it is matrix multiplication, representing composition of the corresponding linear maps.